EXPLORING MATHEMATICAL LITERACY: THE RELATIONSHIP INSTRUCTIONAL PRACTICES

EXPLORING MATHEMATICAL LITERACY: THE RELATIONSHIP INSTRUCTIONAL PRACTICES
EXPLORING MATHEMATICAL LITERACY: THE RELATIONSHIP
BETWEEN TEACHERS’ KNOWLEDGE AND BELIEFS AND THEIR
INSTRUCTIONAL PRACTICES
by
Johanna Jacoba Botha
A thesis submitted in fulfilment of the requirements for the degree of
Philosophiae Doctor (PhD)
Department of Science, Mathematics and Technology Education
Faculty of Education
University of Pretoria
Supervisor: Prof JG Maree
Co-supervisor: Dr GH Stols
October 2011
© University of Pretoria
Acknowledgement
My deepest appreciation to the following:
The Lord, my Provider who promised me that if He brought me to it, He would bring me through it.
“The will of God will never take you where the grace of God cannot protect you”.
My supervisors: Professor Kobus Maree and Dr Gerrit Stols for all their guidance, advice and
motivation.
My family: Jan for all his motivation and strong belief in my abilities, my parents, sisters and brother as
well as all my friends.
My participants: ‘Monty’, ‘Alice’, ‘Denise’ and ‘Elaine’ who allowed me into their classrooms and hearts.
My colleagues: Sonja, Barbara and Ina.
My editors: Mr AK Welman, Mr Tim Steward and Mrs M Labuschagne.
Summary
South Africa is the first country in the world to offer Mathematical Literacy as a school subject. This
subject was introduced in 2006 as an alternative to Mathematics in the Further Education and Training
band. The purpose of this subject is to provide learners with an awareness and understanding of the
role that mathematics plays in the modern world, but also with opportunities to engage in real-life
problems in different contexts. A problem is the beliefs some people in and outside the classroom have
regarding this subject such as teachers believing ML is the dumping ground for mathematics
underperformers (Mbekwa, 2007). Another problem is the belief of some principals that any nonmathematics teacher can teach ML. In practice there is Mathematics teachers who teach ML in the
same way that they teach Mathematics; non-Mathematics teachers who in many cases lack the necessary
mathematical content knowledge and skills to teach ML competently; and Mathematics teachers who
adapted their practices to teach ML using different approaches than those required for teaching
Mathematics. Limited in-depth research has been done on the ML teachers, what they believe and what
knowledge is required to teach this subject effectively and proficiently.
The purpose of this study is to investigate the way in which ML is taught in a limited number of
classrooms with the view to exploring the relationship between ML teachers’ knowledge and beliefs
and their instructional practices. According to Artzt, Armour-Thomas and Curcio (2008) the
instructional practice of the teacher plays out in the classroom where teachers’ goals, knowledge and
beliefs serve as the driving force behind their instructional efforts to guide and mentor learners in their
search for knowledge. To accomplish this aim, an in-depth case study was conducted to explore the
nature of teachers’ knowledge and beliefs about ML as manifested in their instructional practices. A
qualitative research approach was used in which observations and interviews served as data collection
techniques enabling me to interpret the reality as I became part of the lives of the teachers.
My study revealed that there is a dynamic but complex relationship between ML teachers’ knowledge
and beliefs and their instructional practices. The teachers’ knowledge, but not their stated beliefs were
reflected in their instructional practices. Conversely, in one case, the teacher’s instructional practice also
had a positive influence on her knowledge and beliefs. It was further revealed that mathematics teacher
training and teaching experience played a significant role in the productivity of the teachers’ practices.
The findings suggest that although mathematical content knowledge is required to develop PCK, it is
teaching experience that plays a crucial role in the development of teachers’ PCK.
Although the study’s results cannot be generalised due to the small sample, I believe that the findings
concerning the value of teachers’ knowledge and the contradictions between their stated beliefs and
practices could possibly contribute to teacher training. Curriculum decision-makers should realise that
the teaching of ML requires specially trained, competent, dedicated teachers who value the subject. This
exploratory study concludes with recommendations for further research.
Key words: Mathematical literacy; Teachers; Learners; Curriculum; Instructional practice; Tasks;
Discourse; Learning environment; Mathematical content knowledge; Pedagogical content knowledge;
Beliefs.
Table of contents
Chapter 1 .............................................................................................................................................. 1
Introduction and contextualisation ................................................................................................... 1
1.1 Introduction ....................................................................................................................... 1
1.1.1
International perspective on mathematical literacy ................................................. 2
1.1.2
National perspective on mathematical literacy ........................................................ 3
1.1.3
The experiences of ML teachers .............................................................................. 3
1.1.4
Silence in the literature addressed in this study ....................................................... 4
1.2 Rationale for the study ...................................................................................................... 4
1.3 Statement of the problem .................................................................................................. 5
1.4 The purpose of the study ................................................................................................... 6
1.5 Research questions ............................................................................................................ 6
1.6 Methodological considerations ......................................................................................... 6
1.7 Definition of terms ............................................................................................................ 7
1.8 Possible contribution of the study ..................................................................................... 8
1.9 Limitations of the study .................................................................................................... 8
1.10 Summary ........................................................................................................................... 9
1.11 The structure of the thesis ................................................................................................. 9
Chapter 2 ............................................................................................................................................ 11
Literature review and conceptual framework ................................................................................ 11
2.1 Introduction ..................................................................................................................... 11
2.2 Mathematical literacy ...................................................................................................... 11
2.2.1
International perspectives on mathematical literacy.............................................. 12
2.2.1.1
Different conceptions of mathematical literacy ............................................... 12
2.2.1.2 Some contexts in which mathematical literacy can be applied .......................... 16
2.2.1.3 Studies measuring learners’ mathematical literacy skills .................................. 18
2.2.1.4 Defining mathematical literacy .......................................................................... 20
2.2.1.5 The role of mathematical literacy in some international school curricula ......... 21
2.2.1.6 Summary ............................................................................................................ 25
2.2.2
An overview of ML ............................................................................................... 26
2.2.2.1 The history of ML .............................................................................................. 26
2.2.2.2 ML principles ..................................................................................................... 27
2.2.2.3 Pedagogical approaches for teaching ML .......................................................... 31
2.2.2.4 The ML learner profile ....................................................................................... 33
2.2.2.5 Some general concerns about ML...................................................................... 33
2.2.2.6 Comparison between the national and international perspectives on
mathematical literacy ........................................................................................ 35
2.2.2.7 An overview of ML and Mathematics ............................................................... 36
2.2.2.8 Summary ............................................................................................................ 39
2.3 Teachers’ instructional practices ..................................................................................... 40
2.3.1
Tasks ...................................................................................................................... 42
2.3.2
Discourse ............................................................................................................... 43
2.3.3
Learning environment ............................................................................................ 44
2.4 Mathematics teachers’ knowledge and beliefs about mathematics and the teaching
thereof ........................................................................................................................................ 45
2.4.1
Relationship between knowledge and beliefs ........................................................ 45
2.4.2
Overview of the different domains of teachers’ knowledge .................................. 47
2.4.2.1 Shulman’s (1986) categories of content knowledge .......................................... 48
2.4.2.2 Grossman’s (1990) components of PCK ........................................................... 49
i
2.4.2.3
2.4.2.4
2.4.2.5
Borko and Putnam’s (1996) domains of knowledge .......................................... 50
Ball, Thames and Phelps’ (2005) domains of knowledge for teaching ............. 51
Hill, Ball and Schilling’s (2008) domain map for mathematical knowledge
for teaching ....................................................................................................... 52
2.4.2.6 Summary ............................................................................................................ 52
2.4.3
An overview of mathematics teachers’ beliefs about mathematics and the teaching
thereof .................................................................................................................... 53
2.4.3.1 The nature of beliefs .......................................................................................... 53
2.4.3.2 Teachers’ belief systems .................................................................................... 54
2.4.4
The influence of teachers’ knowledge and beliefs on their instructional practices54
2.4.4.1 The influence of teachers’ knowledge and beliefs on the learners .................... 55
2.4.4.2 The influence of teachers’ knowledge and beliefs on their teaching ................. 55
2.4.5
Summary ................................................................................................................ 57
2.5 Conceptual framework .................................................................................................... 57
2.5.1
General view on mathematics teachers’ knowledge and beliefs ........................... 59
2.5.1.1
Mathematics teachers’ MCK ........................................................................... 59
2.5.1.2 Mathematics teachers’ PCK ............................................................................... 60
2.5.1.3 Mathematics teachers’ beliefs ............................................................................ 60
2.5.2
The three domains of PCK and beliefs .................................................................. 61
2.5.2.1 PCK and beliefs regarding content and learners ................................................ 61
2.5.2.2 Knowledge and beliefs regarding content and teaching .................................... 62
2.5.2.3 Knowledge and beliefs regarding the curriculum .............................................. 64
2.5.3
Teachers’ instructional practices ........................................................................... 65
2.5.4
Summary ................................................................................................................ 67
2.6 Conclusion....................................................................................................................... 67
Chapter 3 ............................................................................................................................................ 69
Methodology .................................................................................................................................. 69
3.1 Introduction ..................................................................................................................... 69
3.2 Research paradigm and assumptions .............................................................................. 69
3.2.1
Research paradigm................................................................................................. 69
3.2.2
Paradigmatic assumptions ..................................................................................... 70
3.3 Research approach and design ........................................................................................ 71
3.3.1
Research approach ................................................................................................. 72
3.3.2
Research design ..................................................................................................... 72
3.4 Research site and sampling ............................................................................................. 73
3.5 Data collection techniques .............................................................................................. 73
3.5.1
Observations .......................................................................................................... 74
3.5.2
Interviews .............................................................................................................. 76
3.6 Data analysis strategies ................................................................................................... 77
3.7 Quality assurance criteria ................................................................................................ 80
3.7.1
Trustworthiness of the study.................................................................................. 80
3.7.2
Validity and reliability of the study ....................................................................... 81
3.7.2.1 The Hawthorne effect......................................................................................... 81
3.7.2.2 The Halo effect................................................................................................... 82
3.8 Ethical considerations ..................................................................................................... 82
3.9 Conclusion....................................................................................................................... 83
Chapter 4 ............................................................................................................................................ 85
Presentation and discussion of the findings ................................................................................... 85
4.1 Introduction ..................................................................................................................... 85
4.2 The data collection process ............................................................................................. 86
ii
4.3 Data analysis strategies ................................................................................................... 87
4.3.1
Transcribing the data ............................................................................................. 87
4.3.2
Coding of the data.................................................................................................. 88
4.3.2.1 Theme 1: ML teachers’ instructional practices .................................................. 88
4.3.2.2 Theme 2: ML teachers’ knowledge and beliefs ................................................. 94
4.3.2.3 Inclusion criteria for coding the data ............................................................... 100
4.3.2.4 Exclusion criteria for coding the data .............................................................. 100
4.4 Information regarding the four participants .................................................................. 101
4.4.1
Monty................................................................................................................... 101
4.4.2
Alice..................................................................................................................... 102
4.4.3
Denise .................................................................................................................. 102
4.4.4
Elaine ................................................................................................................... 103
4.5 Theme 1: The ML teachers’ instructional practices ...................................................... 104
4.5.1
Monty’s instructional practice ............................................................................. 104
4.5.1.1
Tasks ............................................................................................................... 104
4.5.1.2 Discourse .......................................................................................................... 107
4.5.1.3 Learning environment ...................................................................................... 109
4.5.2
Alice’s instructional practice ............................................................................... 112
4.5.2.1 Tasks ................................................................................................................ 112
4.5.2.2 Discourse .......................................................................................................... 118
4.5.2.3 Learning environment ...................................................................................... 120
4.5.3
Denise’s instructional practice ............................................................................. 123
4.5.3.1 Tasks ................................................................................................................ 123
4.5.3.2 Discourse .......................................................................................................... 126
4.5.3.3 Learning environment ...................................................................................... 129
4.5.4
Elaine’s instructional practice.............................................................................. 131
4.5.4.1 Tasks ................................................................................................................ 131
4.5.4.2 Discourse .......................................................................................................... 136
4.5.4.3 Learning environment ...................................................................................... 138
4.5.5
Summary of participants’ instructional practices ................................................ 141
4.5.6
Discussion of Theme 1: ML teachers’ instructional practices ............................ 146
4.5.6.1 Tasks ................................................................................................................ 146
4.5.6.2 Discourse .......................................................................................................... 149
4.5.6.3 Learning environment ...................................................................................... 151
4.5.6.4 Summary of discussion on Theme 1 ................................................................ 153
4.6 Theme 2: ML teachers’ knowledge and beliefs ............................................................ 154
4.6.1
Monty’s knowledge and beliefs ........................................................................... 154
4.6.1.1
Mathematical content knowledge (MCK)...................................................... 154
4.6.1.2 Knowledge and beliefs regarding ML learners ................................................ 155
4.6.1.3 Knowledge and beliefs regarding ML teaching ............................................... 156
4.6.1.4 Knowledge and beliefs regarding ML curriculum ........................................... 158
4.6.2
Alice’s knowledge and beliefs ............................................................................. 160
4.6.2.1 Mathematical content knowledge (MCK)........................................................ 160
4.6.2.2 Knowledge and beliefs regarding ML learners ................................................ 163
4.6.2.3 Knowledge and beliefs regarding ML teaching ............................................... 165
4.6.2.4 Knowledge and beliefs regarding ML curriculum ........................................... 167
4.6.3
Denise’s knowledge and beliefs .......................................................................... 169
4.6.3.1 Mathematical content knowledge (MCK)........................................................ 169
4.6.3.2 Knowledge and beliefs regarding ML learners ................................................ 169
4.6.3.3 Knowledge and beliefs regarding ML teaching ............................................... 170
iii
4.6.3.4 Knowledge and beliefs regarding ML curriculum ........................................... 172
4.6.4
Elaine’s knowledge and beliefs ........................................................................... 174
4.6.4.1 Mathematical content knowledge (MCK)........................................................ 174
4.6.4.2 Knowledge and beliefs regarding ML learners ................................................ 175
4.6.4.3 Knowledge and beliefs regarding ML teaching ............................................... 176
4.6.4.4 Knowledge and beliefs regarding ML curriculum ........................................... 178
4.6.5
Summary of the participants’ knowledge and beliefs.......................................... 181
4.6.6.
Discussion of Theme 2: ML teachers’ knowledge and beliefs ............................ 185
4.6.6.1 ML teachers’ mathematical content knowledge (MCK).................................. 185
4.6.6.2 ML teachers’ knowledge and beliefs regarding their learners ......................... 185
4.6.6.3 ML teachers’ knowledge and beliefs regarding the teaching of ML ............... 186
4.6.6.4
ML teachers’ knowledge and beliefs regarding the ML curriculum .............. 187
4.6.6.5 Summary of discussion on Theme 2 ................................................................ 188
4.7 Findings, trends and explanations ................................................................................. 189
4.8 Conclusion..................................................................................................................... 192
Chapter 5 .......................................................................................................................................... 193
Conclusions and implications ...................................................................................................... 193
5.1 Introduction ................................................................................................................... 193
5.2 Chapter summary .......................................................................................................... 193
5.3 Verification of research questions ................................................................................. 194
5.3.1
Question 1: How can ML teachers’ instructional practices be described? .......... 197
5.3.2
Question 2: What is the nature of ML teachers’ knowledge and beliefs? ........... 199
5.3.3
Question 3: How do ML teachers’ knowledge and beliefs relate to their
instructional practices? ........................................................................................ 201
5.3.4
Question 4: What are the possible implications of the findings from Questions 1, 2
and 3 for teacher training? ................................................................................... 202
5.3.5
Question 5: What is the value of the study’s findings for theory building in
teaching and learning ML? .................................................................................. 204
5.3.6
Summary of verification of research questions ................................................... 205
5.4 What would I have done differently? ............................................................................ 206
5.5 Providing for errors in my conclusion .......................................................................... 206
5.6 Conclusions ................................................................................................................... 207
5.7 Recommendations for further research ......................................................................... 208
5.8 Limitations of the study ................................................................................................ 209
5.9 Last reflections .............................................................................................................. 209
References ........................................................................................................................................ 211
Appendices ....................................................................................................................................... 226
iv
List of abbreviations
ACE
Advanced Certificate in Education
BEd
Baccalaureus Educationis
BTech
Baccalaureus Technologiae
CAPS
Curriculum and Assessment Policy Statement
DoE
Department of Education (South Africa)
FET
Further Education and Training
GET
General Education and Training
HED
Higher Education Diploma
MCK
Mathematical content knowledge
ML
Mathematical Literacy (the subject)
NCS
National curriculum statement
OBE
Outcomes-based education
OECD
Organisation for Economic Co-operation and Development
PCK
Pedagogical content knowledge
PISA
Programme for International Student Assessment
QCDA
Qualifications and Curriculum Development Agency
QCE
Queensland Certificate in Education
QG
Queensland Government
QSA
Queensland Studies Authority
RME
Realistic Mathematics Education
TIMSS
Trends in International Mathematics and Science Study
UK
United Kingdom
UP
University of Pretoria
US
United States
v
List of figures
Figure 2.1
Interplay between content, context and problem-solving skills in ML
29
Figure 2.2
Overview and weighting of topics in Grades 10, 11 and 12
30
Figure 2.3
Framework to observe teachers’ instructional practices
41
Figure 2.4
Shulman’s (1986) three categories of content knowledge
48
Figure 2.5
Grossman’s (1990) four components of PCK
49
Figure 2.6
Borko and Putnam’s (1996) three domains of knowledge
50
Figure 2.7
Ball, Thames and Phelps’ (2005) domains of knowledge of mathematics
51
for teaching
Figure 2.8
Hill, Ball and Schilling’s (2008) domain map for mathematical
52
knowledge for teaching
Figure 2.9
Conceptual framework: Instructional practice, knowledge and beliefs
58
framework of analysis
Figure 3.1
The data collection process
74
Figure 3.2
Elucidation of the character of the lesson observations
75
Figure 4.1
Conceptual framework: Instructional practice, knowledge and beliefs
85
framework of analysis
Figure 4.2
ML instructional practices: Tasks
92
Figure 4.3
ML teachers’ instructional practices: Discourse
93
Figure 4.4
ML teachers’ instructional practices: Learning environment
94
Figure 4.5
ML teachers’ PCK and beliefs: ML learners
98
Figure 4.6
ML teachers’ PCK and beliefs: ML teaching
99
Figure 4.7
ML teachers’ PCK and beliefs: ML curriculum
99
vi
List of tables
Table 2.1
A spectrum of pedagogic agendas
32
Table 2.2
The premises of ML and Mathematics
36
Table 2.3
Learning Outcomes for ML and Mathematics
38
Table 2.4
39
Table 2.5
Comparison of the composition of ML and Mathematics across the
different bands
Different terminology used for teachers’ practices
Table 2.6
Relationship between knowledge and beliefs
46
Table 2.7
Overview of different domains of mathematical knowledge
48
Table 2.8
Some beliefs held by learners and their effects in practice
54
Table 2.9
The observable aspects of a lesson
65
Table 2.10
Teacher-centred versus learner-centred instructional practices
66
Table 3.1
Synopsis of methodology
71
Table 3.2
Criteria justifying inclusion and exclusion in the sample
73
Table 3.3
Elucidation of the character of the interviews
76
Table 3.4
Collection, analysis and reporting data
78
Table 4.1
Timeline of the data collection process
87
Table 4.2
89
Table 4.4
Lesson dimensions and dimension indicators as inclusion criteria for
coding the data
PCK and beliefs dimensions and its indicators as inclusion criteria for
coding the data
Exclusion criteria for coding of the data
100
Table 4.5
Biographical information of the four participants
103
Table 4.6
Summary of Monty’s instructional practice
111
Table 4.7
Summary of Alice’s instructional practice
122
Table 4.8
Summary of Denise’s instructional practice
130
Table 4.9
Summary of Elaine’s instructional practice
140
Table 4.10
Snapshot of the four participants and their instructional practices
142
Table 4.11
Summary of Monty’s PCK and beliefs
160
Table 4.12
Summary of Alice’s PCK and beliefs
168
Table 4.13
Summary of Denise’s PCK and beliefs
174
Table 4.14
Summary of Elaine’s PCK and beliefs
180
Table 4.15
Snapshot of the four participants’ PCK and beliefs
182
Table 4.16
Contradictions between teachers’ beliefs and their instructional practices
191
Table 5.1
Summary of participants’ information
196
Table 5.2
Comparison of the participants’ instructional practices
198
Table 5.3
Summary of verification of research questions
205
Table 4.3
vii
40
95
List of appendices
Appendix A
Letter of consent to the ML learners
Appendix B
Letter of consent to the principals
Appendix C
Letter of permission to the department
Appendix D
Ethical clearance certificate
Appendix E
Observation sheet for observing ML teachers’ lessons
Appendix F
Interview schedule 1 (Prior to lessons 2 and 3)
Appendix G
Interview schedule 2 (Final interview)
Appendix H
List of research studies for Literature Control
Appendix I
Analysis of discussions on Theme 1 and Theme 2
Appendix J
Additional information verifying Question 1
Appendix K
Additional information verifying Question 1
Appendix L
Declaration: External coder
viii
Chapter 1
Introduction and contextualisation
1.1
Introduction
When referring to mathematical literacy1 a clear distinction is required between the international and
national perspectives. Internationally mathematical literacy refers to the competence of individuals
(Christiansen, 2006, p. 6), which ranges from a competence demonstrated in word problems to a critical
or democratic competence and whose purpose may be mathematics as a tool in gaining insights into oppression,
inequalities, and exploitation; … to become aware of the effects of applying mathematical models in society … and a third
component has to do with mathematics as a ‘gate-keeper’, i.e., access to further education (p. 6). In South Africa,
according to the Department of Education (DoE, 2005) mathematical literacy on national level refers
to a fundamental subject where learners are provided with learning opportunities to consolidate and
extend their basic mathematical skills.
South Africa is the first country in the world to have Mathematical Literacy (ML)2 as a school subject
(Christiansen, 2007). This subject was introduced in 2006 as an alternative to Mathematics in the
Further Education and Training band (FET)3. The purpose of this subject is to provide learners with an
awareness and understanding of the role that mathematics plays in the modern world, but also with opportunities to
engage in real-life problems in different contexts (DoE, 2003a, p. 9). The DoE (2003a) defined ML as follows:
Mathematical Literacy provides learners with an awareness and understanding of the role that mathematics plays in
the modern world. Mathematical Literacy is a subject driven by life-related applications of mathematics. It enables
learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically
analyse everyday situations and to solve problems (p. 9).
In the implementation of this relatively new subject teachers are crucial as agents of change. Bearing
this in mind, it is essential to plumb the depths of their knowledge and beliefs regarding the subject in
1
The words mathematical literacy (no capital letters) refer to a competency in applying mathematical knowledge.
Mathematical Literacy (ML) refers to the South African school subject.
3 The FET band includes Grade 10 through to Grade 12.
2
1
order to understand their instructional practices. An overview of the international and national
perspectives on mathematical literacy, the experiences of ML teachers and the existing paucity in the
literature now follows.
1.1.1 International perspective on mathematical literacy
With the emphasis on globalisation and the information explosion in mind, mathematical literacy
should imply the empowerment of learners to meet the demands of living in the 21st century (Gellert,
Jablonka & Keitel, 2001; Queensland Government, 2007b; Skovsmose, 2007). Although different
terminology such as ‘numeracy’ and ‘quantitative literacy’ is also used internationally, Jablonka (2003)
prefers to use the term “mathematical literacy” to focus attention on its connection to mathematics and to
being literate, in other words it refers to a mathematically educated and well-informed individual (p. 77).
International comparative studies such as the Organisation for Economic Co-operation and
Development’s (OECD) programme for International Student Assessment (PISA) and the Trends in
International Mathematics and Science Study (TIMSS) have heightened international awareness of the
value and significance of mathematical literacy. It is not just internationally, but also at national level
that there is a growing concern about learners’ mathematical literacy skills (DoE, 2003a). PISA’s
purpose is to measure how well students can apply their knowledge and skills to problems within real-life contexts
while the purpose of TIMSS is to measure the mathematics and science knowledge and skills broadly aligned with
curricula of the participating countries (National Centre for Education Statistics, 2008). At national level there
is also a growing concern about learners’ mathematical literacy skills (DoE, 2003a).
Jablonka (2003) investigated different international perspectives on mathematical literacy and found
that the perspectives basically differ according to the stakeholders’ underlying principles and values. On
the one hand there are researchers who accentuate the formal application of mathematics by
mathematicians to real-world contexts, demanding a high level of mathematical knowledge and the
competence to use and apply that knowledge (Gellert et al., 2001; Hope, 2007; Jablonka, 2003;
Skovsmose, 2007). Other researchers are persuaded that everyone needs some basic level of literacy to
empower them to make well informed decisions in their daily lives, whether personally, to care for their
families or to contribute meaningfully in their workplace or society (McCrone & Dossey, 2007;
McCrone, Dossey, Turner & Lindquest, 2008; Powell & Anderson, 2007; Skovsmose, 2007).
Internationally mathematical literacy in schools refers to a competency or skill to apply mathematical
knowledge and is embedded in the subject Mathematics.
2
1.1.2 National perspective on mathematical literacy
The results of TIMSS 2003 show how poorly mathematics is understood and conceptualised by
learners in South Africa (Bloch, 2009; De Meyer, Pauly & Van de Poele, 2005). In the past learners who
could not perform well in Mathematics in the General Education and Training (GET)4 band usually
stopped studying Mathematics, thus contributing to a perpetuation of high levels of innumeracy (DoE, 2003a, p. 9).
With that in mind ML for Grades 10-12 was introduced in 2003 and implemented for the first time in
2006.
The Constitution of the Republic of South Africa speaks of human rights, social justice and provides a
basis for transformation and development in South Africa (DoE, 2003a). Mathematics as a discipline,
with its inherent potential to develop critical thinking, is a significant role player in the realisation of the
DoE’s (2003a) vision to create internationally competitive and creative learners and thinkers. Guided by
these statements the DoE’s (2003a) purpose with ML was to introduce a subject that would bring
mathematics to all people and to ensure that citizens of the future are highly numerate consumers of mathematics
(p. 9). The emphasis is on the knowledge needed to be a self-managing person, a contributing worker
and a participating citizen. It is clear from the definition of ML on p. 1 that the focus of ML is on the
applicability of mathematics in everyday life situations.
Teachers play a valuable and important role in ensuring the success of a newly introduced subject such
as ML which depends largely on the teachers’ training, experience, knowledge and perceptions of the
subject. In the DoE’s (2009) Report of the task team for the review of the implementation of the
National Curriculum Statement (NCS), it is emphasised that teachers need absolute clarity on what they are
required to teach (p. 16). Teachers need to regain confidence in their practice, and authority as subject specialists in the
classroom (p. 16). Teachers and schools reported that newly qualified teachers have deficiencies in respect to their
subject or learning area specialisations and it would appear that they often have not been adequately prepared in respect to
appropriate methodologies (p. 55).
1.1.3 The experiences of ML teachers
The intentions and purpose of the DoE (2003a) are admirable, but due to implementation problems,
among other things, not all ML teachers share the DoE’s sentiments. In Sidiropolous’ (2008) study on
the implementation of ML in South African schools, she found that the threat experienced by qualified
mathematics teachers regarding their ‘status identity’ undermines the proper implementation of the
4
The GET band includes Grade R through to Grade 9.
3
subject. She also found that teachers do not understand and value this new curriculum which involves
understanding not only the concept of mathematical literacy but also the nature of mathematics, its
transformative purpose and possibilities (p. 254). Teachers need to understand the sudden shift from content, to
context and content as a process (p. 254).
ML requires a different teaching approach to that of Mathematics as the nature of ML is contextualised
and de-compartmentalised (De Villiers, 2007; North, 2005; Venkat & Graven, 2007; Graven & Venkat,
2007). Researchers are concerned about teachers’ knowledge and competency to use and apply an
approach based on mathematical modelling (Brown & Schäfer, 2006; Glover & King, 2009; Vermeulen,
2007). The focus in research on the issue of teachers’ knowledge and competency in ML should
therefore be on teacher education. ML student teachers should experience practically what it means to
develop an understanding of mathematics in context through, for example, an activity and
investigation-based approach (Brown & Schäfer, 2006; Vithal & Bishop, 2006). Glover and King
(2009) regard teacher professional development as an important component of curriculum reform and
draw attention to issues that need to be addressed such as teachers’ beliefs, self-efficacy and knowledge.
They refer to subject knowledge, pedagogical content knowledge (PCK) and curriculum knowledge that
need to be improved.
1.1.4 Silence in the literature addressed in this study
We have a unique situation in South Africa in that ML refers to a subject and not a skill or competency
per se, as is the case internationally. Although the literature informs this study regarding international
views, purposes and definitions of mathematical literacy as competency or skill, there is a gap in the
literature regarding ML teachers’ knowledge and beliefs. Existing literature on national level mainly
focuses on the vision and purpose of the subject, the curriculum implementation process, curriculum
issues such as the relevancy of the outcomes and assessment criteria, the content-context debate and
which approaches to use. The silence in the literature this study attempts to address concerns ML
teachers’ knowledge and beliefs but also the relationship between their knowledge and beliefs and their
instructional practices. It is this gap that this study aims to fill.
1.2
Rationale for the study
A rationale firstly addresses how the researcher developed an interest in the topic and secondly why the
study is worth doing (Vithal & Jansen, 1997). As far as my personal interest is concerned, I have been
involved in teacher training for the past twenty years, teaching Mathematics to undergraduate student
4
teachers. I took a particular interest in the introduction of ML as subject because I value and appreciate
the subject’s vision, purpose and content. As I am interested in how people in and outside the school
environment experience and view ML, I make a point of talking to people about the topic. Through my
involvement in Teaching Practice which forms part of the undergraduate students’ curricula at the
University of Pretoria (UP) and being a member of the data collection team for the FET
Implementation Project, I have been required to visit many local schools where I also have had the
opportunity to talk to principals, teachers and learners. What I learnt and experienced is not at all what
I had anticipated.
As far as the possible value of the study is concerned, I regard it as imperative to conduct a study which
focuses on ML teachers’ instructional practices, as the findings from the study will suggest ways to
improve current teachers’ practices and will also inform and enrich my own teacher training practice.
Koellner et al. (2007) believe that to achieve the vision for school mathematics, no factor is more
important than the teacher. A teacher requires a sound knowledge of subject content, principles and
strategies, needs to believe in the potential of the subject and the learners, and should maintain a
positive attitude. It is also the teacher who is to enthuse and motivate the learners about mathematics
and the role it plays in their lives. Through the study I purpose to contribute to our understanding of
how the subject is currently taught in South African schools and what the nature of the ML teachers’
knowledge and beliefs is, as well as to help people gain insight into the value of ML. Awareness needs
to be created of ML as an important subject and the necessity of having knowledgeable and positive
teachers teaching this subject. As long as negative perceptions prevail regarding any subject, that subject
will not be taken seriously and will not fulfil its purpose. Through the findings of the study I endeavour
to show that ML is not a subject with a lower status, but is a different subject with a different emphasis
and different requirements compared with Mathematics.
1.3
Statement of the problem
Many learners, teachers and parents have negative attitudes towards ML and regard it as an inferior
subject. According to Mbekwa (2007) some teachers regard the subject as a dumping ground for
mathematics underperformers. ML learners are ridiculed for having to take a subject that is considered
a waste of time. ML is a relatively new subject in which an entirely different teaching approach is
required: mathematical content should be taught in terms of real-life situations. For this to occur
successfully, specific skills are required of the ML teacher. This study aims to address the problem
concerning ML teachers’ instructional practices. There are three groups of ML teachers: a) Mathematics
teachers who teach ML in the same way that they teach Mathematics; b) non-Mathematics teachers
5
who in many cases lack the necessary mathematical knowledge, skills and beliefs to teach ML
competently; and c) Mathematics teachers who adapted their practices to teach ML using different
approaches than those required for teaching Mathematics.
1.4
The purpose of the study
Curriculum developers and teachers are still in the process of addressing implementation problems and
determining the required standard of the subject. Limited in-depth research has been done concerning
the ML teachers, what they believe and what knowledge is required to teach this subject effectively and
proficiently. The purpose of this study is to investigate the way in which ML is taught in a limited
number of classrooms with the view to establishing the relationship between ML teachers’ knowledge
and beliefs and their instructional practices. To accomplish this aim, an in-depth study will be
conducted to explore the nature of teachers’ knowledge and beliefs of mathematics and ML as
manifested in their instructional practices. These findings will then be used to investigate the possible
implications thereof for teacher training and theory building.
1.5
Research questions
With the rationale, statement of the problem and purpose as background, the following research
questions were formulated:
Main question:
What is the relationship between Mathematical Literacy teachers’ knowledge and beliefs and their
instructional practices?
Subquestions:
1.
How can ML teachers’ instructional practices be described?
2.
What is the nature of ML teachers’ knowledge and beliefs?
3.
How do ML teachers’ knowledge and beliefs relate to their instructional practices?
4.
What are the possible implications of the findings from Questions 1, 2 and 3 for teacher
training?
5.
1.6
What is the value of the study’s findings for theory building in teaching and learning ML?
Methodological considerations
This study was initiated by my interest in the relationship between teachers’ knowledge and beliefs and
their instructional practices. To answer the research questions a qualitative research approach will be
6
used as it concerns specific meanings, emotions and practices that emerge through the interactions and interdependencies
between people (Hogan, Dolan & Donnelly, 2009, p. 4). The research design is a case study as it observes
effects in real contexts, recognising that context is a powerful determinant of both cause and effect (Cohen, Manion &
Morrison, 2001, p. 181). My research paradigm is social constructivism5 and is based on the
epistemological assumptions that social life is a distinctly human product and that human behaviour is affected by
knowledge of the social world (Nieuwenhuis, 2007, p. 59-60). This study is subjective in nature with the
nominalist position as ontological assumption: reality is understood through words and is the product
of individual consciousness (Cohen et al., 2001). Observations and interviews serve as data collection
techniques to enable me to interpret the reality by becoming part of the lives of the teachers. The data
are analysed according to the categories identified in the conceptual framework.
1.7
Definition of terms
The following are operational definitions of terms used in this study:
•
Beliefs: This term refers to a viewpoint or a way of thinking, or even a preconceived idea a person
holds. Beliefs can be interpreted as mental constructs that represent the codification of people’s experiences and
understandings (Schoenfeld, 1998, p. 19). Beliefs about teaching and learning can be located on a
perspective continuum from traditional (instrumentalist view), to formalist (Platonist view), to a
constructivist perspective (problem solving) (Dionne 1984; Ernest, 1988).
•
Contextualised mathematics: This term is similar to realistic mathematics education (RME) where
mathematics is seen as a human activity that is connected to reality and relevant to society. RME is
founded upon the principles of using real-world contexts, bridging the gap between abstract and
applied mathematics, allowing learners to develop their own problem-solving strategies, and
making connections to other disciplines (Freudenthal as cited in Van den Heuvel-Panhuizen,
1998).
•
Instructional practice: This term refers to the qualitative dimensions of teacher behaviour
regarding their teaching. These dimensions involve teachers’ abilities to model cognitive strategies in
meaningful and purposive activities, promote classroom dialogues, adjust instruction as required and establish
classroom communities in which students collaboratively and cooperatively participate in enquiry-related activities
(Englert, Tarrant & Mariage, 1992, p. 62). A framework used to observe and describe teachers’
instructional practices is built on three observable aspects of mathematical lessons namely tasks,
discourse and the learning environment (Artzt, Armour-Thomas & Curcio, 2008).
•
5
Learners: This term refers to school learners.
Social constructivism is discussed under Section 3.2.1: Research paradigm.
7
•
Pedagogical content knowledge: This term refers to the knowledge teachers need in the teaching
profession that goes beyond having mathematical content knowledge (MCK) only. PCK includes
knowledge of what learners do not understand, why they do not understand it and what can be
done to rectify the situation. It is the knowledge needed to notice, predict and understand learners’
misunderstandings and to assist and guide them to better understanding (Ball, Thames & Phelps,
2005; Shulman, 1986).
•
Productive practice: A practice where the teacher listens to learners’ mathematical thinking and
aims to use it to encourage conversation that revolves around the mathematical ideas in the
sequenced problems (Franke, Kazemi & Battey, 2007, p. 226).
1.8
Possible contribution of the study
Although workshops have been offered by the DoE and papers have been published and presented by
national academics on various issues concerning the implementation of ML, there is no evidence of indepth empirical research that has been conducted on ML teachers’ knowledge and beliefs and the
relationship between ML teachers’ knowledge and beliefs and their instructional practices. This study
will thus contribute to this new field and fill the gap in literature. The study further contributes to ML
theory and practice as the findings will be implemented in undergraduate teacher training programmes
at the University of Pretoria. Furthermore it is important to contribute to the vision and success of the
DoE’s (2003) endeavour to change the current situation where South Africa’s adult population has a
very low level of literacy and mathematical proficiency. To accomplish this, teachers as well as the
community need to realise the place and value of ML in the school curriculum as well as the need for
ML teachers to build their own ‘status identity’. On a more personal level, this study will also contribute
to the development and enrichment of my own practice of preparing ML student teachers. The
personal experiences and findings will provide me with a broader and deeper knowledge and
understanding of the subject ML, of ML teachers’ knowledge and beliefs, and to what extent these
knowledge and beliefs relate to their instructional practices.
1.9
Limitations of the study
The case study, like any other research method, has its weaknesses. In this study a limitation is that the
primary data are gathered from a relatively small number of teachers who will be observed on three
different occasions and will be interviewed three times. Cohen et al. (2001) mentioned that results may
not be generalisable except where other readers/researchers see their application (p. 184). It will therefore not be
possible to generalise the findings, but one may still acknowledge the value of the findings obtained
8
from rich in-depth involvement in specific cases as they provide insights into other, similar situations and cases
(p. 184).
During the data gathering process the Hawthorne effect will be taken into account as teachers naturally
try to impress an observer in class or an interviewer during the time of observation and interviewing.
To a certain extent they may even feel threatened by being observed during their lesson presentations,
despite my reassurances. As they do have busy schedules, some may experience it as extra work and an
intrusion into their privacy and they may not be as dedicated to the project as would be ideal. Despite
the normal human preconceived ideas and perceptions, I will try to guard against being biased and
selective and try to be objective during the data gathering and analysis stages in an effort to avoid the
halo effect during the data analysis. Cohen et al. (2001) describe the halo effect as a cognitive bias in
which the researcher’s knowledge or perception of the person or situation exerts an influence on
subsequent judgement. The fact that more than one data collection technique will be used minimises
this problem. Another way these limitations are minimised is to properly prepare the teachers before
the commencement of the data collection process and the establishment of a positive relationship with
them, emphasising the value of honest and true data and their anonymity during the whole process.
1.10
Summary
This chapter provides an overview of the problem and rationale for the study, the research questions,
methodological considerations and the possible contribution and limitations of the study. The different
meanings attached to mathematical literacy both internationally and nationally have been discussed.
Internationally mathematical literacy refers to the competence of individuals (Christiansen, 2006, p. 6)
whereas nationally mathematical literacy mainly refers to a subject involving mathematical skills for
solving contextual problems. The subject was introduced in 2006 and schools are experiencing a lack of
qualified teachers. The purpose of this study is to investigate, by means of a case study, the way in
which ML is taught with the view to determining the relationship between ML teachers’ knowledge and
beliefs and their instructional practices.
1.11
The structure of the thesis
The thesis consists of five chapters. Chapter 1 is summarised above. Chapter 2 provides an in-depth
analysis of the findings in the relevant literature and also explains the conceptual framework on which
the study is based. In Chapter 3 details regarding the methodology of the study are given. The data
analysis strategies are discussed as well as the trustworthiness and ethical considerations of the study.
9
Chapter 4 includes the presentation of findings from the data obtained through class observations and
interviews conducted with the ML teachers. The findings are also analysed and discussed according to
the research questions based on the conceptual framework and the literature and trends are identified
and explained. Chapter 5 contains the conclusion and implications and comprises a chapter summary,
verification of the research questions, a reflection on the study, the conclusions, recommendations and
limitations of the study.
10
Chapter 2
Literature review and conceptual
framework
2.1
Introduction
This literature study is a critical and integrative synthesis of various researchers’ findings, justifying this
research endeavour. It is imperative to remember that South Africa is the only country offering ML as a
compulsory alternative to Mathematics in Grades 10 to 12. As the study concerns the ML teachers and
the relationship between their knowledge and beliefs and their instructional practices, the literature
review begins with a comparison of the international and national perspectives of mathematical literacy.
Comparisons are made between the different conceptions of mathematical literacy; the contexts in
which mathematical literacy can be applied; international studies measuring learners’ mathematical
knowledge and literacy skills; meanings and definitions of mathematical literacy; and the role
mathematical literacy plays in some school curricula. Following the review on mathematical literacy is a
discussion of the meaning of teachers’ instructional practices and the value of various approaches to
teaching. Moving to the core of the problem, literature regarding teachers’ knowledge and beliefs about
the subject they teach are discussed. Attention is given to the different domains of teachers’ knowledge,
teachers’ belief systems and the relationship between their knowledge and beliefs and their instructional
practices. The literature review concludes with the conceptual framework which is based on concepts
and theories from relevant work in the literature6.
2.2
Mathematical literacy
Mathematical literacy is not a clearly defined term and internationally there exists a range of different
conceptions of mathematical literacy that are discussed in this section. As mathematical literacy (ML) is
a school subject in South Africa, it is important to understand the motivation and purpose of ML in the
South African curriculum and to compare it with the role mathematical literacy plays internationally.
6
Several direct quotations are used in the literature review to avoid nuance chances of meaning to the matter under
discussion.
11
2.2.1 International perspectives on mathematical literacy
In this section I mention the different terminology being used for mathematical literacy, compare
different conceptions of mathematical literacy, discuss different contexts in which mathematics could
be applied and refer to some international comparative studies that measure learners’ mathematical
literacy skills in order to derive a general meaning or definition of mathematical literacy.
There is an expanding body of literature that uses the terms “mathematical literacy” and “numeracy” as
synonyms (Jablonka, 2003). The National Council on Education and the Disciplines however uses the
term “quantitative literacy” to stress the importance of enquiring into the meaning of numeracy in a society that
keeps increasing the use of numbers and quantitative information (Jablonka, 2003, p. 77). Jablonka prefers to use
the term “mathematical literacy” to focus attention on its connection to mathematics and to being literate, in other
words to a mathematically educated and well-informed individual (p. 77).
In a comprehensive study by Jablonka (2003) in which different international perspectives on
mathematical literacy were investigated, she found that the perspectives basically differ according to the
stakeholders’ underlying principles and values. In her opinion there is a direct connection between a
conception of mathematical literacy and a particular social practice. She acknowledges the difficulty of
pointing out the distinct meaning of mathematical literacy as it varies according to the culture and context
of the stakeholders who promote it (p. 76). The different conceptions of mathematical literacy relate to a
number of relationships and factors. One of the relationships is between mathematics, the surrounding
culture, and the curriculum (p. 80) while another is between school mathematics and out-of-school mathematics as
mathematical literacy is about the individual’s ability to use the mathematics they are supposed to learn at school
(p. 97). Varying with respect to the culture and the context four possible perspectives of mathematical
literacy are:
•
•
•
•
The ability to use basic computational and geometrical skills in everyday contexts.
The knowledge and understanding of fundamental mathematical notions.
The ability to develop sophisticated mathematical models.
The capacity for understanding and evaluating another’s use of numbers and mathematical models (p. 76).
With the above-mentioned perspectives as background the different conceptions of mathematical
literacy as found in the literature will subsequently be categorised.
2.2.1.1
Different conceptions of mathematical literacy
The literature revealed different conceptions of mathematical literacy but the resemblances between
mathematical literacy and RME, mathematisation, mathematical modelling, as well as mathematics in
12
action are most evident. As there is opacity as to what each of these conceptions entail and how they
differ a clarification of the concepts and notions will be provided.
Realistic Mathematics Education
Hope (2007) expressed the resemblance of mathematical literacy with the theory of RME. RME uses a
theoretical framework that relies on real-world applications and modelling, a didactical belief propagated by Hans
Freudenthal (Gates & Vistro-Yu, 2003, p. 67). According to Van den Heuvel-Panhuizen (1998),
Freudenthal and his colleagues laid the foundations of RME in the early seventies to address the worldwide need to reform the teaching and learning of mathematics and to move away from mechanistic
mathematics education. Freudenthal’s theory of RME rests upon the following five components:
•
•
•
•
•
Using a real-world context as a starting point for learning.
Bridging the gap between abstract and applied mathematics by using visual models.
Having students develop their own problem-solving strategies rather than memorise rules and procedures.
Making mathematical communication, perhaps in the form of journaling or oral presentations, an integral part of
the lesson.
Making connections to other disciplines using meaningful real-world problems (Hope, 2007, p. 30).
Hope (2007) further believes mathematical literacy is a matter of the appropriate pedagogy that should
be used in teaching mathematics. According to these fundamental pedagogical aspects of teaching
mathematics, it is comprehensible that the traditional school mathematics instruction is too formal, less
intuitive, more abstract, less contextual, more symbolic, and less concrete than the type of instruction
that would expand student thinking and develop mathematical literacy (p. 30).
Mathematisation
Freudenthal believed that the focus should not be on mathematics as a closed system, but on the activity, on the
process of mathematisation (Van den Heuvel-Panhuizen, 1998), and that mathematics should be seen as a
human activity that is connected to reality and relevant to society. Treffers (1978) formulated the idea
of two types of mathematisation, namely horizontal and vertical mathematisation. He stated that in
horizontal mathematisation the students come up with mathematical tools which can help to organise and solve a problem
located in a real-life situation whereas vertical mathematisation is the process of reorganization within the mathematical
system itself, like, for instance, finding shortcuts and discovering connections between concepts and strategies and then
applying these discoveries (Van den Heuvel-Panhuizen, 1998). Freudenthal (1991) explained horizontal
mathematisation as going from the world of life into the world of symbols, while vertical mathematisation means moving
within the world of symbols (p. 24). It would seem that vertical mathematisation refers to the more formal
mathematics while horizontal mathematisation refers to the informal mathematical literacy part.
13
According to Hope (2007) mathematising is a term used by The Organisation of Economic Cooperation and Development (OECD) which involves five elements:
•
•
•
•
•
Starting with a problem whose roots are situated in reality.
Organising the information and data according to mathematical concepts.
Transforming a real-world, concrete application to an abstract problem whose roots are situated in mathematics.
Solving the mathematical problem.
Reflecting back from the mathematical solution to the real-world situation to determine whether the answer
makes sense (p. 29).
Mathematical modelling
ML bears a strong resemblance to mathematical modelling in that both require an application of Polya’s
four basic steps in problem-solving namely: a) understanding the problem; b) designing a plan; c)
carrying out the plan; and d) looking back on the problem. Mathematical modelling can further be
described as a matter of constructing an idealised, abstract model which may then be compared for its degree of similarity
with a real system. (Giere, 1999, p. 50).
Gellert et al. (2001) use mathematical literacy as a metaphor referring to well-educated and wellinformed individuals. According to them different conceptions of mathematical literacy are based on
the relationship between mathematics, reality and the society. Their concept of mathematical literacy
involves gaining a level of mathematical understanding that goes beyond the minimal abilities of calculating, estimating,
and gaining some number sense, and basic geometrical understanding ... by seeing the power of mathematics in its potential
of abstracting from concrete realities by generating concepts and structures for universal application (p. 59). They further
believe these abilities can be developed by experiencing mathematical modes of thinking, such as searching for
patterns, classifying, formalising and symbolising, seeking implications of premises, testing conjectures, arguing, and
thinking propositionally (p. 59) which form the basis of mathematical modelling. Mathematical literacy
requires the mathematical competence to understand the mathematical methods involved and the analytical competence to
demystify the justifications for specific mathematical applications as well as to assess their consequences (p. 66).
Mathematics in action
Although some of the above-mentioned conceptions are formal, involving higher-order mathematical
skills, there are other researchers who regard mathematical literacy as a fundamental requirement for all
people, recognising its essential value to learners in contexts forming part of their everyday living
(McCrone & Dossey, 2007; Powell & Anderson, 2007; Skovsmose, 2007). McCrone and Dossey (2007)
believe mathematical literacy is not about studying higher levels of formal mathematics, but about making
mathematics relevant and empowering for everyone (p. 32). They further call for mathematics to play an even
14
greater part in non-mathematics classes where teachers promote the mathematics embedded in their
subjects.
Skovsmose (2007) refers to mathematical literacy as mathematics in action and considers the role of
mathematical literacy in both mathematicians’ and non-mathematicians’ lives. He based his study on
two types of literacy being either functional or critical, terms introduced by Apple in 1992. Functional
literacy is defined by competencies a person possesses to fulfil a particular job function (p. 4) whereas
critical literacy addresses themes such as working conditions and political issues. Skovsmose prefers to
talk about reflective knowledge with respect to mathematics instead of critical literacy. Reflective
knowledge refers to a competence in evaluating how mathematics is used or could be used (p. 4). Critical literacy is
associated with the skill to create or design models using mathematics whereas functional literacy is the
skill to use and apply those models. To make unambiguous distinctions between these two types is not
that simple and it could have very different interpretations depending on the context of the learner (p. 4).
Clarification of basic concepts and notions
From the discussion above it is clear that the lines between the different concepts such as
mathematisation and mathematical modelling are blurred. Blum and Niss (2010) provided a clarification
of the different concepts and notions when they described the process of applied problem solving. The
process of applied problem solving commences with a real problem situation and through a process
of simplification, idealisation and structuring of the situation, the process ends with a real model of
the original situation. Through the process of mathematisation, the real model is translated into
mathematics. Mathematisation is therefore the process of converting the data, concepts, relations, conditions
and assumptions (p. 208) of the real model into a mathematical model of the original situation.
Mathematical modelling is the entire process leading from the real problem situation to the
mathematical model. Then the mathematical model must be processed to obtain certain mathematical
results. This includes mathematical activities such as drawing conclusions, calculating and checking concrete
examples, applying known mathematical methods and results as well as developing new ones etc. (p. 208). The next
process is to retranslate the results into the real world, i.e. to be interpreted in relation to the original
situation (p. 208). The model is then validated and if discrepancies of any kind occur, they may lead to
the modification of the model or replacement of the model by going through the process cycle more
than once.
Summary
In the light of the above discussion of the different conceptions from the literature, mathematical
literacy cannot adequately be described in terms of skills only, as it involves mathematical problems in
15
contexts that require attributes such as conceptual understanding of formal mathematical knowledge
and problem-solving skills (Gellert et al., 2001). Gellert et al. also believe that the differences between
various conceptions of mathematical literacy consist of the problems to which mathematics is applied (p. 61).
Hope (2007) on the other hand believes mathematical literacy is a matter of the appropriate pedagogy
that should be used in teaching mathematics. All mathematics learners should therefore be provided
with the opportunity to apply their knowledge and logic to real-world situations that form part of their
daily lives. Mathematical literacy implies bridging the gap between abstract and applied mathematics
where the contexts and degree of complexity differ. Jablonka’s (2003) question of: mathematical literacy
for what?, further calls attention to the need for discussing the different contexts in which mathematics
could be applied to further explicate the purpose of mathematical literacy.
2.2.1.2
Some contexts in which mathematical literacy can be applied
Prescribing the different contexts in which mathematical literacy can be applied is as complicated as
conceptualising mathematical literacy. In this section the different categories of contexts are discussed
as well as the role technology plays in determining these contexts.
Context categories according to stakeholders’ demands
Contexts in which mathematical literacy can be applied depend on the stakeholders’ philosophy, view
or principles and could be guided by, among other things, some socio-economic demands (Jablonka,
2003). Jablonka categorised the different, and in some cases, conflicting contexts in which mathematics
can be applied as mathematical literacy for:
•
Developing Human Capital – looking at the world through mathematical eyes where higherorder mathematical skills are applicable, where mathematics is not regarded as culture-bound
and value-driven.
•
Cultural Identity – incorporating ethno-mathematical practices to avoid privileging of Western
academic mathematical knowledge.
•
Social Change – to uncover and communicate aspects of social or political nature (such as
unemployment, life expectancy, national income) in an attempt to overcome the dominance of
academic mathematics in the curriculum.
•
Environmental Awareness – the mathematical content comprises arguments underpinned by
mathematical visualisations, qualitative mathematics that is characterised as not aiming at an
analytical solution but serving as thought experiments and computational mathematics, which
include the use of simulation packages, graphing calculators and spreadsheets.
16
•
Evaluating Mathematics - includes reasoning with condensed measures and indexes, formalising
transactions, reasoning with platonic models, constructing surface-models and numerology.
An example of conflict between some of these conceptions Jablonka referred to is where the
application of mathematics in the Cultural Identity is restricted to contexts situated in a specific culture,
where in the Environmental Awareness the focus is on applying mathematics to contexts of global
nature. Then there is the complex problem of Cultural Identity, where Ethno-mathematics is suggested
as a necessity for addressing cultural conflicts in the classroom. Knoblauch’s (1990) categories are
similar to Jablonka’s, speaking of literacy for professional competence in a technological world, for civic
responsibility and the preservation of heritage, for personal growth and self-fulfilment, and for social and political change
(p. 76). Different contexts in which mathematical literacy can be applied can also be categorised
according to some processes involved in applying mathematics in real-world contexts.
Context categories according to processes
By categorising the content in four processes, Skovsmose (2007) illustrated how mathematics in action can
operate in powerful ways, and power can be exercised through mathematics in action (p. 8). The following categories
show the different conceptions of mathematical literacy categorised as critical and functional literacy as
discussed in par. 2.2.1.1.
•
Construction – includes systems of knowledge and techniques, by means of which technology, in the broadest
interpretation of the term, is maintained and further developed (p. 8) for example in the construction of
the computer.
•
Operating – bringing technology into operation in work practices and job functions. The operator may not be
aware of the mathematical content of the procedures he or she performs (p. 11) for example, ticket
reservations in the travel industry and procedures for buying and selling houses.
•
Consuming – as citizens we are the consumers who need to listen to statements from experts
that are expressed everyday on television and in the newspapers, for example numbers and figures
concerning elections, the economy, exchange rates and investments are mixed with advertising of any number of
special offers (p. 13).
•
Marginalising – a steady growth of favela-like neighbourhoods gloomily testifies that free-growing globalised
capitalism is not an inclusive economy. Instead it marginalises in great measures people as being disposables.
Examples include drugdealing, selling of sunglasses, lighters and other items possible to carry around along
the streets where cars come to a stop (p. 14).
Skovsmose (2007) concluded that mathematical literacy could be either functional or critical but that the
distinction is difficult to maintain, is vague, maybe illusive (p. 17). In many of these processes technology plays a
significant role in the process of context selection.
17
The role of technology
Gellert et al. (2001) pointed out that technology has taken over many processes in society where highly
skilled mathematicians develop mathematical models and processes which people generally do not need
to understand or even be aware of. There is little need for the majority of people to learn more mathematics
in a more successful way as it is based more on common sense than on rational reason or on justifiable evidence (p. 58).
The result is then an increasing mathematisation of our society [which] is complemented by an increasing
demathematisation of its individual members (p. 58). In comparing their view with Skovsmose’s (2007)
categories stated in the preceding paragraph, it is a minority of people who use their advanced
mathematical literacy skills to construct or bring technology into operation while the majority of people
use their basic mathematical skills to operate and consume.
2.2.1.3
Studies measuring learners’ mathematical literacy skills
There are various studies measuring learners’ mathematical knowledge and skills, but in this study only
two international comparative studies will be considered, namely the Organisation for Economic Cooperation and Development’s (OECD) Programme for International Student Assessment’s (PISA) as
well as TIMSS.
The foci of PISA and TIMSS
Every three years PISA assesses 15-year-olds’ reading, mathematical and scientific literacy in different
countries. Its purpose is to measure how well students can apply their knowledge and skills to problems within reallife contexts. PISA is designed to represent a ‘yield’ of learning at age 15, rather than a direct measure of attained
curriculum knowledge (National Centre for Education Statistics, 2008b, p. 3). In 2003, when 45 countries
participated, the focus was on mathematics. Every four years TIMSS assesses fourth- and eighthgraders’ mathematics and science performance in different countries (58 in 2007) to compare U.S.
learners’ performance with that of their peers in other countries. This study’s purpose is to measure the
mathematics and science knowledge and skills broadly aligned with curricula of the participating countries (National
Centre for Education Statistics, 2008a, p. 5). The two studies differ in focus as TIMSS seeks to find out
how well students have mastered curriculum-based scientific and mathematical knowledge and skills whereby the
purpose of PISA is to assess students’ scientific and mathematical literacy, that is, their ability to apply scientific and
mathematical concepts and thinking skills to everyday, non-school situations (Nohara, 2001, p. 11). In the next
section attention is given to PISA’s definition of mathematical literacy and the criteria used in assessing
learners’ mathematical literacy skills.
18
PISA
According to the OECD (2004, p. 37), the content of school mathematics and science in the last
decade was chosen to provide the foundations for the professional training of a small number of mathematicians,
scientists and engineers. With the increased emphasis on the application value of science, mathematics and
technology in modern life to all adults, the objectives of these three subjects changed to personal
fulfillment, employment and full participation in society. Mathematical literacy is concerned with the capacity of
students to analyse, reason and communicate effectively as they pose, solve and interpret mathematical problems in a variety
of situations involving quantitative, spatial, probabilistic or other mathematical concepts. PISA’s (OECD, 2003)
definition of mathematical literacy is:
the capacity to identify, to understand and to engage in mathematics and make well-founded judgement about the role
that mathematics plays, as needed for an individual’s current and future life, occupational life, social life with peers
and relatives, and life as a constructive, concerned and reflective citizen (p. 20).
PISA was originally designed to measure the extent to which learners can apply their mathematical
knowledge in realistic, everyday life situations. It involves the ability to analyse situations in content areas
involving quantity, shape and space, change and relationships and uncertainty (McCrone et al., 2008, p. 35). In
order to be able to assess mathematical literacy, PISA (OECD, 2003) identified three broad criteria to
be used, namely:
•
The content of mathematics – in terms of clusters of relevant, connected mathematical
concepts that appear in real situations and contexts. These include quantity, space and shape,
change and relationships, and uncertainty.
•
The process of mathematics – different skills needed for mathematics such as reproduction –
simple computations; connections – using of ideas and procedures to solve straightforward
and familiar problems; and reflection – using of mathematical thinking, generalisations and
insight to engage in analysis, identify mathematical elements in a situation, formulate questions
and search for solutions.
•
The contexts in which mathematics is used – the kinds of problems encountered in real life vary
in terms of distance from individual, from effecting one directly regarding private life, school
life, work and sports, local community and society and scientific, to scientific problems of more
general interest.
From PISA’s definition and assessment criteria it is clear that the focus is not just on applying routine
procedures but to become cognitively involved in mathematical thought, using and applying formal
mathematics to solve real-life problems. The contexts should involve realistic day-to-day situations
involving people’s personal, occupational and social lives enabling them to become reflective citizens.
19
Bearing in mind the conceptions, contexts and meanings of the concept mathematical literacy discussed
above, I will subsequently discuss the role mathematical literacy plays in the FET school curricula in
Australia and briefly mention the situations in the United Kingdom (UK) and United States (US).
2.2.1.4
Defining mathematical literacy
When referring to mathematical literacy or numeracy, many people in and outside the academic field
tend to believe that only basic mathematical skills are involved or that numeracy refers only to primary
school learners’ mathematics. To conceptualise or define mathematical literacy is unfortunately not that
simple and is far from being well-defined (Gellert et al., 2001; Jablonka, 2003; Skovsmose, 2007).
Different conceptions or definitions are held by researchers ranging from informal mathematics
requiring basic mathematical skills (McCrone & Dossey, 2007; McCrone et al., 2008; Powell &
Anderson, 2007; Skovsmose, 2007) to formal mathematics involving higher-order thinking skills
(Gellert et al., 2001; Hope, 2007; Jablonka, 2003; Skovsmose, 2007). Gellert et al. (2001) pointed to
especially primary school teachers who regard mathematical literacy as informal, defining mathematical
literacy as survival mathematics for all (p. 68) with the exact purpose to propagate that mathematics is not
just formal and complicated, but can be useful and beautiful to all people. Skovsmose (2007) believes
that mathematical literacy can be related to notions such as autonomy, empowerment and globalisation,
whereas Hope (2007) presumes it implies that a person is able to reason, analyse, formulate, and solve problems in
a real-world setting (p. 29).
Gellert et al. (2001) and Jablonka (2003) perceive mathematical literacy in terms of higher-order
mathematical skills. Jablonka is of the opinion that any attempt to define mathematical literacy faces the
problem that it cannot be conceptualised exclusively in terms of mathematical knowledge, because it is about an
individual’s capacity to use and apply this knowledge (p. 78). She defined mathematical literacy as a bundle of
knowledge, skills and values that transcend the difficulties arising from cultural differences and economic inequalities
because mathematics and mathematics education themselves are not seen as culture-bound and value-driven (p. 81). She
conceptualises mathematical literacy in terms of higher-order mathematical skills (p. 97) that are applicable
to all kinds of contexts.
Although a definition of mathematical literacy is elusive, a golden thread running through all attempts
to define mathematical literacy is that mathematical literacy is a valuable competence or skill a person
possesses to put mathematics to work in solving real-life contextual problems. With the emphasis on
globalisation and the information explosion in mind, mathematical literacy should imply the
empowerment of learners to meet the demands of living in a 21st century (Gellert et al., 2001;
20
Queensland Government, 2007b; Skovsmose, 2007). It is informative to investigate the current
situation regarding mathematical literacy in some international school curricula.
2.2.1.5
The role of mathematical literacy in some international school
curricula
Instead of using the term “mathematical literacy” when referring to the competency of applying
mathematical knowledge to life-related problems, Australia and the UK generally refer to the terms
“numeracy”, while the US refers to “quantitative literacy”. A discussion regarding the role mathematical
literacy plays in Australia’s school curricula subsequently follows. I then briefly mention the situation in
the UK and US.
AUSTRALIA
In 2008 all Australian regional governments agreed that instead of the eight different arrangements,
only one national curriculum is to be implemented in 2013, which should play a key role in delivering
quality education (Australian Curriculum, Assessment and Reporting Authority, n.d). Queensland is the
second largest region and a study of the Education Department of Queensland provided insight into
the role numeracy plays in the education system of Australia.
According to the PISA 2003 results when the focus of the study was on mathematics, Australia came
12th out of 41 countries (OECD, 2004). In the TIMSS 2007 they came 14th out of the 58 participating
countries for both Grade 4 and Grade 8 learners (National Center for Education Statistics, 2008). For
the past few years the raising of the numeracy levels of Australian learners received serious attention.
There are numerous documents and guidelines available to teachers on how to develop learners’
numeracy skills in the Mathematics classroom. There are also fact sheets available to parents with
information on numeracy, providing some guiding principles on how to support their children in their
numeracy development.
In a document called Numeracy: Lifelong Confidence with Mathematics - Framework for Action
2007 – 2010, which serves as an action plan to improve numeracy education, the Minister for
Education and Training declared that the Queensland Government (QG) recognises numeracy as a key
pillar of learning and an essential component (QG, 2007b, p. 1) of their curriculum. He also said that teachers
have an important role to play in helping learners to become confident appliers of mathematics in their
everyday lives. A Queensland Certificate in Education (QCE) is awarded at the end of Year 12 to a
person who, in addition to achieving 20 credits in the required pattern of learning has met the
requirements for literacy and numeracy. Learners can meet QCE numeracy requirements by satisfying a
21
number of possible options including a sound achievement in one of their three Mathematics subjects in
school or passing a short course in numeracy developed by the Queensland Studies Authority (QSA, 2009b, p. 1).
Numeracy is clearly an important component in the Queensland school curriculum, but there is no
indication or description of a connection between Mathematics and numeracy in this curriculum.
Mathematics and numeracy
The QSA (2009a) provided a clear explanation of Mathematics and numeracy and said the focus of
Mathematics is on the development of learners’ knowledge and ways of working in a range of situations from real
life to the purely mathematical where numeracy refers to the confident use of mathematical knowledge and problem-solving
skills not only in the Mathematics classroom, but across the school curriculum and in everyday life, work or further
learning (p. 9). In the Queensland Government’s (QG, 2007b) definition of numeracy it is stated that to
be numerate is to use mathematics effectively to meet the general demands of life at home, in paid work, and for
participation in the community and civic life (p. 2). Mathematics and numeracy are interrelated and it is the
responsibility of the Mathematics curriculum to introduce and develop the mathematics which underpins the numeracy
(QSA, 2009a, p. 9). As numeracy refers to the ability to use mathematics in solving life-related
problems, it is essential to determine the contexts in which mathematics could be applied and what the
role of the teacher is in developing learners’ skills in this regard.
The context and teaching of numeracy
In Year 10 to 12 the numeracy work learners relate[s] to a specific context across a broad range of work and study
options (QG, 2007a) and involve:
•
•
•
Applying mathematical skills in new contexts such as: 1) analysing data to inform decision making; 2) deciding
to estimate or calculate an answer depending on the purpose; 3) calculating dimensions and quantities of
materials in vocational tasks such as construction or hospitality.
Selecting, sequencing and evaluating information to understand texts and to communicate with other people.
Using particular communication skills needed to effectively participate in the workplace such as industry terms
and customer services (QG, p. 1).
Teachers are the key role players in selecting contexts relevant to the learner. They need to recognise
numeracy demands and opportunities within the curriculum (QG, 2007b, p. 10) enabling learners to develop
their numerical knowledge, skills and confidence. Teachers should intentionally create opportunities in
which learners can, among other things, explore mathematical ideas with concrete or visual
representations and hands-on activities; experience practical and contextualised learning; communicate
about mathematical issues; develop calculator and computer skills and use multiple solution strategies
(QSA, 2006). According to the Queensland Government (QG, 2007b) teachers’ understanding of
mathematics content needs to be developed with respect to the nature of mathematics as a discipline; the
22
mathematics topics they teach; the relationship of those topics to further learning and everyday life; the impact of
information and communication technologies on the teaching and learning of mathematics (p. 4).
UNITED KINGDOM (UK)
England did not participate in the 2003 or 2006 PISA study, but performed very well in TIMSS 2007,
taking the 7th position for both the 4th and 8th graders out of the 58 participating countries (National
Center for Education Statistics, 2008a). By law all children between ages 5 and 16 must receive a fulltime education. The UK introduced a National Curriculum in 1992 to which state schools need to
adhere until learners reach the age of 16. National Curriculum core subjects are: English, Mathematics
and Science which are offered at different levels.
The UK national curriculum
Within the framework of the National Curriculum, schools are free to plan and organise teaching and
learning in the way that best meets the demands of their pupils. The Qualifications and Curriculum
Development Agency (QCDA) provides guidelines and assistance in this regard. The National
Curriculum is organised in four key stages: Key Stage 1 (5-7 years) and Key Stage 2 (7-11 years) form
part of the Primary curriculum while Key Stage 3 (11-14 years) and Key Stage 4 (14-16 years) form part
of the Secondary curriculum (Government of United Kingdom, 2010a). The aim of the Government is
to address the literacy and numeracy levels of children in the first two Key Stages (5-11 years) in order
to develop pupils’ mathematical thinking and number skills, with a focus on understanding and
application. A document addressed to learners, schools and families, called the The Primary
Framework for literacy and mathematics makes recommendations on how literacy should be
incorporated in daily mathematics lessons (Government of United Kingdom, 2010b). The secondary
curriculum focuses on developing the skills and qualities that learners need not only to succeed in
school, but also in the broader community.
Functional mathematical skills
Numeracy appears in the Early Year Foundation Stage (birth to 5 years) as part of the learning area:
Problem solving, Reasoning and Numeracy. In the Primary (5-11 years) and Secondary (11-16 years)
curricula Mathematics, and no longer numeracy, appears as one of the ten compulsory school subjects.
Functional mathematics frequently appears in the Secondary curriculum referring to functional
mathematical skills the learners should acquire. Learners need these skills and abilities to play an active
and responsible role in their communities, in their everyday life, workplace and in the educational
settings (QCDA, 2010a). Functional mathematical skills are a subset of the key processes set out in the
programme of study. These key processes are representing, analysing, interpreting, evaluating,
23
communicating and reflecting. All teaching needs to contribute to the development of these key
processes. It requires pupils to be introduced to a range of real-life uses of mathematics, including its
role in the modern workplace (QCDA, 2010b). These functional skills need to be developed in the five
strands of Mathematics, namely Mathematical processes and applications; Number; Algebra; Geometry
and measures; and Statistics. Individuals with functional mathematical skills understand a range of
mathematical concepts and know how and when to use these concepts. They have the confidence and
capability to use mathematics to solve increasing complex problems; are able to use a range of tools, including integrated
computer technologies as appropriate; possess the analytical and reasoning skills needed to draw conclusions, justify how
these conclusions are reached and identify errors or inconsistencies; are able to validate and interpret results, judging the
limits of the validity and using the results effectively and efficiently (QCDA, 2010c).
UNITED STATES (US)
In the United States learners take part in both the PISA study and TIMSS. In the PISA 2003 when the
focus of the study was on mathematics, they came 31st out of 41 countries (OECD, 2004). In the
TIMSS 2007 results they took the 11th position for the 4th graders and the 9th position for the 8th graders
out of the 58 countries participating (National Centre for Education Statistics, 2008a).
Although the term “quantitative literacy” is common in the discourse of US mathematics educators, it
does not appear often in their curricula (J. Kilpatrick, personal communication, May 24, 2010).
Kilpatrick is a mathematics expert, advisor, consultant and professor in Mathematics Education,
University Georgia who serves on various mathematical boards and councils. The US does not have a
single national curriculum in Mathematics. In search of the term “quantitative literacy” in National
Curricula in the Departments of Education of Ohio and North Carolina, as suggested by Kilpatrick, a
reference was eventually found on the webpage of the National Council of Teachers of Mathematics
(2010) where it was stated that consumer mathematics should develop a broader quantitative literacy and should
consist primarily of work in informal statistics, such as organizing and interpreting quantitative information.
Comparing the national curriculum documents of Australia, the UK as well as Ohio and North
Carolina in the US, it is evident that Australia accentuates the importance of mathematical literacy in an
education system. Through their national documents for learners, schools and parents they drive an
intensive awareness campaign regarding the raising of the numeracy levels of their learners. Regardless
of the terminology used, numeracy, functional mathematical skills and quantitative literacy are
embedded in Mathematics and involve the competency or skill to use and apply mathematics to solve
contextualised problems.
24
2.2.1.6
Summary
To define, value, position or conceptualise mathematical literacy is a daunting task. Different views
exist but the most common descriptions of mathematical literacy are mathematics in action
(Skovsmose, 2007); mathematics in context (McCrone & Dossey, 2007; Powell & Anderson, 2007);
realistic mathematics education (Hope, 2007); and mathematising (Gellert et al., 2001; Hope, 2007). The
different perspectives of mathematical literacy undoubtedly illustrate how the different conceptions
vary in degree of complexity regarding the required mathematical knowledge and skills where in some
notions advanced and expert mathematical knowledge and higher order cognitive skills are required. It
is however PISA’s definition and criteria for assessment that best describe the requirements of this
study.
Although some researchers accentuate the formal application of mathematics by mathematicians to
real-world contexts demanding a high level of mathematical knowledge and the competence to use and
apply it (Gellert et al., 2001; Hope, 2007; Jablonka, 2003; Skovsmose, 2007), other researchers remain
convinced that all people need some basic level of literacy to empower them to make well informed
decisions in their daily lives, whether personally, to care for their families or to contribute in their
workplace or society (McCrone & Dossey, 2007; McCrone et al., 2008; Powell & Anderson, 2007;
Skovsmose, 2007). The value of being mathematically literate is evident but it remains uncertain to what
extent mathematical literacy could address educational practices and contribute to an individual’s quality
of life or even the development of the country (Gellert et al., 2001; Jablonka, 2003; Skovsmose, 2007).
Nowhere in the literature has mathematical literacy been referred to as a specialised subject. It is rather
regarded as specialised knowledge or a competency or skill embedded in the subject Mathematics.
According to Hope (2007) mathematical literacy is a matter of the appropriate pedagogy that should be
used in teaching mathematics. As mathematical literacy with its focus on the skill of using and applying
mathematical knowledge forms part of Mathematics, the focus of Mathematics teaching should be on
knowledge and the development of skills enabling learners to solve real-life application problems. The
aforementioned perspectives and conceptions are wide and theoretical and to provide only one
international definition of mathematical literacy is not viable as it depends primarily on a particular
social practice and the context involved. With these international perspectives in mind, the South
African perspective on mathematical literacy is discussed below.
25
2.2.2 An overview of ML
In this overview of ML in South Africa the history and principles of the subject are discussed. An
overview of the two subjects ML and Mathematics is given, some general concerns about ML are
discussed and lastly a comparison is made between the national and international perspectives on
mathematical literacy.
2.2.2.1
The history of ML
Background information to ML
One of the reasons behind the implementation of ML as an alternative subject to Mathematics in the
FET band was the low level of learners’ mathematical knowledge and mathematical literacy skills as
shown in the results of international studies (DoE, 2003a). The last time South Africa took part in
TIMSS, an international study, was in 2003 when Grade 8 learners participated and came last out of 46
countries (National Centre for Education Statistics, 2008b). Recently The World Economic Forum
ranked South Africa 120th for Mathematics and Science education, well behind our troubled neighbour
Zimbabwe which was ranked 71st (Maths Excellence, 2009). At present the country’s GET learners are
not participating in any such studies as a four-year Foundations for Learning campaign was introduced
in 2008 in the Foundation and Intermediate phases to improve the reading, writing and numeracy
abilities of all South African children (DoE, 2008a). Another reason for implementing ML was to
address the concern that Mathematics is too abstract, catering primarily to prepare students to proceed to further
mathematically or scientifically oriented studies (Graven & Venkat, 2007, p. 340). They believe ML now offers
an alternative to learners who do not need it for this purpose.
The ML curriculum reform
Curriculum 2005 was introduced in 1998, coinciding with the birth of a new democracy in South
Africa’s post-apartheid era, and was based on the principles of outcomes-based education (OBE)
(DoE, 2009). This curriculum was revised in 2000 and in 2002 the NCS for the FET phase was
developed. In 2009 a task team reviewed the curriculum and apart from problems related to learning
materials and teacher training, the curriculum documents were deemed to be in need of streamlining
(DoE, 2009). The task team found that some of these documents contradicted each other while at
other times there were repetitions. The review supports the DoE’s current move away from OBE and
learning outcomes, which are now replaced with clear content, concept and skill standards as well as
clear and concise assessment requirements (DoE, 2009). The various subject specific documents will be
replaced with a single document called the Curriculum and Assessment Policy Statement (CAPS) (DoE,
2009). The date of implementation should be in 2012.
26
2.2.2.2
ML principles
This section examines the principles of ML such as the purpose, aims, definition, key elements,
composition of the subject as well as the assessment taxonomy on which ML is based.
The purpose of ML
The purpose of ML is to ensure that all learners develop an understanding of mathematics and how it
relates to the world in order to use mathematical information to make valuable decisions affecting their
life, work and society. It is important that learners are able to interpret and critically analyse everyday
situations and solve problems. With this purpose in mind, ML aims to ensure a broadening of the
education of learners, preparing them to meet the demands of a modern world (DoE, 2003a).
According to the DoE (2008b, 2011a), the purpose of the subject includes the ability of a learner to
become:
•
A self-managing person where the focus is on problems that relate to financial issues such as
mortgage bonds, hire-purchase and investments, other personal issues such as the ability to
estimate and calculate length, areas and volumes, to read a map and follow timetables and to
understand house plans, sewing patterns and converting recipes.
•
A contributing worker at the workplace requires the use of fundamental numerical and spatial
skills with understanding in order to deal with work-related formulae, statistical charts and
schedules and to understand instructions involving numerical components.
•
A participating citizen where learners need to acquire a critical stance to mathematical
arguments presented to them in the media or other platforms.
These three abilities as part of ML’s purpose correspond with the international purpose for learner
competence, stating that for learners to be competent means having more than just knowledge, they
must know how to use and apply their mathematical knowledge (Gellert et al., 2001; Hope, 2007;
Jablonka, 2003; Skovsmose, 2007).
The aims of ML
The main aim of ML is to equip learners to be skilled citizens, meeting the demands they will encounter
in their future lives. The process to achieve this aim involves the mastering of mathematical content
through solving contextualized problems. ML aims to develop the following learner abilities (DoE,
2008b):
•
•
•
•
The ability to use basic mathematics to solve problems encountered in everyday life and in work situations.
The ability to understand information represented in mathematical ways.
The ability to engage critically with mathematically based arguments encountered in daily life.
The ability to communicate mathematically (p. 8).
27
The definition of ML
The DoE’s (2003a) national definition of ML reads as follows:
Mathematical Literacy provides learners with an awareness and understanding of the role that mathematics plays in
the modern world. Mathematical Literacy is a subject driven by life-related applications of mathematics. It enables
learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically
analyse everyday situations and to solve problems (p. 9).
There are, according to this definition, three key elements of ML namely 1) the mathematical content,
2) the contexts that should involve everyday life-related problems and 3) the abilities and behaviours
that a mathematically literate person needs to possess which include problem solving through
interpreting and analysing the problem with confidence (Bowie & Frith, 2006). In the new CAPS
document (DoE, 2011a) these three key elements of ML have been extended to five key elements7.
Comparing national and international purposes and definitions of ML, the national purpose and
definition closely relate to that of PISA (National Centre for Education Statistics, 2008b; OECD,
2003). The purpose of PISA is to measure the extent to which students can make use of their mathematical
knowledge in realistic and day-to-day situations (McCrone et al., 2008, p. 35). An international definition of
mathematical literacy according to PISA (OECD, 2003) is:
the capacity to identify, to understand and to engage in mathematics and make well-founded judgements about the
role that mathematics plays, as needed for an individual’s current and future life, occupational life, social life with
peers and relatives, and life as a constructive, concerned and reflective citizen (p. 20).
The interface between the international and national views is the emphasis being placed on the role
mathematics plays in the world and the value of applying mathematics in people’s personal lives, at the
workplace and as participating citizens. Further shared objectives are guiding learners to become
engaged in mathematics and to understand and appreciate how it is embedded in everyday life
situations. A point of difference however is that nationally mathematical literacy refers to both a subject
and competence while internationally it refers to a competence (Christiansen, 2007).
The five key elements of ML
From the purpose, aims and definition of ML the DoE (2011a) lists five key elements involved in ML
namely:
•
The use of elementary mathematical content: The general idea is that the focus is not on
formal abstract mathematical concepts and mathematical content should not be taught in the
absence of context.
7
The five elements of ML are discussed under the next subheading: The five key elements of ML.
28
•
Real-life contexts: These contexts should be authentic and relevant, and should relate to
learners’ daily lives, their future workplace and the wider social, political and global
environments.
•
Solving familiar and unfamiliar problems: Learners should have the ability and skills to
interpret both familiar and unfamiliar real-life contextual problems they encounter in the world.
They should have the ability to apply both mathematical and non-mathematical techniques and
considerations in order to explore and make sense of the context. The interplay between
content, context and solving problems is illustrated in the following figure:
Figure 2.1: Interplay between content, context and problem-solving skills in ML (DoE, 2011a, p.
10)
•
Decision-making and communication: A mathematically literate person should be able to
compare solutions, make decisions regarding the most appropriate choice for a given set of
conditions and communicate their decisions through the use of appropriate terminology.
•
The use of integrated content and/or skills in solving problems: Since most real-life
problems consist of a range of mathematical topics, learners need to use mathematical content
and/or skills drawn from a range of topics and need to identify and use a range of techniques
and skills integrated from a range of content topics.
The composition of ML
The topics in the new CAPS (DoE, 2011a) replace the learning outcomes from the current NCS for
ML. The content, contexts and problem solving skills appropriate to ML are offered in topics and
divided into two sets of topics, namely (DoE, 2011a):
29
•
Basic skills topics: Much of the content in these topics includes the mathematical content and
skills that learners have already been exposed to in Grade 9. Teachers therefore have the
opportunity to revise important mathematical concepts and to provide learners now with the
opportunity to explore and use these concepts in various contexts.
•
Application topics: These topics contain contexts that can be related to situations from
everyday life, the workplace and business environments as well as wider social, national and
global issues that learners are expected to make sense of. A profound understanding of the
content and skills from the Basic skills topics are required to make sense of the contexts and
content from the Application topics. Figure 2.2 below shows an overview and weighting of
the topics according to which the ML curriculum has been organized for Grades 10, 11 and 12.
Figure 2.2: Overview and weighting of topics in Grades 10, 11 and 12 (DoE, 2011a, p. 14)
The ML assessment taxonomy
Assessment should be done at different levels of cognitive demand, from simple reproduction of facts to
detailed analysis and the use of varied and complex methods and approaches (DoE, 2011a, p. 91). The following
assessment taxonomy framework is used:
•
Level 1: Knowing
•
Level 2: Applying routine procedures in familiar contexts
•
Level 3: Applying multi-step procedures in a variety of contexts
•
Level 4: Reasoning and reflecting (p. 84).
30
According to Venkat, Graven, Lampen and Nalube (2009) the emphasis in Levels 1 and 2 is on routine
calculations whereas the key aims of ML are located primarily in Levels 3 and 4. Level 3 refers to the
ability of learners to think numerically and spatially whereas Level 4 refers to critically analysing
everyday life situations.
The DoE (2011a) explicitly states that since ML involves the use of both mathematical and non-mathematical
techniques and considerations in exploring and making sense of authentic real-life scenarios (p. 92), the taxonomy
should be regarded as follows:
This taxonomy should not be seen as being associated exclusively with different levels of mathematical calculations
and/or complexity. In determining the level of complexity and cognitive demand of a task, consideration should also
be given to the extent to which the task requires the use of integrated content and skills drawn from different topics,
the complexity of the context in which the problem is posed, the influence of non-mathematical considerations on the
problem, and the extent to which the learner is required to make sense of the problem without guidance or assistance
(DoE, 2011a, p. 92).
2.2.2.3
Pedagogical approaches for teaching ML
The DoE (2011a) suggests that the focus of ML teaching is the integration of content and skills in reallife contexts. Teachers should provide learners with opportunities to analyse problems and devise ways to
work mathematically in solving them (p. 9) and develop and practice decision-making and communication skills (p. 10).
According to Brown and Schäfer (2006) the emphasis in the ML curriculum is on contextualised
mathematics. These contexts should be realistic and demand real-life authenticity to provide learners
with opportunities to apply and use mathematics in order to make sense of the world, instead of letting
learners do more mathematical content (Bansilal, Mkhwanazl & Mahlaboratoryela, 2010). These
problems should relate to a learner’s daily life, the workplace and the wider social, political and global
environment (DoE, 2011a, p. 12). Brown and Schäfer (2006) found many similarities to that of mathematical
modelling but the differences appeared to be that mathematical modelling is generally described using more advanced
mathematics, in more technical contexts (p. 46) but that the basic principles of modelling can be applied on
elementary mathematics too. ML further focuses on de-compartmentalisation, where mathematical topics are no
longer taught in isolation of each other (North, 2005, p. 35). All ML textbooks are written accordingly with
the initial four learning outcomes being integrated to enable ML teachers to teach ML in a decompartmentalised way.
In a longitudinal study performed by the Marang Centre at the University of the Witwatersrand, Graven
and Venkat (2009) report that the learners who are part of the study, are for the most part positive
about ML, which the researchers attribute to the teachers who substantially changed their pedagogic
practices. This differs from traditional Mathematics teaching in that the nature of tasks in ML (engagement
31
with a scenario rather than application of maths in ‘word problems’) and the nature of interaction in ML (much slower
pace, more discussion and group work) (p. 2). Venkat (2007) argues that if learners are engaged in problems
situated in real-life situations, they will develop valuable skills such as mathematical reasoning, sensemaking, applying different procedures and decision-making.
In Table 2.1 given below, Graven and Venkat (2007) identify a spectrum of pedagogic agendas that
traverses across the question of the nature and degree of integration of context with mathematics within pedagogic situations
(p. 74). This spectrum of agendas is a tool for the ML teachers to think about the nature of their
lessons and may assist the teachers to navigate their teaching along a whole spectrum of pedagogic
agendas.
Table 2.1: A spectrum of pedagogic agendas (Graven & Venkat, 2007, p. 74-75)
1. Context driven (by
learners’ needs)
Driving agenda:
2. Content and context
3. Mainly content
driven
driven
Driving agenda:
Driving agenda:
To explore contexts
that learners need to
interact and engage
with in their lives
(current, future,
citizenship) and to use
maths to achieve this.
Pedagogic demands:
To explore a context so
as to deepen maths
understanding and to
learn maths (new or
GET) and to deepen
understanding of that
context.
Pedagogic demands:
• Involves identifying
contexts/scenarios
needed for the above
agenda.
• Teaching needs
increased discussion
of contexts and
critical engagement
with them and the
mathematics
embedded in them.
• Teaching might
require revisiting or
learning new maths
but largely insofar as
it will service critical
engagement with and
understanding of the
context.
• Involves selecting real
contexts (possibly
edited or adapted) that
enable the above
agenda.
• Teaching needs
discussion about
contexts but this must
be balanced with
revising maths and
learning new maths in
new ways. Contextual
and mathematical
learning need to be
balanced and
connected in a
dialectical relationship
that enables the
agenda.
4. Content driven
Driving agenda:
To learn maths and
then apply it to various
contexts.
To give learners a
second chance to learn
the basics of maths
from the GET band.
Pedagogic demands:
Pedagogic demands:
• Involves selecting
• Involves revision of
contexts that GET
GET maths without
maths can be applied
the need for
to (contrived or more
pedagogic change
real) and editing
except in relation to
these to enable
slower pacing.
application
• Contexts do not
appropriate to the
feature much except
level of learning.
in relation to their
• Teaching focuses on
use in teaching GET
mathematical
basics (e.g. in the
learning and its use in
case of fractions –
applications and does
using cakes for
not necessarily
understanding
require much
fractions).
discussion of
context.
32
Graven and Venkat (2007) analysed ML’s definition and purpose on context as stated by the DoE
(2003a) and propose Agenda 2 to be the core business of ML. They call these four agendas a spectrum
and not a continuum which might imply that teachers move along it in one direction (Graven & Venkat, 2007,
p. 77). The idea is that teachers may use different agendas at different times as required. Although
Agenda 2 is the primary driving agenda, a teacher can adopt other agendas at different points in order to support this
agenda and also to assist in meeting curricula demands (p. 77).
2.2.2.4
The ML learner profile
Although there are positive and enthusiastic ML learners, the majority of learners are less interested and
enthusiastic about mathematics and mathematical activities and many negative feelings result in fear of
anything mathematical (Vermeulen, 2007). According to Vermeulen learners could avoid these negative
experiences and feelings of anxiety in the past by not choosing mathematics, but now they need to
confront them. He argues that it is the parents’ and society’s incorrect beliefs, teachers’ teaching
methods based on their beliefs and attitudes, teachers’ attitudes towards the learners, and teachers’
classroom culture that contribute to learners’ negative feelings. According to Mbekwa (2007) it is a
challenge to teach ML as learners lack understanding and motivation because ML is seen as the dumping
ground for mathematics underperformers (p. 227).
2.2.2.5
Some general concerns about ML
The ML teachers
As ML is a relatively new subject and different to Mathematics, clear guidelines from the DoE
regarding issues such as pedagogical approaches to teach ML should have been a given, but instead the
absence of precedents of what pedagogy and assessment should be like (Graven & Venkat, 2007, p. 67) caused
multifarious interpretations of the curriculum aims. Bowie and Frith (2006) were concerned about a
perception that ML could be interpreted as a slightly toned-downed standard grade Mathematics with word sums
(Bowie & Frith, 2006, p. 32). Experience and research have indicated that Mathematics learners and in
many cases teachers too, find word or application problems requiring conceptual understanding more
difficult than routine problems which require factual recall or the use of routine procedures (Abedi &
Lord, 2001; Grobler, Grobler & Esterhuyse, 2001; Johari, 2003; Schoenfeld, 1988; White &
Mitchelmore, 2002). My own experience like that of De Villiers (2007) confirms that Mathematics
learners cope well with the theory of linear functions, but when it is put in real-life contexts they cannot
solve such problems. Even in ML both teachers and learners find the process of mathematising
contexts complex as a good understanding of both the context and the mathematical content is
required (Bowie & Frith, 2006). A further concern is the number of ML teachers with other
33
specialisations who also teach the subject (Mbekwa, 2007). It is known that in the past, before ML was
introduced, there existed a shortage of appropriately qualified Mathematics teachers (Sidiropoulos,
2008) and the question arises as to the provenance of all the teachers who now teach both ML and
Mathematics. Sidiropolous (2008) further suggested that a change is required not only in pedagogical content
knowledge, but also in understanding the nature and value of Mathematical Literacy (p. 205).
The choice of context
Apart from the complexity of solving contextual problems, the contexts to which mathematics should
be applied in ML are not clear to teachers. A further concern is how mathematical progression is made
through the years regarding the complexity of contexts. A good understanding of the context is
required by both teachers and learners in order to mathematise a context (Bowie & Frith, 2006). For
example when working on personal finances, topics such as budgeting, compound interest, mortgage
payments, and retirement options are not part of all teachers’ and learners’ life experiences. They have
inadequate experiences of banks, interests, risks and return on investments. To teach one mathematical
content topic requires several periods to first explain the context involved. What further complicates
the situation is that in reality banks normally use their own formulae programmes and do not calculate
interest as learners are taught to do (Christiansen, 2007). In choosing contexts, teachers may use the
principle from PISA (OECD, 2003) that categorises contexts according to their distance from the
learner. ML teachers can therefore include contexts from the learner’s private life, school life, work and
sport, and local community and society. It is crucial that contexts be authentic and applicable to the
learners’ environment.
The language issue
In South Africa the majority of learners are taught in English, which is often not their mother tongue.
According to Graven and Venkat (2009) integration with the above-mentioned contexts could be
problematic due to the increased English language demands. Many researchers reported on difficulties
learners experience regarding the contextualised problems and the role language plays in conceptual
understanding (Mbekwa, 2007; Setati, 2005). Maree (2000) posits that insufficient language skills and
language usage play an important role in under-achievement of learners in Mathematics. In his
classification of learners’ mistakes in mathematics, language problems were the most significant
problem identified. He expressed his concern about learners having to unravel problems in
mathematics that require more sophisticated language skills while they actually lack the minimum
language skills to even understand what is being asked.
34
Debate regarding ML as only alternative to Mathematics exists and many teachers expressed their
concern about the existence of two extreme levels of mathematics, especially considering South Africa’s
diverse population (Maths Excellence, 2009). This group of teachers advises a three-level system
consisting of two formal Mathematics subjects and ML as the third option to accommodate the diverse
skills and needs of our learners. Their idea is that ML should then be taken by learners who do not wish
to take either of the formal Mathematics courses. The ML learners can then be equipped with basic
numeracy skills. According to them the current lack of curriculum flexibility could result in the
downgrading of mathematical skills. On the other hand a number of people in the school education
system are against a two-level system as was applicable in the South African schools up to 2007.
According to Kitto (personal communication, February 23, 2011) a reason is that very few township and
rural schools offered higher grade under the old system, so an overwhelming percentage of the higher grade candidates were
white. A huge number of competent black students were denied the chance to demonstrate their ability and get into
engineering and other faculties. She reasons that the policy makers are trying to make sure that everyone who
has the ability to continue with careers in science and engineering has access to the mathematics that is
needed.
2.2.2.6
Comparison between the national and international perspectives on
mathematical literacy
In South Africa the term “mathematical literacy” refers both to a school subject and to the competency of
individuals, where internationally it is mainly the latter (Christiansen, 2007, p. 91). The original NCS Grades
10-12 General (DoE, 2003a) is based on OBE, social transformation and integration, and applied
competence. These principles encourage a learner-centred and activity-based approach.
With the increased international emphasis on the application value of mathematics, science and
technology, the objectives of subjects such as ML changed to personal fulfilment, employment and full
participation in society (OECD, 2004, p. 37). Internationally mathematical literacy as application skills is
embedded in the subject Mathematics. For this purpose real-life contexts are used to re-contextualise
mathematical concepts. From the literature it is evident that mathematical literacy varies in width and
depth and that one needs to interpret it according to the purpose and context being used (Gellert et al.,
2001; Hope, 2007; Jablonka, 2003; McCrone & Dossey, 2007; Powell & Anderson, 2007; Skovsmose,
2007). Jablonka (2003) states that the context in which mathematical literacy is applied, sometimes
demands higher-order mathematical skills, whereas McCrone and Dossey (2007) believe mathematical
literacy should be promoted even in non-mathematics classes to make mathematics relevant and to
empower all learners. Nationally the subject ML focuses on making sense of real-life contexts and scenarios
35
(DoE, 2011a, p. 9) and requires an understanding of only basic mathematical concepts and calculations, and does not
require an understanding of complex and/or abstract mathematical principles (DoE, 2011a, p. 11).
2.2.2.7
An overview of ML and Mathematics
Since ML is a compulsory subject for Grade 10 to 12 learners who do not choose Mathematics as
subject, parents and learners should know what each subject entails and what the implications are for
further studies. For example the DoE (2003a) states that learners who wish to proceed to tertiary
studies of a mathematical nature such as engineering, architecture, natural sciences at tertiary
institutions should not take ML. Issues dealt with in this overview are the subjects’ premises, learning
outcomes and topics to be covered as well as the pedagogical approach for teaching ML and the ML
learner profile.
The premises of ML and Mathematics
In Table 2.2 below the premises of Mathematics and ML are discussed according to the subjects’
purposes, aims, definitions and their educational and career links as set out by the DoE (2003a, 2003b,
2008b, 2011a, 2011b).
ML
Mathematics
Purpose
Provide learner with an awareness and
understanding of the role mathematics
plays in the modern world, enabling
learners to become self-managing people,
contributing workers and participating
citizens (DoE, 2003a, 2011a).
Aim
Table 2.2: The premises of ML and Mathematics
To equip learners to understand
information represented in mathematical
ways and to solve problems encountered in
everyday life and work situations (DoE,
2008b). ML learners should have the ability
or skills to think mathematically, interpret,
analyse and solve problems (DoE, 2003a).
To create an appreciation of the discipline
itself and a deeper understanding and
successful application of knowledge and
skills. This competence contributes not only
to personal and social, but also to learners’
scientific and economic development (DoE,
2003b).
To allow learners to develop into citizens
who are able to deal with the mathematics
that forms part of the society they live in
and on their daily lives. It is more important
for learners to acquire skills such as
investigating, generalising and proving
instead of only acquiring content knowledge
for its own sake (DoE, 2003b).
36
Definition
Career links
Mathematical Literacy provides learners with an
awareness and understanding of the role that
mathematics plays in the modern world.
Mathematical Literacy is a subject driven by liferelated applications of mathematics. It enables
learners to develop the ability and confidence to
think numerically and spatially in order to
interpret and critically analyse everyday situations
and to solve problems (DoE, 2003a).
ML should not be taken by learners who
intend to study mathematically based
disciplines such as natural sciences and
engineering. ML learners proceeding to
Higher Education institutions will have
developed the skills needed to deal
effectively with mathematically related
requirements in disciplines such as the
social and life sciences (DoE, 2003a).
Mathematics enables creative and logical reasoning
about problems in the psychical and social world and
in the context of mathematics itself … is based on
observing patterns, with rigorous logical thinking,
this leads to theories of abstract relations … enables
us to understand the world and make use of that
understanding in our daily lives (DoE, 2003b).
According to CAPS mathematics is a
language that makes use of symbols and notations
for describing numerical, geometric and graphical
relationships. It is a human activity that involves
observing, representing and investigating patterns
and qualitative relationships in physical and social
phenomena and between mathematical objects
themselves (DoE, 2011b, p. 10).
The subject provides a platform for linkages
to Mathematics in Higher Education
institutions. Mathematics is essential for
learners who intend to pursue a career in the
psychical, mathematical, computer, life,
earth, space and environmental sciences or
in technology. Mathematics also plays an
important role in the social, management
and economic sciences (DoE, 2003b).
Studying these premises, it is clear that the two subjects are different in kind and should not be
compared. Sidiropoulos (2008) is also of the opinion that the distinction between ML and Mathematics is
principally not a distinction in level, but a distinction in kind (p. 208). Mathematics on the one hand is regarded
as a purely academic subject with a reputation of being an abstract science, involving mathematical
rigour and a high level of cognitive thinking and reasoning based on sound conceptual understanding
of the content. ML on the other hand does not focus on abstract mathematical concepts but, instead,
primarily on developing practical skills to use elementary mathematical content to find concrete
solutions to numeric, spatial and statistical problems associated with everyday life experiences (DoE,
2011a; Maffessanti, 2009). A shared aim however is the development of competent learners who are
able to use their mathematical knowledge to solve personal and social real-life problems.
The learning outcomes of ML and Mathematics
With the introduction of ML in 2008, the similarities between the learning outcomes for ML and
Mathematics, as seen in the table below, were a major concern as some people thought of ML as a
lower grade Mathematics subject. The learning outcomes as they were applied from 2008 to 2011 for
the Senior Phase in the GET band (Grades 8-9), ML and Mathematics, both in the FET band, are listed
in Table 2.3 below (DoE, 2003a, 2003b; 2010):
37
Table 2.3: Learning outcomes for ML and Mathematics
Learning
outcome 1
Learning
outcome 2
Learning
outcome 3
Learning
outcome 4
Mathematics (GET:
ML (FET)
Senior Phase)
Numbers, Operations Number and Operations in
and Relationships
Context
Patterns, Functions
Functional Relationships
and Algebra
Space, Shape and
Measurement
Measurement
Data Handling
Data Handling
Mathematics (FET)
Number and Number
Relationships
Functions and Algebra
Space, Shape and
Measurement
Data Handling and
Probability
The first two columns show how the learning outcomes for ML build on the learning outcomes for
Mathematics in the GET band (DoE, 2005). Some researchers are of the opinion that ML, being a new
subject with a different focus, should not have used the same content-based learning outcomes as
Mathematics as this scenario ended up being stumbling blocks to the teachers (Bowie & Frith, 2006;
Christiansen, 2007; North, 2005). This concern about similar learning outcomes has been addressed in
the new CAPS (DoE, 2011a) and is discussed in the paragraph below.
Topics covered in ML and Mathematics
Different concerns regarding the content-context issue have been expressed by academics prior to the
new CAPS. There were questions about what content knowledge should be taught by the ML teachers,
which contexts should they use (Geldenhuys, Kruger & Moss, 2009; Julie, 2006; Vithal & Bishop,
2006), and whether the content should determine the context or vice versa (Bowie & Frith, 2006;
Graven & Venkat, 2007). Although these issues were not elucidated clearly in the original NCS for ML
(DoE, 2003a), the new CAPS (DoE, 2011a) addresses these issues.
In ML the topics are divided into two groups, namely the Basic Skills Topics which comprise elementary
mathematical content and skills that learners have already been exposed to in Grade 9 and the Application Topics
which contain the contexts related to scenarios involving daily life, workplace and business environments, and wider
social, national and global issues (DoE, 2011a, p. 13). For this purpose it is necessary to list the content
areas and topics covered in the Mathematics Senior Phase8 as well as the ML and Mathematics in the
FET Phase. Different terminology for content areas and topics is used across the different bands
(DoE, 2011a; DoE, 2011b; DoE, 2010) as indicated in Table 2.4 below:
8
The Senior Phase band includes Grade 7 through to Grade 9.
38
Table 2.4: Comparison of the composition of ML and Mathematics across the different bands
COMPOSITION OF MATHEMATICS AND ML
Senior Phase Mathematics
FET ML
Basic Skills Topics:
Content areas:
1. Number, Operations and 1. Interpreting and
communicating answers
Relations
2. Patterns, Functions and
and calculations
Algebra
2. Numbers and calculations
3. Space and Shape
with numbers
3. Patterns, relationships and
(Geometry)
4. Measurement
representations
5. Data Handling (Statistics) Application Topics:
1. Finance
2. Measurement
3. Maps, plans and other
representations of the
physical world
4. Data handling
5. Probability
Content topics:
Content topics:
Example: Exponents,
A range of content topics
Integers, Fractions etc. under based on Senior Phase content
number 1 above are called
topics only.
the content topics.
2.2.2.8
FET Mathematics
Main content topics:
1. Functions
2. Number patterns,
sequences, series
3. Finance, growth and decay
4. Algebra
5. Differential calculus
6. Probability
7. Euclidean Geometry and
measurement
8. Analytical geometry
9. Trigonometry
10. Statistics
Curriculum statement:
Instead of using Content topics,
Descriptions is used to explain
the content under each main
topic. Example: Practical
problems involving optimisation and
rates of change (DoE, 2011b, p.
11) under number 5 above.
Summary
South Africa was the first country in the world to introduce ML as a school subject in 2006 in the FET
band (Grades 10 to 12) (Christiansen, 2007). A major reason behind the implementation of a
compulsory mathematics subject in the FET band is to improve the low level of learners’ mathematical
knowledge and mathematical literacy skills. One of ML’s purposes is to provide the opportunity for
each learner to become mathematically literate in order to effectively deal with mathematically related
requirements in disciplines such as the social and life sciences (DoE, 2003, p. 11). In comparing the national and
international perspectives the latter refers to various levels of specialised knowledge, skills and
understanding that are required to apply formal mathematics to solve application problems in various
contexts.
The approach to the teaching and learning of ML should provide opportunities to engage with
mathematics in diverse contexts at a level that learners can access logically (DoE, 2003c). However, the
39
teaching of ML in a contextualised and de-compartmentalised manner where the content topics are
integrated, complicates the teaching of the subject as teachers lack the knowledge and skills to do so.
The distinction between ML and Mathematics is principally not a distinction in level, but a distinction in kind
(Sidiropoulos, 2008, p. 208). Mathematics is regarded as a purely academic subject, an abstract science
involving a high level of cognition. ML on the other hand does not focus on abstract mathematical
concepts but primarily on developing practical skills to deal with everyday life experiences (DoE, 2011a;
Maffessanti, 2009).
2.3
Teachers’ instructional practices
The process of teaching and learning is extensive and involves many pedagogical concerns and
influences. Teaching in general involves more than the activities in the classroom and includes activities
such as working with parents, colleagues and engaging in professional development (Franke et al.,
2007). However the instructional practice of the teacher occurs in the classroom where teachers’ goals,
knowledge and beliefs serve as driving forces behind their instructional efforts to guide and mentor
learners in their search of knowledge (Artzt et al., 2008). Different terminology is used in the literature
when referring to teachers’ performances or the act of teaching in the classroom. Terminology such as
teachers’ behaviour, instructional behaviour, instructional practices, classroom practices, classroom
processes, and classroom instruction are frequently used. Table 2.5 below provides a short definition of
four of the frequently used terms when referring to teachers’ practices:
Table 2.5: Different terminology used for teachers’ practices
Classroom
practice
Classroom
instruction
Classroom
processes
Instructional
practice
Focus is on three features, namely discourse, norms and building
relationships (Franke et al., 2007).
Involves interactions among teachers and students around mathematical subject matter
(Kilpatrick, 2001, p. 107).
Interaction taking place between the teacher and learner and all the factors
influencing this interaction (Koehler & Grouws, 1992).
Refers to the qualitative dimensions of teacher behaviour regarding their
teaching (Englert et al., 1992).
The term “instructional practice” best portrays the focus of this study being the ML teachers’
classroom behaviour. Englert et al. (1992) refer to teachers’ instructional practices as teachers’
qualitative dimensions in the teaching and learning process. Qualitative dimensions involve teachers’
abilities to apply appropriate cognitive strategies in meaningful and purposive activities, promote
classroom dialogues and adjust instruction as required, and establish classroom environments in which
40
students cooperatively and collaboratively participate in enquiry-related activities. To examine teachers’
instructional practices, Artzt et al. (2008) use a phase dimension framework that is built on three
observable aspects of mathematical lessons, namely tasks, discourse and the learning environment9
(Figure 2.3).
Figure 2.3: Framework to observe teachers’ instructional practices (Adapted from Artzt et al.,
2008; Englert et al., 1992)
In the light of my research paradigm of social constructivism which suggests that all knowledge is
constructed and based upon not only prior knowledge, but also the cultural and social context
(Ollerton, 2009), the participants’ instructional practices are subsequently discussed. Franke et al. (2007)
recognise a productive instructional practice as a practice creating ongoing opportunities for learning.
There are different perceptions regarding the components of a teacher’s instructional practice. Artzt et
al.’s (2008) dimensions of instructional practices are tasks, discourse and learning environment whereas
Franke et al. (2007) speak of discourse, norms and building relationships as the three features of
classroom practices. From these two views the teachers’ practices could be described as a social
environment where all people in the classroom are in a relationship with one another, have the
opportunity to construct and enhance their knowledge through communicating while solving and
pursuing their conjectures of challenging tasks. For the purpose of my study the dimensions discussed
by Artzt are most appropriate, since they address the practical issues of classroom practice which is
fundamental in ML teaching.
9
The characteristics of the observable aspects of a lesson are further discussed in par. 2.5.5.
41
I will now briefly discuss researchers’ views on tasks, discourse and the learning environment and
report on findings in the literature regarding the three aspects of ML teachers’ lessons.
2.3.1 Tasks
Since knowledge is constructed and based upon, among other things, prior knowledge, the purpose of
tasks is to provide opportunities for learners to connect their knowledge to new information and to build on their
knowledge and interest through active engagement in meaningful problem solving (Artzt et al., 2008, p. 10).
Modes of representation
Franke et al. (2007) believe teaching involves orchestrating the content, that teachers’ planning of their actions
is crucial to enable learners to progress in their cumulative understanding of a particular content area (p.
228). According to Artzt et al. (2008) modes of representation are the forms for representing
mathematical concepts through the use of oral or written language, diagrams, manipulatives, computers, or calculators
(p. 12). Geldenhuys et al. (2009) recommended that teachers should increase the use of resources such
as computers. Bransford, Brown and Cocking (2000) mentioned some people believe technology is
money and time wasted whereas others regard the mere presence of computer technology in schools as
enhancing the learning in the school. When computer technology is used correctly, Bransford et al.
believe it has great potential to enhance student achievement. Since ML is related to real-life situations
such as interest rates of home loans or personal income tax, computer technology could enhance the
learners’ understanding and interest in the subject and its application value as they could find the
specific day’s interest rates or even general information regarding income taxes.
Motivational strategies
The tasks teachers use in their lessons should possess attributes that attract and sustain [the learners’] attention
and emotional investment over time (Artzt et al., 2008, p. 13). Dewey (as cited in Bransford et al., 2000) noted
the following:
From the standpoint of the child, the great waste in school comes from his inability to utilize the experience he gets
outside … while on the other hand, he is unable to apply in daily life what he is learning in school. That is the
isolation of the school – its isolation from life (p. 147).
In my view Dewey’s concern is addressed by the DoE (2003a) when ML was implemented with its
purpose of providing opportunities for learners to experience how mathematics relates to the world,
enabling the learners to use mathematical information to make valuable decisions affecting their life,
work and society (DoE, 2003a). The idea of connecting the school and home environments is
consistent with Moll and Gonzalez (2004) who argued that teachers need to know and understand their
42
learners’ home environments which could be used to understand the learners’ participation in the
classroom. Especially in ML where the emphasis is on content being taught in context and making the
subject applicable to real-life situations (DoE, 2003a), teachers need to take into consideration the
knowledge their learners bring to their classrooms.
Sequencing and difficulty levels
The difficulty levels and sequencing of tasks must allow students to use their past knowledge and experience to
help them understand the requirements of the task (Artzt et al., 2008, p. 13). Bransford et al. (2000) mentioned
that tasks must be at the appropriate level of difficulty in order for learners to remain motivated. They
stated too easy tasks cause learners to become bored while too difficult tasks cause frustration. Hechter
(2011b) reported that the cognitive levels of the assessment tasks set by both teachers in her study were
on a relatively low level. Bansilal (2008), whose study consisted of an analysis of the answers given by
38 ML teachers to various questions taken from a test and the final examination in a module of their
Advanced Certificate in Education (ACE) (ML) programme, revealed that teachers found questions
which had multi-steps, difficult.
2.3.2 Discourse
To contribute to learner understanding, the discourse in class should provide opportunities for learners
to express themselves, to listen to, to question, to respond and to reflect on their thinking (Artzt, et al.,
2008). Franke et al. (2007) believed classrooms involve people who work in social, cultural and political
contexts that shape how they do their work and how that work gets interpreted (p. 227).
Teacher-learner interaction
The teacher plays a critical role in orchestrating discourse in class and should know how to use verbal
and non-verbal strategies to communicate effectively (Artzt, et al., 2008). According to Franke et al.
(2007) teaching is multifaceted and teaching should be seen as deliberate work, where the teacher should
orchestrate the content, the representations of the content, as well as all people in the classroom in
relation to one another. They mentioned that teachers need to have the ability to elicit and interpret
what learners do and know, to act appropriately on that and be able to make decisions emerging from
complex interactions. They do not regard learning as receiving information but rather as engaging in
sense-making as the teacher and learners participate together. Although my study was not concerned
with the influence of teacher-learner interaction on learners’ performance, it is worth noting that
Bansilal et al. (2010) found in their study that the continuous support and feedback the tutors provided
to the practising ML teachers (students) in the ACE (ML) programme improved the students’
43
performances over the semester. Bansilal et al. (2010) regarded this increasing interaction in
communities of practice as providing positive learning opportunities to the students.
Learner-learner interaction
Contributing to learners’ development of conceptual understanding are the opportunities learners have
to interact with each other in such ways that they can support, strengthen and challenge each others’
ideas (Artzt, et al., 2008). Lampert (2004) mentioned that the practice of teaching is not only about the
actions of the teacher but the evolution of relationships between the teacher and learners and among learners
themselves around mathematics and engaging together in constructing mathematical meaning (p. 2). Franke et al.
(2007) expressed their concern that many mathematics classrooms do not provide sufficient
opportunities for learners to develop mathematical understanding. They believe learners must have the
opportunity to become encouraged and curious and talk about and with mathematical expertise (p. 229).
National researchers emphasise the importance of learner-centred approaches where learners are
involved in the lesson, taking part in discussions and group work (Brown & Schäfer, 2006; Venkat,
2007; Venkat & Graven, 2008).
Questioning
The value of proficient oral questioning is that the teacher encourages students to make public their knowledge,
skills, and attitudes in relation to the problem under consideration (Artzt, et al., p. 16). Knowledge of learners’
mathematical thinking will support the teachers to provide opportunities for asking questions which are
linked to the learners’ thinking, will elicit discussion and will draw on connections learners need to
make to comprehend the work (Franke et al., 2007).
2.3.3 Learning environment
In my study I based my rationale for a learning environment on the work of Artzt et al. (2008) who
state that a learning environment comprises a particular social and intellectual climate, the use of
effective modes of instruction and pacing of the content and attending to certain administrative
routines. Bransford et al. (2000) on the other hand, regard a learning environment as involving the
rethinking of what should be taught, how it should be taught and how it should be assessed. When
these two views are compared, a common aspect is how the content should be taught and what should
be taught which forms part of the tasks, and how learners should be assessed does not form part of
Artzt et al.’s learning environment.
44
Social and intellectual climate
The social and intellectual climate defines the tone, style, and manner of the interpersonal interactions in the classroom
and contributes to learners’ social and cognitive growth and development (Artzt et al., 2008, p. 14). Franke et al.
(2007) stated that productive practices occur where learners see themselves as comfortable, confident, and
knowledgeable in their abilities to engage in mathematics (p. 227). Silver, Smith and Nelson (1995) found that
creating an atmosphere of trust and mutual respect was critical for the development of valuable
discourse between the teacher and learners and among learners themselves.
Modes of strategies and pacing
Modes of strategies and pacing are the strategies teachers use in the classroom to help learners attain
the objectives of the lesson and teachers should properly pace the activities so that learners have
enough time to participate and construct new knowledge (Artzt et al., 2008). The use of cognitively
guided instruction is suggested by researchers to support the development of learners’ mathematical
understanding (Carpenter et al., 2000; Bransford et al., 2000; Franke et al., 2007). This approach to
teaching assists learners to overcome their misunderstandings and effectively change conceptual
misconceptions. Another effective strategy mentioned is interactive lecture demonstrations (Franke et
al., 2007). Nationally some researchers proposed effective strategies for teaching ML, namely
mathematical modelling (Brown & Schäfer, 2006); discussions and group work (Venkat & Graven,
2008); co-operative learning (Frith & Prince, 2006) and project work (Vithal, 2006).
Administrative routines
According to Artzt et al. (2008) administrative routines are procedures or activities in classroom
organisation and management. Kounin and Gump (1974) regard these routines as providing an ongoing
sign of organisational and interpersonal behaviour in class.
2.4
Mathematics teachers’ knowledge and beliefs
about mathematics and the teaching thereof
In this section I mention the relationship between knowledge and beliefs, give an overview of different
domains of teachers’ knowledge, and discuss what is meant by teachers’ belief systems. Lastly, I point
out what the influence of teachers’ knowledge and beliefs is on their instructional practices.
2.4.1 Relationship between knowledge and beliefs
There is no agreement on the definitions of knowledge and beliefs, their relationship or even their
influence on teaching (Gess-Newsome, Lederman & Gess-Newsome, 2002). She points out some
45
differences and relationships between knowledge and beliefs (Table 2.6) and emphasises that in practice
the lines between knowledge and beliefs can easily become blurred.
Table 2.6: Relationship between knowledge and beliefs
KNOWLEDGE
BELIEFS
Described as:
Evident, dynamic, emotionally
neutral, internally structured.
Develops with:
Functions:
Age and experience
• Conceptual knowledge
(knowledge that is rich in
relationships) is used in problem
solving situations.
• The amount, accessibility and
organisation thereof distinguish
experts from novices.
Both evidential and non-evidential,
static, emotionally bound, organised
into systems.
Episodically
• Have both affective and evaluative
functions;
• Act as information filters;
• Have an impact on how
knowledge is used, organised and
retrieved;
• Are powerful predictors of
behaviour which can either be
consistent or inconsistent with
beliefs.
Artzt et al. (2008) define teacher knowledge as an integrated system of internalised information acquired over time
about pupils, content and pedagogy and beliefs are defined as an integrated system of internalised assumptions about
the subject, the students, the learning, and teaching (p. 20). They further believe that beliefs function as an
interpretative filter for teachers’ goals and knowledge and strongly affect classroom practice (p. 20). Their views on
knowledge and beliefs correspond with Gess-Newsome et al.’s (2002) except that Artzt et al. (2008)
also describe knowledge as organised into systems.
Liljedahl (2008) strongly believes that any discussion on a teacher’s knowledge cannot be restricted to
knowledge of mathematics and knowledge of teaching mathematics but needs to include a discussion
on teacher’s beliefs. He believes teachers’ actions in the classroom are strongly guided by what they
believe about mathematics and the teaching thereof. He further states that it is a false dichotomy to
distinguish between knowledge and beliefs, as a belief becomes knowledge once the truth criterion is
satisfied (p. 2). Leatham (2006, p. 92) explains this argument as follows:
Of all the things we believe, there are some things that we ‘just believe’ and other things we ‘more than believe – we
know’. Those things we ‘more than believe’ we refer to as knowledge and those things we ‘just believe’ we refer to as
beliefs. Thus beliefs and knowledge can profitably be viewed as complementary subsets of the things we believe.
Borko and Putnam (1996) focus on two interrelated aspects of knowledge and beliefs. They argue that
prospective and experienced teachers’ knowledge and beliefs serve as filters through which their
46
learning takes place and on the other hand knowledge and beliefs themselves are critical targets of
change.
In student teacher training it is important that both students’ mathematical knowledge and beliefs need
to be developed and restructured. In my experience once a student’s mathematical knowledge base is
enhanced, the new or enriched knowledge influences the student’s beliefs about mathematics,
reorganising and broadening the student’s existing belief system10. On the other hand when a student’s
beliefs about mathematics are restructured, they sometimes become more receptive to new
mathematical knowledge.
2.4.2 Overview of the different domains of teachers’ knowledge
The most fundamental aspect in effective and proficient teaching of mathematics is a high level of
knowledge (Kilpatrick, 2001; Taylor, 2008). A teacher needs proper subject matter knowledge and a
high level of PCK to assure effective teaching (Shulman, 1986; Ma, 1999). In Taylor’s (2008) study
short tests in literacy and mathematics amongs others were conducted in primary and secondary
schools throughout South Africa and his finding was that teachers clearly do not have the knowledge
that the curricula require to proficiently teach the learners. To address this problem of teachers’
inadequacy, the school system has to re-establish the emphasis on expert knowledge (Taylor, 2008).
Mathematics teaching is a specialised profession, requiring content knowledge, knowledge of the
curriculum, knowledge about how to teach mathematics and knowledge about how learners learn
mathematics. The question is how these different categories of mathematical knowledge are organised.
Some of the leading mathematics researchers’ categories or domains of mathematical knowledge are
given in Table 2.7 below with a brief summary of each.
10
Beliefs systems are discussed in Section 2.4.3.2.
47
Table 2.7: Overview of different domains of mathematical knowledge
OVERVIEW OF DIFFERENT DOMAINS OF MATHEMATICAL
KNOWLEDGE
SHULMAN
GROSSMAN
Categories of
knowledge
Components of
PCK
1986
1990
BORKO AND
PUTNAM
Domains of
knowledge
1996
BALL, THAMES
AND PHELPS
Domains of
knowledge of
teaching
2005
HILL, BALL
AND
SCHILLING
Domain map for
mathematical
knowledge for
teaching
2008
1. Subject matter
content
knowledge
1. Purposes for
teaching
mathematics
2. PCK
2. Learners’
understanding,
conceptions and
misunderstandings
3. Curricular
knowledge
3. Curriculum and
curricular materials
4. Instructional
strategies and
representations for
teaching topics
2.4.2.1
1. General
1. Subject matter
pedagogical
knowledge
knowledge and • Common
beliefs
knowledge of
2. Subject matter mathematics
knowledge and
content
beliefs
• Specialised
knowledge of
mathematics
content
3. PCK and
2. PCK
beliefs
• Knowledge of
content and
students
• Knowledge of
content and
teaching
1. Subject matter
knowledge
• Common content
knowledge
• Specialised
content
knowledge
• Knowledge at the
mathematical
horizon
2. PCK
• Knowledge of
content and
students
• Knowledge of
content and
teaching
• Knowledge of
curriculum
Shulman’s (1986) categories of content knowledge
Shulman (1986) initiated the debate on different categories of knowledge a mathematics teacher needs.
Figure 2.4 indicates his three categories of content knowledge, namely 1) subject matter content
knowledge; 2) PCK; and 3) curricular knowledge.
CONTENT KNOWLEDGE
Subject matter content
knowledge
PCK
Curricular
knowledge
Figure 2.4: Shulman’s (1986) three categories of content knowledge
48
Subject matter content knowledge as one of the categories of content knowledge goes beyond
knowledge of the facts or concepts of a domain to understand the structures of the subject matter. The
second category PCK refers to pedagogical knowledge that goes beyond subject matter knowledge to
subject matter knowledge for teaching, also called teachers’ professional knowledge. This knowledge
includes the most useful forms of representation of ideas, the most powerful analogies, illustrations, examples,
explanations, and demonstrations – in a word, the ways of representing and formulating the subject that makes it
comprehensible to others (Shulman, 1986, p. 9). The ability to use different representations may be derived
from research or from years of experience in practice. This knowledge further includes a teacher’s
understanding of why certain topics are comprehensible and others not, and what preconceptions
learners have that may be misconceptions that could actually be rectified and reorganised by the teacher
through the use of different strategies. Shulman (1987) further describes PCK as the capacity of a teacher to
transform the content knowledge he or she possesses into forms that are pedagogically powerful and yet adaptive to the
variations in ability and background presented by the students (p. 15), in other words, the knowledge of how to
make the subject comprehensible to others. The third category of knowledge, curricular knowledge,
refers to the knowledge about the full range of programmes designed for the teaching of different
topics at given levels in a subject area. It further includes knowledge regarding the variety of
instructional materials available to teach particular curriculum components. It is imperative for teachers
to be familiar with the topics and their levels being taught in the same subject during the preceding and
subsequent years in school. Teachers also need to be familiar with the curriculum materials studied by
learners in other subjects at the same time (Shulman, 1986). Whereas Shulman’s work is foundational in
this area, other mathematics researchers’ categorisations of mathematical knowledge needed for
teaching are discussed below.
2.4.2.2
Grossman’s (1990) components of PCK
Grossman (as cited in Sowder, 2007) (Figure 2.5) distinguishes between four components of PCK,
namely 1) purposes for teaching mathematics; 2) learners’ understandings, conceptions and potential
misunderstandings; 3) curriculum and curricular materials; and 4) instructional strategies and
representations for teaching particular topics.
PCK
Purposes for
teaching
mathematics
Learners’ understanding,
conceptions and
misunderstandings
Curriculum
and curricular
materials
Instructional strategies
and representations for
teaching topics
Figure 2.5: Grossman’s (1990) four components of PCK
49
Borko and Putnam (1996) believe the first component serves as a conceptual map for the teacher’s
instructional decision-making, and as a basis for making decisions regarding classroom objectives,
instructional strategies, student assignments, textbooks, curricular materials and the evaluation of
student learning. This is a salient component of the professional knowledge base of teachers as it
concerns teachers’ knowledge about the nature of the subject and what is important for students to
learn. The second component is knowledge a proficient teacher has to predict what mathematics
learners will understand, how they will understand it, and what their potential misunderstandings will
be. This knowledge enables a teacher to (Sowder, 2007):
… plan more effectively because they can anticipate learners’ difficulties. They know what prior knowledge must be
present to understand something new. They know how to scaffold knowledge to assist students in developing
understanding. They know how to listen to students. Much of this knowledge comes from practice, but teachers with
poor understanding of mathematics are unlikely to develop this type of knowledge, particularly when the mathematics
in the curriculum becomes more sophisticated (p. 165).
Teachers need to have an understanding of learners’ preconceptions, misconceptions, and alternative
conceptions of specific topics (Borko & Putnam, 1996). The third component includes the ability of
teachers to recognise the particular strengths and weaknesses of textbooks and materials they use.
Competent teachers normally have a collection of materials they use when teaching mathematics. This
component also includes knowledge of how the topics are organised and structured both horizontally
and vertically, i.e. within a grade level and across grades. The fourth component is characterised as a
wide selection of significant representations and the ability to adapt these representations in various
ways in order to meet specific goals for specific learners (Borko & Putnam, 1996).
2.4.2.3
Borko and Putnam’s (1996) domains of knowledge
The framework (Figure 2.6) used by Borko and Putnam (1996) in their study “Learning to teach” was
loosely based on Shulman’s categories of knowledge. The proposed domains are 1) general pedagogical
knowledge and beliefs; 2) subject matter knowledge and beliefs; and 3) PCK and beliefs. These three
domains encompass teachers’ knowledge of teaching, subject matter and learners, the three major
determinants of what teachers do in their classrooms.
DOMAINS OF KNOWLEDGE
General pedagogical
knowledge and beliefs
Subject matter knowledge
and beliefs
Pedagogical content
knowledge and beliefs
Figure 2.6: Borko and Putnam’s (1996) three domains of knowledge
50
The first domain does not form part of this study’s focus but refers to a teacher’s knowledge and
beliefs about teaching and learning in general, which include knowledge of strategies for effective
classroom management, various instructional strategies for specific lesson topics, how to create a
positive learning environment and most fundamental a thorough knowledge of learners, of how they
learn and how learning can be fostered by teaching. Borko and Putnam (1996) do not prescribe a
specific model or set of categories to be used regarding the second domain of knowledge, as long as
the need to know more than just facts, terms and concepts of a discipline is recognised. Key aspects in
this domain include knowledge of how to organise ideas, how to make connections among ideas and
knowing different ways of thinking and argumentation. They briefly refer to the work of Shulman in
1986 as well as Ball in 1990 and 1991. Regarding their third domain, they again discuss the work of
Shulman in 1986 and that of Grossman in 1990 emphasising the importance of PCK for teachers who
want to teach for understanding.
2.4.2.4
Ball, Thames and Phelps’ (2005) domains of knowledge for teaching
According to Silverman and Thompson (2008), Shulman invented the term “PCK” referring to specific
content knowledge as applied to teaching. Since then many researchers within the field of mathematics
teacher education have been developing this notion with special reference to the work of Ball in 1990
as well as Ball and Bass in 2000. Ball et al. (2005) use the term “knowledge of mathematics for
teaching” when referring to the special knowledge needed to teach mathematics for understanding.
Their pioneering work has succeeded in identifying various examples of special ways in which one must know
mathematical procedures and representations to interact productively with students in the context of teaching
(Thompson, 1992, p. 500). Ball et al. (2005) divide knowledge of mathematics for teaching in two
domains, namely 1) subject matter knowledge and 2) PCK where each domain is divided in two subdomains as indicated in Figure 2.7 below.
DOMAINS OF KNOWLEDGE OF MATHEMATICS FOR TEACHING
Subject matter knowledge
Common knowledge
of mathematics
content
Pedagogical content knowledge
Specialised knowledge
of mathematics
content
Knowledge of
content and
students
Knowledge of
content and
teaching
Figure 2.7: Ball, Thames and Phelps’ (2005) domains of knowledge of mathematics for
teaching
51
2.4.2.5
Hill, Ball and Schilling’s (2008) domain map for mathematical
knowledge for teaching
This overview concludes with the domain map for mathematical knowledge for teaching of Hill et al.
(2008) as indicated in Figure 2.8. Similar to Ball et al. (2005), they also divide knowledge into two
domains, namely 1) subject matter knowledge and 2) PCK, but included an additional subdomain under
each domain. Subject matter knowledge now consists of 1) common content knowledge; 2)
specialised content knowledge and 3) knowledge at the mathematical horizon. Common content
knowledge involves knowing central facts, concepts and principles within a relationship while
specialised content knowledge goes beyond common content knowledge. Teachers need to have
specialised knowledge to know more than just explaining the content, but must be able to explain why
it is so, why it is worth knowing and how to relate it to other learning outcomes and other disciplines,
both in theory and practice. Knowledge at the mathematical horizon refers to having knowledge of the
subject beyond the years for which a teacher is responsible for. PCK is now divided into 1) knowledge
of content and students; 2) knowledge of content and teaching and 3) knowledge of the curriculum.
Subject
matter
knowledge
Common
Content
Knowledge
Knowledge
at the
Mathematical
horizon
Pedagogical Content Knowledge
Knowledge
of
Content and
Students
Specialised
Content
Knowledge
Pedagogical
content
knowledge
Knowledge of
Curriculum
Knowledge of
Content and
Teaching
Figure 2.8: Hill, Ball and Schilling’s (2008) domain map for mathematical knowledge for
teaching
My study is based on the PCK domain and is discussed under the heading “Conceptual framework”11.
Although the focus of this study is not on ML teachers’ subject matter knowledge, the value of teachers
having a deep knowledge base is still recognised as part of their complete cognitive knowledge base.
2.4.2.6
Summary
Borko and Putnam (1996) argue that the increased attention to teachers’ knowledge in recent years has
led to multiple schemes for categorising teachers’ mathematical knowledge. One must bear in mind that
any categorisation of teachers’ knowledge is somewhat arbitrary, that there is no single definite system
of categorisation of knowledge and that the boundaries between these categorisations are very vague
11
The conceptual framework is discussed in Section 2.5.
52
(Borko & Putnam, 1996; Hill et al., 2008; Shulman, 1986; Sowder, 2007). Although knowledge is
categorised in different domains, these domains are interwoven in teachers’ instructional practices and
teachers continually draw on all aspects of their knowledge (Koellner et al., 2007).
2.4.3 An overview of mathematics teachers’ beliefs about mathematics
and the teaching thereof
Thompson (1992) emphasises the complexity involved in distinguishing between beliefs and knowledge
and found that in many cases teachers treat their beliefs as knowledge. There is no agreement on how
beliefs are to be evaluated, as beliefs cannot be directly observed or measured, but must be inferred
from what people say, intend and do (Pajares, 1992; Thompson, 1992). In this section I discuss the
nature of beliefs and what is meant by teachers’ belief systems.
2.4.3.1
The nature of beliefs
According to Pajares (1992, p. 316) beliefs are formed through a process of enculturation12 and social
construction and influence a person’s perceptions, behaviour and the processing of new information. He
suggests that beliefs created by individuals years ago are fixed and difficult to change whereas newly
formed beliefs are most vulnerable. Listed below are some of the inferences and generalisations Pajares
made regarding teachers’ educational beliefs:
•
•
•
•
•
•
•
•
Beliefs are formed early and tend to self-perpetuate, persevering even against contradictions caused by reason, time,
schooling, or experience.
The belief system has an adaptive function in helping individuals define and understand the world and
themselves.
Epistemological beliefs play a key role in knowledge interpretation and cognitive monitoring.
Beliefs are prioritised according to their connections or relationship to other beliefs or other cognitive and affective
structures. Apparent inconsistencies may be explained by exploring the functional connections and centrality of
the beliefs.
By their very nature and origin, some beliefs are more incontrovertible than others.
Belief change during adulthood is a relatively rare phenomenon, the most common cause being a conversion from
one authority to another or a gestalt shift. Individuals tend to hold on to beliefs based on incorrect or incomplete
knowledge, even after scientifically correct explanations are presented to them.
Beliefs must be inferred, and this inference must take into account the congruence among individuals’ belief
statements, the intentionality to behave in a predisposed manner, and the behaviour related to the belief in
question.
Beliefs about teaching are well established by the time a student gets to college (p. 324-326).
In Table 2.8 below, Schoenfeld (1988, p. 151) mentions four general beliefs held by learners and their
effects in practice.
12
Enculturation involves the incidental learning process individuals undergo through their lives and includes their
assimilation through individual observation, participation and imitation (Pajares, 1992).
53
Table 2.8: Some beliefs held by learners and their effects in practice
BELIEF
EFFECTS IN PRACTICE
The processes of formal mathematics (e.g. ‘proof’) have
little or nothing to do with discovery or invention.
Students who understand the subject matter can solve
assigned mathematics problems in five minutes or less.
Students fail to use information from formal
mathematics when they are in ‘problem solving mode’.
Students stop working on a problem after just a few
minutes since, if they haven’t solved it, they didn’t
understand the material (and therefore will not solve it).
Only geniuses are capable of discovering, creating, or Mathematics is studied passively, with students
really understanding mathematics.
accepting what is passed down ‘from above’ without the
expectation that they can make sense of it for
themselves.
One succeeds in school by performing the tasks, to the Learning is an incidental by-product to ‘getting the
letter, as described by the teacher.
work done’.
Learners cannot be blamed for holding such beliefs as many teachers and parents also hold these or
similar beliefs. Beliefs play an important role in how people view ML. This is an influential factor in the
success of ML as some teachers and learners have their view of ML influenced by the comments from
people outside the mathematics field, and thus see ML as a worthless and insignificant subject.
2.4.3.2
Teachers’ belief systems
Leatham (2006) argues that the way an individual’s various beliefs are related to each other is just as
important as what the individual believes. A belief system according to Thompson (1992) consists of
conscious and subconscious beliefs, preferences concerning mathematics as discipline, concepts,
meaning, rules, and mental images. In Thompson’s (1992) study on teachers’ conceptions consisting of
beliefs, views and preferences, she points out that some people’s actions and behaviours are influenced
by the nature of their beliefs. She further describes a belief system as a metaphor for examining and
describing how an individual’s beliefs are organised (p. 130). She also typifies such systems as being dynamic in
nature because they undergo change and restructuring as individuals evaluate their beliefs against their
experience. According to Ball (1988) and Thompson (1992) preservice teachers’ beliefs are formed
through the development of a network of interrelated ideas about mathematics, the teaching and
learning thereof and also through their experiences at schools.
2.4.4 The influence of teachers’ knowledge and beliefs on their
instructional practices
Pajares (1992) acknowledges the complexity of a psychological construct such as beliefs, but through
his extensive study of numerous researchers’ findings, he found a strong relationship between teachers’
educational beliefs and their planning, instructional decisions, and classroom practices (p. 326), although the link to
learner outcomes has not been explored extensively. Artzt et al. (2008) refer to teachers’ goals,
54
knowledge and beliefs as teachers’ cognitions and describe them as the driving forces (p. 17) behind
teachers’ instructional practices. In this section I mention the influence of mathematics teachers’
knowledge and beliefs on their learners as well as their teaching of the subject. I also report some
findings from South African studies regarding the influence of ML teachers’ knowledge and beliefs on
their instructional practices.
2.4.4.1
The influence of teachers’ knowledge and beliefs on the learners
Learners’ beliefs were for the most part consistent with the beliefs and views held by their teachers
(Thompson, 1992; Ford, 1994). Ford refers to a study he conducted in which teachers regard good
problem solvers as the smarter learners. He found that this belief was then adopted by learners who
claim that you need to be smart to be able to solve problems. This finding is supported by Mason’s
(2003) study in which learners with low achievement comment that in mathematics, intelligence counts
90% and effort 10% and the intelligence a person is born with, can be exploited but not improved, so a
person either can or cannot do mathematics.
Teachers need to accept and acknowledge their responsibility towards learners and need to provide
learners with opportunities for positive learning experiences. The teacher’s attitude towards the subject
is also significant. The teacher has the responsibility to ensure that mathematics comes alive, that
learners find it constructive and develop a passion for the subject. Ollerton (2009) argues that teachers
cannot force learners to have a positive relationship with their subject but they need to realise that they
have a massive impact (p. 2) on their learners. The teacher has the knowledge and skills to create a
positive learning atmosphere where sufficient opportunities are provided to build this relationship. In
order to do this, teachers need a positive attitude towards the subject and its learners.
2.4.4.2
The influence of teachers’ knowledge and beliefs on their teaching
In practice teachers spontaneously convey their ideas on mathematics to their learners (Ball, 1991).
Teachers’ beliefs about mathematics and the teaching thereof often serve as a foundation on which
their instructional practices are built (Liljedahl, 2008; Pajares, 1992). Liljedahl mentions four
researchers’ notions of teachers’ beliefs which in principle are very similar, each notion consisting of
three different perspectives. Dionne’s (1984) notion is divided into the traditional, formalist and
constructivist perspective. Ernest’s (1988) notion describes three philosophies of mathematics, namely
instrumentalist, Platonist and problem solving while Törner and Grigutsch (1994) name their three
perspectives the toolbox aspect, system aspect and process aspect, which are described as follows:
In the toolbox aspect mathematics is seen as a set of rules, formulae, skills and procedures while mathematical
activity means calculating as well as using rules, procedures and formulae. The system aspect refers to teachers who
55
believe mathematics is characterised by logic, rigorous proofs, exact definitions and a precise mathematical language
and doing mathematics consists of accurate proofs as well as of the use of a precise and rigorous language. The process
aspect refers to teachers who believe mathematics is a constructive process where relations between different notions and
sentences play an important role. Here the mathematical activity involves creative steps, such as generating rules and
formulae, thereby inventing or re-inventing the mathematics. (Liljedahl, 2008, p. 2-3)
Beliefs regarding the nature of mathematics influence a teacher’s choice of teaching approach. Teachers
holding a traditional belief most probably believe that mathematics is an abstract phenomenon that is
far distant from reality. These teachers will then struggle to relate mathematics to real-life situations and
tend to believe mathematics consists of a set of rules and procedures that must be learned mechanically
with little or no connection to each other and hardly any relevance to their everyday lives. They also
tend to separate mathematics from the discipline of discovery and creativity (White & Mitchelmore,
2002; Mason, 2003; Schoenfeld, 1988). Thom (as cited in Golafshani, 2002), also claims that all
mathematical pedagogy, even if scarcely coherent, rests on a philosophy of mathematics (p. 204). He comments on
disparities that do occur where teachers’ conceptions are not reflected in their instructional practices
due to constraints such as fixed curricula, time pressure and other external factors.
Findings regarding the influence of ML teachers’ knowledge and beliefs on their teaching of
ML
Sidiropoulos (2008) found that ML teachers’ instructional practices were neither aligned to the ML
curriculum nor to their alleged beliefs and understanding. She states that external strategies and
interventions that promote the required depth of ML teachers’ understanding are required to change
their instructional practices. She further found that the negative and low expectations those teachers
have of their learners negatively affected their implementation of the curriculum in class. One of the
two teachers believed that everyone could do ML if taught properly, but when that teacher was asked
about his learners’ poor performance, the blame was put on learners’ past history with mathematics.
Mhlolo (2008) believes that ML teachers were not equipped with conceptual skills required for the
implementation of the subject and that they need to re-conceptualise their knowledge and beliefs about
the subject. He further states that there is a problematic relationship between the idealised teacher in
policy documents and teachers’ personal identities. He calls it a mismatch, dislocation or disjuncture
between espoused policy images and the personal identities of teachers. Although there are many ML
teachers who do not meet the requirements as set out by the DoE, there are research studies such as
those of Venkat and Graven (2007), telling stories of successful ML teachers.
Venkat and Graven (2007) report on their longitudinal study performed at an inner city school in
Johannesburg on the difference positive and knowledgeable ML teachers make to learners’ experience
56
of the subject. They suggest learner negativity is associated with a lack of substantive change in
teachers’ pedagogic practice, that is where teachers still incorporate the kinds of tasks and pedagogic practice
that have predominated within learners’ earlier experiences with Mathematics (p. 81).
2.4.5 Summary
Liljedahl (2008) strongly believes that any discussion on a teacher’s knowledge cannot be restricted to
knowledge of mathematics and knowledge of teaching mathematics, but needs to include a discussion
on teacher’s beliefs. Different categories or domains of mathematical knowledge exist but any
categorisation of teachers’ knowledge is somewhat arbitrary as there is no single true system of
categories and the boundaries between these categorisations are usually very vague (Borko & Putnam,
1996; Hill et al., 2008; Shulman, 1986; Sowder, 2007). The different categories of a teacher’s knowledge
are also interwoven in their instructional practices and teachers continually draw on all aspects of their
knowledge (Koellner et al., 2007).
Beliefs consist of conscious and subconscious beliefs as well as preferences concerning mathematics as
a discipline. Beliefs can further be defined as convictions or opinions that are formed either by experience or by the
intervention of ideas through the learning process (Ford, 1994, p. 315). Teachers’ beliefs about mathematics can
be located on a perspective continuum from a traditional to a formalist, to a constructivist perspective
(Dionne 1984).
Knowledge and beliefs are closely related and there is a constant interplay between the two, both
influencing teachers’ instructional practices. Borko and Putnam (1996) argue that on the one hand
prospective and experienced teachers’ knowledge and beliefs serve as filters through which their
learning takes place and on the other hand knowledge and beliefs themselves are critical targets of
change.
2.5
Conceptual framework
The focus of my study is to determine the relationship between ML teachers’ knowledge and beliefs
and their instructional practices. My conceptual framework (Figure 2.9) is based on an amalgamation of
Artzt et al.’s (2008) phase dimension framework13, Franke et al.’s (2007) view of a productive practice
and Hill et al.’s (2008) domain map for mathematical knowledge for teaching14.
13
14
See Section 2.3: Teachers’ instructional practices.
See Section 2.4.2.5: Hill, Ball and Schilling’s (2008) domain map for mathematical knowledge for teaching.
57
Figure 2.9: Conceptual framework: Instructional practice, knowledge and beliefs framework of
analysis (adapted from Artzt et al., 2008; Franke et al., 2007; Hill et al., 2008)
Explanation of my conceptual framework
I believe teachers need to apply appropriate instructional strategies to provide learners with
opportunities to develop their critical thinking and problem solving skills. Figure 2.9 illustrates the
components of, and logic behind my framework. To enable me to determine the relationship between
ML teachers’ instructional practices and their knowledge and beliefs, their practices can be observed in
terms of tasks, discourse and the learning environment15. From these observations the ML teachers’
instructional practices can be described according to the instructional approach used and level of
productivity of their practices. Teacher’s instructional approaches will be described as either teachercentred, learner-centred or a combination of teacher- and learner-centred (Artzt et al., 2008). The level
of productivity of the teachers’ instructional practices will be described based on Franke et al.’s (2007)
view of a productive practice: A practice where the teacher listens to learners’ mathematical thinking
and aims to use it to encourage conversation that revolves around the mathematical ideas in the
sequenced problems. Subsequently I will deal with some of the driving forces behind their lessons,
namely teachers’ knowledge and beliefs concerning content and learners16, content and teaching17 and
15
16
See Section 2.5.5: Teachers’ instructional practices.
See Section 2.5.2: PCK and beliefs regarding content and learners.
58
the curriculum18. The three segments of PCK and beliefs, namely knowledge and beliefs of the ML
learners, ML teaching and the ML curriculum, are strongly influenced by teachers’ idiosyncratic beliefs
about the nature of mathematics as a discipline and ML as subject. These idiosyncratic beliefs can
typically be located on a perspective continuum from traditional to formalist to constructivist19.
Included in my framework is Hill et al.’s (2008) PCK domain (learners, teaching and curriculum in
Figure 2.9) and teachers’ MCK which is similar to Hill et al.’s (2008) common content knowledge as
part of their subject matter knowledge domain. The rationale for this decision is that ML focuses on
solving contextualised problems using only basic mathematics. Notwithstanding the fact that ML
teachers need to have MCK, the focus of my study is not on the assessment of their subject matter
knowledge per se. PCK is defined by Hill et al. (2008) as teachers’ content knowledge intertwined with
(p. 375) knowledge of students; knowledge of teaching; and knowledge of the curriculum. Teachers’
beliefs are integrated in my framework as I believe they are inseparable from teachers’ knowledge.
Incidentally, I excluded teachers’ goals as part of teachers’ cognitions in order to keep the study focused
(Artzt et al., 2008).
Even though some of the headings in this section may come across as repetitive, the previous two
sections focussed on some general views and background from the literature regarding mathematical
knowledge, beliefs and instructional practices. In this section I relate the literature to my study
concerning 1) a general view on mathematics teachers’ PCK and beliefs; 2) PCK and beliefs regarding
the learners, teaching and the curriculum; and 3) instructional practices.
2.5.1 General view on mathematics teachers’ knowledge and beliefs
2.5.1.1
Mathematics teachers’ MCK
Hill et al.’s (2008), domain map for mathematical knowledge for teaching (Figure 2.8) is used in an
attempt to conceptualise and develop measures of teachers’ combined knowledge of content and students (p. 372).
Mathematical knowledge for teaching is divided into two domains, namely subject matter knowledge
and PCK. The subject matter knowledge category consists of three strands, namely common content
knowledge, specialised content knowledge, and knowledge at the horizon. For the purpose of this study
I base ML teachers’ MCK on common content knowledge that can be defined as a basic understanding
17
18
19
See Section 2.5.3: PCK and beliefs regarding content and teaching.
See Section 2.5.4: PCK and beliefs regarding curriculum.
See Section 2.5.1.2: Mathematics teachers’ beliefs.
59
of mathematical skills, procedures, and concepts acquired by any well-educated adult enabling a teacher
to solve mathematical problems in the prescribed curriculum (Ball et al., 2005).
2.5.1.2
Mathematics teachers’ PCK
The conceptual knowledge demanded of teachers to teach school mathematics is different from the
mathematical knowledge mathematicians20 might have of advanced topics (Ball, 1990; Leinhardt et al.,
1991). Dewey (1902) also addressed this issue when he wrote:
Every study or subject thus has two aspects: one for the scientist as a scientist; the other for the teacher as a teacher.
… For the scientist, the subject matter represents simply a given body of truth to be employed in locating new
problems, instituting new researches, and carrying them through to a verified outcome… The problem of the teacher is
a different one… What concerns him as teacher is the ways in which that subject may become part of experience,
what there is in the child’s present that is usable with reference to it; how such elements are to be used; how his own
knowledge of the subject-matter may assist in interpreting the child’s needs and doings, and determine the medium in
which the child should be placed in order that his growth may be properly directed. He is concerned, not with the
subject-matter as such, but with the subject-matter as a related factor in a total and growing experience
(p. 162-163).
PCK is regarded as knowledge that is unique to teachers; knowledge that can only be developed over
time through experience in the classroom or practice and can therefore not be taught (Ball, 1988; Ball et
al., 2005; Koellner et al., 2007; Ma, 1999; Shulman, 1986; Sowder, 2007). Having profound
understanding and knowledge of mathematical subject matter is a prerequisite to develop PCK (Ball,
1990; Van Driel, Verloop & De Vos, 1998). Sowder (2007) is of the opinion that it is only as
mathematics increases in sophistication, that a deep content knowledge base becomes a prerequisite in
developing PCK. Although ML teachers need to have mathematical content knowledge, the DoE
(2003a) stated that the content in ML must not be an end in itself, but must serve the learning outcome
of applying content to certain contexts. Since the emphasis in ML is on solving real-life contextualised
problems using basic mathematics, it is debatable whether a high level of subject matter knowledge is
required by the ML teacher. My belief is nevertheless that ML teachers do need to have conceptual
knowledge of the subject matter involved in the curriculum to enable them to use their knowledge
efficiently in preparing their learners for their future lives.
2.5.1.3
Mathematics teachers’ beliefs
Teachers’ beliefs about mathematics are powerful as they influence their representations of
mathematics (Ball, 1990). She mentions a few beliefs of mathematics teachers that need to be examined
such as their:
20
Mathematicians refer to people using higher levels of formal mathematics in their professions such as engineers and
scientists.
60
understandings about the nature of mathematical knowledge and of mathematics as a field and the substance of
mathematics. What counts as an answer in mathematics? What establishes the validity of an answer? What is
involved in doing mathematics? What do mathematicians do? … What is the origin of some of the mathematics we
use today and how does mathematics change? (p. 458) What do they think an explanation is? How do they sort out
convention from logic with respect to particular principles or ideas? What do they think it means to ‘know’ or to ‘do’
mathematics? (p. 459)
The saying ‘we teach what we believe’ emphasises the importance and far-reaching effects of teachers’
beliefs on their instructional practice (Leatham, 2006). Ollerton (2009) believes that once teachers have
articulated what their pedagogy is and obtain clarity on the beliefs and values they hold and which drive
them, it will help them to strengthen effective practice. Mathematics teachers’ beliefs about
mathematics are located on a perspective continuum from traditional to formalist to constructivist
(Dionne 1984). In my study I regard ML teachers’ knowledge and beliefs as inseparable driving forces
behind their instructional practices. In many cases teachers’ beliefs are established by their knowledge
and changing their knowledge base will change their belief system.
2.5.2 The three domains of PCK and beliefs
In this section I discuss my study’s view on mathematics teachers’ PCK and beliefs regarding the three
domains, namely 1) content and learners; 2) content and teaching; and 3) the curriculum. In my
discussion of each domain, I firstly mention some views from the literature regarding mathematics
teachers in general and then discuss ML teachers’ PCK and beliefs concerning that specific domain.
2.5.2.1
PCK and beliefs regarding content and learners
Knowledge of content and learners includes a teacher’s ability to predict what mathematics learners will
understand and how they will understand it, how learners will probably approach a task, understanding
why certain topics are comprehensible and others not, what alternative conceptions and preconceptions
learners have that could be misconceptions and that should be rectified and reorganised by the teacher
through the use of different strategies (Ball, 1990; Borko & Putnam, 1996; Hill et al., 2008; Shulman,
1986; Sowder 2007). According to Sowder (2007), having this knowledge enables a teacher to:
plan more effectively because they can anticipate learners’ difficulties. They know what prior knowledge must be
present to understand something new. They know how to listen to students. Much of this knowledge comes from
practice, but teachers who have poor understanding of mathematics themselves are unlikely to develop this type of
knowledge, particularly when the mathematics in the curriculum becomes more sophisticated … (p. 165)
This knowledge of content and learners should be taken into account when planning lessons (Koellner
et al., 2007). Teachers must be able to see what learners do, hear what they think and then be able to
act appropriately as mentors to facilitate the learning process (Hill et al., 2008).
61
ML teachers’ knowledge and beliefs regarding their learners
Capturing learners’ attention is particularly significant in the ML classrooms as many ML learners lack
motivation, have negative attitudes and experience anxiety, causing teachers to be discouraged in
teaching this subject (Mbekwa, 2007; Venkat, 2007; Venkat & Graven, 2007; Vermeulen, 2007). A
shortcoming in the implementation process of ML is that teachers were not empowered to deal with and
assist learners with a past history of low attainment in mathematics (Sidiropoulos, 2008, p. 250). To meet these
challenges teachers need a firm knowledge base of the purpose and goal of the subject, its content, the
teaching thereof, its learners and classroom management skills. Some teachers, for instance, only listen
for correct answers and do not use incorrect answers to engage learners in mathematical thinking. Hill
et al. (2007) emphasise the necessity for teachers to rephrase learners’ questions to help them unravel
the problem themselves.
2.5.2.2
Knowledge and beliefs regarding content and teaching
Knowledge regarding content and teaching includes the most useful forms of representation of ideas, the most
powerful analogies, illustrations, examples, explanations, and demonstrations – in a word, the ways of representing and
formulating the subject that make it comprehensible to others (Shulman, 1986, p. 9). He further describes this
knowledge as the capacity of a teacher to transform the content knowledge he or she possesses into forms that are
pedagogically powerful and yet adaptive to the variations in ability and background presented by the students (Shulman,
1987, p. 15).
Teachers should have the ability to recognise the instructional advantages and constraints of using and
adapting various representations depending on the content and needs of the learners and also have the
ability to sequence content to facilitate student learning (Ball, 1990; Borko & Putnam, 1996; Koellner et
al., 2007). Teachers furthermore need to be able to present subject matter in multiple ways like using
story problems, pictures, situations and concrete materials. This knowledge is required to choose the
appropriate pedagogical strategy and instructional material for a lesson, to consider which tasks to set
and which assessment techniques to use. Knowledge of content and teaching further assists teachers to
reflect on their own practice for the purpose of improvement (Koellner et al., 2007). Sowder (2007)
feels teachers need to know how to scaffold knowledge to assist learners in developing understanding.
Hill et al. (2008) concur with this view and assert that teachers need to know different ways of how to
build on student mathematical thinking or how to remedy student errors.
Approaches to teaching ML
Approaches to the teaching and learning of ML should provide extended opportunities to engage with ML in
diverse contexts at a level that learners can access logically (DoE, 2003c, p. 5). The teaching of mathematics in a
62
contextualised and de-compartmentalised way however complicates the teaching of ML as some
teachers lack the knowledge and skills to do so. Sidiropoulos (2008) found that teachers use ML
textbooks where content is embedded in context, but predominantly deliver the algorithmic content to the
learners, and afterwards dress it up with an artificial level of context, maybe using a picture (p. 227).
Principles that guided Frith and Prince’s (2006) curriculum in preparing in-service teachers to teach ML
are the following:
•
•
•
•
•
That material should be context-based and make use of real relevant intrinsically motivating contexts, wherever
possible.
That curriculum tasks should require the exercise of several related competencies, such as writing and using
computers, not just mathematical skills.
That the production of a (mainly verbal) product as an outcome of mathematically literate practice is important
(as well as the understanding and interpretation of existing information).
That students’ confidence should be promoted.
That co-operative learning should be emphasised (p. 55).
ML teachers’ knowledge of different teaching approaches
Sidiropoulos (2008) believes ML teachers’ PCK regarding different teaching strategies or approaches is
inadequate. The success of ML depends largely on the skills of the teachers to apply appropriate
teaching approaches such as discussions and problem solving (Brown & Schäfer, 2006; Venkat, 2007).
Venkat (2007) reports that learners became positive about ML, enjoying the subject and finding it
practical, useful and challenging when teachers changed the nature of tasks and interactions they used
in the ML classroom. Both these shifts provided openings for learners to communicate and participate in classroom
activities, in addition to gaining understandings and make sense of the mathematics being used (Venkat, 2007, p. 30).
They enjoyed being active and focused and coming up with solutions to everyday problems, even
sharing them with their parents at home. Vithal (2006) proposes project work as an approach since the
purpose of project work is to improve effective participation and to provide learners with the
opportunity to ‘read the world’ using mathematics, to develop mathematical power and to change their
orientation towards mathematics. Project work is based on six conceptual principles, namely problem
orientation, participant-directed, inter-disciplinarily, exemplarity, assessment, and practical organisation
(Venkat, 2007). The notion of ML being inter-disciplinary implies that teachers from different
disciplines should work together, drawing on their different disciplines to solve various problems.
ML teachers’ beliefs regarding the teaching of ML
Learners’ positive experiences normally stem from situations in which the teacher has a positive
attitude, believes in the subject and uses approaches applicable to the requirements of the subject.
Mathematics teachers hold a strong belief that teaching ML is a major threat to their Mathematics teacher
63
status-identity (Sidiropoulos, 2008, p. 251). Labels teachers put on ML are lesser maths; it is not real maths; it
is the beginning of maths; it is a maths only better than nothing; it is the maths of oranges and bananas; it is a subject for
the doffies [dim ones] (p. 225). Even the learners and broader community held a similar impoverished view (p.
222) of teachers who teach ML as learners directly asked ML teachers if they are not as bright as the
other teachers or if they are being punished for something they did wrong. Unless teachers undergo
appropriate development programmes to seek a change in their behaviour, PCK and beliefs about the
nature and value of ML, they will continue to fall back on knowledge and beliefs already entrenched in their
instructional practice (p. 205-206). The reality is that deep change is even more difficult to attain on an emotional level
(p. 225), is complex and also personal as new teacher identities will require time to develop and unfold even under
optimal conditions of reform (p. 205). She further found that the ML teachers in her study do not want to
change and they do not want to lower their status in society as Mathematics teachers. She is of the
opinion that the best way to solve this problem is to recruit new ML teachers who do not need to
undergo change in status-identity instead of trying to change the qualified and experienced mathematics
educators (p. 226).
2.5.2.3
Knowledge and beliefs regarding the curriculum
Curricular knowledge refers to the knowledge of the full range of programmes designed for the
teaching of different topics at given levels in a subject area. Teachers need to be familiar with the topics
and level thereof being taught in the same subject during the preceding and later years in school, in
other words how topics are organised horizontally and vertically. Curricular knowledge further includes
knowledge regarding the variety of instructional materials available to teach particular curriculum
components. Teachers need to recognise the particular strengths and weaknesses of textbooks and
materials they are using. Competent teachers normally have a collection of materials they use when
teaching mathematics. They also need to be familiar with the curriculum materials studied by learners in
other subjects at the same time (Borko & Putnam, 1996; Shulman, 1986).
ML teachers and the curriculum
ML teachers need to be informed not only about the ML subject curriculum but all relevant
departmental documents in order to understand what is expected of them to teach this relatively new
subject. For example the DoE (2006) provided a list of resources needed to teach ML such as
advertisements from the media containing contextual problems on percentage and interest rate, graphs
and tables, etcetera. The new CAPS (DoE, 2011a) for ML will hopefully assist teachers concerning the
issue of how to progress from one year to the next.
64
In Sidiropoulos’ (2008) study on the implementation of the ML curriculum, she found that the purpose
of the ML curriculum had not been well understood by the teachers and consequently they did not
value the curriculum and the possibilities it provided for. Negative labels teachers put on ML stem
from the fact that the ML curriculum, which is distinctly different from curricula of the past was diktat on
educators without due consideration on how substantial the required change would be in terms of understanding the
purpose and possibilities of this new curriculum (p. 249). She believes that if the broader purpose and value of
the ML curriculum is well understood by teachers and all stakeholders, this threat to identity may not have
been as prominent as it was (p. 225). She further found that the teachers’ disjointed understandings of the
ML curriculum put emphasis on the complexity of bridging the gap between curriculum as intended and curriculum
as implemented in the context of actual classrooms (p. 225). Other problems are the fact that the curriculum
assumes that all learners can be taught to become mathematically literate and that all educators understood
the concept of mathematical literacy that by its very own nature is distinctly dissimilar from that of mathematics or
numeracy (p. 250), the only known mathematical subjects taught by teachers in South Africa.
2.5.3 Teachers’ instructional practices
In defining instructional practice, Englert et al. (1992) refer to the qualitative dimensions of teachers’
behaviour in their practices. These dimensions involve teachers’ abilities to model cognitive strategies in
meaningful and purposive activities, adjust instruction as required, promote classroom dialogues, and
establish classroom communities in which learners collaboratively and cooperatively participate in
enquiry-related activities. A framework used to observe and describe teachers’ instructional practices is
built on three observable aspects of mathematics lessons, namely tasks, discourse and the learning
environment (Artzt et al., 2008). The characteristics of tasks, discourse and the learning environment
are provided in Table 2.9 below (Artzt et al., 2008, p. 10-12).
Table 2.9: The observable aspects of a lesson
TASKS
Provide opportunities for learners to connect new knowledge to existing knowledge through active
engagement in problem solving activities. Tasks should be motivational, at an appropriate level of
difficulty and sequenced in a meaningful way to help learners clarifying their ideas.
Modes of
Uses different representations such as symbols, diagrams, manipulatives, and
representation computer representations to facilitate content clarity, enabling learners to
connect new knowledge to prior knowledge and skills.
Motivational
strategies
Uses tasks that capture learners’ curiosity, inspiring them to reflect on their
conjectures. The diversity of learners’ interest and experiences should be taken
into account.
65
Sequencing
and difficulty
levels
Sequences tasks in assisting learners to make connections between ideas and
develop conceptual understanding. Uses tasks suitable to what learners already
know and can do and what they need to learn.
DISCOURSE
Describes the verbal exchange among members of the community in the classroom, both teachers
and learners.
TeacherCommunicates with learners in an accepting, non-judgmental manner,
learner
encouraging learner participation. Requires learners to explain and demonstrate
interaction
their thinking while carefully listening to provide clarification.
Learner-learner Encourages learners to listen to, respond to and question one another in order
interaction
to assess each other’s ideas or solutions, and if necessary to rectify or adjust.
Questioning
Poses a variety of types and levels of questions and allowing enough time to
elicit thinking and to follow their reasoning through.
LEARNING ENVIRONMENT
Describe the conditions under which the teaching and learning process unfolds in the classroom
and refer to the circumstances that affect the flow of action in the classroom. This should promote
the development of learners’ conceptual understanding.
Social and
Establishes and maintains a positive culture with and among learners by valuing
intellectual
their ideas and showing respect. Enforces classroom rules to ensure positive
climate
learner behaviour.
Modes of
instruction and
pacing
Uses instructional strategies that encourage and support student involvement
and purposefulness. Attends to time management to ensure learners have the
opportunity to explore mathematical ideas and to express them.
Administrative
routines
Uses effective procedures in organising and managing class activities to
maximise learners’ active involvement in the discourse and tasks.
Flowing from observing the teachers, their instructional practices will be described according to their
instructional approaches and general level of productivity. Table 2.10 below indicates the patterns being
identified in teachers’ instructional practices (Artzt et al., 2008).
Table 2.10: Teacher-centred versus learner-centred instructional practices
Teacher-centred
Tasks
Discourse
Learner-centred
Impede learners’ efforts to build on
prior knowledge; unrelated to learners’
interest; often too easy or too difficult;
illogically sequenced.
Multiple accurate representations to
facilitate content clarity; connect to
learners’ prior knowledge; relevant and
interesting tasks; challenging and
sequenced.
Teacher judges learners’ responses and Teacher has accepting attitude toward
resolves questions without learner
learners’ ideas and encourage learners
input; learners give short responses,
to think and reason; learners explain
lacking explanation and justification; no and justify their responses; learners
interaction among learners; low-level,
listen to and respond to one another’s
66
leading questions are asked.
Learning
environment
Tense and awkward atmosphere;
superficial requests for and use of
learners’ input; use of strategies that
discourage learner participation; pace
too fast or too slow; learners
uninvolved; disorder in class.
ideas; variety of levels and types of
questions.
Relaxed yet businesslike atmosphere;
focus on learner input; strategies focus
on learner involvement; effective
organising and managing of class;
learners actively involved.
To describe the productivity of the teachers’ instructional practices is complex as it is consistently
controversial and will remain controversial to what constitutes good teaching (Franke et al., 2007, p. 226).
2.5.4 Summary
The conceptual framework for my study is based on the domain map for mathematical knowledge for
teaching (Hill et al., 2008) and the categories of an instructional practice, namely tasks, discourse and
learning environment (Artzt et al., 2008). MCK is based on the category common content knowledge as
part of Hill et al’s (2008) subject matter domain. PCK consists of knowledge of content and learners;
knowledge of content and teaching; and knowledge of the curriculum. Knowledge of content and
learners includes teachers’ ability to understand and predict what learners will understand, how they will
understand it, what their preconceptions are, what prior knowledge they need, what possible
misconceptions and alternative conceptions they could have and why some topics are more
comprehensible than others. Knowledge of content and teaching includes different pedagogical
approaches, strategies and representations, use of meaningful sequencing of content and appropriate
instructional material, all depending on the content and learners to make the subject comprehensible.
Curriculum knowledge includes knowledge of the purpose, aim, learning outcomes and assessment
criteria of the subject; the topics to be covered during the preceding, current and later years as well as
the level thereof; the teaching strategies applicable to the subject; and the strengths and weaknesses of
instructional materials.
2.6
Conclusion
Chapter 2 explored the international and national perspectives of mathematical literacy. Although the
emphasis and terminology differ between different countries, researchers unanimously believe in the
importance and value of learners’ mathematical literacy skills that should be developed and enhanced
(Gellert et al., 2001; Jablonka, 2003; Knoblauch, 1990; McCrone & Dossey, 2007; Queensland
Government, 2007b; Skovsmose, 2007). In South Africa ML refers to both a subject and a competency
whereas in other countries it is mainly the latter (Christiansen, 2007). ML may have become stigmatised
67
as a subject having virtually no meaning when it comes to career opportunities, but a closer
investigation showed that this subject has its own demands and requires specialised PCK and a positive
belief system towards ML as a specialised subject. Teachers’ instructional practices are strongly
influenced not only by their MCK, but by their beliefs about the nature of ML as well as their PCK and
beliefs about the learners, the teaching of the subject as well as the curriculum. Beliefs are powerful and
many times the beliefs learners have are for the most part consistent with those of the teachers. The
beliefs teachers hold regarding mathematics as discipline normally varies from a traditional perspective
to a formalist perspective through to a constructivist perspective (Liljedahl, 2007). The next chapter
provides a lay-out of the study’s methodology.
68
Chapter 3
Methodology
3.1
Introduction
This chapter provides a description of the methodology used in this study. In my attempt to understand
the phenomena being studied, I firstly discuss my research paradigm and assumptions as the lenses
through which I view the world. I then explain the rationale for choosing qualitative research as my
approach and case studies as the design used for my study. The research site, sample selection and data
collection techniques are carefully described followed by the data analysis strategies. Lastly I discuss
critical issues such as the trustworthiness of the study and ethical considerations applicable to the study.
3.2
Research paradigm and assumptions
In my research endeavour to obtain knowledge and understanding of the phenomenon being studied, I
need to mention the way in which [I] view the world, by what [I] view understanding to be and by what [I] regard as
the purpose of understanding (Cohen et al., 2001, p. 3). This study’s research paradigm and ontological,
epistemological and methodological assumptions are discussed below.
3.2.1 Research paradigm
My research paradigm is social constructivism which suggests that all knowledge is constructed and
based upon not only prior knowledge, but also the cultural and social context. According to
Nieuwenhuis (2007) the origins of mathematics are social or cultural and the justification of
mathematical knowledge rests on its quasi-empirical basis. Constructivism implies a subjective approach
that is concerned with the uniqueness of each particular situation (idiographic) (Nieuwenhuis, 2007, p. 51). The
focus is on the social construction of people’s ideas and concepts, on how and why they interact with each other, and their
motives and relationships (Nieuwenhuis, 2007, p. 54). What forms the basis for a social constructivist
philosophy of mathematics are the facts that knowledge is not passively received but actively built up by the
cognizing subject and the function of cognition is adaptive and serves the organization of the experiential world, not the
69
discovery of ontological reality (Ollerton, 2009, p. 78). Ernest (1988) also believes knowledge is acquired by
oneself and that it cannot be transferred from one person to another. Ollerton’s (2009) opinion is that
people working individually is not the way most of us operate for the vast majority of the time and that students
might be encouraged to work individually in the first instance and later share and compare their information (p. 77-78),
a view with which I strongly agree. These stated principles of social constructivism have certain
implications on teachers’ approaches to teaching.
According to Koehler and Grouws (1992) teaching based on constructivism as learning theory is
viewed on a continuum between negotiation and imposition, and the teacher’s role is to find and adjust activities for
students (p. 123). Social interaction where learners have the opportunity to communicate and work in
collaboration with their peers is a critical part of knowledge construction (Koehler & Grouws, 1992;
Ollerton, 2009). Social interaction, group work, problem solving and learner-centred approaches play
significant roles in learners’ construction of their own knowledge. I believe that although formal
instruction has some influence on learners’ understanding, learners do not directly assimilate knowledge
or understanding from the teacher, but build their own understanding in the ML classroom through
experience and maturation. The role of the teacher is therefore to guide and mentor the learners in
developing understanding.
3.2.2 Paradigmatic assumptions
The nature of my study is based on three assumptions, namely the ontological, epistemological and
methodological assumptions. Ontology and epistemology have direct implications for the
methodological assumption as it demands different research methods (Cohen et al., 2001). The nature
of my study is subjective as I am personally involved in the process of making sense of the uniqueness
of the situation being studied (Nieuwenhuis, 2007). I hold the nominalist position as ontological
assumption where I understand reality through words and regard reality as the product of individual
consciousness (Cohen et al., 2001). Regarding the epistemological assumption, my study holds an
interpretive position where knowledge is of a softer or transcendental kind and based on experience
and insight of a personal nature. This nominalist and interpretive position demands an idiographic
methodological preference where the focus is on the subjective experience of individuals who create,
modify and interpret the world they are in (Cohen et al., 2001).
70
3.3
Research approach and design
The table below provides a synopsis of the research methodology components of my research.
Table 3.1: Synopsis of methodology
Research
approach
Research
design
Main
question
Research
subquestions
QUALITATIVE
Case study: Exploratory
A case can be a unit or group of people that are analysed and can also consist of
another group(s) to enhance the trustworthiness of a study. This case study consists of
ML teachers as a group. I observe their instructional practices and determine the nature
of their knowledge and beliefs in order to explore the relationship between them. The
nature of the data gathered is qualitative and the nature of the case study is exploratory
(Cohen et al., 2001; Edwards & Talbot 1999; Nieuwenhuis, 2007).
What is the relationship between ML teachers’ knowledge and beliefs and their
instructional practices?
Question 1
Question 2
Question 3
How can ML teachers’
What is the nature of
How do ML teachers’
instructional practices be ML teachers’ knowledge
knowledge and beliefs relate to
described?
and beliefs?
their instructional practices?
• To explore what the
• To determine what
relationship is between
• To comment on the
teachers do in their
teachers’ instructional
teachers’ level of
classrooms with
practices and their knowledge
MCK.
respect to tasks given,
and beliefs.
Objectives
• To further explore
discourse that takes
of the sub• To consider the extent to
teachers’ beliefs
place and the learning
questions
which teachers use PCK in
regarding
ML
learners,
environment which is
their
lessons.
the teaching of ML
established.
and the ML
• To determine why teachers do
curriculum.
what they do in their ML
classrooms.
Participants One Grade 11 ML teacher from five different secondary schools
• Three observations per teacher
Data
collection
• Three semi-structured interviews per teacher: one each before the second and third
techniques
observed lessons and one after the observations.
Observations
Observations
Techniques
Observations
per question
Interviews
Interviews
DEDUCTIVE-inductive approach for data analysis (uppercase denotes the preference
given to the style of analysis)
Data
analysis
•
•
•
•
•
•
Establish units of analysis of the data
Create a ‘domain analysis’
Use ATLAS.ti 6 to analyse the video and audio data
Establish relationships and links between the domains
Making speculative inferences
Summarising
71
3.3.1 Research approach
The research approach for this study is qualitative. Qualitative research seeks … to gain better
understanding of intentionality (from the speech response of the researchee) and meaning (why did this person/group say
something and what did it mean to them?) … to describe and to understand, rather than to explain and predict (Babbie
& Mouton 2001, p. 49). Hogan et al. (2009) point out that qualitative research is about researching
specific meanings, emotions and practices that emerge through the interactions and interdependencies between people (p. 4).
Similarly White (2005) emphasises that qualitative research is concerned with conditions or
relationships that exist, beliefs and attitudes that are held, effects that are being felt and trends that are
developed. It also provides opportunities for marginalised groups to voice their opinions on matters
that are of concern to them and which may have been overlooked in conventional research. The focus
of my study is to describe ML teachers’ instructional practices, their knowledge and beliefs and the
relationship between their knowledge and beliefs and their practices within their naturally occurring context
with the intention of developing an understanding of the meanings imparted by the [ML teachers] – so that the phenomena
can be described in terms of the meaning that they have for the [ML learners] (Nieuwenhuis, 2007).
3.3.2 Research design
This is a case study. According to Cohen et al. (2001) case studies can establish cause and effect and observe
effects in real contexts, recognising that context is a powerful determinant of both causes and effects (p. 181). Edwards
and Talbot (1999) define a case study as a unit of analysis such as an individual or work team where each
case has within it a set of inter-relationships which both bind it together and shape it, but also interact with the world.
The idea of a case study is to allow a fine-tuned exploration of complex sets of inter-relationships (p. 51). Edwards
and Talbot (1999) further distinguish between three uses of case studies, namely explanatory,
descriptive and exploratory cases. This is an exploratory case study where the focus of the study has
already been decided on and explained in the conceptual framework. The focus is the case itself and its own
very particular features, therefore was used to examine complex phenomena (Edwards & Talbot, 1999, p. 53). The
ML teachers are regarded as the ‘unit’ that is studied in order to explore the relationship between ML
teachers’ instructional practices and the driving forces behind their practices. My involvement in the
case study gives a sense of being there (Cohen et al., 2001, p. 79). The analysis of the data enhances my
understanding of the phenomena which will be reflected in an improved ML teacher preparation
programme and will hopefully add value to theory building.
72
3.4
Research site and sampling
A case study requires intensive data collection as well as high quality data and it is preferable to work indepth with a small number of teachers. The inductive approach also requires sampling to be small and
information-orientated, but representative (Edwards & Talbot, 1999). The population consists of the
ML teachers in South Africa which include Mathematics and non-Mathematics teachers from urban
and rural government and private schools. Due to this wide variety of teachers it is not possible to
choose a representative sample. Convenience and purposive sampling were implemented to select five
different secondary schools in Tshwane. Listed below in Table 3.2 are the criteria that justify the
inclusion and exclusion of schools and teachers in the sample:
Criteria
Table 3.2: Criteria justifying inclusion and exclusion in the sample
•
•
•
•
•
•
•
Inclusion
Mathematics teachers
Non-Mathematics teachers
Male and female teachers
Teachers with at least one year’s experience
of teaching ML
Different races
Schools with different performance levels
Section 21, formerly disadvantaged and
independent schools
•
•
•
Exclusion
Private schools
Schools situated far from my work
Not more than one poor
performance school
The sampling is partly convenient as the five schools were chosen from schools in Tshwane that were
easily accessible. Through purposive sampling three traditional black (formerly disadvantaged and
independent), one predominantly white (Section 21) and one predominantly black (Section 21) schools
were chosen. From each school only the Grade 11 teacher participated with the prerequisite that the
teacher had taught ML for at least one year. My rationale for using the Grade 11 teachers in my study is
that I presume some problems or challenges experienced by both teachers and learners either diminish
or increase from Grade 10 to Grade 12. This allows for a kind of ‘middle of the road’ scenario where
the impediment of data, due to possible problems experienced by Grade 10 and 12 teachers and
learners, is reduced. From this sample valuable information was collected regarding the ML teachers’
instructional practices, their knowledge and beliefs.
3.5
Data collection techniques
Case studies are normally time-consuming and not an easy option as the focus is on meanings and the
complexity of interrelations which demand high quality data (Edwards & Talbot, 1999). Cohen et al.
73
(2001) argue that the purposes of case studies are to portray, analyse and interpret the uniqueness of real
individuals and situations … and to catch the complexity and situatedness of behaviour (p. 79). The use of
observations and interviews as data collection techniques improve the quality of this study’s data and
increase the trustworthiness of the study. The classroom observations and personal interviews allowed
me to explore reality by becoming part of the participants’ lives. These data collection techniques were
informed by predetermined categories derived from the study’s conceptual framework.
The process of data collection for each teacher consisted of three observations in an effort to obtain a
relatively true account of the teachers’ instructional practices. Interviews were conducted with the
teachers the period before the second and third observations were made. These interviews were based
on the teachers’ planning of the lessons in order for me to obtain information regarding their PCK.
This was followed by one in-depth interview, based on their lessons presented, as well as their PCK
and beliefs. Following Figure 3.1 below on the data collection process, the observations and interviews
are discussed.
Figure 3.1: The data collection process
3.5.1 Observations
Cohen et al. (2001) believe that case studies are typified by observations as the purpose of observations
is to probe deeply and to analyse intensively the multifarious phenomena that constitute the life cycle of the unit with a view
74
to establishing generalisations about the wider population to which that unit belongs (p. 185). The type of
observation I used was that of the observer as participant, not directly influencing the teaching process
in the class situation (Nieuwenhuis, 2007). The purpose of the classroom observations was to describe
the ML teachers’ instructional practices according to three different dimensions of their lessons, namely
tasks given, discourse and the learning environment21. I also observed the teachers’ classroom
performances with a view to studying demonstrations of their knowledge regarding the ML learners,
the teaching of ML and the ML curriculum.
I decided to undertake three observations, preferably of different classes, to obtain a general impression
of the teacher’s instructional practice. The first observation was before any interviews were conducted
so that the teacher could not be influenced by the questions from the interviews. An observation sheet
was compiled in advance to cover the predetermined categories (Cohen et al., 2001). Figure 3.2 below
provides a clarification of the lesson observations in terms of the teachers’ instructional practices22 and
knowledge and beliefs.
Figure 3.2: Elucidation of the character of the lesson observations
The lessons were video-taped and transcribed afterwards. Field notes were made regarding any
unexpected valuable data that had emerged. Classroom observations are essential since lesson
preparations can provide direction to a lesson, but can never predict exactly what will happen in class,
21
22
See Section 2.5.5: Teachers’ instructional practices.
See Section 2.5.5: Teachers’ instructional practices for complete discussion.
75
as learners’ participation, contribution and interaction with the content, teacher and peers allow for that
dynamic aspect in class from which valuable data can be collected.
3.5.2 Interviews
According to Nieuwenhuis (2007) [t]he aim of qualitative interviews is to see the world through the eyes of the
participant (p. 87) and to learn more about the participants’ behaviours, beliefs and views. In these twoway conversations I tried to remain sensitive to responses of the participants and to identify new
aspects to be discussed (Nieuwenhuis, 2007). Table 3.3 below provides an elucidation of the character
of the interviews.
INTERVIEW 2
A semi-structured and structured interview conducted at
the end of the data collection phase
INTERVIEW 1
A semi-structured interview
conducted prior to the 2nd and
3rd lesson observations
Table 3.3: Elucidation of the character of the interviews
PURPOSE OF INTERVIEW
To gain insight into the participants’ planning of their lessons and providing evidence
of the teachers’ PCK and beliefs regarding the ML learners, the teaching of ML and
the ML curriculum.
EXAMPLES OF INTERVIEW QUESTION CONTENT
• Teachers’ predictions on which content the learners would and would not
understand and the reasons for their understanding or not,
• What possible misunderstandings could occur,
• How they planned their lessons in order to bring learners to understanding the
content and context, and
• What prior-knowledge should be present in the lesson.
PURPOSE OF SECTION A
I used clips from the video recordings from the three lessons presented to guide a
discussion with the participants in order to obtain a better understanding of their
practices. The discussion focussed on the three dimensions of their lessons, namely
tasks, discourse and the learning environment.
PURPOSE OF SECTION B
The questions in this section were divided into three subsections, probing for
teachers’:
• Beliefs about the nature and value of mathematics as discipline and ML as subject,
• Knowledge and beliefs regarding the ML learners, and
• Knowledge and beliefs regarding the teaching of ML.
PURPOSE OF SECTION C
• This section consisted of a set of predetermined questions and allows for
clarification of answers in writing. The questions were based on some of the
official documents such as the NCS Grades 10-12 (General) Mathematical
Literacy (DoE, 2003) as well as the new CAPS (DoE, 2011a).
• The reason for including this section in the interview and not treating it separately
in a questionnaire was to ensure that true data were captured as it was impossible
for the teachers to discuss it with other teachers or to consult the relevant
documents.
76
All interviews were audio-taped and the tape-recordings were transcribed verbatim and coded
afterwards by me. The final aim was to integrate the findings from the observations and interviews to
make sense of the reality and the complexity of the phenomenon, in other words to determine the
relationship between ML teachers’ knowledge and beliefs and their instructional practices.
3.6
Data analysis strategies
According to Cohen et al. (2001) data analysis involves organising, accounting for, and explaining the data; in short,
making sense of the data ….., noting patterns, themes, categories and regularities (p. 147). They further suggest that
early analysis will reduce the problem of data overload as huge volumes of data rapidly accumulate in
qualitative research. Edwards and Talbot (1999) agree to this practice as they believe continuous analysis
of data keeps control of the project and reflects on the approach and design of the project as well as
informing the next data gathering process. To analyse interviews as qualitative data, one has to realise it is
more of a reflexive, reactive interaction between the researcher and the de-contextualised data that are already interpretations of
a social encounter (Cohen et al., 2001, p. 282).
In my study I use DEDUCTIVE-inductive (uppercase denotes the preference given to the style of
analysis) qualitative data analysis as my analysis will initially be deductive and then inductive. My raw
data were analysed according to the categories that have been identified in my study’s conceptual
framework (Figure 2.9). After this deductive phase of analysis, inductive analysis was done where I
studied the organised data in order to explore undiscovered patterns and emergent understandings (Patton, 2002,
p. 454). Edwards and Talbot (1999) believe that although case studies need a theoretical framework,
their strengths are their capacity to reveal new ways of seeing familiar and complex situations (p. 131). Through
inductive analysis new patterns, themes and categories in the data were discovered which contributed
towards possible implications for teacher training and theory building. The inductive approach allows
for correlating the study’s purpose with the findings.
The following research questions guided my analysis process:
1.
How can ML teachers’ instructional practices be described?
2.
What is the nature of ML teachers’ knowledge and beliefs?
3.
How do ML teachers’ knowledge and beliefs relate to their instructional practices?
4.
What are the possible implications of the findings from Questions 1, 2 and 3 for teacher
training?
5.
What is the value of the study’s findings for theory building in teaching and learning ML?
77
For this purpose I adapted Cohen et al.’s (2001, p.148) seven-step analytic strategy. The purpose is to
move from thematically describing the cases to explaining the phenomena to eventually generating
theory:
Step 1: Establish units of analysis of the data, indicating how they are similar and different – ascribing codes to
the data.
Step 2: Create a domain analysis – dividing my data into groups, patterns and themes according to my
conceptual framework.
Step 3: Writing a case study narrative – giving a description of each case, thus providing the reader with
all information needed to understand the case in all its uniqueness (Patton, 2002, p. 450) (Research question 1
and 2).
Steps 1 to 3 are indicated in Table 3.4 below:
Table 3.4: Collection, analysis and reporting data
OBSERVATIONS
Three observations per teacher to obtain a general impression of their instructional practices
DATA
DATA ANALYSIS MODE
REPORTING DATA
COLLECTION
MODE
Observe lessons • Transcribe video data verbatim to text data • Describe teachers’
focussing on:
instructional practices
• Tasks
according to tasks, discourse
• Add field notes to above transcripts
and learning environment.
• Discourse
(Research
question 1)
• Learning
• ATLAS.ti 6 to code video data:
• Describe nature of teachers’
environment
knowledge and beliefs as
using:
CODES:
observed during lesson
I
used
the
codes
from
Artzt
et
al.’s
(2008)
• Video recordings
Framework for the examination of instructional presentations.
• Field notes
practices. They referred to these codes as lesson (Research question 2)
dimensions.
Teaching
Tasks: Representations (TR)
Learners
Tasks: Motivational strategies (TMS)
Teaching
Tasks: Sequence and level (TSL)
Teaching and learners
Discourse: Teacher-learner (DTL)
Teaching and learners
Discourse: Learner-learner (DLL)
Teaching and learners
Discourse: Questioning (DQ)
Teaching and learners
Learning environment: Climate (LEC)
Learning environment: Strategies, pace (LESP) Teaching
Learning environment: Administrative (LEA) Teaching
INTERVIEW 1
Semi-structured interview conducted prior to observations 2 and 3 and based on the teacher’s
planning of that day’s lesson.
78
DATA COLLECTION
MODE
Finding evidence of PCK and
beliefs regarding the:
• ML learners
• Teaching of ML
• ML curriculum
using:
• Clips from the video
recordings
• Tape recordings
DATA ANALYSIS MODE
• Transcribe audio data verbatim
to text data
• ATLAS.ti 6 to code audio data:
CODES for PCK and beliefs:
ML learners (L)
Teaching of ML (T)
ML curriculum (C)
REPORTING DATA
Describe nature of teachers’
PCK and beliefs as new
information from interviews
could be compared with
findings from observations.
(Research question 2)
INTERVIEW 2
Semi-structured interview conducted after all three observations and based on teachers’ personal
experiences to gain deeper insight in their practices
DATA COLLECTION
REPORTING DATA
DATA ANALYSIS MODE
MODE
Section A:
Further describe nature of
For all three sections:
Discuss outstanding incidents • Transcribe audio data verbatim teachers’ PCK and beliefs as
from their lessons to obtain a
new information emerges
to text data
better understanding thereof. • Use ATLAS.ti 6 to code audio from the second interview in
using:
addition to observations and
and written data (using same
• Tape recordings
codes as with observations and first interview.
(Research question 2)
interview 1)
Section B:
Discussion according to a set of
predetermined questions
based on:
• Beliefs about nature of
mathematics as discipline and
ML as subject
• PCK and beliefs regarding the
ML learners
• PCK and beliefs regarding ML
teaching
using:
• Tape recordings
Section C:
A set of predetermined questions
answered in writing based on the
NCS and CAPS to determine
teachers’ knowledge
of the curriculum.
After the process discussed above, I continued with Cohen et al.’s (2001, p. 149) steps of data analysis,
namely:
Step 4: Establish relationships and linkages between the domains – the data were put in context by
establishing relationships and links between the domains and also between the sets of data from the
79
observations and interviews. This was done by identifying confirming cases, by seeking ‘underlying associations’
and connections between data subsets. (Research question 3)
Step 5: Summarising – reporting on the main features of the research so far indicating the major
themes, issues and problems that have arisen from the data, also seeking negative and discrepant cases.
Step 6: Making speculative inferences – from the analysis I could draw certain conclusions and could
consider the implications of those findings for teacher training. (Research question 4 and 5)
All video and audio data were transcribed verbatim to text data immediately after the data were
collected. Following the transcribing process, I coded the transcriptions by using ATLAS.ti 6 which
allows for codes to be easily accessed, sorted and merged. My transcripts are synchronised with
associated files in order to jump from a particular part in the transcript to the original recording.
3.7
Quality assurance criteria
To conform to the quality assurance criteria for qualitative research, I considered aspects such as the
trustworthiness, validity and reliability of my study and also bore in mind the Hawthorne and Halo
effect. Being aware of the use of different terminology (trustworthiness, validity and reliability) by
different researchers, I use the terms interchangeably as all these terms are referring to valuable aspects
of quality assurance applicable to my qualitative study.
3.7.1 Trustworthiness of the study
Nieuwenhuis (2007) uses the term trustworthiness and states that when qualitative researchers speak of
research ‘validity and reliability’ they are usually referring to research that is credible and trustworthy (p. 80). By using
multiple data collection strategies such as multiple observations and interviews, the researcher as data
gathering instrument, enhances the trustworthiness of the study. I acquired the services of a peer
researcher with years of experience to assist me with the coding and interpretation of the data to
further enhance trustworthiness (Nieuwenhuis, 2007).
Two factors affecting the trustworthiness of the study are the small sample and number of lessons
observed influencing the extent to which the sample is representative. There is also no agreement on
how PCK and beliefs are to be evaluated. Nespor (1987) reasons that belief systems often include affective
feelings and evaluations, vivid memories of personal experiences, and assumptions about the existence of entities and
alternative worlds, all of which are simply not open to outside evaluation or critical examination in the same sense that the
components of knowledge systems are (p. 321).
80
3.7.2 Validity and reliability of the study
Cohen et al. (2001) refer to validity and reliability in qualitative research and do not use the terms
“credibility and trustworthiness”. They regard validity as an important aspect of both quantitative and
qualitative research to ensure that a particular instrument measures what it is supposed to measure. A
study may be declared reliable if findings from a particular group are replicated when a similar group in
a similar context is investigated. Reliability then refers to the consistency and re-applicability over time, over
instruments and over groups of respondents. It is concerned with precision and accuracy (Cohen et al., 2001, p. 117).
Prompted by these views I came to the conclusion that the validity of my qualitative study was
addressed through the … honesty, depth, richness and scope of [my study’s] data … (Cohen et al., 2001, p. 105).
Factors that contributed to a degree of bias were the subjectivity of respondents, their opinions,
attitudes and perspectives. I enhanced the reliability of my study by the stability of observations meaning
that I would have made the same observations and interpretations [if the ML teachers] had been observed at a different
time or in a different place (Cohen et al., 2001, p. 119). I facilitated inter-rater reliability by inviting a
researcher with many years’ experience in analysing qualitative data to act as my external coder.
3.7.2.1
The Hawthorne effect
During the data collection stage I took the Hawthorne effect (Cohen et al., 2001) into consideration:
the credibility of my data may be influenced due to my presence in class possibly influencing teachers’
behaviour during observations. To reduce this effect, the first observation was done without a prior
interview or discussion as the interview questions prior to the second and third observations could
influence teachers’ behaviour in the classroom. I emphasised the fact that I was interested in the
uniqueness of each teacher and my purpose was not to report their performances in class to their
superiors. To further enhance the trustworthiness of the observations the lessons were video-taped,
field notes were taken and after each observation the teacher had to verify my field notes
(Nieuwenhuis, 2007).
To enhance the trustworthiness23 of the interviews, it was important that the interviewees be honest
and open in their responses. The data from the two interviews prior to the lessons were compared with
the classroom observations. The same interview schedules, including the same questions and sequence
thereof were used for all interviewees. The questions were short and concise in order to avoid
confusion or misunderstanding. Section C of the last interview where teachers answered the questions
in writing was completed in my presence as part of the interview. This was to ensure that the data
23
Note that for the purpose of my study I use trustworthiness, validity and reliability interchangeably as discussed
under Section 3.7.
81
obtained were credible as the teachers were not able to consult another teacher or the relevant
documents. I considered the fact that some teachers might have preferred to complete it in writing
instead of orally as they might have felt less threatened or pressured. This allowed for more time to
think about the questions and to provide valuable responses.
3.7.2.2
The Halo effect
The Halo effect also needs to be considered during the data collection stage: where the researcher’s
knowledge of the person or knowledge of other data about the person or situation exerts an influence on subsequent
judgements (Nieuwenhuis, 2007, p. 116). To ensure trustworthiness of the observations, a pilot study was
conducted during the assessment period of one of my internship ML students to ensure that the
observational categories themselves are appropriate, exhaustive, discrete, unambiguous and effectively operationalise the
purposes of the research (Nieuwenhuis, 2007, p. 129). To enhance the trustworthiness of the interviews, I
avoided the tendency to seek answers that would have supported my preconceived ideas. The peer
researcher who assisted me with the coding and transcribing of the data pre-empted this problem. The
interviewees were asked exactly the same questions and after each interview I gave a summary of my
interpretation of the interview for them to verify or modify. The interviews were also piloted to refine
contents, wording, length etc., to ensure that questions were interpreted the same way by different teachers
and that it did not take too much time, as the teachers’ demanding schedules had to be taken into
account (Nieuwenhuis, 2007, p. 129).
3.8
Ethical considerations
Ethics involves the moral issues implicit in the research work with respect to people directly involved in
or affected by the project. To ensure that the study adhered to the research ethics requirements,
application for ethical clearance was requested from the Ethics Department at the University of
Pretoria as well as the Gauteng Department of Education. These applications were submitted after the
proposal was successfully defended at faculty level and before fieldwork was conducted. Issues
addressed in the application involve the sensitivity level of the research activities, the research
approach, design and methodology, including full detail regarding the participants, voluntary
participation, informed consent, confidentiality, anonymity and risk.
The participants were invited to take part in the study and were informed of the purpose of the study
and their participative roles. They were not obliged to take part in the study but instead had a choice to
participate knowing that they could withdraw at any stage. After joining the study, the participants
signed a letter of informed consent. The letter explained the purpose of the study, the procedures to be
82
followed during the investigation, the possible advantages and disadvantages as well as information
regarding confidentiality, anonymity and possible risks involved in taking part in the study.
This study has a medium level of sensitivity as the participants were video-taped during their lesson
presentations and the interviews were audio-taped in order to have a clear and accurate record of all
events and verbal communication. It is highly unlikely that any of the participants was physically or
psychologically harmed during this research. The only possible harm participants might have
experienced is the invasion of their privacy by video-taping the lessons they presented or feelings of
anxiety and discomfort in sharing their knowledge and beliefs during the interviews that were audiotaped. To lower the level of discomfort when questions were asked about their PCK of the curriculum,
I gave them the option to rather answer the questions in writing. In this manner they had sufficient
time to think and to avoid a situation where they could have felt threatened or even embarrassed if they
could not answer a question. I am also aware that teachers are very busy and seldom have free periods,
and time used for the interviews was also part of the intrusion into their lives.
It is important to notice that a respondent may be considered anonymous when the researcher cannot identify a given
response with a given respondent (White, 2005). Confidentiality means that although researchers know who has
provided the information or are able to identify participants from the information given, they will in no way make the
connection known publicly; the boundaries surrounding the shared secret will be protected (Cohen et al., 2001, p. 62).
To ensure anonymity and confidentiality the participants were not expected to identify themselves
publicly and if their names were known, it was kept confidential at all times. No names were mentioned
of any school or participant during the dissemination phase of the study when the research report was
written, but instead the schools were numbered and pseudonyms were used. The signed consent letters
served as a further guarantee to the participants regarding the anonymity and confidentiality of the
study. The interviews took place in a private environment. The video-tapes of the observations and
audio-tapes of the interviews by means of which the participants can be identified are accessible only to
me. Participants were asked to review the draft report before it was finalised.
3.9
Conclusion
In this chapter I discussed social constructivism as my research paradigm, stated the nature of my study
being subjective and taking up an interpretive position. A qualitative research approach is used and the
research design is an exploratory case study. The research site is secondary schools in Tshwane and the
sample comprises five grade 11 ML teachers. Observations were used to access teachers’ instructional
practices and to determine the nature of their MCK and the extent to which they apply their PCK
83
during their instructional practice. Interviews were used to describe the ML teachers’ beliefs about
mathematics as a discipline as well as their PCK and beliefs about the ML learners, the teaching of the
subject and the ML curriculum. ATLAS.ti 6 was used to analyse the video and audio data in order to
establish a relationship between teachers’ instructional practices and their knowledge and beliefs. I lastly
discussed the trustworthiness of the study and ethical considerations that were taken into consideration.
In the next chapter the results of the study are presented and discussed.
84
Chapter 4
Presentation and discussion of the
findings
4.1
Introduction
Since the data collection process and data analysis strategies are discussed in Chapter 3, I briefly rapport
on the data collection process24 and the data analysis strategies25 while providing a comprehensive
elucidation regarding the coding of the data. Based on my conceptual framework (Figure 4.1) I
thematically present and discuss the findings from each participant, relate the findings to the literature
and lastly explain the identified trends. The two themes are: 1) ML teachers’ instructional practices and
2) ML teachers’ knowledge and beliefs.
Figure 4.1: Conceptual framework: Instructional practice, knowledge and beliefs framework of
analysis (adapted from Artzt et al., 2008; Franke et al., 2007; Hill et al., 2008)
24
25
A detailed description of the process is provided in Section 3.5: Data collection techniques.
For more detail, see Section 3.6: Data analysis strategies.
85
The research questions are:
Main question:
What is the relationship between Mathematical Literacy teachers’ knowledge and beliefs and their
instructional practices?
Subquestions:
1.
How can ML teachers’ instructional practices be described?
2.
What is the nature of ML teachers’ knowledge and beliefs?
3.
How do ML teachers’ knowledge and beliefs relate to their instructional practices?
4.
What are the possible implications of the findings from Questions 1, 2 and 3 for teacher
training?
5.
4.2
What is the value of the study’s findings for theory building in teaching and learning ML?
The data collection process
The data collection took place in Pretoria during the second quarter (May and June) of 2011. I initially
contacted the principals of five schools telephonically to discuss my study and request their
participation in my study. Two letters of invitation were sent to the schools, one addressed to the
principal and the other to the Grade 11 ML teacher and a meeting was scheduled at each school
between the principal, teacher and me. The initial participants26 were Monty, Elaine, Alice, Edith and
Denise. During the data analysis process I realised that Edith’s case did not add value to my study since
her practice was similar to three of the four teachers’ practices. I also reached a point of data saturation
and decided to continue with the other four cases only. During the data collection period all
communication and arrangements were made directly with the participants except for Alice’s school
where I worked through the principal.
I kept to the data collection process27 of three observations with an interview conducted prior to the
second and third observations, followed by a last interview some time after the last observation. The
duration of the two interviews prior to the observations was approximately half an hour each and was
conducted during the period or break before the specific lesson. These interviews were based on the
teachers’ planning of their lessons. The duration of the last in-depth interview was approximately 50
minutes per interview. I only observed Grade 11 ML lessons and all participants had had at least one
year experience of teaching ML, one of the selection criteria28.
26
Pseudonyms were used for ethical purposes.
The data collection process is discussed in Section 3.5: Data collection techniques.
28 The other selection criteria are discussed in Section 3.4: Research site and sampling.
27
86
In Table 4.1 a timeline is given indicating the dates all five participant’s lessons were observed and
interviews conducted.
Table 4.1: Timeline of the data collection process
4.3
Data gathering instrument
Participants29
Date in 2011
Observation 1
Interview & Observation 2
Observation 1
Interview & Observation 3
Observation 1
Interview & Observation 2
Interview & Observation 2
Interview & Observation 3
Observation 1
Observation 1
Interview & Observation 3
Interview & Observation 2
Interview & Observation 2
Interview & Observation 3
Interview & Observation 3
Last interview
Last interview
Last interview
Last interview
Last interview
Monty
Monty
Elaine
Monty
Alice
Alice
Elaine
Alice
Edith
Denise
Elaine
Edith
Denise
Edith
Denise
Edith
Denise
Alice
Monty
Elaine
3 May
4 May
6 May
9 May
9 May
10 May
12 May
16 May
16 May
17 May
19 May
19 May
20 May
25 May
26 May
27 May
30 May
1 June
1 June
2 June
Data analysis strategies
The study’s DEDUCTIVE-inductive approach and analytic strategies used in analysing the data are
discussed in Chapter 330. In this section I only discuss the transcribing and coding of the data. The
inclusion and exclusion criteria for coding the data are also discussed and presented in table form.
4.3.1 Transcribing the data
I transcribed my video and audio-taped data verbatim to text data immediately after the data had been
collected. Care was taken not to interpret the data already during the transcribing phase. After each
29
30
Pseudonyms were used to protect the participants’ true identities.
See Section 3.6: Data analysis strategies.
87
observation all hand-written field notes made during the observations as well as insights that were
thought of afterwards which had not been noted were typed on a template form. Uncertainties that
emerged were cleared by watching the video-tapes of the lessons or listening to the audio-tapes again.
Transcripts were read afterwards to ensure the transcripts were true accounts of the actual observations
and interviews.
4.3.2 Coding of the data
In coding the data, I used a deductive approach based on my conceptual framework. According to the
conceptual framework two themes, namely 1) the ML teachers’ instructional practices and 2) their
knowledge and beliefs were identified while the subthemes for each theme were chosen according to
the work of Artzt et al. (2008) and Hill et al. (2008). Codes have been ascribed to the different lesson
dimension indicators of each subtheme according to which the raw data were analysed. By using the
software programme ATLAS.ti 6, I coded the transcripts according to a set of pre-determined lesson
dimension indicators and their associated codes as given in Table 4.231 and Table 4.332.
After the data were coded I created coding families (Archer, 2009) which are clusters comprising codes
related to one other. According to my conceptual framework, families were created by selecting from
the list of all codes those codes that were related to one another. A specific code could belong to more
than one family and families were therefore not exclusive. The data were analysed according to two
themes, namely 1) ML teachers’ instructional practices and 2) ML teachers’ knowledge and beliefs. Subthemes for each of these themes were created using Atlas.ti 6. Networks for these sub-themes were
created afterwards where the connections between the different codes assigned to the families were
indicated.
4.3.2.1
Theme 1: ML teachers’ instructional practices
The three subthemes (also called the lesson dimensions) which could best describe the teachers’
practices are 1) tasks; 2) discourse; and 3) learning environment (Artzt et al., 2008). The first column in
Table 4.2 below indicates the three subthemes or lesson dimensions with their different categories. In
the second column are the descriptions of the lesson dimension indicators with the codes created for
them. All data were collected from the observations only.
31
32
Table 4.2 is given under Section 4.3.2.1.
Table 4.3 is given under Section 4.3.2.2.
88
Table 4.2: Lesson dimensions and dimension indicators as inclusion criteria for coding the data (Adapted from Artzt et al., 2008)
LESSON DIMENSIONS
DESCRIPTION OF LESSON DIMENSION INDICATORS (CODES)
TASKS
Modes of representation
(TR)
Motivational strategies
(TMS)
Sequencing and difficulty
levels
(TSL)
TR1. Uses representations such as oral or written language, symbols, diagrams, graphs, tables, manipulatives, and
computer or calculator representations to accurately facilitate content clarity.
TR2. Provides multiple representations that enable learners to connect their prior knowledge and skills to the new
mathematical situation such as graphs, tables, formulae.
TMS1. Uses tasks that capture learners' curiosity and inspires them to participate in the lesson, but also to speculate on and
pursue their conjectures, such as tasks which elicit a class discussion or an interesting context used.
TMS2. Takes into account the diversity of student interests, experiences and abilities, such as when the teacher provides
additional tasks for the more advanced learners.
TMS3. Points out the value of the mathematics being learned so that learners will appreciate and understand the value of
mathematics, such as informing them about real-life situations or even other subjects where the mathematical
content is used.
TSL1. Sequences tasks and learning activities so that learners can progress in their cumulative understanding of a
particular content area and can make connections between ideas learned in the past and those they will learn in the
future such as working from easy to difficult and known to unknown tasks.
TSL2. Uses tasks, including homework that is suitable to what the learners already know and can do and what they need
to learn or improve on. Tasks should involve past work, reinforce current work and set the stage for future work
such as tasks where opportunity is given to practice identified or predicted learners’ misunderstandings.
TSL3. Tasks should reflect quality, not quantity. Should be appropriate and on the learners’ level such as increasing the
level of difficulty from Grade 10 and applying it to more complex contexts.
DISCOURSE
Teacher-learner interaction
(DTL)
DTL1.Communicates with learners in a non-judgmental manner and encourages the participation of each student, such as
addressing a large number of learners or working on a learner’s incorrect answer to lead the learner to
understanding.
DTL2. Requires learners to give full explanations and justifications or demonstrations orally and/or in writing such as
89
Learner-learner interaction
(DLL)
Questioning
(DQ)
learners explaining their work on the board.
DTL3. Listens carefully to learners’ ideas and makes appropriate decisions regarding when to offer information, provide
clarification, model, lead, and let learners grapple with difficulties. In response to a learner’s question, instead of
telling how or doing it for the learner, rather provides scaffolding to support solving a problem or encourage
learners to share ideas for carrying out a task.
DTL4. When giving feedback to learners’ answers, rather accept and praise instead of criticising and rejecting answers.
DTL5. Recognises and clarifies learners’ misunderstandings and misconceptions.
DLL1. Encourages learners to listen to, respond to, and question each other so that they can evaluate and, if necessary,
discard or revise ideas and take full responsibility for arriving at mathematical conjectures and/or conclusions.
DLL2. Avoids situations in which a group of learners dominate in the verbal communication in class.
DQ1. Teacher needs to pose a variety of levels and types of questions using appropriate wait times that elicit, engage and
challenge learners’ thinking. Cognitive levels:
i)
Memory: factual questions such as: What is the mean?
ii)
Convergent: narrow questions such as: What does it mean to write 12,5% as a decimal? Also complete the
word/sentence questions such as: We call it the co? and learners then need to complete the word:
coefficient.
iii)
Divergent: broad and open-ended questions such as: Why did you decide to use the compound interest
formula?
DQ2. The teacher should listen to learners’ ideas and ask them to clarify and justify their ideas such as asking: Why do you
say that? or How did you solve that?
DQ3. Should contribute to the verbal communication and participation of learners creating opportunities where learners
listen to, respond to, question and answer the teacher and one another.
DQ4. Learners’ responses: i) chorus; ii) volunteered; iii) teacher-selected.
LEARNING
ENVIRONMENT
Social and intellectual climate LEC1. Establishes and maintains a positive rapport with and among learners by showing respect for and valuing learners’
(LEC)
ideas and ways of thinking such as no one laughing when an incorrect answer is given.
LEC2. Enforces classroom rules and procedures to ensure appropriate classroom behaviour such as one person talking at
a time.
LEC3. Should have a positive attitude towards the learners and the subject, such as praising learners’ attempts and being
proud and well-prepared ML teachers.
Modes of strategies and
LESP1. Uses various instructional strategies that encourage and support student involvement as well as facilitate goal
pacing
attainment such as cooperative learning, learners explaining work at the board, direct instruction (lecturing),
90
(LESP)
Administrative routines
(LEA)
abstract procedural, group work, active learning, discussion, problem solving, inquiry and team-teaching.
LESP2. Provides and structures the time necessary for learners to express themselves and explore mathematical ideas and
problems such as enough opportunities to discuss or do group work.
LESP3. Effective use of class time to accommodate all three phases of a lesson: initiation, development and closure.
LESP4. There should be a logical flow in the lesson such as revising prior knowledge before introducing new content and
assess whether learning occurred.
LEA1. Uses effective procedures for organization and management of the classroom so that time is maximized for
learners' active involvement in the discourse and tasks such as allowing time for learners to practice what has been
explained by the teacher and not to rush them while working on a problem.
LEA2. Classroom arrangement should be appropriate to the lesson style used such as learners sitting in groups if group
work is applied.
LEA3. The position of the teacher in the classroom should contribute to a positive learning atmosphere such as working
between the learners, having eye contact with individual learners.
LEA4. The written information on the board/transparencies should be correct and ordered in order to contribute to
learners’ conceptual understanding.
Source: Adapted from: A Cognitive Model for Examining Teachers' Instructional Practice in Mathematics: A Guide for Facilitating Teacher Reflection, by
A.F. Artzt and E. Armour-Thomas, 1999, Educational Studies in Mathematics, 40(3), p. 217. Copyright © 1999 by Kluwer Academic Publishers.
Adapted with kind permission from Kluwer Academic Publishers.
91
Using Atlas.ti 6, networks of the code families (Archer, 2009) are now illustrated and explained. As I have
mentioned, the data for the three code families under Theme 1 were collected from the lesson
observations only. The code family created for the first subtheme Tasks appear in Figure 4.2 below. The
broken line arrows indicate the three different lesson dimensions being linked to the code family
Instructional practices: Tasks. Atlas.ti 6 uses solid line arrows with double equal signs to indicate the
codes associated with the different lesson dimensions (Archer, 2009). For example codes TR1 and TR2
are associated with lesson dimension Tasks: Modes of representation (TR). A full description of
each code such as TR1 and TR2 is provided in Table 4.2 above.
Figure 4.2: ML instructional practices: Tasks
At the end of each code, for example TMS133, there is a pair of numbers in parentheses {2-3}. The 2
refers to the groundedness, in other words the frequency with which the code was attached to
quotations in the observation transcripts for a specific participant. This means that there were two
incidents during the three lessons observed from a specific participant where there was evidence of the
teacher capturing the learners’ curiosity. The 3 is the density, indicating the number of times a code has
been linked to codes in all the networks that were created. In this example it means the code TMS1 was
also associated with two other codes T2 and T5 (see Table 4.3) in the subtheme ML teaching under
Theme 2: ML teachers’ PCK and beliefs. Notice that at the end of the three categories in Figure 4.2,
namely: Tasks: Modes of representation (TR), Tasks: Motivational strategies (TMS) and Tasks:
Sequencing and difficulty levels (TSL) the numbers in parentheses are {0-2}, {0-3} and {0-2}. This
indicates that the codes TR, TMS and TSL were not associated with quotations in the transcripts. These
33
According to Table 4.2 TMS1 refers to Tasks: Motivational strategies: Teacher uses tasks that capture learners’ curiosity
and inspire them to participate.
92
are subthemes that were not coded as such. Instead their different lesson dimension indicators were
coded.
The code family created for the second subtheme Discourse is given in Figure 4.3 below. The broken
line arrows again indicate the three different lesson dimensions being linked to the code family
Instructional practices: Discourse. The solid line arrows with double equal signs indicate the codes
associated with the different lesson dimensions. For example codes DTL1, DTL2, DTL3, DTL4 and
DTL5 are associated with the lesson dimension Discourse: Teacher-learner interaction (DTL). The
full descriptions of the codes are in Table 4.2.
Figure 4.3: ML teachers’ instructional practices: Discourse
The code family created for the third subtheme Learning environment appears in Figure 4.4 below. The
broken line arrows again indicate the three different lesson dimensions being linked to the code family
Instructional practices: Learning environment. The solid line arrows with double equal signs
indicate the codes associated with the different lesson dimensions. For example codes LEC1, LEC2 and
LEC3 are associated with the lesson dimension Learning environment: Social and intellectual
climate (LEC). The full descriptions of the codes are in Table 4.2.
93
Figure 4.4: ML teachers’ instructional practices: Learning environment
4.3.2.2
Theme 2: ML teachers’ knowledge and beliefs
The four subthemes of the ML teachers’ knowledge and beliefs are 1) MCK; 2) PCK regarding ML
learners; 3) PCK regarding ML teaching; and 4) PCK regarding the ML curriculum (Hill et al., 2008).
The first column in Table 4.3 below indicates the four subthemes or dimensions. In the second column
are the different descriptions of the dimensions with the codes created for them. These are the codes
that appear in the coding families on 1) ML learners (Figure 4.5), 2) ML teaching (Figure 4.6); and 3) ML
curriculum (Figure 4.7). The data were collected from both the interviews and the lesson observations.
94
Table 4.3: Knowledge and beliefs dimensions and its indicators as inclusion criteria for coding the data (Adapted from Artzt et al., 2008; Ball, 1990;
Borko & Putnam, 1996; Hill et al., 2008; Shulman, 1986; Shulman, 1987)
PCK AND
BELIEFS
DIMENSIONS
Mathematical
content
knowledge
(MCK)
ML learners
(L)
INSTRUMENT:
Interview (Int)
Observation (Obs)
Correspond
with
indicators in
Table B
Interview
question
numbers
Report on teachers’ mathematical content knowledge. Record on the accuracy of
teachers’ content, mathematical errors made by teachers, teacher’s misconceptions or
misrepresentations.
Observation
L1. 1st Interview
2a, 2b
L2. 1st Interview
3a, 3b
L3. 1st Interview
L4. 1st Interview
L5. 1st Interview
4c; 5
6
8
L6. Observation
DTL2/3/4/5
T1. 1st Int & Obs TR2|TSL1
T2. Observation TR1|TMS1
ML teaching
(T)
T3. Observation
DESCRIPTION OF TEACHERS’ PCK AND BELIEFS INDICATORS
(CODES)
TMS2
7
Teacher’s ability to:
L1. predict what mathematics learners will understand; but also understand why that
mathematics is comprehensible to the learners;
L2. predict what mathematics learners will not understand; but also understand why
that mathematics is incomprehensible to the learners;
L3. predict how they will come to understand it;
L4. predict how learners will probably approach a task;
L5. understand what alternative conceptions and preconceptions learners have that
could be misconceptions and that should be rectified and reorganised by the
teacher through the use of different strategies;
L6. see what learners do, know how to listen and hear what they think and then be
able to act appropriately as mentors to facilitate the learning process.
Teachers should:
T1. know what prior knowledge must be present to understand new work;
T2. know useful forms of representation of ideas, the most powerful analogies,
illustrations, examples, explanations, and demonstrations – in a word, the ways of
representing and formulating the subject depending on the content and learners’
needs, in order to make it comprehensible to them;
T3. have the capacity to transform the content knowledge he or she possesses into
forms that are pedagogically powerful and yet adaptive to the variations in ability
and background presented by the learners;
95
T4. 1st Int & Obs TSL1|LESP4
T5. 1st Int & Obs TR3|TSL2|
TMS1|TSL3
T6. 2nd Int
5
6a
Section A
(1-4)
Section B
T4. have the ability to sequence content to facilitate student learning;
T5. to choose the appropriate instructional strategy and instructional material for a
lesson, consider tasks to set and assessment techniques to use;
T6. reflect on their own practices for the purpose of improvement.
1(6);2(2-4,6)
3(1-4)
C1-C6: 2nd Int
Section C
(5)
Section C
(6)
Section C
(7)
Section C
(8)
Section C
ML Curriculum
(C)
(1,2,4,9)
Section C
(10)
C7. 1st & 2nd Int
& Obs
Context
C8. 2nd Interview
Nature 1
Nature 2
Int.1
(4a,4b)
Int. 2(3)
Section B
(1,3)
Teachers:
C1. should have knowledge regarding the variety of resources/instructional materials
available to teach particular curriculum components;
C2. need to recognise the particular strengths and weaknesses of textbooks and
materials they are using and should have a collection of materials they use when
teaching mathematics;
C3. need to be familiar with the curriculum materials studied by learners in other
subjects at the same time and how it integrates with ML;
C4. should be informed of the various departmental ML documents, providing info
regarding the purpose and value of ML, resources to use and how to progress from
one year to the next;
C5. should have knowledge about the definition, purpose, learning outcomes and the
new CAPS;
C6. need to be familiar with the topics and level of different topics being taught in the
same subject during the preceding and later years in school, in other words how
topics are organised horizontally and vertically.
C7. should teach content in context. The context should be applicable to the content
and the teacher needs to know the context and be able to apply meaningfully.
C8. View of mathematics as discipline: From traditional to formalist to constructivist
(Includes the role of the teacher in his/her instructional practice)
• A traditional view refers to teachers who belief that mathematics is an abstract
phenomenon unrelated to reality. These teachers will then struggle to relate
mathematics to real-life situations and tend to believe mathematics consists of a set
of rules and procedures that must be learned mechanically with little or no
96
C9. 2nd Interview
Nature 3
Section B
(2)
Section B
(4,5
connection to one another and hardly any relevance to their everyday lives. They
also tend to separate mathematics from the discipline of discovery and creativity
and an abstract procedural approach is used.
• The formalist view refers to teachers who believe mathematics is characterised by
logic, rigorous proofs, exact definitions and a precise mathematical language and
doing mathematics consists of accurate proofs as well as of the use of a precise and
rigorous language.
• The constructivist view refers to teachers who believe mathematics is a constructive
process where relations between different notions and sentences play an important
role. Here the mathematical activity involves creative steps, such as generating rules
and formulae, thereby inventing or re-inventing the mathematics.
• View of ML as subject being a lower grade Mathematics and/or a life skill.
• Possible definitions of the nature of mathematics: i) mathematics is the language of
science; ii) is the study of patterns; iii) is a system of abstract ideas.
C9. The value of mathematics and ML for people in their daily lives.
Source: Adapted from: A Cognitive Model for Examining Teachers' Instructional Practice in Mathematics: A Guide for Facilitating Teacher Reflection, by
A.F. Artzt and E. Armour-Thomas, 1999, Educational Studies in Mathematics, 40(3), p. 217. Copyright © 1999 by Kluwer Academic Publishers.
Adapted with kind permission from Kluwer Academic Publishers.
97
Using Atlas.ti 6 (Archer, 2009), networks of the code families are now illustrated and explained. The data
for the three code families under Theme 2 were collected from both the teachers’ interviews and the
lesson observations. The code family created for the first subtheme ML learners appears in Figure 4.5
below. The broken line arrows indicate the six different codes (indicators) being linked to the code
family ML learners, namely L1, L2, L3, L4, L5, L6. A full description of each code is provided in
Table 4.3 above. The data linked to these six codes were collected from the interviews conducted with
the teachers. The solid line arrows with double equal signs indicate other codes from the observations
that are associated with the codes from the interviews (Archer, 2009). For example, codes DTL2, DTL3,
DTL4 and DTL5 (see Table 4.2) are associated with L6 (see Table 4.3).
Figure 4.5: ML teachers’ PCK and beliefs: ML learners
The code family created for the second subtheme ML teaching appears in Figure 4.6 below. The broken
line arrows indicate the six different codes (indicators) being linked to the code family ML teaching,
namely T1, T2, T3, T4, T5, T6. A full description of each code is provided in Table 4.3 above. The data
linked to these codes were collected from the interviews conducted with the teachers. The solid line
arrows with double equal signs indicate other codes from the observations that are associated with the
codes from the interviews. For example, codes TMS1, TR3, TSL2 and TSL3 (see Table 4.2) are
associated with T5 (see Table 4.3).
98
Figure 4.6: ML teachers’ PCK and beliefs: ML teaching
The code family created for the third sub-theme ML curriculum is given in Figure 4.7 below. The
broken line arrows indicate the nine different codes (indicators) being linked to the code family ML
curriculum, namely C1, C2, C3, C4, C5, C6, C7, C8 and C9. A full description of each code is
provided in Table 4.3 above. The data linked to these nine codes were collected from the interviews
conducted with the teachers. The solid line arrows with double equal signs indicate other codes from
the interviews too that are associated with the initial codes. For example, codes Nature 1 and Nature 2
(see Table 4.3) are associated with C8 (see Table 4.3).
Figure 4.7: ML teachers’ PCK and beliefs: ML curriculum
99
4.3.2.3
Inclusion criteria for coding the data
There are two tables indicating the inclusion criteria for coding the data. Table 4.2 was used to analyse
the ML teachers’ instructional practices. The table consists of the different lesson dimensions, namely
tasks, discourse and learning environment and the respective lesson dimension indicators. The
descriptions of the lesson dimension indicators serve as inclusion criteria for coding the data from the
observations. Examples of each code are provided. These codes were used to analyse the raw data and
reporting on the data.
Table 4.3 was used to analyse the ML teachers’ knowledge and beliefs. The codes have been assigned
according to MCK as well as the three PCK and beliefs regarding the ML learners, ML teaching and the
ML curriculum. In Table 4.3 the descriptions of teachers’ knowledge and beliefs indicators serve as
inclusion criteria for coding the data. Each code has been linked with the corresponding interview
question(s) as well as cross referencing to specific code(s) in Table 4.2 of the observations.
4.3.2.4
Exclusion criteria for coding the data
In the process of coding the observations, some of the activities and discourse were not relevant to my
study and did not form part of my predescribed lesson indicators whereas others were inaudible. These
were excluded when the data were coded. During the interviews some participants did not always keep
to the question asked or sometimes used the chance to raise personal points of concern. In Table 4.4
below I listed these exclusion criteria as well as examples of text that were excluded from coding.
Table 4.4: Exclusion criteria for coding of the data
Exclusion criteria
Incidents during class observations when I
could not hear what was said
Interruptions
The question in the second interview
regarding the ML learners’ abilities and
motivation as the question was included for
personal interest only
Elaborations when questions have been
misunderstood (The misunderstanding is
included in the coding but not the elaboration
part of the answer)
Examples of text excluded from coding
This occurred when the teacher attended to
individual learners’ at their desks or when they
had private conversations at the board or at the
teacher’s desk. Some of the data were inaudible
when I did the transcribing.
Teachers that needed to attend to people who
knocked on the door.
Describe your Grade 11 ML learners in terms of their
a) mathematical abilities and b) motivation. How does
their motivation compare with that of the Mathematics
learners?
How do you think will the learners
approach these tasks?
All they have to do is most times when I try to give
them the tasks on data handling and try to make them
set the questionnaire and then try to see make it look
real-life, you try to make it what people think about the
100
Providing detailed examples to illustrate their
answers.
4.4
like a … and then you get a questionnaire and then
would give that to your friends and then you collect those
data and then you try to sort them out and then present
them using a bar graph or line graph.
Describe the ideal ML classroom in terms
of instructional strategies used.
I try to involve them as much as I can, that they
understand that. They must learn through doing it,
trying on their own. Try doing it, drawing the Cartesian
plane, those are the points, plot them, OK join the
points and what do you see? Oh OK, its curved, it’s
called parabolic function, how do we use it in everyday
life. Because if I just draw and plot it myself they look
at it but the constructing, they construct while they are
doing.
Information regarding the four participants
In the next section biographical information regarding the four participants Monty, Alice, Denise and
Elaine is provided as well as some background information regarding the observed lessons.
Pseudonyms were used to protect their identities.
4.4.1 Monty
Monty is a novice teacher in his second year of teaching Grades 10, 11 and 12 ML and one year of
teaching Grade 10 Mathematics. He is 24 years old and completed his Baccalaureus Educationis (BEd)
degree with Mathematics as major in 2010. Apart from the six ML courses he attended during 2010,
organised and presented by the DoE and the District Office, he had had no formal training for
teaching ML. He teaches at an inner city school of 500 learners with 18 and 35 learners in the two
Grade 11 classes respectively.
The topic of Monty’s first two lessons I observed was solving simultaneous equations using the
substitution method, only mentioning the elimination method (Learning Outcome 2). The first two
observations were done on 3 and 4 May 2011. The third lesson was on data handling where the mean,
mode, median and range as measures of dispersion were discussed (Learning Outcome 4). The last
lesson was observed on 9 May 2011.
101
4.4.2 Alice
Alice grew up in Nigeria, is 30 years old and in 1995 she obtained a Baccalaureus Technologiae (BTech)
Management Accounting degree at a University of Technology with second year Financial
Mathematics. She did not take any Mathematics Education or Mathematics Methodology courses. She
has no experience of teaching Mathematics and it is her second year of teaching ML. She is teaching at
an independent inner city school with 350 learners where the number of learners in her Grade 11 ML
classes ranges from 20 to 30.
The first lesson I observed with Alice was based on the use of the quadratic formula to solve quadratic
equations (Learning Outcome 1), a sequential or follow-on lesson to reinforce the work done the
previous period. A student teacher was responsible for the previous lesson34. I observed this lesson on
9 May 2011. The second lesson was on graphing the parabola (Learning Outcome 2) using a table
method and finding the intercepts but without determining the turning point. This lesson was observed
on 10 May 2011. The third lesson was on data handling: the cumulative frequency, relative frequency,
standard deviation and the ogive (Learning Outcome 4). This lesson was observed on 16 May 2011.
4.4.3 Denise
Denise is 42 years old and completed a BEd degree in 2003 with Mathematics and Methodology of
Mathematics as two of her major subjects and also completed her BEd Honours in 2009. She obtained
both degrees from the University of Witwatersrand. She completed a 40-hour course based on ML and
the teaching thereof at the University of South Africa. She has seven years’ experience of teaching
Mathematics and it is her fourth year of teaching ML. She is teaching at a Section 21 (former model C)
school in Pretoria with 908 learners where 92% of the learners are black while the other 8% consist of
White, Indian, Coloured and Asian learners. The number of learners in her Grade 11 classes ranges
from 20 to 32.
The first two lessons I observed were the same lessons presented to two different classes and
concerned conversions from metric units to imperial units, involving capacity and mass problems
(Learning Outcome 3). These two lessons were observed on 17 May 2011 and 20 May 2011
respectively. The third lesson was a follow-on lesson where more advanced conversions within the
metric system only were done on capacity, mass, length, area and volume. This lesson was observed on
26 May 2011. Denise has experience and knowledge of her subject and her lessons were coloured with
34
The situation that occurred when the student teacher introduced the topic and Alice took over the next period is
discussed under Section 4.4.2.3: Mathematical content knowledge.
102
humorous comments so that learners participated and enjoyed her classes. Learners were involved by
solving problems in class, writing on the board and answering questions.
4.4.4 Elaine
Elaine is 44 years old and completed her Higher Education Diploma (HED): Senior Primary with
Mathematics and Mathematics Didactics as two of her major subjects in 1989 at the Normaal College
of Education. During those years the Mathematics I-IV, offered as a major subject at colleges, was
equivalent to a first year Mathematics offered at universities. She did not attend any courses on ML or
the teaching thereof. She has eight years experience of teaching Mathematics and it is her third year of
teaching ML. She is teaching at a Section 21 (former model C) school in Pretoria with 1 300 learners
where 95% of the learners are white whereas the other 5% are black. She is responsible for only one
Grade 11 ML class and there are 25 learners in the class.
During the period of observation, Elaine was busy with her revision programme. The first and third
lessons I observed were based on calculating circumference, area, volume and surface-area of two and
three-dimensional shapes (Learning Outcome 3). These lessons were observed on 6 May 2011 and 19
May 2011. The second lesson was based on time (Learning Outcome 3) and interest (Learning
Outcome 1) and was observed on 12 May 2011. Elaine is the ML coordinator at her school and her
goal is not only to equip her ML learners with knowledge and skills they can use in their lives, but also
to promote the purpose and value of ML everywhere she goes. Elaine believes that the notion of
contextual mathematics makes ML a valuable and interesting subject as learners’ general knowledge is
enriched through their experiences with contextual mathematical problems. She mentioned that her ML
learners are often envied by the Mathematics learners because of their interesting lessons.
To summarise: The most relevant information appears in the Table 4.5 below:
Table 4.5: Biographical information of the four participants
Age (years)
Highest qualification
Mathematics teaching
experience (years)
ML teaching
experience (years)
Monty
24
BEd: FET
Alice
30
BTech:
Management
Accounting
Denise
42
Elaine
44
BEd Honours
HED: Senior
Primary
1
0
7
8
2
2
4
3
103
4.5
Theme 1: The ML teachers’ instructional practices
In this section I present and discuss the findings from the observations of Monty, Alice, Denise and
Elaine. All discussions on the sub-themes Tasks, Discourse and Learning environment are
structured strictly according to the specific order of the different lesson dimension descriptors (codes)
as indicated in Table 4.235 (Artzt, et al., 2008). The language of all quotes from Monty, Alice and Denise
has not been edited. Since Elaine’s classes were conducted in Afrikaans I translated her quotations into
English. Background information regarding the observed lessons of the participants is given. A
summary is provided at the end of this section in table form analogous to Table 4.2.
4.5.1 Monty’s instructional practice
4.5.1.1 Tasks
Tasks: Modes of representation (TR)
In the first two lessons on solving simultaneous equations, Monty represented the mathematical
concepts by means of written examples on the board using the variables x and y as well as calculators
for basic calculations (TR1). For example Monty asked the learners to write an answer of −
12
10
as a
decimal (TR1). He told the learners they could also use their calculators to change a decimal answer to a
fraction form and taught them how to operate their Sharp and Casio calculators (TR1). In the
introduction of the third lesson on measures of dispersion, Monty called a learner to the front and
asked the class to name characteristics that describe him as a boy and not a girl. He used a manipulative
(a learner) to demonstrate that even in mathematics concepts have certain characteristics and said:
So even if you go to this thing of Maths or ML, you need to have a picture of it, what are the characteristics of it?
What must I identify to be able to say this is that thing?36 (TR1).
Monty followed through with the characteristics idea by asking the learners to provide another word for
some of the concepts, for example that average is another word for mean and middle number is
another word for median (TR1). He told them how to calculate the measures of dispersion but did not
tell them or ask them why and when we use these measures of dispersion and did not apply these
concepts to a real-life situation (TR1). He did not use multiple representations to enable learners to
connect prior knowledge to the new content, but only verbally referred to prior knowledge when he
said:
35
36
Table 4.2 is discussed under Section 4.3.2.1: Inclusion criteria for coding the data.
The language of all quotes from Monty, Alice and Denise has not been edited. I translated Elaine’s quotes from Afrikaans
to English.
104
We learned how to solve an unknown. We learned how to draw graphs of linear function, how to draw the graph of
linear function (TR2).
Tasks: Motivational strategies (TMS)
The tasks in both the first and second lessons (Picture 4.1) were based on pure mathematics and not on
interesting and applicable real-life situations in which learners could do problem solving in groups or
discuss the meaning of the solutions (TMS1).
Picture 4.1: The two tasks from Lesson 1
The tasks did not capture the learners’ curiosity nor inspire them to speculate on their conjectures
(TMS1). The learners did not appear interested while they were listening, looking and copying work
from the board (TMS1). After the first example (
x + 2y = 4
) of solving simultaneous equations during
7x − 5y = 9
the first lesson, Monty said:
OK, now you try this one:
y=x+3
(pause). OK, you tell me, I write for you.
y = 3x − 7
But at the end he treated this second example the same way as the first one, telling them what to do
and not allowing the learners to solve the problem themselves. This second example was an easier
problem since y was already the subject (TMS1). The examples used in the data handling lesson were
not motivating to the learners (TMS1). These examples based on measures of dispersion were meant
for learners with low ability and no experience (TMS2). To illustrate the mean (Picture 4.2), Monty said:
For example we have numbers 1, 2, 3, 4, 5, 6 and you want to calculate the mean (TMS2).
Picture 4.2: Monty example of how to calculate the mean
105
The example for the median (Picture 4.3) was where numbers from 3 to 13 were given in random order
and he said:
The median. What’s a median? A middle number. So for you to find a correct middle number you have to arrange
the numbers in an ascending order… (TMS2).
Picture 4.3: Monty’s example of how to calculate the median
The example for the mode (Picture 4.4) was:
OK, let’s move on to mode. Please, if you don’t understand something, please ask. OK, the mode (erases work on
board). You are given this data: 1 2 2 3 4 5. What’s the mode? OK first, what’s a mode? A mode is a number
that appears? Girl: more than once (TMS2).
Picture 4.4: Monty’s example of how to calculate the mode
His motivation is examination driven, preparing the learners for the examination by frequently telling
them what is expected of them in the examination, how to use their time sensibly and that all steps
should be shown otherwise they will lose marks (TMS3). He once asked in a lesson:
What was difficult? Learner: Nothing. Teacher: So what I am saying to you, when you say to yourself something is
difficult, it is, but once you did it right you see nothing is impossible (TMS3).
In his introduction to the data lesson when he had a comprehensive description of how research is
done, he said:
Why am I telling you this? But in Grade 11 I must just tell you, remember guys, life doesn’t end here at school, life
doesn’t end at school. Outside you will need that information to use it.
He wanted to emphasise the value of ML for learners’ future lives, but did not discuss how the content
is applicable to real-life situations and no such example or homework was done (TMS3).
106
Tasks: Sequencing and difficulty level (TSL)
Not much attention was given to sequence the tasks in order for the learners to obtain cumulative
understanding of the content (TSL1). He worked from a more difficult example
x + 2y = 4
, where x or
7x − 5y = 9
y first needed to be made the subject of the equation in order to continue with the solution of the
problem, to an easier example
y= x+3
, where y was already the subject of the equations (TSL1). He
y = 3x − 7
explained the elimination method prior to the two lessons I observed in which he explained the
substitution method, but he did not expect them to use both methods in order to practise them
simultaneously (TSL1). In the data handling lesson the examples discussed in the above paragraph were
not sequenced but connections were made with ideas learned in the past (TSL1). The tasks given in the
lessons on simultaneous equations provided opportunities where learners could reinforce current work
whereas the tasks in the data handling lesson were meant to revise basic Grade 10 work (TSL2). The
homework for the data handling lesson was not carefully selected, instead the first four problems from
the exercise were chosen, questions that were basic and easy. (TSL2) The questions based on
simultaneous equations were appropriate but not set in context and therefore on Level 1 (Knowing) of
the ML assessment taxonomy37 (TSL3). These examples were basic, did not reflect quality and were on
a level far below Grade 11, even for a revision lesson38 (TSL3). The homework was suitable for what
Monty did in class, but did not require the learners to do proper revision of their Grade 10 work
(TSL3).
4.5.1.2
Discourse
Discourse: Teacher-learner interaction (DTL)
Monty ensured that all learners were quiet while listening, attending and copying the work from the
board during his presentations (DTL1). He was non-judgmental but did not encourage learner
participation, with the result that there was no evidence of the learners’ ideas (DTL1). He never asked
them to explain or justify themselves, instead asking lower order and basic calculation questions
throughout the lessons (DTL2). There was no evidence of the teacher listening to learners’ ideas and
providing scaffolding to support their attempts (DTL3). Some learners mumbled answers to Monty’s
questions (DTL3). After the incident in which a learner wrote her solution on the board Monty asked:
Anyone who can do the second one? (Silence). Anyone? …… have you tried it? Anyone? ……. (All just look at
him). OK first, let’s go to our notes. Firstly we said we need to label them…
37
38
Discussed in Section 2.2.2.2: ML principles.
Examples have already been given under Tasks: Motivational strategies.
107
He continued to explain in exactly the same way the solution to the first problem (DTL3). When
learners answered his questions, he did not comment on their answers. He either repeated the answer
or if the answer was wrong, provided the correct answer for example:
The mode is 2 and? Girl: 4. Teacher: 4 (DTL3).
In another example
3x − y = 0...............1
, Monty asked:
4 x + 2 y = −12.........2
Is x or y the subject of the formula? Is x or y the subject of the 2 given equations? Boy: y. Teacher: Is x the subject of
the formula in the 2 equations? Another boy: No. Teacher: What you do? What do you have to do? Some learners:
Third equation. Teacher: A third equation (DTL3).
When he did the example
x + 2y = 4
, he asked: Is x or y a subject of an equation?
7x − 5y = 9
He initially got no response and later some learners told him that x was the subject (DTL4). In the
discourse that followed Monty became irritated, looked troubled and laughed as he could not
understand why the learners did not know the answer (DTL4). There was no evidence of the teacher
recognising or clarifying learners’ misunderstandings (DTL5). He assisted most of the learners while
they were busy with classwork (DTL5).
Discourse: Learner-learner interaction (DLL)
He did not encourage learners to listen to, respond to or question one another although there were
opportunities during classwork where they could do so (DLL1). On one occasion he said:
If you did not understand it well yesterday you can now work with your friend. Don’t allow your friend to just copy,
he must ask and talk.
Only three groups were formed with two learners per group (DLL1). One group just worked from the
same textbook while the other two groups discussed the work. The rest of the learners worked
individually (DLL1). There was not a learner or group of learners who dominated the verbal
communication in class (DLL2).
Discourse: Questioning (DQ)
Most of the questions Monty asked were factual and of lower order, such as complete the
word/sentence and calculation type of questions and in many cases he answered the questions himself
(DQ1). In one lesson there were 97 such questions. Examples of such questions are:
•
When 2x 7 was written on the board during lesson 1, the following was asked:
108
•
T39: OK we call this one a co…? (Referring to 2) efficient (and completed the word while some learners mumbled an
answer)
T: This one we call it…? (Indicating to the x)
L: Variable.
T: The variable (and writes it on the board).
An example of answering his own questions was during the introduction phase of lesson 1:
T: What are simultaneous equations? It’s a combination of two linear functions.
•
Factual questions such as the following were asked during the lesson on data handling:
•
T: What is meant by mean? What comes to your mind? When you see mean, what comes to your mind?
L: Bigger, smaller.
T: Average. It’s average. And once you see that average, you see lots of numbers adding each other and dividing by
the number of them, that is the picture you must have
Lower order questions during simultaneous equations :
T: OK now we know x=-1,2. We go and substitute x=-1,2 in…?
L: 2.
T: Into…?
L: 2.
T: Into equation…?
L: 2.
T: 2. (Teacher did it.) What is the answer there of 4(-1,2)? Huh? What’s the answer?
L: -4,8.
T: -4?
L: -4.8.
T: -4.8 (DQ1).
On only one occasion Monty asked a learner to explain her work but did not follow it through:
T: Explain your answer, here use this (He gives her a large triangle. She is shy and cannot look at the class and put
her head in her hand.) OK people (and he takes it out of her hand) if you are given these 2 equations: Equation 1,
x is the subject of the formula …. (and he continues to explain her answer) (DQ2).
The questions did not contribute to the verbal communication and participation of learners and did not
create opportunities when they could listen to and respond to each other’s answers (DQ3). In general
his questions were addressed to the whole class, who mumbled answers or sometimes responded in
chorus (DQ4). A few times some learners volunteered to answer and on only three occasions he called
on particular learners to answer his questions (DQ4).
4.5.1.3
Learning environment
Learning Environment: Climate (LEC)
To comment on Monty’s relationship with and among the learners based on how they valued each
other’s ideas and ways of thinking is difficult as his lessons were teacher-centred with minimum
interaction (LEC1). He generally did not value or seek their ideas (LEC1). When I commented in the
39
When discourse was quoted, I used the following abbreviations: Teacher (T); Learner (L); and Researcher (R).
109
last interview on the class discipline I observed, Monty replied that he has classroom rules and values
discipline in order for learning to take place (LEC2). He further mentioned that the principal highly
values discipline in their classes (LEC2). The learners respected Monty and were well behaved, sat
quietly in class listening to him and copying the work from the board. Monty only needed to discipline
them once during the three lessons when he said:
OK hey hey hey, listen (pause), listen, the break is over. Please don’t disturb my class. Shhh. So, if I see you talking,
I am going to chase you out and then you will come back next term (LEC2).
On more than one occasion he reminded them that they are not Mathematics people and this may either
be comforting to the learners or a matter of degrading them (LEC3). Monty is confident and
enthusiastic about ML and has a positive attitude towards the subject and the learners (LEC3).
Learning Environment: Strategies and pacing (LESP)
Monty used direct instruction (lecturing) as instructional strategy and his teaching style varied between a
traditional and formal authority style (LESP1). Learners were involved copying work from the board,
listening to his explanations and answering basic low level questions (LESP1). During the second
lesson he gave the learners two problems to complete in class, but allowed only a few minutes to solve
the problems before he asked a learner to write her solution on the board. Since she was too shy to talk,
he again explained her solution in detail as he had done with the previous examples (LESP1). Near the
end of this lesson he was running out of time and hurried through the last example by saying: OK let’s go
faster there is no more time (LESP2). I observed only two phases of a lesson in the first two lessons, namely
the initiation and development phases (LESP3). Since there was no closure phase there was no
opportunity to summarise or assess learners’ knowledge and understanding (LESP3). Only the last six
minutes of the first lesson were allocated for a class activity of which the learners used five minutes to
get settled (LESP3). His second and third lessons flowed logically (LESP4). For example, in his third
lesson he used the following discussion as part of his introduction:
T: Where can we solve this simultaneous? OK, remember we are approaching the Election Day and we need to
support the campaign. Don’t you think the results can be solved simultaneous, how? Remember he has to sit on the
parliament and the province and we have 9 provinces né? Remember for the vote of the 18th they are going to take
the result because remember people voted for this particular party or this party or organisation. They are going to add
all those results and what information do you think we can get out of that? We can convert it into equations and
solve simultaneous. That will tell us how many positions that party is going to get in…? Parliament, né? So you see
we solve it simultaneous. Once you know the number of how many people voted for the party you know how many
will sit in parliament.
This was followed by more examples of the work they had done the previous day and he again referred
to contexts at the end of the lesson (LESP4). It was unclear how the learners were supposed to connect
the context to the content (LESP4).
110
Learning Environment: Administrative routines (LEA)
Time was not used efficiently to maximise learners’ involvement as Monty spent most of the period
using direct instruction (LEA1). Monty did not involve the learners in his explanation of how research
is done and did all the explaining of the examples on his own; meaning no active learner involvement
(LEA1). There was no opportunity for the learners to be part of an active learning process in which
they could learn from each other and improve their understanding (LEA1). The time at the end of the
lesson when learners were supposed to start doing classwork, was not well spent as they used five of
the six minutes to settle down (LEA1). They were seated at their individual desks during content
presentation, but could work in groups during classwork time if they preferred (LEA2). When Monty
explained work he was always in front of the class but when the learners were working at their desks, he
walked up and down the isles assisting learners with their work which contributed to a positive learning
atmosphere (LEA3). The written information on the board was correct but not always well-organised
(LEA4).
Summary
Table 4.6: Summary of Monty’s instructional practice
LESSON
DIMENSIONS
Tasks
Modes of representation
(TR)
Motivational strategies
(TMS)
Sequencing and difficulty
levels
(TSL)
DESCRIPTION OF LESSON DIMENSION INDICATORS
Monty used representations such as written examples on the board,
variables, calculators and a manipulative. He seldom connected
learners’ prior knowledge to the new mathematical situation.
The tasks he chose were not motivational to the learners. He talked
about contexts during two of his lessons but the contexts were not
applicable and valuable. He attempted to point out the value of
mathematics so that learners would value the work they were doing
but his explanation was vague and learners could not relate to it.
Not much attention was given to sequencing the tasks and no
connections were made with ideas learned in the past. The tasks on
simultaneous equations were on a Grade 11 level (Level 1) but the
tasks in the data handling lesson were on Grade 10 level (Level 1).
Discourse
Teacher-learner
interaction
(DTL)
He was non-judgmental but did not encourage learner participation
with the result that there was no evidence of the learners’ ideas,
occasions when learners’ thinking was challenged or even when he
recognised or clarified learners’ misunderstandings. He assisted most
of the learners while they were busy with classwork
Learner-learner
interaction
(DLL)
There was minimum learner-learner interaction.
Questioning
(DQ)
The questions were of lower order. The type of questions was mainly
complete the word/sentence and calculation questions. The
questions were addressed to the class and the responses were
111
volunteered.
Learning environments
Social and intellectual
climate
(LEC)
Modes of strategies and
pacing
(LESP)
Administrative routines
(LEA)
There was a positive rapport between the teacher and the learners.
He maintained good discipline throughout the lessons in order for
learning to take place and had a good relationship with his learners.
His teaching style varied between a traditional and formal authority
style. He used direct instruction (lecturing) as instructional strategy
and once allowed a learner to work on the board. Typical of teacherdirected lessons, learners were involved copying work from the
board, listening to explanations of the teacher and answering basic
low level questions. Monty did not get to the closure phase of a
lesson where learners’ knowledge could be assessed. There was a
logical flow to his lessons.
Time was not used efficiently to maximise learners’ involvement as
Monty used most of the period lecturing to the learners. He stayed in
contact with the learners as he moved between them which
contributed to a positive learning atmosphere.
4.5.2 Alice’s instructional practice
4.5.2.1
Tasks
Tasks: Modes of representation (TR)
In the first lesson on the use of the quadratic formula, Alice used representations40 such as written
examples on the board, symbols, the formula, tables, graphs and calculators (TR1). In her next lesson
on graphing parabolas, she used a table as well as the formula to calculate the x-intercepts (TR1). In the
third lesson on data handling the following was taught within a single 35 minute period: 1) the mean,
mode, median and range (when a set of data were given and when instead of a set of data a table with
frequencies were given); 2) pie charts (interpreting a given pie chart and drawing a pie chart), bar graphs
and histograms; and 3) tables with tallies, frequencies, ∑ symbol, cumulative frequencies (TR1). She
did not do the standard deviation or the ogive that she also planned to do (TR1). She started the lesson
by revising the bar graph and histogram, continued with another example on how to set up a table with
marks of learners, tallies and frequencies when a set of data were given, then mentioned they were
going to do the pie chart, but when she turned around to clean the board she continued with different
elementary examples on the mode, mean and median. Afterwards she did a comprehensive example on
a pie chart. She then introduced the symbol ∑ to the learners and continued her lecture on cumulative
frequencies. When the learners did not understand her explanation, she drew a table on the board with
three columns for the marks, frequency and cumulative frequency. She then mentioned the drawing of
40
Examples of all these representations are given in the text to follow.
112
the ogive but told them they would get back to this as she first wanted to explain how a table could be
constructed when only the frequencies rather than a set of data were given. She then explained how to
find the range and median from the above table (TR1). The learners complained throughout the lesson
that they did not understand the work (TR1). She used these various representations in an attempt to
have the learners connect their prior knowledge with the new content, but the extent of the content
and the way she presented the work was too much for the learners to absorb and led to confusion
(TR2). Some learners just withdrew from all activities during the lesson (TR2).
Tasks: Motivational strategies (TMS)
As I have said, on the day before the first observation, a student teacher taught the learners how to use
the quadratic formula to solve quadratic equations. In the first lesson I observed, Alice decided to do
three more such examples (Picture 4.5) with the learners.
Picture 4.5: The three tasks Alice gave the learners during lesson 1
She wrote three quadratic equations on the board of which only the first equation was not given in
standard form (TMS1). They only did the first one ( 15 x 2 = 11x − 2 ) and much later during the lesson
took another example from the textbook ( x 2 − 10 x + 25 = 0 ). The tasks provided the learners with an
opportunity to pursue their conjectures (TMS1). A group of learners were motivated to take part in the
lesson and to pursue their conjectures as the teacher was making many mistakes41 on the board, but
neither the teacher nor the learners could correct the errors in the teacher’s work (TMS1). Only a few
learners in front of the class participated in the lessons by answering questions or asking questions that
Alice most of the times could either not hear or understand. The rest of the learners only copied all
written work from the board and some were lying on their arms or talking to each other (TMS1). In the
data handling lesson some of the learners were inspired to take part in searching for ways to correct
their work or to try and find meaning and understanding (TMS1). There was no evidence that Alice
took into account the diversity of learners’ interests, abilities and experiences (TMS2). She generally
presented her lessons on a level suitable for Mathematics and not ML learners as many of the tasks
41
Examples of these mistakes are provided later on under Mathematical content knowledge.
113
were on Mathematics level as will be discussed under the next subheading (TMS2). Alice did not point
out the value of the mathematics being learned, which could have contributed to learners’ appreciation
of the subject (TMS3).
Tasks: Sequencing and difficulty levels (TSL)
There were incidents for which Alice sequenced her activities. For example, during the first lesson she
revised the standard form of quadratic equations and the meaning of the variables a, b, and c (the
coefficients) before using the formula (TSL1). For most parts of the lessons Alice tried to link the
content with other relevant content or even prior knowledge, but it was not done in a sequential and
meaningful way. This resulted in the learners being confused (TSL2). For instance, after she finished
the first example of using the formula to solve a quadratic equation, the following dialogue followed:
T: You remember when we draw the graph (and she erases part of the board and draw a table for x and y
coordinates) Quiet guys! (She writes: y = x 2 ) Guys! Now you are not given any formula. We need to start at a
negative.
L: Why do you start with -2?
T: Because they don’t give it. I am just assuming this is a problem. (She completes the table). This is now where you
draw your graph. (She draws two graphs below the table). This is your positive and this is your negative (indicating
to the first and second graph). I am not drawing this one (Pointing to example they did). Quiet!
L: Shhhh.
T: Let’s draw (and she draws a set of axes and labels them. Learners talk and teacher looks at example and erases
the set of axes before she could even draw something on the axes) Shhh shhh. OK.
L: Mam, where’s my textbook?
T: OK, we have 6 x 2 + x = 12 . Quiet please! If you don’t want to learn, you can leave the class (and she continues
to solve 6 x 2 + x = 12 ).
This is but one example of Alice jumping between examples and incomplete explanations (TSL2). In
the second lesson when learners had to draw the parabola, they started to draw y = ( x − 2) 2 − 1 for
− 1 ≤ x ≤ 4 using a table method (Picture 4.6) followed by the intercept method using the quadratic
formula (Picture 4.7). Here she stated that they did not need to calculate the turning point as the graph
(Picture 4.8) would automatically go through the correct turning point if they worked accurately (TSL2).
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Picture 4.6: Using a table method to draw the graph of a parabola
Picture 4.7: Using the formula to draw the graph of the same parabola above
Picture 4.8: The graph of the parabola
The tasks were on Grade 11 level and suitable for what the learners were supposed to know and be able
to do, but would only need to practise (TSL3).
In the data handling lesson she sequenced the concepts and content to be covered during the lesson as
she revised, in this order, the mode, mean, median and range (Picture 4.9) before she introduced the
cumulative frequencies and ogives (TSL1).
115
Picture 4.9: Calculating the measures of dispersion from a list of data
Although the lesson was well sequenced, the tasks within the presentation of a concept were not
sequenced so that learners were not able to progress in their cumulative understanding of the content
(TSL1). She gave an appropriate and interesting example using a pie chart was given (Picture 4.10) and
learners had to answer questions based on the pie chart.
Picture 4.10: Task given based on the pie chart
While the learners were still struggling with the task the following discourse took place:
T: But sometimes they don’t give you this. You are given this information (pointing towards answers) and then you
are asked to represent it like this (pointing to the pie chart). Now take for example (erases the pie chart) you are
given this, you are only given the number of learners, the answers and then they ask you to represent this information
in a pie chart. What do you do? (Silence.)
L1: divide by 540.
L2: 135 over 540.
T: Remember it’s a fraction. This is a fraction of this (pointing to 135 and 540), so it’s this divided by this times
what? 360, which gives you 90 degrees which was given here. So sometimes you are given data to represent in a pie
chart, sometimes you are given the pie chart and you have to try and get the number of learners. OK, I am sure you
are OK now with your pie graph and the frequency.
After this verbal explanation she immediately continued with another example of using a table and
frequencies to calculate the mean (Picture 4.11) and median (Picture 4.12) (TSL1).
116
Picture 4.11: Using a table and frequencies to calculate the mean
Picture 4.12: Using a table and frequencies to calculate the median
The extract from the observation given below and based on Picture 4.12 above is another example
where learners could not progress in their cumulative understanding of different methods for finding
the mean and median (TSL1.
T: OK, what about if you want to find your median using this table, what do you do?
L1: You write it in ascending order.
T: But now your marks are in ascending order and you have your frequency and your total frequency is what?
L2: 25.
T: 25 and you know that you have 25 data’s. So it’s an odd number. So you have to get the middle number, so how
do you do that?
L3: You say 18+1 over …
L4: No.
L3: You said 18+1. (The boy takes 18 as middle number in the table).
L4: Mam, you said there are 5 numbers, so it is 5+1 over 2.
T: Remember this is the data, this is the frequency (pointing to the table), so it means you have two 10’s, four 15’s,
eight 18’s; ten 22’s and one 25. The frequency is 25. So its 25 plus 1 divided by 2 its 13. So your 13th data is
going to be your median. So this plus this gives you 6 and this plus this gives you 14 so your 13th data is 18.
L3: But that is what I said (TSL1).
117
The learners did not appear confident about finding the measures of dispersion using two methods as
she never applied both methods using the same example and they also needed clarity concerning
aspects of the pie chart (TSL1). The learners were not involved in individual class work as the teacher
did several examples either verbally or in writing on the board throughout the lesson (TSL2). Many of
these examples were not relevant to the new content she actually planned to introduce and by the time
she got to the new content the learners were exhausted and confused (TSL2). The curriculum does not
require learners to use the formula
n +1
to find the median and the symbol
2
∑ F to find the sum of the
frequencies (TSL3). All tasks were on Level 1 (Knowing) of the ML assessment taxonomy except for
the only contextual example, the one on the pie chart that was on Level 2 (Applying routine procedures
in familiar contexts) (TSL3).
4.5.2.2
Discourse
Discourse: Teacher-learner interactions (DTL)
During the first lesson Alice used the quadratic formula incorrectly ( x = −b ±
b 2 − 4ac
2a
) and also
omitted brackets during the substitution, causing confusion and chaos in class as several learners talked
at once to Alice and one another in attempts to clarify the problems. At first Alice ordered them to
keep quiet and later shouted at them as she tried to find the problems herself. Since she could not
identify her mistakes, she allowed participation from the group of learners sitting in front while the rest
of the class was ignored (DTL1). During the second lesson she did not involve any of the learners and
seldom looked back at the learners in the class while working on the board (DTL1). In the third lesson
she encouraged a little participation from the learners in front by asking questions. The rest of the boys
at the back were still not involved (DTL1). A girl once attempted to rectify the mistakes on the board
but Alice ignored her and she went back to her desk (DTL2). On two other occasions Alice asked a
boy and later a girl to come and write on the board but she did not require the learners to explain their
work. Instead Alice corrected the boy’s work by telling him what to write (DTL2).
Although Alice allowed the learners to become involved in the first lesson, she was too anxious to
maintain control of the lesson and the learners and as a result she could not listen carefully to their
ideas in order to support their thinking (DTL3). Instead of listening to the learners’ ideas to direct her
instruction, she used a formal authoritative style and told the learners what to do while demonstrating
on the board. In the second lesson she kept strictly to this style to avoid the chaos of the first lesson.
Alice said to the learners: Don’t ask me how to get this; you have to look at me (DTL3). When she was busy
118
with the example discussed above, she did not prompt the learners to justify their answers, instead she
re-explained the content the same way she did before. Many times she would not comment on their
answers, would make a face and re-explain (DTL3). The girl who drew the bar graph and histogram
(Picture 4.13) on the board during the third lesson asked the teacher:
L1: Mam, can I draw it? (Girl comes to the board and draws both graphs on the board).
T: Shh, quiet.
L1: It’s a bar graph (draws the one on the right saying).
L2 No, it’s a histogram. (L1 writes ‘histogram’ and then draws the graph on the left). It’s the bar graph.
T: OK, you know sometimes you get information that you can represent on the bar graph or histogram. OK, then it’s
your tables, your table always consist of the marks, the tallies, you still remember your frequency.
Picture 4.13: Learner’s work on the board during the introduction of the third lesson
The teacher did not follow up on the girl’s sketches by discussing the difference between the bar graph
and histogram or the use of these graphs to provide scaffolding to support the learners in their
conceptual understanding (DTL3).
In the data handling lesson where Alice was more comfortable with the content, she frowned and made
a face when the learners gave incorrect answers to her questions and also when they mentioned that
they did not understand the work (DTL4). She became irritated when she could not understand their
misunderstandings or misconceptions (DTL5). She said it was impossible not to understand data
handling especially since all content of the lesson was supposed to be easy and well-known to the
learners (DTL5). There was no evidence that she recognised learners’ misunderstandings. For example
when she had discussed the pie chart and learners complained that it was complicated and she just
replied that it was not and continued with the solution of the problem. When she completed the pie
chart she said: OK, I am sure you are OK now with your pie graph and the frequency. Ok, turn to p. 37 (and she
starts writing: ∑ …. (DTL5).
Discourse: Learner-learner interactions (DLL)
ML classrooms are supposed to be learner-centred. Instead of applying this approach, Alice wanted the
learners to keep quiet (DLL1). The learner-learner interactions were learners talking to each other about
119
non-mathematics issues and discussions of possible ways of correcting the teacher’s mistakes on the
board (DLL1). The discourse between learners was therefore not as a result of opportunities Alice
created enabling them to discuss the work (DLL1). She allowed a group of learners to dominate the
verbal communication in the class and it seemed that she depended on their assistance (DLL2). She
ignored the boys at the back who did not participate at all (DLL2).
Discourse: Questioning (DQ)
The questions Alice asked while writing on the board were generally memory questions and of a low
level such as basic calculation, and complete the word/sentence type of questions (DQ1). Examples
taken from the different lessons are:
•
•
•
•
•
T: What do you have to do? You have to get it in standard form so you need to take it over to this side and she
writes and the sign is going to change when you take it over.
T: So with this, all you have to do is substitute into your formula, a is? 15, b is? -11 and c is? 2.
T: How do you find the minimum?
L: You just see it.
T: Where do you get it?
L: It’s at -1.
T: This is the minimum value (showing at turning point and continues with other work).
T: Do you understand this?
T: Your frequency is what?
L: It’s the number.
T: OK, it’s the number (and she continues to do a new example on the board) (DQ1).
When Alice listened to learners’ answers she did not ask them to clarify their answers. Instead she
would provide the answer:
T: To find your mean, your mean is always the?
L: Middle number.
T: No, mean it’s the sum of the data divided by the number of data (DQ2).
Although she did not create many opportunities for learners to contribute to the verbal
communication, the group of girls who did take part in the lessons had the opportunity to question the
teacher and their peers (DQ3). Learners’ responses were mostly volunteered or chorus. Once or twice
she did call on a specific learner (DQ4).
4.5.2.3
Learning environment
Learning Environment: Social and intellectual climate (LEC)
From what I have observed Alice did not maintain a positive relationship with and among the learners
(LEC1). She seems to care about them as she claims to sacrifice her breaks to be available for learners
to come to her for help (LEC1). Not much evidence could be found when Alice valued her learners’
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ideas or even the student teacher’s ideas and ways of thinking (LEC1). Alice struggled to control the
learners because of the mistakes she made in the first lesson and the overload of content she covered in
the third lesson and frequently had to shout at them: Quiet please! If you don’t want to learn, you can leave the
class. The reason for the learners’ misbehaviour was that learners were confused and discouraged and
Alice could not handle the situation (LEC2). Generally she did not create enough opportunities for
learner participation which is typical of a teacher-centred strategy (LEC2). From the beginning of the
second lesson she was strict and enforced discipline ensuring they kept quiet, sat at their own tables and
copied the work from the board and therefore needed to discipline the learners only five times (LEC2).
If Alice had had a positive attitude towards the learners and the subject as she stated in an interview, it
was not evident in her lessons (LEC3). She rather appeared bored, irritated and un-enthusiastic, never
giving the learners any accolades (LEC3).
Learning Environment: Modes of instruction and pacing (LESP)
Alice’s teaching style varied between a traditional and demonstrative style (LESP1). She used direct
instruction (lecturing), a teacher-centred approach in her lessons as well as a little discussion with a few
learners in front of the class (LESP1). On two occasions learners worked on the board: the one learner
drew the bar graph and histogram and the other wrote a solution on the board (LESP1). The direct
instruction strategy Alice used did not always support learner involvement and goal attainment and she
was not aware of the learners’ lack of knowledge and skills regarding the two topics she covered. Alice
assumed that if she understood the work, the learners would understand it too (LESP1). No assessment
of the learners’ knowledge was done and there was no evidence that Alice’s goals had been reached
(LESP1). The only time Alice provided time for the learners to express themselves was during the first
lesson when the learners tried to rectify Alice’s work and to discuss the problems and the nature of the
solutions with one another (LESP2). There was no other occasion when Alice structured the time
necessary for learners to express themselves and to explore the mathematical content (LESP2). In most
of the lessons she did not use her time effectively to accommodate all three phases of the lesson
(LESP3). She did not have time for closure at the end of the lesson when she could have summarised
or assessed the learners’ knowledge and understanding (LESP3). As far as the logical flow of the
lessons is concerned, Alice could sequence the content in the lesson, but failed to have a logical flow in
her explanations of specific concepts as she attempted to provide the learners with too much
disorganised information and incomplete explanations (LESP4).
Learning Environment: Administrative routines (LEA)
During the first lesson Alice did allow time for learners’ involvement in discourse, but it was not the
result of effective procedures and management of her classroom (LEA1). There was not enough time
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allocated for learners to be actively involved in working on tasks in class. Most of the time she ordered
them to keep up with her as she worked on the board and would then say:
T: Can I erase this?
L: No.
T: Why? What is taking you so long to write (and she erased it and wrote the following example) Quiet! (LEA1).
The classroom arrangement was appropriate to the lesson style as learners were seated in three long
rows in this very wide classroom, seated at desks with no space between them (LEA2). Alice’s position
in class during the first and third lesson did not contribute to learners’ conceptual understanding as she
was in front of the class the whole period, busy working on the board, frequently looking things up in
her textbook and talking only to the learners in the front (LEA3). During the second lesson she once
walked through the class (LEA3). The work she did on the chalkboard was not organised, she cleaned
wherever she needed space to write, causing learners to be confused as they needed to listen and copy
the work from the board (LEA4). Some of the examples on the board were incomplete since she often
completed her explanations verbally. Sometimes information was missing or unrealistic, like a table
without a heading (Picture 4.14) or the angle measurements of the pie chart42.
Picture 4.14: Example of no headings given in a table
Summary
Table 4.7: Summary of Alice’s instructional practice
LESSON
DIMENSIONS
Tasks
Modes of representation
(TR)
Motivational strategies
(TMS)
42
DESCRIPTION OF LESSON DIMENSION INDICATORS
Alice used representations such as written examples on the board,
symbols, the formula, tables, graphs and calculators. She could not
proficiently use the various representations to connect learners’ prior
knowledge with the new mathematical situation.
Alice treated the ML learners as if they were Mathematics learners,
not taking into account that they were of lesser ability. The only time
Alice did an example that was set in context was in the third lesson.
Although a small group of learners took part in the lessons, the
majority of the learners did not. She did not point out the value of
the mathematics being learned.
See Figure 4.3 discussed under Tasks: Sequencing and difficulty levels.
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Sequencing and difficulty
levels
(TSL)
Discourse
Teacher-learner
interaction
(DTL)
Learner-learner
interaction
(DLL)
Questioning
(DQ)
The tasks she chose were appropriate and on Grade 11 level (Level 1)
but were not presented logically or in context to ensure that learners
were motivated.
Her interaction with the learners was at times judgmental and it could
not be said that she encouraged participation or even created
opportunities where learners’ thinking was challenged. She did not
recognise or acknowledge her own and the learners’ mistakes and
misunderstandings.
Learner-learner interaction was observed during the first lesson but
this was not as a result of opportunities Alice created for learners to
take part in discussing the work. Instead the learners discussed
possible ways to correct Alice’s mistakes on the board while others
talked about non-mathematics issues.
The types of questions asked were memory, calculation, and
complete the sentence questions. Learners’ responses were
volunteered or chorus.
Learning environments
Social and intellectual
climate
(LEC)
Modes of strategies and
pacing
(LESP)
Administrative routines
(LEA)
Alice did not establish a positive relationship with and among the
learners by valuing the learners’ ideas and ways of thinking. She at
times appeared bored, irritated and unenthusiastic, not praising the
learners’ work.
Since Alice used direct instruction (lecturing) as instructional strategy,
a large amount of information was shared verbally. Some of the
explanations were done incompletely on the board. This strategy did
not always support learner involvement and goal attainment.
Generally she planned too much content per period causing her to
lose the logical flow of her lesson.
During the first lesson Alice did allow time for learners’ involvement
in discourse but it was not the result of effective procedures and
management of her classroom. She needed their input to correct her
mistakes on the board. The classroom arrangement was appropriate
to the lesson style but Alice’s position in class did not contribute to
learners’ conceptual understanding as she was in front of the class
most of the time.
4.5.3 Denise’s instructional practice
4.5.3.1
Tasks
Tasks: Modes of representation (TR)
In the first two lessons on conversions from metric to imperial units, Denise used representations43
such as written work on the board, conversion tables, the variable x to find the unknown values,
calculators and during the third lesson she used a diagram (Picture 4.15 below) to explain conversions
between different units of length (TR1).
43
Examples of these representations are discussed under Discourse.
123
Picture 4.15: Diagram used for conversions between different units of length
She used this representation to enable the learners to connect their prior knowledge of different units
of length to the conversions between the different units of length (TR2). Most of the learners used
equations and cross multiplication to solve the unknown value and she reminded them that ratios could
also be used to solve the unknown value (TR2).
Tasks: Motivational strategies (TMS)
Denise treated the lessons as Mathematics and not ML lessons where the tasks were not set in a context
as prescribed by the DoE (2003a), but were instead asked directly (TMS1). The following tasks (Picture
4.16) were given in the first and second lessons of which only the first three were completed during the
first period (TMS1):
Picture 4.16: The tasks during lesson 1 and lesson 2
Denise’s learners were inspired to participate in the lesson which might be due to Denise’s teaching
style and not necessarily as a result of the nature of the tasks (TMS1). From the following it appeared as
if Denise took account of the diversity of learners’ abilities (TMS2). She knew all her learners by their
names and randomly called on learners to come and work on the board. On one occasion when a
learner made mistakes, she called on another learner to come and rectify the work. Another time she
also called a specific learner as she knew he used a method different from the rest of the learners who
worked on the board (TMS2). All learners who had worked on the board used equations to solve the
problems. She then asked another learner and when he started to use ratios instead of equations
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(Picture 4.17), she said: [y]ou know why you are doing the problem, you can see I have purposed it, my boy (indicating
to the ratio he uses) (TMS2).
Picture 4.17: Learner using ratios instead of equations
These given tasks were set in context in the textbook: a South African company exporting food
products to the UK (TMS3). Denise did not mention this context at all, not even to elicit a class
discussion on export in order to increase the learners’ interest in the lesson and their appreciation of
the value of mathematics in everyday life situations (TMS3). The only time she pointed out the value of
the mathematics they were learning was when she told them: you must understand so that if somebody is absent
from class you can explain it to him (TMS3). Her motivation is examination driven, preparing the learners
for examination through practice (TMS3). She further advised them to work faster as they would not be
able to finish the coming examination paper in time at their current working pace. She timed the
learners while working on their classwork and after one problem she said: you must take three minutes and
you used 10 minutes, using the mark allocation of the problem as a time guide (TMS3).
Tasks: Sequencing and difficulty levels (TSL)
Denise sequenced the tasks over the two different lessons she did on conversions which enabled the
learners to progress in their cumulative understanding of the content (TSL1). The prior knowledge of
these conversion lessons was the different units of measurement as well as the meaning of the different
concepts (TSL2). It was only at the end of the third lesson that Denise asked the learners to identify the
different concepts according to the unit of measurement. She did not ask them to explain or define the
concepts:
T: Number 1? Length, mass or capacity?
L: Capacity.
T: Can you see it? Right. and number 2? What is it?
L: Mass.
T: Mass and number 3 Jenny is doing now?
L: Capacity. (Learners complained that they wanted to leave since the bell had rung) (TSL2).
125
These two lessons were revision lessons so learners were supposed to know how to perform the
conversions but still needed to improve on their skills (TSL2). Denise used this opportunity to identify
and correct learners’ common errors and misunderstandings (TSL2).
The content of the tasks were appropriate and on Grade 11 level but since no task was set in a context,
her tasks were only on Level 1 (Knowing) of the ML assessment taxonomy (TSL3). In the first lesson
she gave the learners conversions within the metric system such as change 340ml to litres as well as
conversions from metric to imperial units for example: complete 500kg = ____ lb. The tasks were
based on capacity and mass only. In the second lesson the tasks were based on conversions within the
metric system but included not only mass and capacity but length, area and volume too. Some of the
learners complained about the complexity of the area and volume tasks (TSL3).
4.5.3.2
Discourse
Discourse: Teacher-learner interactions (DTL)
Denise encouraged participation from the learners during all three lessons as she walked through the
class attending to learners’ work, questioning and explaining to them (DTL1). She further called
specific learners to work on the board and involved the rest of the learners by asking them to comment
on the work on the board (DTL1). Learners demonstrated their work in writing on the board (Picture
4.18) but were not asked to explain or justify their work (DTL2).
Picture 4.18: The work on the board of two learners
Denise based her instruction on what she saw the learners wrote on the board but did not listen to their
explanations. She pointed out the errors and through questioning she involved the learners to take part
in doing the corrections (DTL3). Once Denise asked a learner to come and correct another learner’s
126
work on the board and when he started to do that she said: Hey David44, leave Cindy’s business, go and write
your own stuff so that we can compare (DTL3). There was little evidence of Denise listening to learners’ ideas.
Instead she looked at their written work in order to provide scaffolding to support their thinking
(DTL3). When they converted an area problem, the following discussion ensued:
L: Must I say the amount to the power of 2?
T: (Denise immediately starts writing on the board and sings the following song:) King Henry died a miserable death
called measles. This should be in your computer all the time (pointing to their heads), when to multiply and when
to divide.
… (Denise continues to explain Picture 4.15 to the class).
L: No mam, I want to know do you square the amount?
T: No!! But OK, it’s a good question (and Denise continues with next task) (DTL3).
Denise applauded the learners’ answers and made comments such as: You did excellent so far, guys and I
am happy the way she is doing it, not looking at her textbook, because that is what we assess, understanding (DTL4).
Denise marked some of the learners’ work on the board as she would have marked a problem in a test
(DTL5). Based on the following work from three different learners (Picture 4.19) the following
discourse took place serving as proof of her recognition and clarification of learners’ misunderstandings
and misconceptions:
Picture 4.19: Work from three different learners
T: So now your conversion, it’s squared metre to squared km. What’s wrong there?
L: You’re supposed to divide.
T: You’re supposed to divide because it’s from metre to kilometre. And you have to?
44
Pseudonyms were used instead of learners’ true names.
127
L: Multiply.
T: Multiply. OK? So that is where your problem is. So when you move from m to km, what do you do? Multiply or
divide? So come and correct it. (She erased the step from the board and moved to the next problem on the board.)
So when it’s m to km, it’s correct? (She encircled the division sign.) So from metre to km it’s a 1000, you divide
by a 1000, so it’s correct and this is correct. Those are the basics. Now the problem is here (underlying the 1,04)
two significant figures. So two significant figures, it gives you 1,0cm, right? (She went to the next problem.) So
here it’s mm to what? m, so we divide by what? a 1000 (and marks it right), gives you 24,96m. Two significant
figures, you did not answer that. 24,9 what? 24,96 I said, but now 2 significant figures? Anyone to help? (She
called on a learner.) What is the answer?
L: 25.
T: 25. Don’t forget to round off. (She goes to Learner 3’ work.) You said divide by 1000000 why? Because moving
from m to km you divide by 1000, now it’s 1000 to the power of 2 (and she wrote (1000) 2 , so you get 1
million. So you divide by million and this is the answer. It’s correct. Now you’re answer to 2 significant figures?
(The learner did not do that yet). Rectify your results. (Learner corrected her work on the board) (DTL5).
Discourse: Learner-learner interactions (DLL)
She did not encourage the learners to listen to or question each other but instead most of the discourse
was between Denise and the learners (DLL1). There was one incident during the second lesson when
Denise left the class for a few minutes. Suddenly most of the learners wanted to help the learner who
was at that stage making a mistake on the board, but he just became more confused. By the time
Denise entered the classroom the learners became quiet and she continued to correct his work (DLL1).
There was no learner or group of learners who dominated the verbal communication in class (DLL2).
Discourse: Questioning (DQ)
The lessons were characterised by questions directed at the learners (DQ1). Most of the questions were
calculation, memory and convergent questions (DQ1). Many questions required the learners to look up
conversions from the table in the textbook such as: 1kg equals to how many pounds? (DQ1). Generally
Denise did not allow enough time for the learners to become engaged in the discourse. Instead she
provided the answers herself (DQ1). Denise did not ask learners to clarify or justify their ideas and two
such examples are given below (DQ2):
•
•
T: Is the area one now correct? Is this one correct? (She looked at specific learners.) Is this now correct?
L: (mumbles something no one could hear).
T: If you think it’s not, then you just say NO, because it is as if you’ve got doubts. Is it correct that one? (She
pointed to another learner). 3 251 squared meter is how many squared km? So now your conversion, its squared
metre to squared km. What’s wrong there?
L: You’re supposed to divide.
T: You’re supposed to divide because it’s from metre to kilometre (DQ2).
T: So here it’s mm to what? m, so we divide by what? A 1000 (Denise marked it right), gives you 24,96m. Two
significant figures, you did not answer that. 24,9 what? 24,96 I said, but now 2 significant figures? Anyone to
help? (She called on a learner.) What is the answer?
L: 25.
T: 25 (DQ2).
128
Denise created opportunities for learners to communicate and participate by answering questions she
posed throughout all three lessons but she did not create such opportunities among the learners (DQ3).
Learners’ responses were mostly teacher-selected and at times volunteered (DQ4).
4.5.3.3
Learning environment
Learning environment: Social and intellectual climate (LEC)
Denise maintained a positive rapport with the students as she valued their attempts (LEC1). There was
one incident when the learners objected to a learner’s answer who then appeared puzzled. Denise then
told the class to calm down and to notice that the learner had worked accurately but only misread the
value from the conversion table (LEC1). Although Denise could be humorous at times which the
learners absolutely enjoyed, she was also very strict and did not hesitate to remind the learners of the
appropriate classroom behaviour (LEC2). Comments such as the following were frequently heard:
• Take your hands out of your pocket
• Stop talking
• T: Is Nelius correct?
L: No.
T: Don’t say no under the table please, speak up
• Shhh, we are just checking, don’t fight. We are not fighting (LEC2).
Denise had a positive attitude towards the subject and learners and made comments such as:
• You are just writing like a professor there né? (Everyone laughed)
• Keep on practising till we don’t see that minor mistakes. Keep on practising.
• So it’s nice when you say you are ready, it’s nice (LEC3).
A more negative incident occurred when she said: Anyone who does not have a calculator, you sit on the floor
and then you complain to your parents, so that they can give you a calculator. (One boy sat on the floor) and she said:
Oh, we have one customer today (and she let him sit there) (LEC3).
Learning environment: Modes of strategies and pacing (LESP)
Denise’s style of teaching was that of a facilitator. She had a learner-centred approach where discussion
and learners writing on the board were used as instructional strategies (LESP1). These strategies
supported learner involvement throughout all three lessons and learners had sufficient time to express
themselves when answering questions either in their books or on the board (LESP1). In the first lesson
Denise did one example on the board to demonstrate and explain what was expected of them (LESP1).
She then gave the learners problems to solve individually and after each problem they did the
corrections together. In the other two lessons the learners had already completed the tasks at home, so
129
Denise used the entire period to do corrections (LESP1). The corrections were done by asking learners
to write their solutions on the board which Denise then used to guide her instruction (LESP1). She
pointed out the learners’ errors and misunderstandings and involved the learners by asking questions in
order for that the learners could understand (LESP1). After a problem had been discussed, the learners
had to assess their own solutions (LESP1).
In the first lesson Denise gave the learners five problems to solve individually in class and after each
problem they did the corrections together (LESP2). In the other two lessons the learners already
completed the tasks at home, so Denise used the entire period for corrections (LESP2). The learners
also knew that anyone of them could be asked at any time and they needed to be prepared at all times
(LESP2). She ensured participation of the learners through continual questioning and ensuring that
they were doing their own corrections (LESP2). She encouraged the learners to explore and use their
textbooks efficiently by reminding them that the textbook consisted of activities, worksheets,
assessments, projects and reviews and that they were currently busy with a worksheet (LESP2). Her
lessons were not typical lessons consisting of the initial, development and closure phases as she was
busy with revision when learners needed to practice their skills (LESP3).
Learning environment: Administrative routines (LEA)
She organised and managed the class effectively to ensure that time was maximised for the learners to
develop conceptual understanding (LEA1). The learners were seated at individual desks which were
appropriate for the lesson style (LEA2). Denise was in contact with her learners as she continually
moved between the desks when she was not explaining and demonstrating in front of the class (LEA3).
She also attended to learners individually at their desks assisting them with the work (LEA3). The work
Denise did on the board was correct and all corrections to the learners’ work were indicated so that
learners could do their own corrections (LEA4).
Summary
Table 4.8: Summary of Denise’s instructional practice
LESSON
DIMENSIONS
Tasks
Modes of representation
(TR)
Motivational strategies
(TMS)
DESCRIPTION OF LESSON DIMENSION INDICATORS
She made use of representations such as written work on the board,
conversion tables, calculators and a diagram to illustrate the different
units of measurement of length. These representations allowed her to
link learners’ prior knowledge with the new content of the day.
The learners were motivated and inspired to take part in the lesson,
not necessarily due to the nature of the tasks but to the fact that they
wanted to show their work on the board. She did not point out the
130
Sequencing and difficulty
levels
(TSL)
Discourse
value of mathematics in everyday life.
The given tasks were sequenced over the different lessons and were
appropriate, although not set in context and the content was on
Grade 11 level (Level 1).
Teacher-learner
interaction
(DTL)
Denise verbally encouraged the learners as she praised their efforts
that were written on the board. Most of the times she did not expect
the learners to explain their thinking. The lessons were characterised
by the discourse between Denise and the learners and the number of
questions she posed to the learners.
Learner-learner
interaction
(DLL)
No discourse based on the content was observed among the learners.
Questioning
(DQ)
Most of the questions were calculation, memory and convergent
questions. Many times Denise did not allow enough time for the
learners to become engaged in the discourse and just provided the
answers herself.
Learning environments
Social and intellectual
climate
(LEC)
Modes of strategies and
pacing
(LESP)
Administrative routines
(LEA)
The social and intellectual climate in the class can be described as
positive as Denise had a positive rapport with the learners valuing
their ideas and praising their efforts.
Denise used a teacher-learner-centred approach with discussion, and
learners’ writing on the board as instructional strategies. These
strategies were effective to ensure learner participation. She worked
at a manageable pace throughout.
The administrative routines such as management of time to maximise
learner involvement, classroom arrangement and the information on
the board were effective.
4.5.4 Elaine’s instructional practice
4.5.4.1
Tasks
Tasks: Modes of representation (R)
To facilitate content clarity Elaine used representations such as written work on transparencies, tables,
symbols, formulae, calculators, a demonstration calculator and sketches of manipulatives (Picture 4.20)
in all her lessons (TR1).
131
Picture 4.20: Examples of sketches used in Elaine’s discussions of solutions
In the second lesson on time she used a table (See Question 5A on the next page) for parking tariffs
(TR1). One of the formulae the learners used during the second lesson was: F =
x[(1 + i ) n − 1]
i
(TR1).
Elaine expected the learners to explain the meaning of each variable in the formula enabling them to
proficiently apply their knowledge to other similar unknown formulae (TR1). Below is an example of a
sketch of a goal box (Picture 4.21) Elaine gave the learners to assist them in solving a problem based on
a soccer field (TR1).
Picture 4.21: Elaine’s drawing of the given goal box to explain the solution
Initially most of the learners calculated the time problems without the use of their calculators until a
learner asked Elaine how she could use her calculator to find the answer. Elaine used a large CASIO
demonstration calculator (Picture 4.22) which she put on the board to demonstrate the use of the
calculator.
Picture 4.22: Casio demonstration calculator
132
The other learners were eager to master their calculators in calculating the answers to the time
problems. Learners then assisted one another while she assisted specific learners who still could not
manage their calculators. Since many of the learners were either not aware of or not able to use their
calculators, they were very pleased afterwards with their accomplishment. Elaine generally drew the
learners’ attention to the required prior-knowledge needed to understand the content of the specific
tasks (TR2). To connect the learners’ prior knowledge with the new knowledge she alternated between
discussions with questioning and class tests of which the answers were afterwards discussed in class and
self-assessed by learners (TR2).
Tasks: Motivational strategies (TMS)
The tasks that captured the learners’ curiosity were especially those based on time and interest (TMS1).
The learners manually calculated the answers of the following question but enjoyed checking their
answers using their calculators (TMS1).
Question 5A
Sam parks her car every day in a parking area at her work. The table below shows the cost of parking for specific
time periods:
Parking tariffs
Hours
Cost
0 – 1 hour
1 – 3 hours
3 – 5 hours
5 – 7 hours
7 – 9 hours
More than 9 hours
Saturdays
Sundays
Free
R4,00
R6,00
R8,00
R10,00
R12,00
R5,00
Free
5.1 On Monday morning Sam arrives at the parking area at 07:50 and leaves the parking area at 17:15. How
much does she pay for parking?
5.2 On Tuesday she arrives at 08:01 and leaves the parking area again at 08:45. She goes back to work at 12:15
and leaves for home at 17:30. How much does she pay in total for the parking?
5.3 On Wednesday Sam parks her car at the parking area. She has to pay R8,00 because she parked there from
08:45 to 13:25. Now use the table to determine whether she paid the correct amount. Show all steps and give a
reason for your answer.
The following interest problems (Question 5B below) were set in a context of buying a house and the
impact thereof on an individual’s, or even their parents’ budgets (TMS1).
From the learners’
participation in the discussions it seemed as if the learners took an interest in these tasks.
Question 5B
5.1 James wants to buy a house that is in the market for R780 000. If he pays a deposit of R78 000 and then R7
800 per month for 20 years, how much will he pay in total for the house?
5.2 James decides to rather first save money to increase his deposit. He invests R7 800 at 15% per year, interest
compounded semi-annually. How much money will he have saved after 7 years? ( A = P(1 + i ) n )
133
5.3 James’ parents also invested R450,00 monthly in a savings account for 8 years. The interest rate was 11% per
year compounded monthly. How much money does he have in that account? ( F =
x[(1 + i ) n − 1]
)
i
5.4 James can afford to pay R6 500 each month for 18 years on a home loan at 17% per year, interest compounded
monthly. How much money can he borrow from the bank? ( P =
x[1 − (1 + i ) − n ]
) (TMS1).
i
Elaine had a very demanding learner in class who had, according to her, been diagnosed with attention
deficit hyperactivity disorder. She successfully kept him involved and focussed throughout the lessons
(TMS2). To a few hard working learners she said: The people who already completed the work, I will come and
assess your work and will then give you the next tasks to be done45 (TMS2). These were two examples where
Elaine took the diversity of learners’ abilities and experiences into account (TMS2). Since all questions
were based on realistic everyday life situations the learners were able to relate to the tasks (TMS3). An
interesting discussion followed when Elaine asked: Let us talk a little about why a person would rather wait to
buy a house until he increased his deposit. She then referred to the learners’ personal lives where she discussed
a typical household’s budget and their parents’ expenses so that they could understand what their
parents sometimes had to tell them: There is no money for whatever you wanted at that stage (TMS3). Elaine
wanted the learners to gain understanding and to be able to apply their knowledge to other similar
problems and situations that might arise in their future lives (TMS3). She emphasised that they needed
to show all calculations at all times, also in the examinations since marks are specifically allocated to
their calculations and not just the final answers (TMS3).
Tasks: Sequencing and difficulty levels (TSL)
Elaine sequenced the tasks in all three lessons by progressing in a lesson from easier to more complex
tasks (TSL1). She also sequenced her class activities: for example during the second lesson on time and
interest she first checked their homework and together they did Question 5A46 on time. Before
discussing the next Question 5B47, she gave them the following class test based on prior-knowledge
needed to answer that question.
Class Test: Banking matters
A = P(1 + i ) n
1. What do I calculate with this formula?
2. Write in words the meaning of each of the following:
A=
P=
i=
3. What does the following mean to you? BODMAS
45
n=
Since Elaine’s classes were not presented in English, her texts were translated by me.
Question 5A is given under Tasks: Motivational strategies.
47 Question 5B is given under Tasks: Motivational strategies.
46
134
After the test they discussed the answers and then proceeded with the questions. In this lesson she also
proceeded from the easier task on time to the more complex task on interest. With the task on interest
she progressed from discussing the meanings of the unknown values in the formula to discussing the
context, the formulae to solve the problems in the given task. She concluded the lesson by interpreting
the solutions.
During the third lesson she also first walked through the whole class checking the learners’ homework,
then introduced the topic for that day’s lesson, followed by a discussion of a contextual task the
learners completed at home and then the lesson was rounded off by giving the learners the following
class test:
Class Test: Perimeter, area, volume
1. What does the following mean to you?
a) Perimeter
b) Area
c) Volume
2. If the given figure’s measurements are given in centimetres, what will be your unit of measurement for your answer
when the following need to be calculated?
a) Perimeter
b) Area
c) Volume
3. The following figure is a rectangle.
7,8cm
A
12,6cm
a) Calculate the perimeter of figure A.
b) What is the area of figure A?
c) Calculate the volume of figure A (TSL1).
Except for surface area all concepts were already introduced in Grade 10 but she appropriately applied
the content to more complex situations (TSL2). Elaine was busy with her revision programme and
during the first interview she stated that approximately three quarters of the learners knew the work by
then. Half the remaining learners knew half the work while the other half did not have any idea of the
work (TSL2). She stated that the purpose of her revision lessons was to either reinforce or enhance
learners’ current knowledge, to highlight learners’ mistakes and to allow them to practise their
knowledge and skills in order to prepare them for the coming examination (TSL2). The tasks Elaine
covered in class were appropriate, on Grade 11 level, and reflected quality (TSL3). Based on the ML
assessment taxonomy, Elaine selected tasks on Level 1 (Knowing) and Level 2 (Applying routine
procedures in familiar contexts), but most of the tasks were on Level 3 (Application of multi-step
135
procedures in a variety of contexts) and Level 4 (reasoning and reflecting) which required more
advanced levels of thinking skills (TSL3). An example of such a task was a task based on a goal box:
The distance from the lower edge of the crossbar to the ground is 2,44 m. The distance from the goal post to the back
of the goal is also 2,44 m. An extra pole is to be welded from the top corner of the goal post to the back of the
supporting base.
• How long must the support pole be (in metres)?
• Convert this length to mm.
• Write this answer in scientific notation.
• If the diameter of the new pole is 4 cm, what will its circumference be?
4.5.4.2
Discourse
Discourse: Teacher-learner interactions (DTL)
Elaine involved most of the learners in her class by asking them questions, clarifying their uncertainties
and assessing their classwork (DTL1). She knew her learners by name and posed specific questions to
specific learners such as:
I want to know, Hennie48, did you do it like that?
Cecil, are you OK now, tell me what does it mean there?
Lindy, what do you have there?
Kevin, what did you say about monthly interest?
Martie, do you agree on this? (DTL1).
She communicated in a non-judgmental manner especially with the learner who was diagnosed with
attention deficit hyperactivity disorder and who at times could be annoying. She remained calm and in
control of the situation, attended to his comments or questions and continued with the lesson. An
example was the following discourse between her and the specific learner:
T: In ML they give you all the formulae and information.
L: It’s not always like that. Another time I thought they normally give you everything and they did not. They
changed everything just as they wanted, changed this and that!
T: Is that true? Oh, then I must look into what happened there (DTL1).
Whether learners were doing or discussing classwork or even after writing class tests, Elaine required
them to give explanations and justifications orally or in writing (DTL2). She said on several occasions:
You must show me where you get that and Yes, but what does it actually mean? (DTL2).
The following example is one of several incidents during Elaine’s lessons where she listened carefully to
learners’ ideas and provided scaffolding to support their thinking:
48
Pseudonyms were used instead of learners’ true names.
136
T: What does it mean there?
L1: Compound interest.
T: Compound interest. Why did you choose compound interest?
L1: It’s not simple interest.
T: Right, but what tells you that it’s not simple interest, but compound interest?
L1: The bracket and the part below.
L2: But there is no fraction.
L3: A is the final amount.
T: In simple interest A is also the final amount. I told you earlier that you know it’s compound interest when you
see ‘n’ written as a power, then we reason this is more complicated than the normal formula, then it’s compound
interest. So I don’t want you to just guess that it’s compound, you must be able to give a reason why you say it is
compound interest (DTL3).
She accepted their answers without criticising their efforts (DTL4). On several occasions Elaine said:
Good or Well done and even thanked them for doing their homework as there were times they did not do
their homework, telling her they could pass ML without doing homework (DTL4).
Elaine recognised and clarified the learners’ common errors and misunderstandings (DTL5). For
example there was a misunderstanding when the discussion was about the initial value and end value
when interest was calculated. Instead of saying initial value a learner said present value and Elaine
explained how present value could be interpreted as being the value after a certain period or could even
mean the end value. She emphasised that the initial value is the value you began with (DTL5).
Discourse: Learner-learner interactions (DLL)
A number of times Elaine encouraged the learners to listen to or respond to other learners’ ideas and
answers and would say: Listen, here Simon49 is saying … Ernest, can you respond to Simon’s statement? (DLL1).
On another occasion she asked: Who agrees with her? Who wants to argue with her? I know you sometimes like to
argue. So Celeste, do you agree with her? … (DLL1). Most of the discourse was not among the learners but
between Elaine and the learners (DLL1). While doing classwork learners had the opportunity to discuss
the work with each other and discourse occurred that I could unfortunately not hear (DLL1). An
example where Elaine guided a discussion among learners was when the different variables of the
formula were discussed:
T: If we calculate interest semi-annually, by what must n be divided? How many times will interest be calculated per
year?
L1: Two.
T: Why not six?
L2: It is six.
L1: Because you will get interest in the middle of the year and then the end of the year.
49
Pseudonyms are used in all quotes.
137
T: Good, so semi-annually means every half of the year interest will be calculated, so at the end of the sixth month
you get your first interest and then up to the twelfth month it will be the second six months period when I will get
my next interest. So, if semi-annually I divide by two (DLL1).
The only learner who tried to dominate the verbal communication was the learner who was diagnosed
with attention deficit hyperactivity disorder, but she treated him in a firm and calm manner and once
said: OK, you had your moment (DLL2).
Discourse: Questioning (DQ)
Elaine is a well-prepared and confident teacher who allowed enough time for learners to respond to her
questions (DQ1). She posed questions on all three levels, namely memory, convergent and divergent
questions (DQ1). Examples of convergent questions were: What does it mean to write 12,5% as a decimal?;
and
T: What kind of a triangle is formed?
L1: Right-angled triangle.
T: Then I can indicate the right angle on my drawing and know that I can work with which theorem?
L1: Theorem of Pythagoras
T: Who can give me the theorem of Pythagoras?
L2: r 2 = x 2 + y 2
T: Good, also tell me in words what the theorem means … (DQ1).
Examples of divergent questions were: Explain in detail to an ignorant person the meaning of each variable in the
interest formula; and when an interest formula was given to the learners and they were asked what kind of
interest was represented, she asked: Why did you decide on compound interest? (DQ1).
Elaine consistently listened to learners’ ideas and in many instances she asked them to clarify and/or
justify their answers (DQ2). Her questions contributed to the verbal communication and participation
of the learners and she created opportunities which the learners could listen to, respond to and
question her as teacher or even their peers (DQ3). Learners’ responses were mostly teacher-selected but
also volunteered (DQ4).
4.5.4.3
Learning environment
Learning environment: Social and intellectual climate (LEC)
Elaine was well-prepared, made her lessons interesting and continually involved the learners in class
discussions (LEC1). Just as important to her was mutual respect of one another and she maintained a
positive rapport with and among learners by emphasising the importance of people valuing each others’
ideas and ways of thinking (LEC1).
138
Discipline and classroom rules played a major role in her classroom ensuring learners’ positive
behaviour (LEC2). At the beginning of each period she completed the attendance register and then
walked through the class to control their homework and made a note of those who did not complete
their homework (LEC2). There were only a few times when it was necessary for her to discipline the
learners and these were some comments:
I again ask you, put your suitcase next to your desk
If I asked you to discuss it with your friend, my choice of words was wrong
Thank you, you had your moment
That was rude (LEC2).
Elaine appreciates both the subject and her learners and praised them by saying: I am fond of you; and I
really appreciate your cooperation (LEC3).
Learning environment: Modes of strategies and pacing (LESP)
Elaine’s teaching style varied between being a facilitator and mediator of learning (LESP1). She
proficiently used instructional strategies such as class discussions and direct instruction (LESP1). The
use of these strategies provided opportunities for the involvement of the learners and facilitated goal
attainment (LESP1). She structured her lessons in such a way that learners had enough time to express
themselves and explore their ideas and solutions (LESP2). She never rushed through the work or put
pressure on them to work faster (LESP2). Her lessons did not consist of an initial, development and
closure stage as they were revision lessons (LESP3). She made valuable use of her class time and
completed what she had planned for the day (LESP3). There was a logical flow in her lessons as she
worked from easier to more complex tasks and from familiar to less familiar concepts saying:: Now let’s
go a little bit further … (LESP4).
Learning environment: Administrative routines (LEA)
Elaine believed enough time should be allowed for learners to practise their knowledge and skills and
therefore made provision for a revision period in her year plan to allow learners to prepare themselves
for the examination (LEA1). She allowed a certain amount of time for learners to solve a problem but
at the end of that time would still ask: Who needs more time? (LEA1). Elaine arranged the class so that
learners were seated in pairs, which was appropriate for the particular lesson style (LEA2). When she
explained work she was in front of the class facing the learners because she used the overhead
projector, but would otherwise move between the learners attending to their needs (LEA3). The
written information on the transparencies and blackboard was very neat and organised with no
mistakes (LEA4). Permanently visible on the right hand side of the blackboard was the work they
139
already completed so that the learners could take note of their progress (LEA4). The following was
written on the board:
Chapter 4
• Units 1,2,4,5,6,7,8,9
• Test papers: B1; A2; B2, Class Activity (out of 21)
• Taxation: Unit 10
Book: Paper F1; F2
Summary
Table 4.9: Summary of Elaine’s instructional practice
LESSON
DIMENSIONS
Tasks
Modes of representation
(TR)
Motivational strategies
(TMS)
Sequencing and difficulty
levels
(TSL)
DESCRIPTION OF LESSON DIMENSION INDICATORS
To facilitate content clarity Elaine used representations such as written
work on the board, tables, symbols, formulae, calculators, a demonstration
calculator and sketches of manipulatives in the three lessons I observed.
Her class tests consisted of oral or written questions in order to connect
learners’ prior knowledge to the new mathematical situation.
The learners were interested in the tasks as they spontaneously took part
in the class discussions, especially in those tasks that were based on buying
a house and working out parking tariffs using their calculators. They
enjoyed taking part in the discussions Elaine led. She took learners’ diverse
abilities into account and accommodated the learner with attention deficit
hyperactivity disorder as well as a few hard-working learners. The value of
mathematics was frequently emphasised as Elaine discussed real scenarios,
also applying the work to their personal lives.
Elaine sequenced her tasks to enable the learners to progress in their
cumulative understanding of the content and they were able to make
connections with ideas learned in the past. The lessons were revision
lessons to improve the learners’ knowledge and skills. The tasks reflected
quality and were on Grade 11 level (Levels 1-4).
Discourse
Teacher-learner
interaction
(DTL)
Learner-learner
interaction
(DLL)
Questioning
(DQ)
Learning environments
Social and intellectual
Elaine involved most of the learners, either by attending to their needs at
their desks or posing questions. She communicated in a non-judgmental
manner at all times. She required learners to give explanations and
justifications orally and in writing. She recognised and clarified the
learners’ common errors and misunderstandings.
Elaine encouraged the learners to listen to or respond to other learners’
ideas. The discussions were not necessarily among the learners but in most
cases between her and the learners. During class work the learners
discussed the work with one another.
Elaine is a confident and well-prepared teacher and allowed enough time
for learners to respond to her questions. She asked a variety of questions
and posed questions on all three levels, namely memory, convergent and
divergent. Learners had to clarify and/or justify their answers. Their
responses were mostly teacher-selected but also volunteered.
She made her lessons interesting, involved the learners in discussions and
140
climate
(LEC)
Modes of strategies and
pacing
(LESP)
Administrative routines
(LEA)
maintained a positive rapport with and among learners. Discipline and
classroom rules played a major role in her classroom to ensure learners’
positive behaviour. Elaine was very fond of the subject and her learners
and praised them for their efforts.
She used instructional strategies such as discussions and direct instruction.
The use of discussions provided opportunities for the involvement of
learners and facilitated goal attainment. Learners had enough time to
express themselves and explore their ideas and solutions and there was a
logical flow in her lessons.
She allowed enough time before the examination for learners to improve
their knowledge and practise their skills. Learners were seated in pairs
which was appropriate for the lesson style. Elaine was in front of the class
when she explained the work but otherwise moved between the learners.
4.5.5 Summary of participants’ instructional practices
Table 4.10 below provides a snapshot of the four participants’ background information and their
instructional practices.
141
Table 4.10: Snapshot of the four participants and their instructional practices
PARTICIPANTS
MONTY
ALICE
DENISE
ELAINE
BEd Honours degree in
Mathematics Education with
seven years’ experience of
teaching Mathematics and three
years of teaching ML.
HED: Senior Primary with
Mathematics and Methodology
of Mathematics as major
subjects. She had eight years’
experience of teaching
Mathematics and three years of
teaching ML.
BACKGROUND
BEd degree with Mathematics
and Methodology of
Mathematics as major subjects.
Novice teacher with one year
experience of teaching
Mathematics and two years of
teaching ML.
BTech Management
Accounting degree with no
Mathematics Education
training. Novice teacher with
only one year’s teaching
experience, teaching ML only.
Modes of
representation
(TR)
• Used representations such as
written examples on the
board, variables, calculators
and a manipulative.
• Seldom connected learners’
prior knowledge to the new
mathematical situation.
• Used representations such as
written examples on the
board, symbols, the formula,
tables, graphs and calculators.
• The various representations
did not contribute to
connecting learners’ prior
knowledge with the new
mathematical situation.
Motivational
strategies
(TMS)
• Only mathematical content
was taught. The nature of the
tasks did not capture the
learners’ curiosity or inspire
• Except for the one task being
set in a context, the lessons
consisted of mathematical
content only. When she made
Qualifications and
experience
TASKS
142
• Used representations such as
written work on the board
• Used representations such as
and transparencies, tables,
written work on the board,
symbols, formulae,
tables, calculators and a
calculators, a demonstration
diagram.
calculator and sketches.
• These representations
• Through tests and oral
allowed her to link learners’
questioning she connected
prior knowledge with the new
learners’ prior knowledge to
content of the day.
the new mathematical
situation.
• Pure mathematical content
• The learners were interested
was taught. Learners were
in the tasks as they
motivated and inspired by the
spontaneously took part in
teacher and not necessarily by
the class discussions. She
them to pursue their
conjectures.
• He only mentioned contexts
to which content could be
applied to point out the value
of mathematics but the
explanations were vague.
Sequencing and
difficulty levels
(TSL)
• Not much evidence of the
sequencing of tasks.
• The lessons on simultaneous
equations were on Grade 11
level (Level 1) but the data
handling lesson on Grade 10
level (Level 1).
mistakes on the board, some
of the learners were
motivated to pursue their
conjectures.
• Did not point out the value
of mathematics in every-day
life.
• Most of the times the tasks
were not successfully
sequenced to enable learners
to progress in their
cumulative understanding of
the work.
• Tasks were on Grade 11 level
(Level 1).
the nature of the tasks.
• Did not point out the value
of mathematics in every-day
life.
took learners’ diverse abilities
into account.
• She frequently reminded
them of the value of
mathematics in their lives.
• Tasks were sequenced over
the different lessons, were
suitable to what the learners
already knew but needed to
improve on.
• Tasks were on Grade 11 level
(Level 1).
• Tasks were sequenced in the
lessons to enable the learners
to progress in their
cumulative understanding of
the work, set in context.
• Tasks were applicable and on
Grade 11 level (Levels 1-4).
• Non-judgmental and verbally
encouraged the learners as
she praised their efforts.
• Required learners to give
demonstrations of their work
in writing but did not expect
them to explain their work.
• She provided scaffolding to
support learners’
understanding.
• She recognised and clarified
learners’ misunderstandings.
• Non-judgmental and all
learners were involved
through questioning and
discussions.
• Learners had to give
explanations and justifications
of their thinking, both orally
and in writing.
• She provided scaffolding to
support learners’
understanding.
• She recognised and clarified
learners’ misunderstandings.
DISCOURSE
Teacher-learner
interaction
(DTL)
• Communicated in a nonjudgemental manner but did
not encourage learner
participation except for
posing basic questions.
• Did not require learners to
give full explanations.
• Re-explained the work
instead of providing
scaffolding to support
learners’ thinking.
• Could not recognise learners’
misunderstandings.
• She was judgmental in her
communication and did not
encourage the participation of
learners except where she
needed help with her own
mistakes and
misunderstandings.
• Did not require learners to
give full explanations.
• She did not listen to learners
to determine where and how
she could provide scaffolding.
• Did not recognise learners’
misunderstandings as she had
misconceptions herself.
143
Learner-learner
interaction
(DLL)
Questioning
(DQ)
• Did not encourage learners to
listen to, respond to and
question one another.
• Types of questions were
complete the word/sentence
and calculation questions.
• Did not contribute to
learners’ participation in
discussions.
• Responses were volunteered.
• The observed interaction
during the first lesson only
was a result of the mistakes
Alice made that were
discussed and not because
she created positive
opportunities for learners to
discuss the work
• She did not encourage
learners to listen to, respond
to or question each other’s
ideas.
• Elaine encouraged the
learners to listen to and
respond to other learners’
ideas. The discussions were
mainly between her and the
learners.
• Types of questions asked
were memory, rhetoric,
calculation, and complete the
word/sentence questions.
• Did not contribute to
learners’ participation in
discussions.
• Learners’ responses were
volunteered or chorus.
• Types of questions were
calculation, memory and
convergent questions. Many
times Denise did not allow
enough time for the learners
to become engaged in the
discourse and just provided
the answers herself.
• Contributed to learners’
participation in discussions.
• Learners’ responses were
teacher-selected.
• Types of questions were
memory, convergent and
divergent questions. She
allowed enough time for
learners to respond to her
questions.
• Contributed to the verbal
communication of learners
during discussions.
• Learners’ responses were
mostly teacher-selected but
also volunteered.
• Not a positive relationship
with and among learners.
• She did not ensure
appropriate classroom
behaviour.
• Seemed bored, irritated and
un-enthusiastic at times.
• Had a positive rapport with
the learners as she valued
their ideas and praised their
efforts.
• She was confident and strict
and applied classroom rules.
• She had a positive attitude
towards the learners and the
subject.
• Positive rapport with and
among learners and praised
their efforts.
• Good discipline.
• Confident, well-prepared and
enthusiastic. A calm and
relaxed atmosphere.
LEARNER
ENVIRONMENT
Social and
intellectual climate
(LEC)
• Positive rapport between him
and the learners.
• Good discipline and in
control of the learners and
lessons.
• Confident and enthusiastic
about teaching ML.
144
Modes of strategies
and pacing
(LESP)
Administrative
routines
(LEA)
• His style varied between
• Her style varied between
traditional and formal
traditional and demonstrative.
authority. He used direct
She used direct instruction as
instruction as instructional
instructional strategy.
strategy and once a learner
•
Not enough learner
wrote on the board.
involvement.
• Not enough time was
• In the third lesson too much
provided for learners to
content was covered with
explore mathematical ideas.
little logical flow.
• Good pacing and logical flow.
• Not enough time was
allocated to learner activities.
• Learners’ used the time to
• Her position in class did not
copy work from the board
contribute to learners’
and listened to the teacher.
conceptual understanding as
• Moved between the learners
she was mostly standing at a
when he was not explaining
specific desk in front of the
work on the board.
class attending to her
textbook and the few learners
• The written work on the
in front of her.
board was disorganised and
incomplete at times.
• There were mistakes on the
board and the work was not
always organised.
145
• She was a mediator. She used
discussions and learners
working on the board as
instructional strategies.
• These strategies ensured
learner participation.
• Lessons were presented at a
manageable pace.
• Her style varied between
mediator and facilitator. She
used discussion and direct
instruction as instructional
strategies.
• Enough time for learner
involvement and goal
attainment.
• Logical flow in lessons.
• Used time effectively to
maximise learner
involvement.
• She moved between the
learners to assist them and to
ask questions.
• Information on the board
was correct and ordered.
• Managed time effectively for
maximum learner
involvement.
• Her position in class
contributed to a positive
learning atmosphere.
• Very neat, correct and
organised work on
transparencies and the board.
4.5.6 Discussion of Theme 1: ML teachers’ instructional practices
I again conducted a comprehensive, advanced electronic search after presenting the data in order to
establish a basis from which I could execute a literature control of my findings in this chapter. My
search covered the period January 2008 to September 2011 as I wanted to correlate my study’s findings
with the most recent research studies conducted in ML classrooms50. Of the 32 studies on ML, 17 were
based on discussions, analysis, critiquing, developing frameworks or investigating certain theoretical or
curriculum aspects of ML in South Africa. Seven were concerned with in-service ML teachers’
experiences and their development through the ACE in the ML programme at different universities.
Eight studies investigated ML teachers’ instructional practices and the classroom experiences of ML
learners. Two of the eight studies were intervention studies conducted with one teacher, and six of the
eight studies were conducted at one school only. An experienced academic information specialist at the
University of Pretoria, Ms Clarisse Venter, also conducted an advanced electronic search for the period
January 2008 to September 2011 but could not add any studies to my existing list. Her reply was: Most of
the studies were discussions, analysis etc., which you do not want (C. Venter, personal communication, September
12, 2011).
In the next section, I will conduct a literature control where the findings from this study are compared
with the findings from other research studies on ML teachers’ instructional practices. I base the
discussion on Artzt et al.’s (2008) three dimensions of a lesson, namely tasks, discourse and learning
environment51.
4.5.6.1
Tasks
Since I believe knowledge is constructed and based largely on prior knowledge, I support the view that
the purpose of tasks such as examples given on the board, problems, activities and projects being given
to the learners is to provide opportunities for learners to connect their knowledge to new information and to build on
their knowledge and interest through active engagement in meaningful problem solving (Artzt et al., 2008, p. 10). The
tasks used by ML teachers to facilitate learning in their instructional practices are discussed in this
section in terms of Artzt et al.’s (2008) categorisations of such tasks52: modes of representation;
motivational strategies; sequencing; and difficulty levels of tasks.
50
See Addendum H for a list of studies conducted on ML for the period January 2008 to September 2011.
See Table 4.2 under Section 4.3.2.1.
52 See Table 4.2 under Section 4.3.2.1.
51
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Modes of representation
•
Use of various representations by ML teachers
All four participants in my study used various representations during their classes as was also found
with all the participants in the other research studies (Sidiropolous, 2008; Venkat & Graven, 2008;
Venkat, 2010). Elaine was the only participant in my study who expressed a need to increase the use of
technology such as computers in her ML classroom. As far as I could establish, no previous study has
reported this finding.
•
Linking learners’ prior knowledge to new situations
In my study, novice teachers Monty and Alice, unlike experienced teachers Denise and Elaine, neither
determined nor used their learners’ prior knowledge to facilitate the assimilation of new content
knowledge. This finding strongly confirms Sidiropolous’ (2008) finding that one of the two teachers in
her research group did not determine his learners’ prior knowledge or use his learners’ prior knowledge
to facilitate the assimilation of new content knowledge.
Motivational strategies
•
Use of tasks to motivate learners to reflect on and pursue their conjectures
Only Elaine in my study used contextual tasks that inspired the learners to reflect on their answers. In
ensuing class discussions, the learners’ conjectures were explored and expanded, enhancing their
understanding of the work. Conversely, the other three teachers in my study did not use tasks that
would motivate the learners to reflect on or to pursue their conjectures. My finding can therefore
probably be regarded as consistent with Sidiropolous’ (2008) findings where both teachers in her study
did not ask the learners to reflect on or discuss their solutions. My finding that only one of the four
teachers in my group used tasks to motivate learners is, however, inconsistent with the research results
obtained by Buytenhuys, Graven and Venkatakrishnan (2007) and Venkat (2010) who found that the
teachers in their studies used contextual tasks and discussions and succeeded in inspiring the learners to
reflect on their answers and explain and justify their arguments. It should, however, be mentioned that
the two teachers in the latter two studies were extremely dedicated teachers – not unlike the teacher in
my group who used contextual tasks and discussions to inspire the learners to reflect on their answers
and to explain and justify their arguments.
•
Pointing out the value of mathematics through the use of life-related tasks
Part of the definition of ML (DoE, 2003a) is the use of the life-related applications of mathematics to
make learners aware of and understand the role of mathematics in the modern world. In my study, only
Elaine based her lessons on solving contextual problems on discussions that related the learners’ newly
acquired knowledge to the outside world and home situations. This finding is inconsistent with the
147
findings of a number of researchers such as Buytenhuys et al. (2007), Hechter (2011a), Venkat and
Graven (2008) and Zengela (2008) who found evidence of the successful use of life-related application
problems in pointing out the value of mathematical literacy in everyday life. Three of the four teachers
in my research group taught mathematical content only and did not once refer to its value in everyday
life situations. My finding that only one of the four teachers in my study realised the value of teaching
mathematics through the use of life-related tasks in everyday life situations is consistent with the
finding of Sidiropolous (2008) who established that only one of the two teachers in her study realised
the value of teaching mathematical literacy in real-world contexts and accordingly taught the subject on
the basis of expecting the learners to solve real life-related problems.
Sequencing and difficulty levels
•
Sequencing of tasks enabling learners to progress in their cumulative understanding
Three of the four teachers in my research group were not able to sequence their tasks proficiently to
enable the learners to progress in their cumulative understanding of a particular task and to make
connections between ideas learned in the past and those they will encounter in their future lives. There
appears to be a gap in the literature in this regard.
•
Grading of classroom tasks according to the ML Assessment Taxonomy53
My finding in this regard was that three of the four teachers selected tasks only from Level 1 (Knowing)
according to the ML Assessment Taxonomy while Elaine selected tasks from all four levels (Knowing;
Applying routine procedures in familiar contexts; Applying multi-step procedures in a variety of
contexts; and Reasoning and reflecting). My finding is inconsistent with that of Govender (2011) who
reported that the only teacher in her study asked the learners to perform tasks on all four levels.
Interestingly, whereas the learners in the class of the one teacher (Elaine) who did select tasks from all
four levels understood the problems and could solve them, apparently because of the support given by
this teacher, in Govender’s (2011) study, the learners could not understand the problems and, despite
the support given by the teacher, could not solve the problems. Govender stated that the learners
found these kinds of problems difficult, were not used to such questions and did not understand the
contexts. The difference in the latter set of findings can be explained by the fact that at the time of the
study Elaine was teaching in a traditional white school in Pretoria where the learners were familiar with
the contexts while the teacher in Govender’s study was teaching in a black township school in Port
Elizabeth where the contexts were not part of the learners’ real-life experiences.
53
See Section 2.2.2.2: ML principles.
148
4.5.6.2
Discourse
As a way of contributing to learner understanding, the discourse in class should provide opportunities
for learners to express themselves, to listen to, to question, to respond to and to reflect on their
thinking (Artzt, et al., 2008). I will now conduct a literature control on the discourse in ML classrooms
based on Artzt et al.’s (2008) perspective of discourse54, namely teacher-learner interaction, learnerlearner interaction and the use of questioning to enable learners to build on their existing knowledge.
Teacher-learner interaction
•
Nature of teachers’ communication and learner participation
Except for Alice, the other three teachers in my study communicated with the learners in a nonjudgmental manner thus contributing to a positive relationship between the teachers and the learners.
This finding (only one of the four teachers in my study was judgmental) is inconsistent with that of
Sidiropolous (2008) where both teachers in her study judged their learners’ abilities and expressed low
expectations of the learners.
Monty and Alice did not encourage learner participation apart from posing low-level oral questions
where the answers were often provided to the learners before they could try to answer the questions.
The finding that two of the four teachers in my group did not encourage learner participation
moderately confirms Sidiropolous’ (2008) finding that both teachers in her study did not generally
encourage learner participation. Apart from one occasion, no time was allowed by the teachers in
Sidiropolous’ (2008) group for discussion on solutions or any critical engagement with mathematical
arguments in their instructional practices. Denise ensured learner participation by requesting the
learners to write their solutions on the board and involving them in discussions afterwards whereas
Elaine involved the learners only in class discussions. The finding that only two of the four teachers in
my group encouraged the use of discussions to enhance learner participation is moderately consistent
with the findings of Venkat and Graven (2008) and Venkat (2010). The two experienced teachers in
their studies encouraged class discussions, which were used to stimulate enhanced participation and
communication.
•
Opportunities for learners to explain and demonstrate their work
Only Elaine in my study asked the learners to explain their thinking and solutions, a strategy that
elicited further discussion between her and the learners in class. Denise did not request the learners to
explain their thinking but merely required them to demonstrate their work on the board. The other two
teachers did not require the learners to explain or justify their work at all. The finding that two of the
four teachers in my group provided opportunities for the learners to explain or demonstrate their work
54
See Table 4.2 under Section 4.3.2.1.
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is moderately consistent with the finding of Venkat (2010) that the (dedicated and experienced) teacher
in her group (she had only this person in her research group) asked the learners to explain their
thinking and solutions by giving them opportunities to explain and demonstrate their work (this teacher
insisted on justification and explanation). Since there is evidence in the literature that collaboration
from learners in the ML classroom contributes to the development of positive mathematical identities
(Graven, 2011), my findings should be of interest to the Department of Education. Seemingly, during
teacher training, more emphasis should be placed on the importance of providing opportunities for
learners to express their ideas and thinking and to explain and justify their work.
•
Use of scaffolding to support learner understanding
Even though Denise’s lessons were based on mathematical content only, both she and Elaine provided
scaffolding to support learners in solving problems and understanding the tasks instead of merely
telling them how to solve the problem or doing the problem for them. In contrast to the instructional
practices of Denise and Elaine, Monty either re-explained the work or solved the problem for the
learners while Alice was not concerned about the learners’ ideas and thinking. My finding that two of
the four teachers in my study provided scaffolding is moderately consistent with the results obtained by
Hechter (2011a) who found evidence of pedagogical support and scaffolding in the practices of both
ML teachers who were part of her study. It should, however, be mentioned that these teachers were
students enrolled for the ACE (ML) programme where they had to plan lessons according to certain
guidelines (including ways to accommodate scaffolding in their instruction) and then implement those
lesson plans in their instructional practices. In other words, they were required to facilitate scaffolding.
It is not clear whether they would have done so had they not been required to do so.
Learner-learner interaction
Except for the very limited evidence of learner-learner interactions in Elaine’s class, interactions
between learners where they had the opportunity to support, strengthen and challenge each other’s
ideas were absent in the other teachers’ instructional practices. This finding is inconsistent with that of
Venkat and Graven (2008) where the single ML teacher in their research, being experienced and
dedicated, used extensive communication and discussion of tasks during her lessons. My finding,
however, concurs with the finding of Sidiropolous (2008) where both teachers in her study did not
encourage critical engagement by learners in mathematical arguments. Given the belief (Brown &
Schäfer, 2006; Venkat, 2007; Venkat & Graven, 2008) that a learner-centred approach is of the utmost
importance in teaching ML, that is, where learners are actively involved in the lessons by taking part in
discussions and group work but also by using their knowledge outside the classroom, my findings
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(albeit based on the actions of a limited sample of participants) should be a source of concern to
education authorities.
Oral questioning
In my study, I found that three of the four teachers asked low-level questions such as complete the
sentence and simple calculation and memory questions, which did not allow enough time for the
learners to respond, and where in most cases the teachers provided the answers themselves. Elaine,
however, asked various types of oral questions on different levels and gave the learners sufficient time
to respond. As far as I could establish, these two findings regarding ML teachers’ use of oral
questioning in their classes have not been reported before and should be a source of concern to
education authorities.
4.5.6.3
Learning environment
Artzt et al. (2008) use the term learning environment to describe the conditions under which the
teaching-learning process unfolds in the classroom. I will now discuss the learning environments of the
participants’ ML classrooms in terms of Artzt et al.’s (2008) categorisations of a learning environment55,
comprising a social and intellectual climate, modes of strategies and pacing, and administrative routines.
Social and intellectual climate
•
Maintaining a positive relationship with and among learners in the classroom
Monty’s formal authoritative style of teaching restrained the building of positive relationships with and
among the learners, and Alice focused only on the mathematical content instead of building
relationships. However, Denise and especially Elaine created an atmosphere in the class where the
learners were comfortable and confident as they engaged in the tasks. My finding that two of the four
teachers in my study did not maintain a positive relationship with and among the learners is moderately
consistent with that of Venkat and Graven (2008) who found that the teacher in their study was patient
and that the learners could therefore work in a relaxed environment. This gave the learners a sense of
exploration, of working without fear or failure or ridicule, and of learning with enjoyment (p. 40).
•
Use of classroom rules
All four teachers in my study mentioned that they had to apply classroom rules to ensure appropriate
classroom behaviour since learner misbehaviour could be a problem in ML classrooms. Monty and
Denise were very strict; Alice at times could not apply her rules effectively while Elaine was more
relaxed in applying her classroom rules as the learners seemed to know what was expected of them. As
55
See Table 4.2 under Section 4.3.2.1.
151
far as I could establish, findings on ML teachers’ application of classroom rules have not been reported
before.
•
Teachers’ attitudes towards the subject and the learners
All four of the teachers in my study had a positive attitude towards the subject ML. My finding is
strongly consistent with Fransman’s (2010) finding where the four ML teachers in her focus group,
who were enrolled for the ACE (ML), experienced the training to become some kind of mathematics teacher,
i.e. to be trained as a Mathematical Literacy teacher, as a challenging experience in which they were developing some
sort of status-embraced identity (p. 175). My and Fransman’s (2010) findings are strongly inconsistent with
the finding of Sidiropolous (2008) where both teachers in her study regarded the teaching of ML as a
threat to their ‘status identity’ (p. 221) as Mathematics teachers. Sidiropolous (2008) surmised that her
finding could have been influenced by the fact that her study was conducted only one year after the
subject had been introduced. At that stage, negative attitudes towards the teaching of ML were
common (Sidiropolous, 2008).
Modes of strategies and pacing
•
Use of appropriate instructional strategies
I found that Monty and Alice, the two novice teachers in my study, seemed to believe that learners
learn through direct transfer of information (traditional approach, as defined in Section 2.4.4.2). Both
Denise and Elaine, however, based their instruction on their learners’ knowledge − Denise by using her
learners’ written solutions on the board to elicit discussion and Elaine by mainly using class discussions
as an instructional strategy. Since two of the four teachers in my study used appropriate instructional
strategies (as defined in Section 2.2.2.3), this finding is moderately consistent with that of Graven and
Venkat (2009) who reported that all the teachers in their research changed their pedagogical approaches
to teaching ML by using discussions and group work. Conversely, since two of the four teachers in my
study did not use appropriate instructional strategies, this finding is also moderately consistent with
Sidiropolous’ (2008) finding − she established that both teachers in her study kept to a traditional
teacher-centred approach by not using group work or discussions in their ML classrooms.
On the basis of Graven and Venkat’s (2007) proposed spectrum of pedagogic agendas56 ranging
from Context; to Content and context; to Mainly content; to Content driven, only Elaine’s pedagogic
agenda was Content and context driven. The pedagogic agendas of Monty, Alice and Denise were
Content driven. My finding in this regard is consistent with that of Hechter (2011a) who found that the
pedagogic agenda of one of the two teachers in her study closely matched the Mainly content driven
56
See Table 2.1 under Section 2.2.2.3: Pedagogical approaches for teaching ML.
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agenda while the other teacher’s pedagogic agenda partially matched the Context and Mainly content
driven agendas.
•
Effective structuring of available time
Three of the four teachers in my study worked at a manageable and slower pace compared to the pace
in Mathematics classes thus allowing the learners more time to understand the work. This finding is
consistent with that of Venkat and Graven (2008) where the only teacher in their study was willing to
‘wait’ in ML in contrast to the imperatives to rush ahead in Mathematics (p. 38).
Administrative routines
•
Maximise time for learners’ active involvement in tasks and discourse
I found that too much time was spent in Monty’s class on learners who copied work from the board;
while in Alice’s class, the learners spent most of their time looking at Alice’s demonstrations on the
board. Denise and Elaine, on the other hand, managed their time to maximise learner involvement. My
finding (two of the four teachers maximised the time available for learners to be actively involved in
tasks and discourse) is moderately consistent with that of Venkat and Graven (2008) where the teacher
in their study waited for the learners to understand before moving on.
•
Classroom arrangement, position of teacher in class, written information on the board
My finding that the classroom arrangements of the four teachers in my study were appropriate for the
lesson styles they used is moderately consistent with the finding of Sidiropolous (2008) that one of the
two teachers in her study arranged the desks appropriately in his classroom in groups. Except for Alice,
who worked on the board or from her textbook at a table in the front of the class, the other three
teachers moved between the learners’ desks engaging with the learners and their work. My finding that
three of the four teachers in my study moved among the learners is consistent with Sidiropolous’ (2008)
finding that one of the two teachers in her study moved among his learners. Regarding the written
work of the teachers on the board or on transparencies, I found that the work of Denise and Elaine
was organised whereas Monty’s work was disorganised, and Alice made mistakes or did incomplete
work on the board. As far as I could establish, this finding has not been reported before.
4.5.6.4
Summary of discussion on Theme 1
To summarise: Denise adopted a teacher-learner approach, Elaine a learner-centred approach while
Monty and Alice used a teacher-centred approach. Denise and Elaine used discussions and had the
learners working on the board in order to encourage learner participation, allowing the learners to
explain and/or justify their thinking. They used various representations and scaffolding to guide the
learners to conceptualise new knowledge. Elaine selected tasks on all four levels according to the ML
Assessment Taxonomy. The learning environments created by both Denise and Elaine were more
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relaxed with enough time for learner participation. Monty and Alice used direct instruction and thus
discouraged learner participation by not allowing the learners to explain and/or justify their thinking.
They used various representations, but not scaffolding, to guide the learners to conceptualise new
knowledge. Both these teachers selected tasks on the lowest level of the ML Assessment Taxonomy.
Monty’s learning environment was formal while Alice’s learning environment was relatively tense and
awkward, and she did not allow enough time for learner participation.
I agree with Graven and Venkat’s (2009) view that ML teachers need to make a substantive change in
their instructional (pedagogic) practice in terms of the nature of the educational tasks, the agenda
driving their teaching (on the continuum from Content to Context driven) and the way they interpret
the subject ML. I also concur with Venkat and Graven’s (2008) view that the nature of tasks and the
nature of interactions are two key concepts contributing to positive change in ML classrooms that
should be considered by ML teachers. In my research study, only Elaine’s instructional practice
conformed to these two key concepts.
4.6
Theme 2: ML teachers’ knowledge and beliefs
In this section I present and discuss the findings from the interviews and observations of Monty, Alice,
Denise and Elaine. All discussions on the subthemes MCK, and PCK regarding: ML learners, ML
teaching and ML curriculum are structured strictly according to the guidelines in Table 4.357.
Background information regarding the observed lessons of the participants is given in Section 4.5. The
language of all quotations from Monty, Alice and Denise has not been edited. Since Elaine’s classes
were conducted in Afrikaans, I translated her quotes from Afrikaans to English. The subthemes of each
participant are now discussed.
4.6.1 Monty’s knowledge and beliefs
4.6.1.1
Mathematical content knowledge58 (MCK)
I wanted to know from Monty how important it was for a ML teacher to have sufficient MCK and he
replied that:
ML needs mathematics knowledge because there are many things you need to know from maths, the basics, for
example the chapter on calculating angles or areas, surface area or volume. If you have never done this thing before,
how are you going to understand it?
57
58
Table 4.3 is discussed under Section 4.3.2.1: Inclusion criteria for coding the data.
Since there is only one indicator or code in Table 4.3 regarding the teacher’s mathematical content knowledge, this whole
paragraph’s code is: MCK.
154
Since Monty did not make any mathematical errors in his oral explanations or board work, it appeared
as if his MCK regarding the specific content covered in the three lessons is sufficient. Since basic
mathematical content was taught, there was no opportunity to observe whether he understood more
than just the procedure of solving simultaneous equations. The same applies to the data handling lesson
where I could not observe the extent to which he understands why and when we use the different
measures of dispersion. Because learners were not expected to explain why different measures of
dispersion are used, their conceptual understanding of the concepts also could not be determined.
When Monty discussed the median he only said:
OK now, a median is a middle number né? I don’t have another definition for that, it’s a middle number.
Monty did not use the glossary in the NCS for ML to define the terms properly, so that learners do not
just have a synonym but can explain the meaning of the term or as he stated it, know the characteristics
of the terms. Some minor mistakes I observed were:
•
With simultaneous equations, after finding the solution for x and y, he did not put them in an
ordered number pair or emphasise that the answer represents a point;
•
he should have used parentheses next to the two equations in order to emphasise that it was a
system of simultaneous equations;
•
in the data handling lesson he was not consequent as he sometimes said from 3 up, above, no mode
other times he said: more than 3, no mode.
4.6.1.2
Knowledge and beliefs regarding ML learners
For the second lesson on simultaneous equations, Monty predicted the learners would understand how
to approach the problem (L1). For the data handling lesson on the four basic measures of dispersion, he
predicted that the learners would understand how to collect data, organize and summarise them and to present
them at the end of the lesson (L1). He later mentioned that the learners just need to know the mode, mean, median
and range, but we don’t go in details but I have to give them a definition and how to gather information for future
purposes, because they will need it even after they completed the school (L1). Monty’s reason why the learners would
have understood those aspects was that all of the mentioned aspects were known to the learners, they
were familiar with them and also because it forms part of living, it is part of their lives (L1).
What he predicted they would not understand is the variable x because once they see x they get anxious, because
what comes to their mind is once they see x they think it is Maths (L2). He thought the learners would come to
understanding through many examples whereby I can say look at it, this is how it can be done and everything (L3).
The more you have examples, the more they can see how to do it, but also by giving the learners more sums because
the more they practice maths the more they understand it. Especially if they do it individually, that is when they learn
155
(L3). Monty predicted learners would approach the tasks by asking a friend or looking it up in their
notes (L4). He encouraged the learners to work individually since they were approaching examinations
(L4).
According to him, the learners reveal misconceptions when doing substitution. The following is such
an example:
x=4-2y must be substituted in 7x. They tend to forget 7x means 7 multiplied by x so they just say 7 multiplied by
4 then -2y, which is wrong. That 4-2y is one thing like 7x. They have to multiply 7 by 4-2y (L5).
In the data handling lesson the only possible misunderstanding according to him is that they forget to
arrange the data in ascending form (L5). I could not actually determine whether his prediction was correct in
this regard, since the learners did not participate in the lesson (L5).
In the following example the learners thought that when the coefficient of a variable (say x) is one, it
means that x is then the subject of the equation. So when Monty did the example
x + 2y = 4
, he asked:
7x − 5y = 9
T: Is x or y a subject of an equation? (No response). Is x or y a subject to the equation?
L: x.
T: Huh?
L: x.
T: Yes?
L’s: x.
T: OK. I am asking: Is x or y a subject in the formula? Huh?
L’S: x. (Teacher looks very troubled and learners laugh). OK, give me an example where x is a subject of the
equation (wait, no response). OK, people remember it must be? L = something, so L is the subject of the formula.
So, do you see x or y is a subject here? (Teacher is irritated).
L’s: No.
T: No. What do you do? You get x or y alone.
Monty was not perceptive as to what the learners were thinking (L6). Most of the learners worked
individually and Monty attended to them by looking at their work and talking to them (L6).
Unfortunately I could not assess whether Monty acted appropriately to facilitate learning as I could not
hear the discourse taking place. I did however, notice that as he looked at their work he did not ask
them to explain what they did; instead he was explaining again to them (L6).
4.6.1.3
Knowledge and beliefs regarding ML teaching
Monty regarded the following as prior knowledge for the lessons on simultaneous equations: the
coefficient, variable and index. That’s the best knowledge and the sign comes before the number (T1). During the
introduction phase of the lesson, Monty revised the terminology as planned as well as the two methods
they used to solve systems of equations (T1). He did not discuss like terms or the multiplicative inverse
156
during the introduction as prior knowledge, but mentioned that later as part of the solution (T1). For
the data handling lesson Monty said: everything is prior knowledge (T1).
In the examples and explanations on solving systems of simultaneous equations, Monty emphasised the
steps to follow which he believed would simplify the work and make it easier for the learners to
understand (T2). He did not use graphs as another form of representation to contribute to the learners’
conceptual understanding of the work and the meaning of the solutions (T2). At times his explanations
confused the learners because they took the form of lectures in which he made careless mistakes such
as saying: dividing with the multiplicative inverse instead of multiplying with the multiplicative inverse and
forgetting a sign in front of the value (T2). In the data handling lesson Monty demonstrated that it is
important to be able to know the characteristics of mathematical terms, but he did not ask the learners to
explain the different measures of dispersion and tell when and why these measures are used. This could
have improved learners’ understanding of the work (T2). During this lesson he verbally explained most
of the examples without illustrating solutions on the board so that learners could see what he was
talking about (T2).
There was no evidence that learners’ different abilities and backgrounds were taken into account in
presenting the content to the learners (T3). It is difficult to comment on his ability to sequence the
content in order to facilitate learning since basic examples were used through all the lessons except for
the example on simultaneous equations when he proceeded from a more difficult to an easier example
(T4). Monty’s choice of an instructional strategy to present his lessons was not in line with the purpose
of ML as ML learners are supposed to be actively involved in solving contextual problems (T5).
ML teaching: Reflecting on his practice 59 (T6)
When Monty reflected on his instructional practice he said he used direct instruction because our learners
are different from other school learners so we need to use the direct instruction, referring to discipline problems.
Commenting on the discipline in his class he said: I do have classroom rules in my class whereby I say we must
respect one another. He views his role in the ML classroom as being the facilitator where he helps the
learners to understand the work. To improve his learners’ appreciation of ML, he will keep on motivating
them about real life and what you may do with ML. To improve the learners’ participation in the lessons he
gives them questions from previous examination papers to prepare at home and present and explain the
next day. When asking him how he feels about teaching ML as this may influence the way he
approaches his ML lessons, he said:
59
Only one code was used to report on the teacher’s reflection regarding his own practice namely T6.
157
If I can get a chance I will go for Maths. I like when I am being challenged. I am not really challenged. It is
something that I am not enjoying. It is not working with hard working people who always ask questions, who want
to learn like the Maths group. Here they must do the subject, it is part of their packages, it is compulsory.
His goal is to get 100% pass rate and at least 5 distinctions. Last year I had 97% with 1 distinction and 3 B’s and
one learner could not make it and most of them got above 40%.
During the last interview I asked Monty to describe an ideal ML classroom in terms of, among other
things, the instructional strategies used. He believes that the teacher should use direct instruction
initially when new content is introduced, followed by group work because the group work it goes with problem
solving strategy. There should be discussion with writing something down. They must ask me questions. Regarding
the learning environment, he believes one learner per desk facing the chalkboard. If they need to do group work
they can combine… I don’t believe too much in rules because I believe the educator can make environment good or nice for
learning. He stated that his classroom is not like this ideal classroom because it’s hard to change things if you
are still a new teacher. You find them sitting like that, doing things like that, although you impose all the rules like that,
they still do it, for them to cooperate it will take you long. He mentioned that although he actually prefers group
work he encourages individual work now because they are approaching the examinations. I use the group
work just to show them how cooperative work is productive. He also believes that when the learners
communicate in peer groups they start to understand and they feel free to ask anything. Monty values the idea of
learners writing their solutions on the board and explaining it afterwards because then it is going to be
stored in your memory for always. He believes the difference in approach used between ML and
Mathematics is that fewer examples are done in ML, the pace is slower and the teacher does not need
to go the extra mile because the people you are working with in ML are not like the people you are working with in
Maths.
4.6.1.4
Knowledge and beliefs regarding ML curriculum
The DoE (2006) recommends a list of resources or instructional materials needed to teach ML (C1).
The resources Monty used during the three lessons I observed were a textbook and blackboard as (C1).
He used a textbook: Mathematical Literacy for the Classroom (Laridon et al., 2006) and previous
examination papers. He explained the strengths and weaknesses of the textbook as: information is clear
and understandable, lots of examples and exercises but some topics have little information and few examples (C2). He
was not aware of the curriculum content being studied in other school subjects that integrate with ML
(C3). Other departmental documents he knew of were the memorandums and circulars of which the
latter was useful and valuable to him (C4). As far as the NCS: ML is concerned, he knew that there are
four learning outcomes but could not name them (C5). According to Monty the DoE defines ML as a
subject aimed to enhance learners’ skills of counting though they are not doing maths and that the DoE’s purpose for
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the subject is to give every learner an opportunity to learn how to count because in real-life situation counting is a norm
(C5). Monty did not know which contexts the DoE suggests teachers should use in the teaching of ML
(C5). Regarding the new CAPS for ML he only knew that the teacher is a facilitator not like previously when
the learner was a centre of every learning (C5). At the end of the last interview I provided Monty with a list of
concepts and contents to be covered in Learning Outcome 4: Data handling (NCS, 2003a) and Monty
could only place seven out of 25 concepts in the correct grade in which they should be introduced (C6).
The lessons were presented as Mathematics lessons whene content was not situated in a context,
although he did mention a few examples of contexts where the mathematical content could be applied
(C7). During the first interview before the second lesson on simultaneous equations I asked Monty
about the context in which the lesson was set (C7). He was startled and took a few seconds to come up
with the idea of the elections. There was a 15-minute break between the interview and the class and I
assumed he used that time to think about this context. During the introduction he talked about
elections and the parties’ campaigns but it was not clear how the given information was applicable to
that day’s lesson (C7). In the data handling lesson he talked about how research is done but did not link
this to the learners’ experiences or the content of that lesson. At the end of the data handling lesson he
said:
In real-life situation, where can we use data handling? Census or SARS, for SARS to see how many people owe money,
they have to get data, they have to have people registered to SARS, those that are only in business, they can see how many
people are paying their taxes and so on and so on. Another one, remember guys we have elections of RCL elections ne?
We said the class has two representatives ne? But you were able to vote for more than two people. But at the end of the
day, they managed to get two representatives per class, ne? How? So that we can say this is our? RCL. So we had many
people on the valid paper but at the end of the day we have a certain number of people to represent RCL. You see how we
use data.
The context of elections was appropriate since municipal elections were to be held eight days later, but
he could not apply the context appropriately and meaningfully to the content. The same applied to the
SARS context he talked about. I doubt whether the learners were able to tell how data handling was
applied in SARS and elections (C7). Later in the same lesson he gave this example of where and how
statistics could be applied:
OK, now another story, when you do athletics, remember we use a stopwatch ne? A stopwatch helps us to record the
time. So for example for one particular learner let’s say we have athletics, we can record different times and we can
calculate that data and you present it using a pie chart or whatever and you can use that data to arrange all things
by recording that time during the events. OK now its fine (C7).
According to Monty mathematics is a tool used for solving problems (C8). I asked Monty how he views
mathematics as a discipline compared to ML as subject and his answer was:
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I view it as constructivism because you have to be constructive if whatever you are doing especially in these days so that
you can be successful. ML is viewed as a mathematics, but not lower grade and not challenging like Mathematics …
ML is like a life skill because you learn how to divide things, how to add things, things like you are always doing
when you are going to shopping, more of a life skill than a Mathematics subject (C8).
He described the value of mathematics and ML as:
It’s for logical thoughts because you learn to do things step by step and it gives you that strength as a person or
individual to reason and think outside the box. The value of ML is that in a few years’ time most of them will be
doing Mathematics because now they can notice from ML that they can do Mathematics. The learners learn about
counting, structures, angles, everything, so they can use that in their working place (C9).
Summary
Table 4.11: Summary of Monty’s knowledge and beliefs
KNOWLEDGE
AND
BELIEFS
DIMENSIONS
Mathematical
content
knowledge
DESCRIPTION OF TEACHERS’ KNOWLEDGE AND BELIEFS’
INDICATORS
Monty regarded mathematical knowledge as a prerequisite to teach ML. It
appeared as if his MCK regarding the specific content covered is sufficient.
(MCK)
ML learners
(L)
ML teaching
(T)
ML Curriculum
(C)
He believes learners gain understanding by looking at various examples on the
board and through much practice. Although he regards group work as
important where learners have the opportunity to talk to one another and learn
from each other, he did not apply group work in class. He sees individual work
as vital before the examinations.
He did not always enable learners to connect their prior knowledge to the new
content. He chose very basic examples in his data handling lesson and did not
take learners’ different abilities into account.
He knew about the value of ML but could not provide the required information
from the NCS (2003a). He views Mathematics as logical and constructive,
valuable to all people. ML is viewed as a kind of mathematics, but not a lower
grade of Mathematics.
4.6.2 Alice’s knowledge and beliefs
4.6.2.1
Mathematical content knowledge60 (MCK)
Alice believes that ML teachers need to have sufficient MCK and that no non-mathematics teacher can
teach this subject. Regarding her own MCK many mathematical errors were observed. In the first
lesson she used the formula incorrectly (See Picture 4.23). Later in the lesson she changed the formula
60
Since there is only one indicator or code in Table 4.3 regarding the teacher’s mathematical content knowledge, this whole
paragraph’s code is: MCK.
160
and put the denominator 2a under the root sign (Picture 4.24). Near the end of the solution she again
changed
1
30
to
1
30
(Picture 4.25).
Picture 4.23: Formula used incorrectly
Picture 4.24: Formula is changed
Picture 4.25: Another change in formula during further calculations
After erasing work a few times from the board, Alice wrote: 11 +
1 1
=
. A girl corrected the previous
30 30
step to:
11 + 1
. In many cases Alice omitted to put the values in brackets when she substituted:
30
− (−10)
± − 10 2 − 4 × 1 × 25
. The learners were confused as this formula differed from the formula
2 ×1
(x=
− b ± b 2 − 4ac
) they used the previous day when the student teacher was responsible for the
2a
lesson. Alice believed that the student teacher used a wrong method as his work did not correspond to
hers. Then there was the issue of having two solutions with the same sign which the teacher and
learners believed were not supposed to happen (Picture 4.26).
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Picture 4.26: An attempt to get a positive and negative answer
The following discourse took place:
T: Am I right? (She checks again her calculations on a calculator.) You are supposed to get a negative and a positive
answer. So, what happened here? I am sure there is something wrong because here we have two positive answers.
Guys please! (The same girl from earlier who wanted to show the teacher on the board during example one brings her
book to the teacher and talks to her but nobody could hear.) Yes but you have a negative and a negative so it should
change.
L: Oh OK!
T: Quiet please! We are right, it’s OK, as long as you know the method. So that’s an exception. This is an
exception. Let’s do another one.
In the second lesson where they drew the graph of the parabola, she did not attend to the given
restriction in the example y = ( x − 2) 2 − 1 for − 1 ≤ x ≤ 4 . She also mentioned that they must have two
intercepts with the X-axis, which is not necessarily true. Only at the end of the second lesson did she
write the x and y value of a point as an ordered number pair for the first time, saying: Here you have
(0,3) , this is your x and this is your y. When using the method of intercepts, they only had three points to
plot the parabola with because they did not calculate the turning point and the following discourse took
place:
T: You now join these three points.
L: How do I know where to go?
T: You can go anywhere, now you don’t have this point (pointing to turning point). If you do a proper job, you will
find your graph will go exactly through that point (turning point) … If you have to find the minimum, use the graph
or the maximum value of y. How do you find the minimum?
L: You just see it.
T: Where do you get it?
L: It’s at -1.
T: This is the minimum value (showing at turning point).
L: There are two graphs now.
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T: This is the same graph, we used two methods. (Everyone laughed as the graphs did not look the same). It’s the
same graph. We used the table method and then the formula.
L: OK.
T: Are you sure you do understand?
L: Ja.
T: OK.
During the third lesson she calculated the median without arranging the data (Picture 4.27), then told
the learners to arrange the data in ascending order saying: So it might not be that answer.
Picture 4.27: Alice calculating the median incorrectly
The pie chart (Picture 4.28) was not drawn accurately – even though it was a rough sketch, 60° should
have looked like an acute angle.
Picture 4.28: Unrealistic drawing of the pie chart
4.6.2.2
Knowledge and beliefs regarding ML learners
In the lesson on how to draw the parabola using two methods, Alice predicted that the learners would
understand that they can substitute that equation in that formula (L1). When I asked her how the learners
would understand the work, she told me about the preparation of the lesson and after a prompt she
complained about the large classes that needed to be divided in two groups and the 40-minute periods
that are not sufficient to do what she planned and therefore never answered the question (L1). She
predicted that the learners would not understand plotting the graph as they have difficulty doing the
following:
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Maybe your x = 0 and then your y, make it -3, so when plotting the graph, where you have the -3, they go plotting it
at the point where x is 0 and y is 0. It’s always a confusing thing, plotting the graphs (L2).
On predicting why it is difficult to the learners she replied:
It shouldn’t be difficult but I don’t know …. I cannot say this is why. Sometimes it can be confusing because now
when you have 0 as the x and then you’re trained to plot the y, you’re thinking, you know you have to let both points
meet, then you’re thinking maybe you should, and this is the -3 and your x is 0 here. They are thinking maybe they
put it here or here (L2).
She predicted that the learners would approach the tasks by coming to her during break so that she
could assist them, but offered no other strategy to assist them: they want individual help but aside from that I
don’t know (L4). She could not predict any other misconceptions learners might have (L5). In reality
neither the learners nor the teacher understood why there were two positive answers when the
quadratic equation was solved and as she predicted the learners made mistakes when point (0,−3) was
plotted (L5). When the table was completed in order to draw the graph of y = ( x − 2)2 − 1 , some learners
did not understand how she obtained her answers when she worked directly from x to ( x − 2) 2 because
she omitted the steps where ( x − 2) should have been calculated (L5).
Regarding the data handling lesson, she mentioned that the learners would understand everything as
they had done all the concepts in Grade 10 and according to Alice data handling is one of the simplest aspects
of ML so I think they should understand it all (L1). She stated: it’s going to go smoothly with the chart, the tables and
even the graphs (L1). Regarding the new work, the cumulative frequency and drawing of the ogive, she
said they should understand it and they know it but it is just a bit advanced … it is just like a continuation of work
they did (L1). When asked how the learners will understand it she said:
I will tell them about the data and the raw information which you have to try to present, make presentable maybe
using the graphs which is the pie graph or line graph or the charts which is the pie charts or maybe using the tables
where they have to use the tallies or frequency, but there are different ways in which you can present your data and
then we are going to try and refresh their memories on the mean, the mode, the median and then we are going to talk
about the cumulative frequency, how to plot an ogive, we are going to work out the standard deviation (L3).
She predicted that the learners would not understand the actual mathematics involved:
They used to have a problem with the pie chart, I don’t know why. You know the pie chart? I don’t know, they find
it difficult to allocate those degrees even though from percentage to the grades of the pie chart. It is always a big
problem (L2).
When I asked her why the learners did not understand the pie chart, she said: I don’t know (L2). Alice
stated that the learners would not have a problem doing the tasks and then talked about group tasks set
by the department (L4). She could not predict any other misconceptions the learners might have (L5).
Although Alice said they would understand everything except the pie chart, they in fact did not
164
understand how she calculated the measures of dispersion using the table method and one learner who
was very involved in the lessons mentioned how difficult the pie chart was, but Alice disagreed and
continued with the lesson (L5).
There was little evidence of what the learners understood, how they understood it, what they did not
understand and how they approached tasks because they were not required to do any tasks in class
(L1-L4). She told the learners that they would understand the content if they practised enough and if
they watched her. She said: Don’t ask me how to get this, you have to look at me (L3). The chaos of the first
lesson showed that Alice did not have the ability to understand the learners in terms of their thinking
(L6). When Alice was asked to comment on the first lesson she said:
Because when one person says this, and other person says this, then they get confused and they know it’s just
everything you tell them, they go the same way, they don’t want to think out of the box. If you say it’s -(-) they want
to know this can just change to plus, so you must do it the sáme way every time. If I am the first person that taught
them, then they want to go the same way as me. And I got confused too (L6).
4.6.2.3
Knowledge and beliefs regarding ML teaching
In the second lesson on drawing parabolas using two methods, she regarded simplification, factorization
and the simultaneous equations as prior knowledge (T1). She did not revise any prior knowledge during the
introduction stage of the lesson, but immediately started with the table method when she reminded
them to use both negative and positive values. Thereafter she used the formula which they had learned
during the previous periods (T1). In using the formula she revised the values of a, b and c in the
formula (T1). Regarding the data handling lesson, she said everything she planned was prior knowledge.
In her introduction she revised the different kinds of graphs and charts which the learners already knew
(T1).
Alice used various useful forms of representing ideas such as graphs, tables, formulae, calculators and
symbols, but these were often overrepresented (T2). Although she used various representations, instead
of creating opportunities for learners to develop conceptual understanding, the learners were confused
(T2). She could not always transform the content knowledge into forms that are pedagogically powerful
to attend to learners’ diverse needs (T3). She has the ability to sequence the different content of her
lessons, but it is when dealing with the particular concepts that too much information is shared using
incomplete and varied representations. This happened especially in the data handling lesson (T4). When
the learners did not understand or when they gave incorrect answers, she re-explained by using the
same words as before, apparently not being capable of reformulation of her explanations (T4). After
she did the example in which the pie chart was given, she told them that a question could be asked
165
where they needed to draw the pie chart using given information. Unfortunately she did not follow this
through by asking the learners to undertake such a task (T4).
Regarding the class and homework she said: when we give the class work, we should make it look more practical,
not just the theory and the x thing (T5). About grading of the problems she commented: They are based on this
topic. It’s practically the same thing, the same level (T5). For the data handling lesson, she planned to use
exercises from the textbook and at the end of the week a task from the DoE (T5). Alice generally
selected too much content for one period so that there was no time for her to assess her learners’
knowledge and understanding (T5).
ML teaching: Reflecting on her practice61 (T6)
Alice claimed that she really loves to teach ML and found it quite interesting. She tries to motivate the
learners and invites them to come to her class during break time if they do not understand the work,
but still they are not motivated and would rather play pool during break, which distresses her. To
improve her learners’ appreciation of ML she plans to get down to their level and make them understand what
they really need to know, to make it quite simple. To improve the learners’ participation in the lessons she tries
to pose questions to all, but this is not uniformly successful: The boys at the back are always (pause), they just
want to be in class, they don’t participate. The ones in front want to take part. She believes learners master new
knowledge as follows: To learn new things it is all about making up your mind, I want to know this. The learner
must go back to what he is taught, or come to the teacher, try to do your own research with books and take it up from
there.
Alice chose direct instruction as strategy because the first lesson we have you must teach them the basics. I try to
introduce them, make sure they understand the basics of what they are doing, or what the topic is about, it is more like
formal teaching. Later on the discussion comes. Her view on calling learners to the board is: it’s all about
practising in the presence of everyone and it gives them confidence and then their friends can learn from them and then try
to assist them … and it forces them to come to class prepared. She does not favour discourse between the
learners:
Actually I don’t like to encourage that [discourse between learners] because most times when I am talking I found
discussions among themselves and then sometimes they miss out when they do that. Some learners speaking in their
dialect and vernacular, it’s a bit difficult when you are teaching and when a learner asks you, mam, what is that
word? Please can you repeat that and then I am too busy teaching. So, I try to make them listen not to miss out.
She does not believe that the teaching of ML differs from that of Mathematics. Regarding her role in
the classroom she said: I try to be in charge but give them the chance to talk to me and go and read because ML asks
for practising, so I would say I am the mediator.
61
Only one code was used to report on the teacher’s reflection regarding her own practice namely T6.
166
During the last interview I asked Alice to describe an ideal ML classroom in terms of, among other
things, the instructional strategies used. She described her ideal ML classroom as one where various
instructional strategies are used such as group work and discussion in class and then active learning and learners
coming to the front. In her description of the ideal discourse in her ML class, she mentioned she wants
learners to speak up when they do not understand and to ask intelligent questions. I really want them to ask: why did
you do this, how did you do this? Is this how you are supposed to do it? As far as the learning environment is
concerned, she wants all learners to have their own textbooks so that she does not need to waste time
by writing the whole question on the board. Her goals in teaching ML are that all learners do well in
their examinations and that they understand that ML is quite simple if you put your mind to it.
4.6.2.4
Knowledge and beliefs regarding ML curriculum
The DoE (2006) recommends a list of resources or instructional materials needed to teach ML (C1).
The resources Alice used in the three lessons I observed were the Oxford Successful ML (Pretorius,
Potgieter & Ladewig, 2006) textbook, the whiteboard and calculators. Although the textbooks are
available at her school for learners to buy, she said that the learners want the school to give it to them and that
the learners asked her to give them the money to buy the textbooks, so most learners do not have their
own textbooks (C1). According to Alice a point of strength of the textbooks is the large number of
practice questions but a weakness is that the textbooks do not provide a step-by-step method of answering
questions (C2). She had no knowledge of the curricula of learners’ other subjects and how those curricula
integrate with ML (C3). The only departmental document she knew of is the work schedule which she
finds useful as it guides her teaching (C4).
The DoE’s definition of ML according to Alice is: The department of education makes it look like ML is a
lower substitute for Mathematics and their stated purpose of ML is to help learners have a basic knowledge in
mathematical related issues (C5). She knows ML has four learning outcomes but could not state them and
she does not know anything about CAPS (C5). At the end of the last interview I provided Alice with a
list of concepts and contents to be covered in Learning Outcome 4: Data handling (NCS, 2003a) and
Alice placed 16 out of 25 concepts in the correct year that they are to be introduced (C6).
During the last interview I asked Alice to which contexts the content should be applied according to
the DoE and Alice’s answer was: Teacher-learners participating (C7). In the interview before the second
lesson on drawing the parabola, I asked about the context she was about to use and she talked about all
the mathematical content to be covered. After a prompt she replied that no context is going to be used
(C7). Although she mentioned in her interview before the third lesson that elections would be used in
167
the lesson, she did not once refer to elections during the lesson. Pure statistics were done except for the
one example that was based on the ‘musical group’ which the learners found interesting (C7).
To Alice mathematics is a logical subject and it has to do with constructivism (C8). She does not see the
difference between Mathematics and ML and said:
ML is just a little bit easier … it’s still part of the Maths, it’s just of a lower grade. One of them said I don’t see
the difference, I said it’s just a make believe and then they keep passing the same belief as they come. Everyone comes
with the mentality it’s difficult, and when they come here they say it is still Maths, it is difficult (C8).
According to Alice mathematics is quite important because you find it in everything; it is vital (C9). The value of
mathematics and ML are the same as both cover Financial Mathematics and since all people deal with
calculations every day all people need a basic knowledge of mathematics (C9).
Summary
Table 4.12: Summary of Alice’s knowledge and beliefs
KNOWLEDGE
DESCRIPTION OF TEACHERS’ KNOWLEDGE AND BELIEFS
AND
BELIEFS
INDICATORS
DIMENSIONS
Mathematical
Alice regards MCK as an important prerequisite in teaching ML. She however
content
made
numerous mathematical errors in class and her MCK appeared to be
knowledge
insufficient
regarding the specific content covered in the three lessons.
(MCK)
ML learners
(L)
ML Teaching
(T)
ML Curriculum
(C)
She did not have sufficient knowledge of learners as she predicted they would
understand all content she dealt with them but in reality they did not understand
all the work. Apart from her own misunderstandings and misconceptions, she
could not understand the learners’ misunderstandings.
She struggled to have a logical flow in presenting the different concepts in her
lessons and to sequence her activities. Too much content was covered in some
of the lessons, which caused learners to become confused and frustrated.
She does have sufficient curriculum knowledge but needs to know more about
the DoE’s vision for the subject. Alice views mathematics as a logical subject
and believes that it has to do with constructivism. She does not see a difference
between Mathematics and ML, ML is just a bit easier. The value of mathematics
and ML is that both cover Financial Mathematics.
168
4.6.3 Denise’s knowledge and beliefs
4.6.3.1
Mathematical content knowledge62 (MCK)
Denise believed that MCK is a prerequisite to teach ML. She said:
If you don’t know the maths, you cannot understand the practicality of this ML. You must have content knowledge,
because then at least you can build on that. Especially to take the maths and put it in context form and vice versa. If
you don’t have maths, I don’t know where you will start.
From the lessons I observed, Denise’s MCK is good and no mistakes were made. To prepare the
learners for examination, she emphasised the importance of showing all steps and adding the units at all
times in their calculations and final answers.
4.6.3.2
Knowledge and beliefs regarding ML learners
For the first lesson63 on conversions from metric to imperial units based on capacity and mass, Denise
predicted that the learners would understand the conversion of metrics to imperials and then straight metrics
conversions because she did a lot of drilling in class and they had to practise it at home too (L1). She also
mentioned if they don’t keep on doing that [practising conversions], in revision you will see those errors (L1). For the
third lesson on conversions within the metric system based on capacity, mass, length, area and volume,
she predicted that they will easily understand the distance (L1). Denise’s predictions regarding what learners
would understand and would not understand were in line with what happened in class (L1). Denise did
not predict that the learners would misunderstand any content in the second lesson but concerning the
third lesson she predicted that the learners would not understand conversions regarding area and
volume because:
The distance is easy. Between the conversions from mm to km, the 10, the exponential is to the power of one. Coming
to area, now it turns to the power of two, so before they divide or even multiply they start now forgetting, because the
10 is to the power of two. Then the same applies to the volume, the 10 is to the power of three. And then they
usually multiply or either divide it without considering that the 10 is to the power of three before they can divide or
multiply. There they got difficulties, but with length it is straight forward for them (L2).
Denise believes that learners come to understand once they can see their mistakes on the chalkboard …
respond to questions … correspond and check what their misunderstanding was (L3). She also believes that learners
develop an understanding through individual practice but also during
formal assessment when I am done with new work, the feedback to them is how they will learn and rectify where they
don’t understand”. She believed the learners will approach the tasks all by themselves, not even referring to the
textbook as they know how to do these conversions (L3).
62
Since there is only one indicator or code in Table 4.3 regarding the teacher’s mathematical content knowledge, this whole
paragraph’s code is: MCK.
63 The first lesson was repeated the next day to another class, so only two different lessons were observed.
169
Denise believes that another possible misunderstanding the learners could have is when 250ml must be
converted to pints and they search for millilitre to pint in the conversion table not realising millilitre must first
be converted to litre and then the conversion from litre to pints is provided in the table (L5).
Sometimes, she said, they also confused distance with area (L5). Another example Denise gave of
learners’ misunderstanding was:
I give the measurements of the cube and I say they must calculate the volume in cubic mm, then the second question
now I say the answer they got for the cube, they must convert it to cubic metre, but I said to them one cubic metre
equals to 1 million cubic mm. Then you know, they can’t actually do this, it’s in context now, I don’t know why …
they want it straightforward (L5).
As Denise discussed the learners’ work on the board, it appeared that she understood the learners’
alternative conceptions. There was only one incident in which learners had to do a conversion problem
based on area and one learner wanted to square the answer too64. Denise misunderstood the learner’s
question and after an elaborate answer, the learner told Denise that was not what she asked and the
learner repeated her question. Denise then understood the question but still did not address the
learner’s problem, just replying no to the question (L5). Denise provided opportunities for the learners
to express themselves in writing on the board so that she was able to see what they did, but not many
opportunities were created for her to listen to their thinking (L6). Based on the work she saw on the
board, she acted appropriately to facilitate the learning process by discussing the learners’ work with the
individual as well as the rest of the class. She even involved learners to correct other learners’ work
(L6).
4.6.3.3
Knowledge and beliefs regarding ML teaching
According to Denise the prior knowledge needed to be present to understand the work for the second
lesson was metrics [as] they did that in Grade 10, also to solve other problems from metrics to imperials and ratios
(T1). Denise did not revise the prior knowledge at the beginning of the lesson but integrated it in and
across her lessons (T1). Since the lesson was on conversion from the metric to imperial system, some
problems required an initial conversion within the metric system (prior knowledge) before the actual
conversion could be made (T1). During the second interview I questioned Denise about the prior
knowledge needed for the third lesson to which she replied: Not applicable. It is a revision lesson. (T1).
Denise used different forms of representation to make the content comprehensible to the learners such
as written demonstrations and oral explanations where the use of either equations or ratios to solve the
conversion problems were demonstrated (T2). She also used a diagram to revise prior knowledge based
on conversions of length within the metric system (T2). These examples and tasks motivated the
64
This example is discussed under Discourse: Teacher-learner interactions.
170
learners to understanding their own solutions and thinking (T2). The way Denise sequenced her tasks
and explanations was proof of her ability to transform her own knowledge into forms that are
pedagogically powerful (T3). She sequenced the tasks from the first to the third lesson: the tasks
became more demanding and also included a wider variety of concepts, but only within the metric
system (T4). Within the lessons there was no sequencing of the tasks from easy to difficult (T4).
Denise chose an appropriate instructional strategy for her revision lessons (T5). She used discussions in
which she built her instruction on the learners’ knowledge, their common errors and misunderstandings
(T5). This strategy provided opportunities for informal assessment of the learners’ knowledge (T5). The
tasks were chosen to include conversions within the metric system but also conversions between the
metric and imperial systems (T5). These tasks were based on length, area, volume, mass and capacity
(T5). Denise chose the tasks from the learners’ textbook but for assignments and assessments she
chose tasks from other resources too so that they can get exposed to other authors questions and approaches (T5).
ML teaching: Reflecting on her practice65 (T6)
Denise’s experience of teaching ML was:
It [ML] is not challenging. I feel a little bit bored, because I have done the pure maths. But for the sake of them [the
learners] so that they must understand why we learn maths, I start to be a little bit of motivated. In pure maths you
enjoy it throughout.
It is important for her to motivate the learners by telling them the value of ML:
It is going to help you throughout your life where you are able to work out your own things, if you have your own
business, your work one day, you are able to work with percentage, unlike not having maths at all. Basically it’s
personal as well as the reading of the stats, doing the inflation, price increase, petrol increase, you are able to calculate
that.
She ensured learner participation by giving the learners’ tasks to complete individually and afterwards
allowed some of them to write their solutions on the board so that corrections could be done. She
believes that learners learn from the feedback she gave on their work done on the board.
Her reason for choosing the strategy of learners working on the board was: Because I just want to see what
they misunderstood, the content and the context. It then becomes easier for me to rectify any misunderstandings. When
introducing a new topic she believes she needs to explain the content and concepts that are a little bit new, new
words they are not familiar with and explain it thoroughly how its application work. Then from there I start doing the
65
Only one code was used to report on the teacher’s reflection regarding her own practice namely T6.
171
drilling. Although I did not observe learner-learner interaction, she supported the idea thereof as she
said:
They feel very comfortable when they talk to one another unlike with me (pause), some they find it very comfortable
between peer and peer because they can understand the same group … So when they talk to one another it’s not a
problem … sometimes you find they talk together and once they argue, they come to me. That’s what I like about
them.
Experience has taught her that teaching ML is different from teaching Mathematics as most of the ML is
on contexts whereas the basics are not well aligned like Mathematics is for me. I compare it with natural science and
technology, technology is the root of science. Literacy is the root of pure maths. After two prompts to direct her
answer towards the teaching of the two subjects, she still continued to talk about the difference
between the two subjects. She sees her role in her ML classroom as being the one who is there to teach
them … and then I facilitate whether they know the content or topics of the contexts are well understood.
During the last interview I asked Denise to describe an ideal ML classroom in terms of the instructional
strategies used. She described an ideal ML classroom as one in which the learners are involved in the
lesson, where they learn through doing it, trying on their own. She emphasised that it is meaningless if they
just look at her doing the work on the board; they should practise it themselves so that they can
discover and construct their own meaning. A vital point was to have opportunities where she could
communicate with the learners in order to determine learners’ misconceptions and errors and have
sufficient time to rectify it through conversations. Regarding the learning environment she believes:
It goes back to motivation, why they are learning this. We must come to a point where we can show them the realistic
part and the value of it in everyday life. The lessons are contextual. The learners say where am I going to use ML in
tourism? I don’t see it. So then I must show them that wherever you are going to work, you are going to use it.
When I asked her how this ideal classroom compared with her own classroom, she said:
Even though the time restricts us, I try. They must also be involved in the learning process. It must not be always me
telling them this is how it is done, take it or leave it. They must also contribute, they must come up with example,
they sometimes tell you of things from their world. So give them that freedom to participate.
Her personal goal in teaching ML is that these learners can be able to use this ML in everyday life; it will be perfect
to me.
4.6.3.4
Knowledge and beliefs regarding ML curriculum
The DoE (2006) recommends a list of resources or instructional materials needed to teach ML (C1).
The instructional materials Denise used to teach her lessons on conversions were the Oxford
Successful ML (Pretorius et al., 2006) and Classroom ML (Laridon et al., 2006) textbooks (C1). She did
not mention any advantages of the textbooks but experienced the textbooks as not so good and effective and
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the questions are sometimes confusing for the learners and language is difficult for them (C2). She was not aware of
the curricula of other school subjects which integrate with mathematics (C3). Departmental documents
she knew of were the Assessment policy guideline; and Learning outcomes with assessment standards of which she
experienced the Assessment policy guideline as useful and valuable (C4).
When I asked Denise in the last interview how the DoE defines ML, she stated: There is no definition for
this subject only the implementation is important (C5). According to her, the DoE’s purpose with ML
according to Denise is for the learners to know how to use maths in their everyday life (C5). She did not know
anything about the new CAPS document as they had had no training yet (C5). She knew ML has four
learning outcomes, but could not mention them (C5). At the end of the last interview I provided
Denise with a list of concepts and contents to be covered in Learning Outcome 3: space, shape and
measurement (NCS, 2003a) and Denise placed seven out of 19 concepts in the correct year that they
are to be introduced (C6). In the interview before the second lesson, I asked about the context she was
about to use and she replied: Like the cuboid, to find the volume of a cuboid in mm and then convert this into cubic
metres (C7). For the third lesson she said the context to be used was a table with matching tables, matching
column A and column B, so that they can see one kilogram is how many pounds and so on (C7). She did not mention
any contexts in one of her lessons and also did not mention a cuboid in the second lesson.
Denise views mathematics as follows:
It’s not formal. If this is the formula, I can also change it as long as I know I can prove it, and make arguments
why I change it, as long as I can justify it … For me it’s flexible. Then I can construct my own beliefs and my own
understandings on what I am working on (C8).
She summarised her view of mathematics by saying mathematics is flexible and creative (C8). Her perception
of ML is that it is
not a higher grade or standard grade maths … more like a life skill … it’s a maths on its own [and learners] must
know at least the origin which is in pure maths, the origin of the curved graph, the origin of the parabola, the origin
of the straight line graph, not just the application.
The value of mathematics is endless to her, but among other things, she mentioned the following:
Whatever I do, it’s maths. Personal, work, everywhere it’s found. I used to say to my learners, if you walk, you count
the steps you make, its maths, 1,2,3, it’s natural numbers. I say go and buy zero bread, I ask what will you bring?
Nothing, and zero is a whole number. You see now maths is everything (C9).
To her the value of ML is that learners who are not capable of doing Mathematics can do ML. She said:
These learners don’t know how to handle their personal lives and they are now able to work or manipulate with what
is outside. For example if they want to read a pie chart, statistics in general, so now unlike before the learners who
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are not capable to do pure maths or do not have maths, they can do this. So for me this is better than nothing because
they can use it somewhere (C9).
Summary
Table 4.13: Summary of Denise’s knowledge and beliefs
KNOWLEDGE
AND
BELIEFS
DIMENSIONS
Mathematical
content
knowledge
(MCK)
ML learners
(L)
ML teaching
(T)
ML Curriculum
(C)
DESCRIPTION OF TEACHERS’ KNOWLEDGE AND BELIEFS
INDICATORS
Denise regarded MCK as a prerequisite to teach ML. She made no errors in her
examples or corrections of the learners’ work, but also did not elicit any
discussions regarding the conceptual meaning of the different units of
measurement or their application value in everyday life situations.
She correctly predicted what learners would and would not understand. She did
not allow learners to explain their thinking so that she could listen to their
thinking. She acted appropriately to the work she saw on the board.
She correctly identified the prior knowledge for the second lesson. Denise used
various appropriate representations to make the content comprehensible to the
learners, applied appropriate instructional strategies and sequenced her tasks
over the different lessons to enable learners to progress in their cumulative
understanding. According to Denise the ideal ML classroom compared well
with her own except for allowing the learners more time to discover the content
as time is always a restriction.
Regarding the NCS, Denise could not provide the DoE’s definition, purpose
and learning outcomes or even place half of the topics in the correct year that
they are to be introduced. Denise views mathematics as flexible and creative,
constructing one’s own understanding and ML as a type of mathematics, but
unique. According to her, both mathematics and ML are valuable as people use
both in their personal and work environments.
4.6.4 Elaine’s knowledge and beliefs
4.6.4.1
Mathematical content knowledge66 (MCK)
When Elaine was asked about the extent to which MCK is a prerequisite to teach ML, she said:
A motivated teacher who is prepared to work hard will manage the teaching of ML but there are some details of
mathematical knowledge a ML teacher needs to know. You will not be able to just explain rates of change and
ratios, you will need to learn about the finer things, how to explain it, what is important. It is only now, after three
years of teaching ML that I feel confident in front of my classes, that I know what is important and what should be
done in each grade.
From the lessons I observed, Elaine’s MCK is very good. She did not make any mathematical errors
and I did not observe any misconceptions.
66
Since there is only one indicator or code in Table 4.3 regarding the teacher’s mathematical content knowledge, this whole
paragraph’s code is: MCK.
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4.6.4.2
Knowledge and beliefs regarding ML learners
Regarding the second revision lesson on time and interest, Elaine predicted during the interview just
before the lesson that the learners would recognise the formula and be able to tell that the formula is
used for compound interest (L1). Regarding the third lesson on perimeter, area and volume, she
predicted that the learners would know the units and how to calculate the perimeter, area and volume
as they did that in Grade 10 (L1). She realised some learners would still struggle in the second lesson to
convert the variable in the formula to a decimal as well as with the concept of interest being calculated monthly and semiannually (L2). The task she used in the third lesson was more complex than the previous year and [i]t is
possible that not all learners will be able to verbally explain the concepts. Not all learners have conceptual knowledge of
the concepts or sometimes they have the concept in their mind but do not have the ability to verbalise that concept (L2).
She believed that the learners would understand the tasks in the second lesson once they can explore and
have that aha feeling, but she needed to make the lesson applicable and link the new work with their personal lives
such as personal tax, start at home, their parents’ salaries … (L3). Concerning the third lesson, the learners
would understand the work once they learnt to carefully read through the problem, study the drawing and indicate
all the information with coloured pens on the drawing as the visual representation simplifies it (L3). Her predictions
about what learners would and would not understand were realised in her lessons.
During the interview just before the lesson she predicted that the learners would approach new tasks by
discussing the work with their peers (L4). She frequently noticed that they do not just ask anyone, they will
not ask a friend who knows less than themselves but will ask someone they know is able to explain the work to them
(L4). In her experience they really listen to and learn from one another. She said people joke about the buzz in
her class but they [the learners] buzz about the work (L4). Many times she preferred to put a weak and strong
performer together and then the one learns to explain and the other one learns to understand, thus using cooperative
learning as instructional strategy (L4). At her school all teachers need to remain after school till 14h30
and many learners make appointments with her for individual support (L4). They also use their
textbooks and scripts as sources of reference (L4).
Regarding other possible misconceptions learners might have, she mentioned a problem with cubic
centimetres: they do not always have the concept of length x breadth x height, so it is three units multiplied with each
other, the same with area (L5). Elaine required learners to give explanations and justifications orally and in
writing (L6). A fundamental part of Elaine’s lessons was learners’ abilities to explain the meaning of
concepts as these explanations provided her with proof of learners’ conceptual understanding of these
concepts (L6). This enabled Denise to see what individual learners do, listen to what they think and to
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act appropriately (L6). She recognised and clarified the learners’ common errors and misunderstandings
(L6).67
4.6.4.3
Knowledge and beliefs regarding ML teaching
As far as the prior knowledge of the observed lessons was concerned, Elaine said all mathematical
content had been introduced either in Grade 10 or Grade 11 as the lessons were revision lessons (T1).
In these three lessons the learners needed to practise their skills and use their existing knowledge by
solving unfamiliar and more complex contextual problems (T1). In class there were numerous
situations when Elaine revised prior knowledge before presenting a particular concept (T1). Elaine
applied representations such as tables, symbols, formulae, calculators, a demonstration calculator and
sketches of manipulatives to make the tasks comprehensible to the learners (T2). These representations
proved to be useful as learners participated and comprehended the work (T3). The tasks were
sequenced and presented in a pedagogically powerful way to facilitate learning (T4).
In preparing her lessons Elaine realised some learners could become bored as they repeated similar
work a few times. She also needed to attend to those learners who only knew half of the work or even
less (T5). She then planned to discuss the work she knew they struggled with, such as explaining the
significance of the different variables in the formula. She involved them in discussions during which
their errors and misunderstandings were corrected. Her planning was based on preparing the learners
for the examination, so she provided guidance on the type of questions they could expect and how
those questions should be approached (T5). Elaine’s choice of instructional material was appropriate as
she chose tasks from previous examination papers to enable learners to solve typical examination
questions in different contexts (T5). She encouraged the learners to use coloured pencils to indicate
given information on their sketches (T5).
ML teaching: Reflecting on her practice68 (T6)
When Elaine was asked two years ago to be the coordinator for ML, she initially felt that she was being
demoted, but she claims that ML had grown on her since then. She enjoys being involved in ML and
never wants to go back to Mathematics. To improve her learners’ appreciation of ML she bases her
lessons on real-life situations as the learners need to recognise what the subject is about and where
mathematics could be used. She refers her learners to the yearbooks of tertiary institutions to familiarise
them with the requirements of possible future studies and how ML can add value to their studies. She
67
68
More detail is given under Elaine’s Discourse: Teacher-learner interactions.
Only one code was used to report on the teacher’s reflection regarding her own practice namely T6.
176
also mentioned that she wanted to improve her learners’ participation in the lessons. She would like to
use two of the seven periods a week to do something out of the ordinary like taking them on an
excursion or watching a DVD where they can discover the role of mathematics in specific situations or
events, which could then be followed by a class discussion. She said she could give them a worksheet
before the excursion or DVD which the learners could complete during and/or after such an event.
She also wanted to use games and newspapers and magazines to further enrich the learners’
appreciation of mathematics. She believes such activities contribute towards proficient learning.
I asked Elaine to reflect on the three lessons I had observed. According to her she chose discussions as
one of her instructional strategies since it gave her a platform to work from. She determined the gaps in
their knowledge, which became the focus of the lesson. She believes discourse between the learners
indicates that the learners are involved and interested in the lesson. To elicit such discussions she enjoys
throwing in a question with an impossible answer such as the question in the class test (Picture 4.29)
during the third lesson where she asked: The following figure is a rectangle. Calculate the volume of the figure.
Picture 4.29: A question asked in a class test
According to Elaine her classroom rules are less rigid and structured than other classes in the school.
She wants her learners to have fun in a relaxed atmosphere. She said she did not want the subject to
have a negative stigma and therefore made an effort to make the subject alive and interesting. She asked
the principal and colleagues to respect her subject as she respects their subjects and not to make fun of
ML by referring to it as a very low level of Mathematics. Elaine believes ML should not even be
compared with Mathematics, but should be regarded as a subject on its own. Evidence that her learners
enjoy and value ML lies in the fact that learners who had to change from Mathematics to ML asked to
be in her class. One of Elaine’s most important rules is that learners should respect one another and
value other people’s thinking and ideas.
Elaine believed that the teaching approach of ML is totally different to Mathematics because ML is
presented in a more relaxed way where the learners are not pressed to finish in time. According to her
in ML the high ability learners can continue with additional work and there is enough time to further
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attend to the slower learners. There is always time to go back to their Grade 9 content and do revision,
which is not the case when teaching Mathematics. She wished that a standard grade of Mathematics
could be implemented again to cater for those learners who do not actually belong in the ML class but
at the same time do not wish to take Mathematics. She regarded her role in class as being the mediator
between the content and learners. Often learners had the correct ideas but did not know how to
formulate them. She then needed to guide the learners in the process of discovery and provide
scaffolding to assist them in moving from their uncertain and disorganised thinking processes to
conceptual understanding. She believes that the learners sometimes need to struggle through the
process of problem solving and once they experience success, it serves as a lesson in life that there are
times one needs to struggle through solving a problem, but it is possible to arrive at a solution.
During the last interview I asked Elaine to describe an ideal ML classroom in terms of the instructional
strategies used. She described an ideal ML classroom as one in which a teacher uses instructional
strategies that are effective to her as individual. She also stated that apart from the discourse between
the teacher and learners there should be enough opportunities for communication between the
learners. An ideal is having computers with internet access in such a class to enable learners for
example to find the present exchange rate and as such make the tasks more realistic. She believes
information you search for on your own and read it by yourself is more valuable and will be remembered longer. Elaine
planned to have newspapers available in class to enable the learners to work with news of the day. In
comparing this ideal ML classroom with her own practice she believes it is the same except for not
having computers in her classroom. Her purpose in teaching ML is to equip the learners with life skills
and she hoped that one day when they think back they would remember her and what she taught them,
but more importantly, the life skills and values she taught them.
4.6.4.4
Knowledge and beliefs regarding ML curriculum
Regarding her curriculum knowledge she knew about the appropriate instructional material that could
be used for the lesson she did on perimeter, area and volume as she used textbooks, previous
examination papers, models of two-dimensional figures and three-dimensional objects as well as
transparencies (C1). She used the Mathematical Literacy for the Classroom (Laridon et al., 2006)
textbook, a book containing previous examination papers and the internet (C1). She valued the fact that
each learner has his/her own copy and is able to use the textbook at home (C1). She mentioned that a
weakness of textbooks is the fact that changes made by the DoE regarding the curriculum cannot be
accommodated C2). She found previous examination papers valuable as the questions prepare the
learners for examinations (C2). The value of the models was the opportunities for learners to discover
knowledge through visual experiences (C2). According to Elaine the ML curriculum should include
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mathematical content that integrates with the curricula of other school subjects. She provided the
following list of subjects that integrate with ML:
Business economic – graphs
Accountancy – salaries
Engineering graphical design – scales and drawings
Mechanical technology – trigonometry
Tourism – map work (C3).
In her opinion trigonometry that was initially in the curriculum but later omitted should be put back as
trigonometry is required by Mechanical Technology (C3). Departmental documents she was aware of
were circulars and the CASS document which she found useful (C4).
According to Elaine the DoE defines ML as equipping all learners with mathematical skills by using problems
from real-life situations (C5) in order to equip learners with basic mathematical skills. They implemented it based on
the recommendation of the private sector that required workers to become mathematically literate (C5). She had not yet
perused the new CAPS document but believed this document will be more distinct, comprehensible
and user-friendly (C5). She knew that the four learning outcomes were mathematical concepts, financial
mathematics, measurement, data handling and probability (C5). At the end of the last interview I
provided Elaine with a list of concepts and contents to be covered in Learning Outcome 3: Space,
shape and measurement (NCS, 2003a) and she placed 11 out of 19 concepts in the correct year in
which they were to be introduced. The curriculum required that content should be set in context which
Elaine appropriately did (C6).
As prescribed by the DoE (2003a), Elaine taught the content in all three her lessons in context (C7).
The contexts were applicable to the content and she had sufficient knowledge of the contexts she
shared in discussions with the learners (C7). An example of such a discussion was:
T: Let us quickly discuss, why would somebody rather wait to buy a house and first increase his deposit before doing
so?
L1: Isn’t it to lower his instalment?
T: Yes, to decrease his instalment. Did you also know that if one day you have bought your own house you can
decrease the period of paying back the money you have borrowed by several years if only you pay all extra money
you might have available in a month into your home loan account. By doing that, what would you save on?
L2: Money.
T: Money yes, but you will save on interest you need to pay on that home loan account.
Most of the contexts she used in her class were unfamiliar to the learners (C7).
She believes mathematics is about rules and formulae that need to be discovered. Mathematics should
not be presented on a blackboard to learners as a rigid discipline with fixed rules (C8). She concluded
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by saying: Mathematics is a discovering experience (C8). Learners should be able to experiment with the
mathematics available to them. She believes ML is for everyone, even for Mathematics learners as they
too would increase their general knowledge on life-related issues as well as their reading skills (C8). She
believes mathematics has an unbelievable value as mathematics stimulates one’s brain and practice higher order
thinking … it expands your vision and you generally think better. You don’t do rote learning only, you have to reason too
(C9). The value of ML is incredible as a learner returned from his holiday one day telling her in excitement that he
used the mathematics she taught them (C9). One of her learners told her that recently he had amazed his
father by telling him about what he needed to take in account before buying a new car.. These were
proof of the value of ML and the life skills ML learners acquire (C9). Elaine explained that she and her
husband had bought a house a few years ago and had to learn for the first time in 42 years about all the
costs and implications involved in buying a house and arranging finance, whereas her Grade 12 learners
had already learnt about this in school (C9).
Summary
Table 4.14: Summary of Elaine’s knowledge and beliefs
KNOWLEDGE
DESCRIPTION OF TEACHERS’ KNOWLEDGE AND BELIEFS
AND
BELIEFS
INDICATORS
DIMENSIONS
Mathematical
Elaine has very good MCK and no errors or misconceptions were observed.
content
She believes it is possible, but not ideal, to teach ML without having MCK as
knowledge
long as the teacher is hard working and motivated.
(MCK)
ML learners
(L)
ML teaching
(T)
ML Curriculum
(C)
She had the ability to predict what learners would and would not understand
and why and how they would understand the new content. She predicted that
they would approach new tasks by discussing the work with peers from which
they could learn, but that they would also come to ask her or consult their
textbooks.
Elaine taught the content in context and knew the prior knowledge needed to
explain new concepts. She used relevant examples, illustrations and explanations
to make the work comprehensible to the learners. She transformed her content
knowledge into forms that were pedagogically powerful. She also efficiently
sequenced the content to facilitate learning. The instructional material was
chosen appropriately. Elaine likes teaching ML and believes the subject should
be taught in an interesting and practical way so that learners can enjoy and value
the subject.
She is well informed regarding the curriculum since she correctly answered most
of the curriculum questions I asked during the last interview. She knew about a
variety of instructional materials to use in the lessons she presented and was
familiar with other school subjects that integrate with ML. She was not aware of
all available departmental documents or even the content of CAPS, but knew
the definition and purpose of ML according to the NCS. Most of the time the
contexts were unknown to the learners. She regarded mathematics as being
flexible and that learners should discover and experiment with it. She values
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mathematics as a discipline that stimulates the brain, allowing higher order
thinking and reasoning. ML is for everyone, even for Mathematics learners
because their general knowledge on life-related issues should also be increased.
4.6.5 Summary of the participants’ knowledge and beliefs
Table 4.15 provides a snapshot of the four participants’ MCK and PCK and beliefs regarding the ML
learners, the teaching of ML and the ML curriculum.
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Table 4.15: Snapshot of the four participants’ knowledge and beliefs
PARTICIPANTS
Mathematical
content knowledge
(MCK)
PCK and beliefs
regarding the ML
learners
(L)
PCK and beliefs
regarding the ML
MONTY
He believed MCK is a
prerequisite to teach ML. He
made no mistakes and it
appeared as if his MCK
regarding the specific content
covered is sufficient.
• His predictions about the
content the learners would
and would not understand
did not correspond with what
happened in class.
• Some comprehension of
learners’ misunderstandings
but when he did not
understand their
misunderstandings he became
irritated.
• Limited evidence of learners
expressing themselves to
determine if he could act
appropriately on their ideas.
• He believed learners gained
understanding from looking
at examples and practising the
work.
• He was aware of prior
knowledge needed for
ALICE
DENISE
ELAINE
She believed MCK is a
prerequisite to teach ML. She
made several mistakes and it
appeared as if her MCK is
insufficient regarding the
specific content covered in the
three lessons.
Denise believed MCK is a
prerequisite to teach ML. She
made no errors in her
examples or corrections on the
board and it seemed as if she
had sufficient MCK regarding
the specific content covered in
the three lessons.
Elaine believed MCK is a
prerequisite to teach ML. No
errors or misconceptions were
observed and it seemed as if
she had sufficient MCK
regarding the specific content
covered in the three lessons.
• She predicted learners would
find all content easy and
could not understand their
common errors or
misunderstandings.
• She proved to have certain
misconceptions herself so it
would be difficult for her to
predict possible learner
misconceptions.
• No evidence of learners
expressing themselves to
determine if she could act
appropriately on their ideas.
• She believed learners learn
from practising in the
presence of someone who
gives them confidence.
• She correctly predicted what
learners would and would not
understand.
• She realised learners’ possible
misconceptions and rectified
them in class.
• She did not allow learners to
explain their thinking so that
she could hear their thinking
but as they demonstrated
their work on the board, she
could act appropriately with
regard to their written work
on the board.
• She believed learners learn by
explaining the work to others
in small groups.
• She had the ability to predict
what learners would and
would not understand and
how they would understand
the new content.
• She was aware of learners’
possible misconceptions and
rectified their
misunderstandings in class.
• She looked at learners’ work,
listened to their thinking and
acted appropriately.
• She believed learners learn
once the teacher builds on
their existing knowledge and
they could talk about their
thinking.
• She did not connect learners’
prior knowledge with new
• Prior knowledge was
integrated in her revision
• Knew what prior knowledge
should have been revised at
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teaching
(T)
PCK and beliefs
regarding the ML
Curriculum
(C)
learners to gain
understanding.
• He chose very basic and
similar examples and did not
use multiple representations.
• Not much evidence of
sequencing the content.
• His choice of direct
instruction as instructional
strategy and use of textbook
mainly were not appropriate
to teaching ML.
• He believed direct instruction
should initially be used to
introduce new content,
followed by group work and
discussions for solving
problems.
• He believed the difference in
approach between ML and
Mathematics is the use of
fewer examples and working
at a slower pace.
situations.
• The demonstrations and
explanations used did not
make the content
comprehensible to the
learners. She did not use
various representations.
• She mostly sequenced the
content but the amount and
pace made it difficult for the
learners to comprehend the
content.
• Her choice of direct
instruction as instructional
strategy and use of textbook
only were not appropriate to
teaching ML.
• She believed in using formal
teaching when introducing
the topic and discussions
later.
• She believed that the teaching
of ML does not differ from
teaching Mathematics.
lessons.
• She used varied and
appropriate representations to
make the content
comprehensible to the
learners, applied appropriate
instructional strategies and
sequenced her tasks.
• Some evidence of sequencing
the tasks was observed.
• Discussions and learners
working on the board were
appropriate strategies for her
revision lessons.
• She believed she initially had
to explain the content,
concepts and application
thereof, followed by drilling.
The learners should also be
involved by sharing their
ideas but time is a restriction.
• She believed the teaching of
ML differs from that of
Mathematics as ML is based
on contexts.
• He had no knowledge of
other subjects’ curricula that
integrate with ML.
• Knew about definition,
purpose and learning
• She had no knowledge of
other subjects’ curricula that
integrate with ML.
• Knew about definition,
purpose and learning
• She had no knowledge of
other subjects’ curricula that
integrate with ML.
• She did not know the
definition and could not state
183
certain stages of lesson.
• She taught the content in
context and used powerful
examples, illustrations and
explanations and various
representations to make the
work comprehensible to the
learners.
• She sequenced the content to
facilitate learning.
• Discussions and using
textbooks and previous
examination papers were
appropriate for her revision
lessons.
• She believed ML should be
taught by basing her class
discussions on real-life
situations and the learners’
prior knowledge.
• She believed teaching ML
differs from teaching
Mathematics as there is
enough time for learners to
discover new situations
through discussions and
problem solving.
• She knew how ML integrates
with the curricula of five
other subjects in school.
• She knew the definition,
purpose, learning outcomes
outcomes but was not aware
of all departmental
documents.
• Did not teach content in
context. He referred to reallife scenarios but it was
unclear how the content
should be applied in those
contexts.
• He viewed mathematics as
constructivism and logical
although this view was not
implemented in practice. He
believed ML is a kind of
mathematics, but not a lower
grade of Mathematics.
outcomes but were not aware
the learning outcomes but
of all departmental
knew the purpose of ML and
documents.
the various departmental
documents.
• Did not teach content in
context except for the one
• Did not teach content in
life-related example used.
context.
• She viewed mathematics as
• She viewed mathematics as
being logical and having to do flexible and creative and
with constructivism, the latter
regarded ML as a unique
not being observed. She
form of mathematics.
believed there was no
difference between
Mathematics and ML and
that ML is just a little easier.
184
and relevant departmental
documents.
• In all her lessons content was
taught in context as ML
should be taught.
• She viewed mathematics as a
flexible discipline that should
be used to discover and
experiment with. She believed
ML is a subject on its own
which is meant for all learners
(Mathematics learners too).
4.6.6. Discussion of Theme 2: ML teachers’ knowledge and beliefs
In this section I will conduct a literature control where the findings from this study are compared with
the findings from other research studies on ML teachers’ knowledge and beliefs regarding the ML
learners, ML teaching and the ML curriculum is based on the indicators in Table 4.369.
4.6.6.1
•
ML teachers’ mathematical content knowledge (MCK)
Teachers’ belief that MCK is a prerequisite to teach ML
My finding that all four teachers in my study believed MCK is a prerequisite to teach ML is strongly
consistent with Fransman’s (2010) finding that all four teachers in her focus group believed ML
teachers should know the mathematics content well enough to be able to do the mathematics
themselves. The four teachers in my study also believed that all ML teachers should have some form of
tertiary training in mathematics, a finding that is moderately consistent with Sidiropolous’ (2008)
finding that one of the two teachers in her study believed that educators teaching ML should be
qualified and have some form of tertiary training in mathematics.
•
ML teachers’ level of MCK
Except for Alice, the teachers in my study appeared to have sufficient MCK regarding the topics they
were teaching at the time of the observations. Alice’s MCK was not always coherent, and she made
several mistakes in the written examples on the board as well as during her verbal explanations. This
finding, where one of the four teachers in my study had insufficient MCK of the ML topics taught, is
inconsistent with Hechter’s (2011a) finding that both the teachers in her study had insufficient MCK of
the ML topics taught (their knowledge was not coherent and some errors were made with respect to the mathematical
content dealt with in the classrooms, p. 149). My finding is also inconsistent with Bansilal’s (2008) finding that
most of the ML teachers in her study had insufficient knowledge of the ML topics taught.
4.6.6.2
ML teachers’ knowledge and beliefs regarding their learners
Denise and Elaine (as experienced former Mathematics teachers) demonstrated specific knowledge of
the ML learners’ prior knowledge and what content should be emphasised and how the content would
be understood by the learners. They were able to predict learners’ common errors and misconceptions
and could act appropriately to facilitate learning. Compared to Denise and Elaine, Monty and Alice (as
novice teachers) demonstrated superficial knowledge. As far as I could establish, no study has reported
this finding before.
69See
Table 4.3 under Section 4.3.2.2: Theme 2: ML teachers’ PCK and beliefs.
185
Denise and Elaine once again had similar beliefs on how learners come to understand mathematical
content as did Monty and Alice. Denise believed that learners learn by explaining the work to each
other while Elaine believed they learn when the teacher builds on their existing knowledge, and they
then talk about their thinking. Conversely, Monty believed that learners gain understanding by studying
several examples and through a lot practice while Alice believed that learners learn by practising the
work in the presence of someone who gives them confidence. My finding that two of the four teachers
in my study believed that learners reach understanding through active involvement in the lessons is
strongly consistent with the finding of Sidiropolous (2008) that one of the two teachers in her study
believed that learners reach understanding through critical and creative engagement in the lessons.
However, apart from this observation, my literature control did not yield any other reportable findings,
that is, findings that I could realistically compare with my own.
4.6.6.3
ML teachers’ knowledge and beliefs regarding the teaching of ML
Knowledge regarding the teaching of ML refers to teachers’ ability to know what prior knowledge
should be present for learners to understand new work; to use various representations and resources to
facilitate learner understanding; to transform their own knowledge into forms that are pedagogically
powerful; to sequence content; and to choose appropriate instructional strategies and materials (Artzt,
et al., 2008; Ball, 1990; Borko & Putnam, 1996; Hill et al., 2008; Shulman 1986; Shulman, 1987).
In the interviews prior to the observed lessons, only Denise and Elaine could identify the prior
knowledge that should have been present for the learners to understand the work. Monty and Alice said
everything was prior knowledge as they believed all the content had already been done in Grade 10.
Regarding the various representations such as tables, figures and graphs that teachers can use to
facilitate learners’ understanding, both Alice and Elaine conformed to this requirement. Only Alice did
not have the ability to transform her own knowledge into forms that were pedagogically powerful.
During the interview prior to the lesson on data-handling, it seemed she had sufficient knowledge of
the topic, but her lesson presentation was incoherent, and the learners were confused and frustrated. As
far as I could establish, these findings have not been reported before.
It is difficult to comment on the teachers’ ability to use various resources as three of the four teachers
in my study taught content only when only a textbook and calculators were used. Elaine was busy with
her revision programme at the time of the study and used various textbooks, previous examination
papers and calculators as resources. This finding, namely that three of the four teachers in my study
used a textbook and calculators only, is consistent with the finding of Sidiropolous (2008) where both
teachers in her study used only a textbook and calculators. The teachers’ ability to sequence the
186
content70 and choose appropriate instructional material71 as well as their ability to choose appropriate
instructional strategies72 has already been discussed.
4.6.6.4
ML teachers’ knowledge and beliefs regarding the ML curriculum
• Content-context issue
Denise and Elaine believed that ML teaching differs from teaching Mathematics. This finding is
strongly consistent with Sidiropolous’ (2008) finding that one of the two teachers in her study believed
that ML teaching is different from teaching Mathematics. Elaine demonstrated that her belief and
instructional practice conformed to the requirement of the DoE (2003a) of engaging with contexts rather
than applying mathematics already learned to contexts (p. 43). The other three teachers believed that teachers
should initially explain the content and then apply the content to contexts using discussions if time
permitted. My finding that only one of the four teachers in my study used relevant contexts in order for
the learners to explore the content is inconsistent with the finding of Venkat (2010) where the teacher
in her study used contexts but consistent with the finding of Sidiropolous (2008) as one of the teachers
in her study did not use contexts. My finding that all four teachers in my study however believed
contexts should be used is strongly consistent with both Sidiropolous’ (2008) and Hechter’s (2011a)
finding where both teachers in each study believed that contexts should be used to facilitate learning.
•
Integration of ML with other subjects
Apart from the fact that ML requires a different teaching approach to that of Mathematics, the ML
curriculum also requires ML to be taught in a de-compartmentalised manner and to be integrated with
other school subjects (DoE, 2003a; De Villiers, 2007; North, 2005; Venkat & Graven, 2007). All four
teachers in my study believed ML should be integrated with other school subjects. My finding is
strongly consistent with both Sidiropolous’ (2008) and Fransman’s (2010) findings where all the
teachers in their studies believed ML should be integrated with other disciplines. Only Elaine could
identify other school subjects that ML should be integrated with, and only she had knowledge of their
curricula. She believed the ML curriculum should address the mathematical needs of those subjects. As
far as I could establish, this finding has not been reported before.
•
Other ML curriculum issues
Only Elaine had sufficient knowledge of other ML curriculum issues such as knowledge of the
appropriate use of various instructional materials, the strengths and weaknesses of textbooks, the use of
departmental documents as guidelines as well as knowledge of topics that were taught in preceding and
would be taught in subsequent years, i.e. Grades 10 and 12. Sidiropolous (2008) reported that both the
70
See Discussion of Theme 1: Tasks.
See Discussion of Theme 1: Tasks.
72 See Discussion of Theme 1: Learning environment.
71
187
teachers in her study had insufficient knowledge of other curriculum issues (as explained above). My
finding that three of the four teachers in my study had insufficient knowledge of other ML curriculum
issues is consistent with Sidiropolous’ finding.
•
Teachers’ beliefs about the nature of mathematics as a discipline
All four teachers in my study had a constructivist perspective on teaching and learning mathematics.
Monty and Alice believed mathematics is a logical discipline while Denise and Elaine considered
mathematics as being flexible and creative. As far as I could establish, no study up to now has reported
on ML teachers’ beliefs about the nature of mathematics as a discipline.
•
Teachers’ beliefs about the nature of ML as a subject
Only Alice considered ML similar to Mathematics but at a lower level − the other three teachers viewed
ML as a unique subject. This finding that only one of the four teachers in my study considered ML to
be similar to Mathematics but at a lower level provides some (albeit limited) support for Fransman’s
(2010) finding (only 2 out of the 58 teachers in her study viewed ML as similar to Mathematics but at a
lower level) and is consistent with Hechter’s (2011a) finding where half of the teachers in her study
viewed ML as similar to Mathematics but at a lower level, but is inconsistent with Sidiropolous’ (2008)
finding that all the teachers in her study viewed ML as similar to Mathematics but at a lower level.
•
Teachers’ beliefs about the value of ML as a subject
All four teachers in my study believed ML has great value as learners obtain knowledge they can use in
their everyday lives and work situations in the future. My finding that all four teachers in my study
valued ML is strongly inconsistent with Sidiropolous’ (2008) finding that neither of the two teachers in
her study believed ML has great value in helping learners obtain knowledge they can use in their
everyday lives and work situations in the future.
4.6.6.5
Summary of discussion on Theme 2
To summarise: All four teachers in my study believed that MCK is a prerequisite for teaching ML, and
three of them appeared to have sufficient MCK of the topics they were teaching at the time of the
observations. Compared to Monty and Alice who demonstrated superficial knowledge of the learners’
prior knowledge, Denise and Elaine demonstrated specific knowledge of the learners’ prior knowledge,
which content to emphasise and how it would be understood by learners to the extent that they could
accurately predict the learners’ problems with the content. Monty and Alice believed learners gain
understanding by studying several examples while Denise and Elaine believed a teacher should build on
learners’ prior knowledge and involve learners in discussions. Only Elaine based her teaching on liferelated problems while both she and Denise used appropriate instructional strategies to facilitate
learning. Monty and Alice believed the teaching of ML is no different to teaching Mathematics.
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Only Elaine knew which subjects are integrated with ML although all four teachers believed that ML
should be integrated with other disciplines. Apart from Elaine, the teachers in my study had a
superficial understanding of the ML curriculum. All four teachers had a constructivist perspective on
teaching and learning mathematics. Only Alice viewed ML as a lower level of Mathematics while the
other teachers viewed ML as a unique subject. All four teachers regarded ML as valuable because,
through it, learners could obtain knowledge they could use in their daily lives.
4.7
Findings, trends and explanations
An analysis of the discussions on Theme 1 and Theme 2 was done and resulted in the following
summary in which the findings, trends and explanations are delineated.
•
Experiences and beliefs shared by all four teachers in my study
I found that all four teachers were positive about teaching ML and all shared the following beliefs:
Mathematics is a subject that is best mastered by implementing a constructivist approach to teaching
and learning; ML is a valuable subject; ML should be integrated with other school subjects; and MCK is
a prerequisite for teaching ML. Many teachers in recent studies share this positive attitude towards
teaching ML and attach value to the subject. Sidiropolous (2008), however, found that both the
teachers in her study had negative attitudes towards ML. This discrepancy in findings can perhaps be
attributed to teachers becoming aware of the uniqueness of the subject and starting to realise its value
during the past four years − Sidiropolous conducted her study only a year after ML had been
introduced at a time when teachers tended to be negative about this unfamiliar subject. Research
indicates that almost all ML teachers believe that MCK is a prerequisite for teaching ML and that all
ML teachers should have some form of tertiary mathematics training. With Alice having some form of
tertiary mathematics training but still making several errors and not being able to transform [her] own
knowledge into forms that are pedagogical powerful (Shulman, 1987, p. 15), it seems that not only general
training in the content of mathematics but also mathematics teacher training (i.e. training in the
teaching and learning of mathematics) is required to teach ML proficiently. A concern is what happens
in ML classes where teachers from other disciplines with no formal mathematics teacher training teach
ML.
•
Two differing cases
In my study, two highly differing cases were Alice and Elaine. Alice, as a novice teacher with no
mathematics teacher training, was the only teacher who communicated judgmentally with the learners;
did not work at a slower pace as stipulated in the ML curriculum (DoE, 2003a); did not have the ability
to transform her own knowledge into forms that were pedagogically powerful; and viewed ML as
similar but inferior to mathematics. Elaine, a former mathematics teacher with years of experience, was
189
the only teacher who used contextual tasks effectively; pointed out the value of mathematics to the
learners; selected tasks from all four levels of the ML Assessment Taxonomy; required the learners to
explain their answers; posed a variety of oral questions on different levels; and had sufficient curriculum
knowledge. From this comparison, it seems experience and mathematics teacher training play a crucial
role in the instructional practice of the ML teachers.
•
Role of teaching experience in ML teachers’ instructional practice
In comparing the instructional practices of experienced and inexperienced teachers from my own and
other research studies, I found that not all experienced teachers taught ML in a satisfactory manner.
For example, the experienced Denise in my study did not comply with all the requirements regarding
the instructional approach to teaching ML as set out by the DoE (2003a). Another example is the study
of Sidiropolous (2008) where the instructional practices of the two experienced, but negative, teachers
were not aligned with the curriculum or with their claimed beliefs. Conversely, there were
inexperienced teachers in the studies (one from each study) of Fransman (2010) and Hechter (2011a)
whose practices were aligned with the curriculum and with their claimed beliefs − teachers who had
developed a new status identity (Fransman, 2010, p. 184) of being a ML teacher. It seems that apart from
having sufficient teaching experience, the success of a ML teacher’s practice can be improved by being
positive about and taking ownership of teaching ML as well as developing a new status identity of being
a ML teacher.
•
Value of teacher and ML training
Apart from Monty and Alice both being novice teachers, a major difference in their instructional
practices is that Alice had no teacher training but had completed a mathematics course as part of her
Management degree whereas Monty had completed a BEd degree with Mathematics as a major. In
contrast to Alice, Monty communicated with the learners in a non-judgmental manner; assisted them
individually; worked at a slower pace; demonstrated sufficient MCK; and viewed ML as a unique
subject. These findings suggest that teacher training plays a role in teacher-learner interactions and in
establishing a manageable pace of work. Only Monty attended short courses for in-service ML teachers.
A comparison of the teachers in my study with the teachers who were part of the ACE (ML)
programme of Fransman (2010) and Hechter (2011a) revealed that ML teacher training had a positive
influence on the instructional practices and beliefs of the teachers who were enrolled for the ACE (ML)
course. Examples of this include their ability to provide scaffolding and to teach the mathematical
content using contextualised tasks.
•
On a continuum from teacher-centred to learner-centred
If I could place the teachers in my study on a continuum from teacher-centred on the left to learnercentred on the right, this would be the order from left to right: Alice, Monty, Denise and Elaine. Monty
190
and Alice as novice teachers adopted a teacher-centred approach. Denise as a former Mathematics
teacher, with many years’ experience of teaching Mathematics, leant more towards a learner-centred
approach. Elaine, also a former Mathematics teacher with many years of experience in teaching
Mathematics, also adopted a learner-centred approach. It appears that teacher training and experience
influenced the teachers’ ability to facilitate learning effectively in the ML classroom in my study.
Although both teachers in Sidiropolous’ (2008) study had mathematics training and teaching
experience, they nevertheless still used a teacher-centred approach. This can perhaps be attributed to
factors such as their having had no ML teacher training or their beliefs regarding the teaching of ML.
Venkat and Graven (2008) and Venkat (2010) reported that the teachers in their studies were trained
and experienced, but it was after making an actual decision, as Elaine did, to change their pedagogic
practices to a learner-centred and activity-based approach, that the teachers as well as the learners could
experience the value of ML.
Contradictions between teachers’ beliefs and their instructional practices
•
The following beliefs held by the teachers in my study were contradicted in their practices.
Table 4.16: Contradictions between teachers’ beliefs and their instructional practices
Teacher
Belief
Monty, Alice,
Monty, Alice
Mathematics is a constructivist discipline
ML should be integrated with other
subjects
Monty, Alice,
Denise
Real-life application problems should be
used
Monty,
Denise
Denise
ML is a unique subject
Denise
Practice
Learners learn by explaining the work to
each other in small groups
ML teaching differs from Mathematics
teaching
Direct instruction (lecturing) was used.
No integration was evident in their
lessons. Could not explain how ML
could be integrated with other subjects.
With the exception of one real-life
example in one of Alice’s classes, only
content was taught.
Lessons were typical Mathematics
lessons
She used their solutions to discuss the
work with them.
Content was lectured and if there was
enough time, discussions took place.
Such contradictions were also evident in Sidiropolous’ (2008) study where both teachers believed ML is
a maths only better than nothing and even the maths of oranges and bananas (p. 225) while, in later interviews,
they complained about the difficulty level of the subject. They also believed that a teacher could work
at a much slower pace, but, during the interviews, they complained about not having enough time to
teach the way they knew they were supposed to teach ML. A possible reason for this contradiction is
that the theory they espoused was difficult to carry out in practice.
191
To summarise: The following are some issues that need attention by ML teachers in their instructional
practices.
1. Knowledge of the pedagogic approach in which teaching content is integrated with contexts.
2. Knowledge of varied, applicable instructional strategies and material.
3. Selection of tasks according to the ML Assessment Taxonomy.
4. Different types and level of oral questioning in the classroom.
5. Increasing learner participation by giving them the opportunity to verbalise their thinking.
6. How to facilitate learner-learner interactions.
7. Establishing and maintaining a positive learning atmosphere.
8. Having a positive attitude towards the subject and the learners.
9. Choice and efficient use of appropriate instructional strategies.
To ensure proficient ML teaching, it seems that teachers require the following: ML teacher training; a
certain level of MCK (it was not the purpose of the study to determine the required level of MCK);
experience; and a positive attitude towards and the desire to change their instructional practices.
4.8
Conclusion
In this chapter, I discussed the data collection process that took place in Pretoria during the second
quarter of 2011. I initially had five participants, but, during the data analysis process, I realised that the
one case did not add value to my study, and so I decided to continue only with the other four cases.
Data were collected by means of three lesson observations per teacher with interviews conducted prior
to the second and third observations and a third and last in-depth interview conducted after the
observations based on the observed lessons as well as their knowledge and beliefs. I adopted a
deductive approach to coding the data as I had identified two themes: ML teachers’ instructional
practices and ML teachers’ knowledge and beliefs prior to the data collection stage. Different categories
for each theme were chosen according to the work of Artzt et al. (2009), Ball (1990), Borko and
Putnam (1996), Hill et al. (2008), Shulman (1986) and Shulman (1987) apropos of which the raw data
were analysed. In this chapter I also presented the data of the four participants and discussed my
findings on the basis of a literature control. I finally identified trends and possible explanations for the
trends.
In the next chapter, the research questions are answered, and I reflect on my research study and draw
conclusions from the case study. I also discuss the limitations and significance of the study and make
recommendations for further research.
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Chapter 5
Conclusions and implications
5.1
Introduction
In this chapter I provide a summary of the previous four chapters, answer the research questions that
guided this study and reflect on my research as to what I would have done differently and make
provision for the fact that I may have been wrong in my interpretation of the teachers’ instructional
practices and knowledge and beliefs. This is followed by the conclusions, recommendations and
limitations of the study. A final reflection is done on the research study.
5.2
Chapter summary
In Chapter 1 I introduced and contextualised the research study. The purpose of this study was to
investigate, by means of a case study, the way in which ML is taught with the view to determining the
relationship between ML teachers’ knowledge and beliefs and their instructional practices. The different
meanings attached to mathematical literacy both internationally and nationally have been discussed. I
discussed the problem and the rationale for the study, formulated the research questions, and discussed
the methodological considerations and the possible contribution and limitations of the study.
Chapter 2 presented an in-depth analysis of the findings in the relevant literature as well as the
conceptual framework on which the study is based. Comparisons were made between the different
conceptions of mathematical literacy internationally and nationally, followed by a discussion on ML as a
compulsory alternative to Mathematics in Grades 10 to 12 in South Africa. Following this was a
discussion of the meaning of teachers’ instructional practices and the value of various approaches to
teaching. Attention was given to the different domains of teachers’ knowledge, teachers’ belief systems
and the relationship between their knowledge and beliefs and their instructional practices. The
conceptual framework, which is based on concepts and theories from relevant work in the literature,
was then discussed.
193
A description of the qualitative methodology used in this study was reported in Chapter 3. I discussed
social constructivism as my research paradigm, and the nature of my study as subjective and
interpretive. This is an exploratory case study. Observations were used to examine teachers’
instructional practices and to study demonstrations of their MCK and knowledge regarding the ML
learners, the teaching of ML and the ML curriculum. Interviews were used to determine why teachers
do what they do in class and to determine how they apply their PCK during their instructional practice.
ATLAS.ti 6 was used to analyse the video and audio data. I lastly discussed the trustworthiness of the
study and ethical considerations that were taken into consideration.
In Chapter 4 I briefly reported on the data collection process, presented and discussed the findings and
lastly identified trends and possible explanations for those trends. A DEDUCTIVE-inductive
(uppercase denotes the preference given to the style of analysis) approach to coding the data was used
as I identified two themes: ML teachers’ instructional practices and ML teachers’ knowledge and beliefs
prior to the data collection stage. After this deductive phase of analysis, inductive analysis was done
when I studied the organised data in order to explore new patterns and trends. I presented the findings
from the data obtained through class observations and interviews according to the different categories
provided in Table 4.2 and Table 4.3. The findings were then related to the findings in the literature and
trends were identified and subsequently explained.
5.3
Verification of research questions
Based on the rationale that ML is a significant subject which may positively influence the lives of many
learners, and the problem that many teachers have negative views and experiences of the subject, I
decided to explore the relationship between ML teachers’ instructional practices and their knowledge
and beliefs. In order to do so, the following research main question was formulated: What is the
relationship between ML teachers’ knowledge and beliefs and their instructional practices? To address
this main question, the following five subquestions guided the enquiry:
1.
How can ML teachers’ instructional practices be described?
2.
What is the nature of ML teachers’ knowledge and beliefs?
3.
How do ML teachers’ knowledge and beliefs relate to their instructional practices?
4.
What are the possible implications of the findings from Questions 1, 2 and 3 for teacher
training?
5.
What is the value of the study’s findings for theory building in teaching and learning ML?
I will subsequently utilize social constructivism as research paradigm (the epistemological approach
which guides my own teaching and learning practice and orientation in mathematics) to verify these
194
questions. In short: the social constructivist holds that all knowledge is constructed and based upon not
only prior knowledge, but also the cultural and social context (Ollerton, 2009).
The following table (Table 5.1) regarding the four participants’ experience, teacher training,
instructional approach, productivity of instructional practice, MCK, PCK and beliefs was prepared to
facilitate the discussion on the verification of the research questions.
195
Table 5.1: Summary of participants’ information
Monty
Alice
Keys used in the table:
MCK en PCK: Sufficient knowledge: and Insufficient knowledge: Beliefs versus practice: Corresponds: and Contradicts: Paragraph numbers in the thesis are indicated in brackets.
Experience (years)
(3)
(2)
Maths teacher training
• Approach used
• Productivity of practice
PCK
MCK
Learners
Teaching
Curriculum
Beliefs versus practice
(Only imperative aspects)
Learners
• Teacher-centred
• Somewhat unproductive
(11)
• Learner-centred
• Productive
(4.6.2.1)
(4.6.3.1)
(4.6.4.1)
(4.6.1.2)
(4.6.1.3)
(4.6.1.4)
(4.6.2.2)
(4.6.2.3)
(4.6.2.4)
(4.6.3.2)
(4.6.3.3)
(4.6.3.4)
(4.6.4.2)
(4.6.4.3)
(4.6.4.4)
Learn by teacher’s examples
Learn by looking at teacher
[4.5.1.3 (LESP1) & 4.6.1.2]
[4.5.2.3 (LESP1) & 4.6.2.2]
Learn by demonstrating work
and teacher building on that
Learn by discovery and
discussions
[4.5.3.3 (LESP1) & 4.6.3.2]
[4.5.4.3 (LESP1) & 4.6.4.2]
ML teaching is different to
Mathematics teaching
ML is the same as Mathematics
teaching
ML teaching is different to
Mathematics teaching
ML teaching is different to
Mathematics teaching
[4.5.1.1 (TMS3) & 4.6.1.3]
[4.5.2.1 (TMS3) & 4.6.2.3]
[4.5.3.1 (TMS3) & 4.6.3.3]
[4.5.4.1 (TMS3) & 4.6.4.3]
Curriculum
(11)
• Teacher- and learnercentred
• Somewhat productive
Elaine
(4.6.1.1)
Teaching
• Teacher-centred
• Unproductive
Denise
Constructivist perspective;
Content in context; ML has
application value
Constructivist perspective;
Content in context; ML has
application value
Constructivist perspective;
Content in context; ML has
application value
Constructivist perspective;
Content in context; ML has
application value
(4.5.1.3; 4.5.1.1 & 4.6.1.4)
(4.5.2.3; 4.5.2.1 & 4.6.2.4)
(4.5.3.3; 4.5.3.1 & 4.6.3.4)
(4.5.4.3; 4.5.4.1 & 4.6.4.4)
196
5.3.1 Question 1: How can ML teachers’ instructional practices be
described?
I used an adapted version of the theoretical framework provided by Artzt et al. (2008) on teachers’
instructional practices as well as Franke et al.s’ (2007) definition of a productive practice to
contextualise and interpret my results. To answer this question, the participants’ instructional practices
were described according to the lesson dimensions as indicated in this study’s conceptual framework,
but, to avoid repetition in the thesis, a detailed description is provided in Appendix J.
The two novice teachers in my study
Monty and Alice’s instructional practices can be described as teacher-centred in that they believed their
role as teachers was to transmit mathematical content, demonstrate procedures for solving problems
and explain the process of solving sample problems. However, Artzt et al. (2008) suggests that this
approach is not ideal as the teacher-centred approach can serve as a mask for teachers who do not fully
understand the content, the learners or the pedagogy, as was found in the practices of these two
teachers in my study.
Monty and Alice’s instructional practices can also not be described as productive (Franke et al.
(2007) consider a productive practice as a practice where the teacher creates ongoing opportunities for
learning). Alice’s instructional practice was less productive than Monty’s. In fact, her practice was
largely dysfunctional as it was characterised by inattentive learners and ineffective teaching. She also
failed to connect the learners’ prior knowledge with new mathematical situations. Given that both
Monty and Alice were novice teachers, an explanation of the differences between their instructional
practices could be that Alice had no formal mathematics education training while Monty had completed
a BEd with Mathematics and Methodology of Mathematics as major subjects.
The two experienced teachers in my study
Denise’s instructional practice can be described as a combination of teacher- and learner-centred,
leaning more towards learner-centred, while Elaine’s instructional practice can be characterised as
learner-centred. Denise and Elaine believed that learners should develop both procedural and a
conceptual understanding of the mathematical content. A learner-centred approach to teaching requires
the teacher to create opportunities for learners to achieve understanding through active engagement
with each other and the problem-solving process (Artzt, et al., 2008). What appears to have made
Elaine’s practice more productive than Denise’s was Elaine’s use of contexts to explore the
mathematical content; her pointing out the value of mathematics in everyday-life situations; her
197
selection of tasks from Levels 1-4 of the ML Assessment Taxonomy; her allowing her learners to
explain their answers; and her asking various types and different levels of oral questions.
Comparison of the participants’ instructional practices
The key differences between the two novice teachers and the two experienced teachers are listed in
Table 5.2 below. I found that the instructional practices of the four teachers in my study could be
described as predominantly teacher-centred: the practices of two of the four teachers were exclusively
teacher-centred; one teacher’s practice could be described as a combination of teacher- and learnercentred, leaning more towards learner-centred; and the fourth teacher’s practice could be described as
exclusively learner-centred.
Table 5.2: Comparison of the participants’ instructional practices
Two experienced teachers
Two novice teachers
(Monty and Alice)
Approach
Tasks
Discourse
Learning
environment
Denise
Elaine
Teacher- and learner Learner-centred
centred
• Content only
• Content in contexts
• Content only
• On Level 1 only
• On Levels 1-4
• On Level 1 only
Build lessons on learners’ prior knowledge
Learners
demonstrated
Learners demonstrated and
• Learners did not express
their thinking
explained their thinking
their thinking
• No scaffolding
• Did not recognise
• Provided scaffolding
learners’ typical
• Recognised learners’ typical misunderstandings
misunderstandings
• Formal atmosphere
• Discussions
• Direct instruction
• Logical flow in lessons
• Not enough logical flow
• Positive classroom atmosphere
• Minimum learner
• Maximum learner participation
participation
Teacher-centred
However, certain common trends in all four cases were also observed the most significant of which was
their collective failure to encourage learner-learner interaction. Again, this is far from ideal: The DoE
(2003a) clearly states that learners need to develop the ability to communicate mathematically and that
teachers should create opportunities for classroom dialogue where learners can listen to, respond to and
question each other so that they can discard or revise their own ideas (Artzt et al., 2008).
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In summary: My study seems to provide evidence that confirms the existence of a relationship between
the teaching approach used in ML classrooms and the productivity of the teacher’s instructional
practice. The four practices observed in my study ranged from Elaine’s learner-centred approach,
yielding a productive instructional practice, to Alice’s teacher-centred approach, yielding an
unproductive instructional practice.
According to my findings, it seems that teaching experience as well as mathematics teacher training
(Table 5.1) may play a significant role in the productivity of the instructional practices of the four
participants: both Denise and Elaine (with 11 years’ experience of either teaching Mathematics or ML)
had productive practices; and, comparing the practices of the two novice teachers (with three years and
less of teaching experience), the teacher with mathematics teacher training had a more productive
instructional practice than the teacher without teacher training.
5.3.2 Question 2: What is the nature of ML teachers’ knowledge and
beliefs?
As stated in Question 1, the participants’ PCK and beliefs were described according to the study’s
conceptual framework, but, to avoid repetition in the thesis, a detailed description is provided in
Appendix K.
Teachers’ level of MCK
The purpose of this study was not to assess ML teachers’ content knowledge but rather to comment on
the accuracy of their mathematical content and the occurrence of their misconceptions. Against this
background, I found that the three teachers who had prior teacher training in mathematics appeared to
have sufficient MCK regarding the topics they were teaching at the time of the observations as no
mathematical errors (except for the two minor omissions and single mistake of one teacher) were made
or misconceptions observed. The only teacher with no mathematics teacher training was guilty of
several mathematical errors, and some of the mathematical content she taught indicated her own
misconceptions regarding the content. All four teachers believed MCK was a prerequisite to teach ML.
Teachers’ level of PCK
Regarding the nature of the ML teachers’ PCK, one of the experienced teachers (11 years’ experience)
illustrated sufficient knowledge of all three domains of PCK: the ML learners, the teaching of ML and
the ML curriculum. The other experienced teacher (also 11 years’ experience) had sufficient knowledge
of two of the three domains of PCK: the ML learners and the teaching of ML. The other two teachers
had only superficial knowledge of all three domains. My finding that the two experienced teachers had
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developed PCK confirms the findings of Ball (1988), Ball et al. (2005), Koellner et al. (2007), Ma
(1999), Shulman (1986) and Sowder, (2007) that PCK can be developed only over time through
experience in the classroom and that it cannot be taught. Some researchers (Ball, 1990; Van Driel,
Verloop & de Vos, 1998) also believe that solid understanding and knowledge of mathematical subject
matter are prerequisites for developing PCK. I also found that the two teachers who had developed a
certain level of PCK also had adequate MCK. However, Monty’s instructional practice indicates that
sufficient MCK (teacher training) does not guarantee PCK (Table 5.1).
Teachers’ beliefs
All four teachers claimed that they had a constructivist perspective on mathematics as a discipline; that
ML involves the teaching of mathematics in context; that learners should realise the application value
of mathematics; and that learner-learner interaction in the form of group work and discussions is
required for learners to develop understanding of mathematics. The main differences between the two
novices’ and the two experienced teachers’ beliefs were in their beliefs on how learners learn: The
novice teachers believed that learners learn through information received from the teacher while the
experienced teachers believed that learners should be active participants in their own learning. Only one
teacher in my study, Alice, believed that ML is similar to Mathematics but on a lower level while the
other three teachers believed that ML is a unique subject in its own right.
To summarise (see Table 5.1): Both teachers with sufficient PCK also had sufficient MCK, which
suggests that MCK is required to develop PCK. Furthermore, the three teachers with mathematics
teacher training (one novice teacher and two experienced teachers) had sufficient MCK, but since the
novice teacher still lacked PCK, it could be suggested that although MCK is required to develop PCK,
it is teaching experience that plays a crucial role in the development of teachers’ PCK. These findings
will hopefully contribute to this new field and fill the gap in literature regarding ML teachers’
knowledge and beliefs.
Based on the interviews with the four participants in my study, it seems that mathematics teacher
training is required to enhance teachers’ MCK and that, although MCK is required to develop PCK, it
is through teaching experience that teachers develop PCK.
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5.3.3 Question 3: How do ML teachers’ knowledge and beliefs relate to
their instructional practices?
Teachers’ knowledge
Alice was the only teacher in my study who had insufficient MCK, and, because only Alice’s
instructional practice was described as unproductive, it seems that ML teachers’ level of MCK strongly
influences the productivity of their practices (Table 5.1). This finding supports Kilpatrick’s (2001) view
that proficient teaching demands, among other things, teachers’ conceptual understanding and
procedural fluency. The two novice teachers in my study had insufficient PCK and unproductive
instructional practices in contrast to the two experienced teachers who had sufficient PCK and
productive instructional practices. This finding suggests that PCK influences the productivity of
teachers’ practices. Since PCK influences teachers’ teaching approach, which, in turn, influences the
productivity of teachers’ practices, it can be deduced that PCK influences ML teachers’ practices.
I realise that other factors also play an important role in the productivity of teachers’ instructional
practices, but it seems as if MCK and PCK have a definite influence on such practices.
Teachers’ beliefs
Trends were found in the correspondences and contradictions between the teachers’ stated beliefs and
their instructional practices. A common belief expressed by all four teachers was how learners learn.
The following important contradictions were noted between three of the four ML teachers’ stated
beliefs about the nature of mathematics and the teaching of ML and their instructional practices. All
four teachers claimed that they taught mathematics from a constructivist perspective, but, in practice, it
was only Elaine who created opportunities for the learners to discover, experiment and reason in order
to achieve understanding. Monty and Alice’s perspective was traditional while Denise’s was formalist
(Dionne, 1984).
•
All four teachers believed that ML involved the teaching of mathematics in context and that
learners should realise the application value of mathematics, but only Elaine used relevant
contexts to enable the learners to explore the mathematical content and to appreciate the value
of mathematics.
•
All four teachers believed that learner-learner interaction, such as learners explaining their work
to each other in small groups, was important in providing opportunities for learners to develop
conceptual understanding of mathematics. However, it was only in Elaine’s instructional
practice that some evidence of learner-learner interaction was observed.
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Contradictions between three of the four teachers’ stated beliefs and their instructional practices were
found in the practices of Monty and Alice, who used a teacher-centred approach, and Denise who used
a combination of teacher- and learner-centred approaches. A reason for these inconsistencies could be
that the teachers had heard about a constructivist perspective on mathematics or knew how ML should
be taught, but their existing cognitive structures were not ready to accommodate the required changes
(Artzt et al., 2008).
According to the literature, teachers’ knowledge and true beliefs strongly influence their practices (Artzt
et al., 2008; Ball, 1990; Liljedahl, 2008; Pajares, 1992). It was only Elaine in my study who adopted a
learner-centred approach and whose knowledge and stated beliefs corresponded with her instructional
practice. Not only did her knowledge and beliefs influence her instructional practice positively, but
conversely her instructional practice (experiences with the subject and its learners) also positively
influenced her knowledge and beliefs regarding the ML learners and the teaching of the subject. This
was evident from my last interview with Elaine during which she told me that when she had been asked
two years ago to be the coordinator for ML, she had initially felt that she was being demoted, but she
claimed that ML had grown on her since then. She enjoyed being involved in ML and never wanted to
return to Mathematics.
In summary: Based on the findings of this study, it can be tentatively assumed that, except for the one
teacher who used a learner-centred approach, the teachers’ stated beliefs about teaching ML and the
ML curriculum did not influence their instructional practices whereas knowledge had a strong
influence. It is possible that the stated beliefs did not reflect the true beliefs of the teachers in this
study. In the case where a learner-centred approach was used, not only did the teacher’s knowledge and
beliefs influence her practice, but her practice also influenced her knowledge and beliefs. These findings
will hopefully also contribute to this new field, also filling the gap in literature regarding the relationship
between ML teachers’ knowledge and beliefs and their instructional practices.
5.3.4 Question 4: What are the possible implications of the findings
from Questions 1, 2 and 3 for teacher training?
Effective and purposeful training of pre- and in-service ML teachers is of critical importance in South
Africa, a finding that was also reported by, among others, Bansilal (2008) and Sidiropolous (2008). The
instructional practice of Alice (the only teacher in my study without mathematics teacher training)
proved to be unproductive resulting in discouraged and uninvolved learners. Knowledgeable,
competent and dedicated ML teachers such as Elaine in my study and the findings from other studies
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(Graven & Venkat, 2009; Hechter, 2011a; Venkat & Graven, 2008) reveal that the aims and purposes
of ML are realistic and achievable. My findings also indicate that MCK alone is insufficient; ML
teachers need knowledge regarding the teaching and learning of ML to teach the subject effectively.
Teaching ML
Key factors that should be part of all ML instructional practices:
•
Having a learner-centred approach and using appropriate instructional strategies.
•
Engaging with contexts rather than applying mathematics already learned to contexts.
•
Selecting tasks at all four levels of the ML Assessment Taxonomy − the emphasis in Levels 1
and 2 is on routine calculations while the key aims of ML are located primarily in Levels 3 and
4.
•
Engaging learners in discussions thereby enabling them to communicate their thinking through
the use of appropriate terminology.
•
Using various instructional resources to connect learners’ knowledge with new situations.
•
Teachers having sufficient general knowledge of the contexts in which the lesson is situated,
enabling them to engage the learners in meaningful mathematical discourse.
These key factors were also confirmed by Artzt et al. (2008), DoE (2003a, 2011), Graven and Venkat
(2007) and Venkat et al. (2009).
Other didactical and methodological issues that were identified during this study and should be
addressed include logical sequencing of tasks; efficient oral questioning; managing discipline in ML
classrooms; creating a positive learning atmosphere in class; effective board work; valuing teachers who
are enthusiastic about ML and its learners; engender an understanding of the theory and practice of
PCK; enhancing teachers’ curriculum knowledge and their knowledge of how ML is integrated with
other subjects.
ML teachers’ knowledge and beliefs
Similar to the training of Mathematics teachers, the focus of ML teacher training should be the
development of sufficient MCK to enhance conceptual understanding of various mathematical topics.
ML teacher training should however differ from Mathematics teacher training regarding the following
aspects: The level of the mathematical content in the ML programme need not be on a second-year
BSc level, but should include (apart from a generic component being offered to both Mathematics and
ML student teachers) specialised training regarding the teaching and learning of ML. In particular,
teachers should learn how to structure mathematical content to enable learners to progress in their
203
cumulative understanding of the content and to link learners’ prior knowledge with new content. The
theory and practice of PCK should be incorporated in all student teacher training programmes during
teaching practice and internship. Only a few tertiary institutions have to date developed such
programmes − the teachers who participated in this study, for example, had no such exposure.
ML teachers’ instructional practices were strongly positively influenced by their knowledge and not
their stated beliefs. The distinction between what teachers say they know and believe and is played out
in their classrooms emerged as a pivotal aspect of teaching and learning success. Addressing this
distinction during teacher training seems to be important.
In summary: ML teacher training is specialised and implies that teachers should be equipped with MCK
and skills to facilitate the learning process. ML teacher training has to a large extent been neglected as
the subject was introduced in 2008 yet it was only in 2011 that a ML teacher training programme was
introduced at the University of Pretoria. This programme consists of a three-year mathematics content
component and a methodology of ML component in the fourth year. It is furthermore recommended
that all practising ML teachers should be required to complete an ACE (ML) programme, which is in
line with the DoE’s (2009) requirement that all intermediate phase teachers should have completed a
mathematics course by 2014.
5.3.5 Question 5: What is the value of the study’s findings for theory
building in teaching and learning ML?
Flowing from the finding that ML teachers’ knowledge, but not necessarily their beliefs, influences their
instructional practices, the following matters warrant the attention of curriculum decision-makers:
•
My research revealed that teachers who had formal training or experience in the teaching of
mathematics in classrooms also displayed evidence of having adequate MCK. In addition,
adequate MCK impacted positively on teaching and learner understanding.
•
The ML teachers who had productive instructional practices had sufficient knowledge of both
mathematical content and the teaching and learning of ML.
•
Competent, dedicated teachers who value the ML curriculum are needed to teach ML. New
student-teachers should be recruited to become ML teachers so that they can develop a new
status identity.
To answer my main question: There is a dynamic but complex relationship between ML teachers’
knowledge and beliefs and their instructional practices. Firstly, their knowledge, but not their stated
beliefs were reflected in their practices. Secondly, some of the teachers’ classroom practices belied their
204
stated beliefs. Conversely, in one case, the teacher’s practice also had a positive influence on her
knowledge and beliefs.
5.3.6 Summary of verification of research questions
In Table 5.3 below I provide a summary of the research questions, data collection techniques used,
objectives of the questions and research findings.
Table 5.3: Summary of verification of research questions
Research questions
(Data collection
techniques)
Objectives of the questions
• To determine what teachers
do in their classrooms with
respect to: tasks given,
discourse that takes place and
1. How can ML
the learning environment
teachers’ instructional
which is established.
practices be described?
(Observations)
• To describe teachers’ practices
according to the teaching
approach used and level of
productivity of their practices.
2. What is the nature • To comment on the teachers’
level of MCK.
of ML teachers’
knowledge and beliefs? • To explore teachers’ PCK and
(Observations &
beliefs regarding ML learners,
interviews)
the teaching of ML and the
ML curriculum.
3. How do ML
teachers’ knowledge
and beliefs relate to
their instructional
practices?
(Observations &
interviews)
4. What are the
possible implications
of the findings from
Questions 1, 2 and 3
for teacher training?
(Observations &
interviews)
• To explore what the
relationship is between
teachers’ instructional
practices and their knowledge
and beliefs.
• To investigate how teachers
use PCK in their lessons.
• To improve my own practice.
• To inform current teacher
training and development
programmes.
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Research findings
Predominantly teacher-centred: two teachers
used a teacher-centred, one teacher a
combination of teacher- and learner-centred
and one teacher a learner-centred approach.
In all practices there is a positive relationship
between the approach used and the level of
productivity, ranging from a productive
practice in which a learner-centred approach
was used to an unproductive practice where a
teacher-centred approach was used.
The two more experienced ex-Mathematics
teachers proved to have sufficient knowledge
while the two novice teachers still lack PCK.
Mathematics teacher training is required to
enhance teachers’ MCK and although MCK
is required to develop PCK, it is through
teaching experience that teachers develop
PCK.
Teachers’ beliefs did not influence their
instructional practices, but knowledge
strongly influences teachers’ instructional
practices. In the case where a learner-centred
approach was used, not only did the teacher’s
knowledge and beliefs influence her practice,
but her practice also influenced her
knowledge and beliefs.
ML teacher training is specialised and implies
that teachers should be equipped with
specific MCK; skills to integrate content and
context in their teaching in order to facilitate
the learning process; and knowledge of the
ML curriculum. An understanding of the
theory and practice of PCK and its
importance should be engendered in all
student teachers training.
5. What is the value of
the study’s findings for • To add to the body of
knowledge regarding the
theory building in
relatively newly introduced
teaching and learning
subject and to make
ML?
suggestions to the curriculum
(Observations &
stakeholders.
interviews)
5.4
ML teachers’ instructional practices are
strongly positively influenced by their
knowledge, but not their stated beliefs. The
distinction between what teachers say they
know and believe and is played out in their
classrooms emerged as a pivotal aspect of
teaching and learning success. Addressing
this distinction during teacher training seems
to be important.
What would I have done differently?
During the data presentation stage, I realised that I had missed valuable communications between the
teacher and the learners at their desks as I did not want to intrude by moving around in class with a
video camera. More information regarding the teachers’ PCK would possibly have emerged from this
discourse. With the insight of hindsight, I would have employed a research assistant to videotape all my
sessions for careful perusal and analysis.
In discussing my findings another aspect I wished I could have done differently was to include ML
teachers from other non-mathematics disciplines. Alice’s practice already informed this study about
teachers who were teaching without having teacher training, but it would have been valuable to
investigate the practices of teachers who did have teacher training but no formal mathematics training.
5.5
Providing for errors in my conclusion
I engaged with five (although only four actually participated in the research) ML teachers who allowed
me in their classrooms and also shared some of their knowledge and beliefs with me. I have made some
decisions on their instructional practices and the nature of their knowledge and beliefs, and I have to
accept the fact that I may have been wrong in some of my conclusions, albeit unknowingly and
unintentionally. I attempted to enhance the credibility and trustworthiness of my study through
triangulation by using three observations and three interviews at different stages of the data collection
process. Since there is no agreement among academics on how knowledge and beliefs are to be
evaluated, I acquired the services of a peer researcher to assist me with the coding and interpretation of
the data to further enhance the trustworthiness of my study. I also verified my findings with findings
from the literature.
To reduce the Hawthorne effect the first observation was done without a prior interview or discussion
because the interview questions prior to the second and third observations could influence teachers’
206
behaviour in the classroom. I emphasised to the teachers the fact that I was interested in the
uniqueness of each teacher and my purpose was not to report their performances in class to their
superiors. I furthermore used the same interview schedules, including the same questions in the same
sequence for all interviewees. Section C of the last interview was based on the teachers’ knowledge of
the ML curriculum. I gave the teachers the option to answer the questions orally or in writing. I
mentioned the advantage of providing the answers in writing: that they might have felt less threatened
or pressured and that it also allowed them more time to think about the questions and to provide
valuable responses. In choosing to provide the answers in writing, they were requested to complete the
section in my presence as part of the interview. This was to ensure that the data obtained were credible,
as the teachers were not able to consult another teacher or the relevant documents.
5.6
Conclusions
Some conclusions regarding the relationship between ML teachers’ knowledge and beliefs and their
instructional practices appear below.
ML teachers’ instructional practices:
•
Given that the use of contexts should be the focus of ML lessons, the puzzling absence of the
use of contexts in three of the four teachers’ practices was notable and should be addressed.
Not just learners but teachers too need to understand the contexts in order to have informative
and enlightening class discussions.
•
ML teachers’ instructional practices should be predominantly learner-centred, including the use
of active learning instructional strategies such as cooperative learning and discussions.
•
The following aspects of ML lessons in particular deserve to be emphasized, not only during
teacher training but especially in post-teacher training: tasks should be logically sequenced; tasks
should not be too easy or too difficult; learners’ understanding should be monitored; learnerlearner interactions should be optimized; there should be variety in levels and types of oral
questioning during instruction; learners’ ideas and ways of thinking should be acknowledged
and appreciated consistently; and a positive learning atmosphere should be instilled.
Knowledge and beliefs:
•
ML teachers need to attain a certain level of MCK as well as knowledge of the teaching and
learning of ML before being allowed to teach the subject.
207
•
ML teachers need to attain an adequate sense of procedural knowledge and conceptual
understanding of the ML learners, the teaching of ML and the ML curriculum as well as
experience to develop PCK.
The relationship between ML teachers’ knowledge and beliefs and their
instructional practices:
•
ML student teachers as well as in-service ML teachers should be afforded ample opportunity to
enhance their MCK but also to engender an understanding of the theory and practice of PCK
as knowledge strongly influenced the ML teachers’ instructional practices.
•
ML student teachers (during initial teacher training) as well as practising ML teachers (during inservice training) should be sensitized to the importance of ensuring that their instructional
practices are consistent with their true beliefs and not their stated beliefs only.
•
Instead of merely claiming to be teaching in a constructivist manner, ML teachers should be
taught how to actually teach in a constructivist manner.
5.7
Recommendations for further research
Several aspects of the teaching and learning of ML require further research in order for ML to come
into its own as a viable and valuable subject in its own right. These include investigation into:
•
The knowledge required to engage learners in such a manner as to explore the depths of their
prior knowledge during teaching.
•
The nature and level of content knowledge that is required to teach ML effectively.
•
The ways in which ML teachers can transform their own MCK into learning facilitation
strategies that are pedagogically powerful, through the choice of appropriate teaching and
learning strategies and supporting materials.
•
Identification of authentic and relevant contexts that not only relate to learners’ daily lives, their
future workplace and the wider social, political and global environment, but how such contexts
can be applied effectively to the required lesson content.
•
The viability of developing PCK during teacher training.
•
ML teachers’ true and stated beliefs and the influence thereof on their instructional practices.
•
The development of effective questioning techniques and assessment strategies in the ML
classroom.
208
•
The guidance of ML teachers to becoming au fait with the extent to which language potentially
influences ML learners’ achievements and acquire the skills needed to deal with these issues in
their classrooms adequately.
•
5.8
ML learners’ expectations and experiences of the subject.
Limitations of the study
Data were gathered from a very small number of ML teachers and generalization of the results is
impossible. However, generalization was not an aim of the study. Another limitation is the fact that the
observations were all done in the second part of Term 2 and two of the four teachers were busy with
revision. Furthermore, more data regarding the teachers’ knowledge of the learners could have been
gathered during the observations if I had been party to the discourse between the teachers and
individual learners sitting at their desks. As my presence in class already influenced the teaching
process, I did not want to intrude furthermore on the learners’ learning process. I am also acutely aware
that different researchers may interpret my data differently. My own perspective is bound by space,
time and personal experience. Even though my conclusions were carefully scrutinized and confirmed or
refuted by my supervisors, my external coder as well as my participants, the possibility that subjectivity
may have influenced my findings cannot be ruled out.
5.9
Last reflections
It is with mixed feelings that I am making this attempt to bring my ‘doctoral journey’ to its conclusion.
I realize, among many other things that it has been a time of accelerated growth and learning for me,
both professionally and personally. I began this journey with a strong assumption that ML has a rightful
place in the school curriculum as a valuable subject and that this subject should be taught by a
mathematics teacher. Furthermore, I assumed that the success of this relatively new subject depended
strongly on the input, training, experience and perceptions of these (mathematics) teachers.
Initially I wanted to explore how ML teachers’ PCK influence their instructional practices, but as
my study developed, I realised that MCK and teachers’ beliefs also play a crucial role in teachers’
practices. This study allowed me the opportunity to become part of the lives of four ML teachers
whom I have observed and listened to as they shared their knowledge and beliefs with me. However,
even though I benefited a great deal from my exchanges with all four teachers, I wish to state that it
was especially during the time I worked with Elaine that I gained insight into how a teacher’s practice
could influence her knowledge and belief. My interactions with her changed the focus of the study
209
from exploring the influence of ML teachers’ knowledge and beliefs on their practices to exploring the
relationship between teachers’ knowledge and beliefs and their practices.
During the data analysis stage I became aware of the complexity involved in analysing teachers’
practices, knowledge and beliefs. I realized that the boundaries between these categorisations of
knowledge are very vague. More particularly, it dawned on me that not all stated beliefs are true beliefs
and that the boundaries between knowledge and beliefs are vague and blurred. I found the process of
conducting literature control a most exhausting and emotionally draining exercise. This was mainly the
case because of the paucity of research relating to my study, which made it extremely difficult to
compare my study with other studies. The vast majority of studies that could be related to mine were
conducted on very small samples –, in many cases involving one or two teachers only.
I hope that my findings will contribute to teacher training and theory and that this study will
contribute to the building of a mathematically literate nation. Maree (2011) voices my thoughts also
when he stated the following:
Since what happens in the classroom will eventually determine whether or not lasting change can be effected, the role
of the school [teacher] is crucial in creating an optimal learning environment … Learners should leave school better
equipped to cope with the challenges of university study and life itself. They need to be empowered to choose
appropriate careers, enter society and make meaningful social contributions.
210
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Appendices
Appendix A
Letter of consent to the ML teachers
Appendix B
Letter of consent to the principals
Appendix C
Letter of permission to the department
Appendix D
Ethical clearance certificate
Appendix E
Observation sheet for observing ML teachers’ lessons
Appendix F
Interview schedule 1 (Prior to lessons 2 and 3)
Appendix G
Interview schedule 2 (Final interview)
Appendix H
List of research studies for Literature Control
Appendix I
Analysis of discussions on Theme 1 and Theme 2
Appendix J
Additional information verifying Question 1
Appendix K
Additional information verifying Question 1
Appendix L
Declaration: External coder
226
Appendix A: Letter of consent to the ML teachers
FACULTY OF EDUCATION
Mrs. J.J. Botha
Natural Science Building 4-13
Groenkloof campus, UP
[email protected]
Tel: 082 475 6096
19 April 2011
Dear Ms/Mr ………………..………………
Letter of consent to the Mathematical Literacy teacher
You are invited to participate in a research project aimed at investigating the influence of Mathematical
Literacy teachers’ knowledge and beliefs on their instructional practices. This research will be reported upon
in my PhD thesis conducted at the University of Pretoria.
Your participation in this research project is voluntary and confidential. It is proposed that you form part of
this study’s data collection phase by being observed three times when teaching your Mathematical Literacy
class(es) and being individually interviewed twice. The lessons will be video recorded and the interviews will
be audio-taped by me in order to have a clear and accurate record of all the activities and communication
that took place.
The process will be as follows: during the third term of this year I would like to observe you teaching three
Grade 11 Mathematical Literacy lessons during school hours, preferably to different Mathematical Literacy
classes. I would like to conduct a short interview with you prior to the second and third lessons and another
interview at the end of the three observations. The duration of the interviews prior to the lessons will not be
more than 20 minutes and can be conducted during break or a free period you have. The duration of the
third and final interview will take a maximum of an hour and will be scheduled at a time convenient to you.
The focus of the questions is your knowledge and beliefs regarding Mathematical Literacy as subject, the
teaching thereof and the Mathematical Literacy learners. The interviews will be scheduled at a place
convenient to you.
Should you declare yourself willing to participate in this study, confidentiality and anonymity will be
guaranteed at all times. You may decide to withdraw at any stage should you not wish to continue with your
participation. Your decision to accept/decline involvement in this research will not influence your teaching
career in any way, nor will your participation be reflected in your performance appraisal.
If you are willing to participate in this study, please sign this letter as a declaration of your consent, i.e. that
you participate in this project willingly and that you understand that you may withdraw from the research
project at any time.
Yours sincerely
………………………………………………………….
Date: ………………………………
Researcher: Mrs. J.J. Botha
………………………………………………………….
Date: ………………………………
Co-supervisor: Dr. G. Stols
______________________________________________________________________________________
I the undersigned, hereby grant consent to Mrs. J.J. Botha to observe my classes and conduct
interviews with me for her PhD research.
Participant’s name ……………..…..………… Participant’s signature ………………………Date: ………………
E-mail address ……………………………………………….
227
Contact number ……………………………..
Appendix B: Letter of consent to the principals
FACULTY OF EDUCATION
Mrs. J.J. Botha
Natural Science Building 4-13
Groenkloof campus, UP
[email protected]
Tel: 082 475 6096
19 April 2011
Dear Dr/Ms/Mr ………………..………………
Letter of consent to the Principal
I hereby request permission to use your school for my research project. I would like to invite a Mathematical
Literacy teacher to participate in this research project aimed at investigating the influence of Mathematical
Literacy teachers’ knowledge and beliefs on their instructional practices. This research will be reported upon
in my PhD thesis conducted at the University of Pretoria.
Your participation in this research project is voluntary and confidential. It is proposed that the teacher forms
part of this study’s data collection phase by being observed three times when teaching Mathematical Literacy
class(es) and being individually interviewed three times. The lessons will be video recorded and the
interviews will be audio-taped by me in order to have a clear and accurate record of all the activities and
communication during the lesson.
The process will be as follows: during the third term of this year, should you look favourably upon my
request, I would like to observe the teacher teaching three Grade 11 Mathematical Literacy lessons,
preferably to different Mathematical Literacy classes during normal school hours. I would like to conduct a
short interview with the teacher prior to the second and third lessons and another interview at the end of the
three observations. The duration of the interviews prior to the lessons will not be more than 20 minutes and
can be conducted during break or a free period the teacher has. The duration of the third and final interview
will take a maximum of an hour and will be scheduled at a time convenient to the teacher. The focus of the
questions is on the teachers’ knowledge and beliefs regarding Mathematical Literacy as subject, the teaching
thereof and the Mathematical Literacy learners. The interviews will be scheduled at a time and place
convenient to the teacher.
Confidentiality and anonymity will be guaranteed at all times. Your decision to accept involvement in this
research will hopefully contribute to the improvement of Mathematical Literacy teachers’ practices. If you are
willing to allow a member of your staff to participate in this study, please sign this letter as a declaration of
your consent.
Yours sincerely
………………………………………………………….
Date: ………………………………
Researcher: Mrs. J.J. Botha
………………………………………………………….
Date: ………………………………
Co-supervisor: Dr. G. Stols
______________________________________________________________________________________
I the undersigned, hereby grant consent to Mrs. J.J. Botha to conduct her research in this school for
her PhD research.
School principal’s name ……………………………………..
School principal’s signature ...............................................
Date:………………………………
E-mail address ……………………………………………….
Contact number ……………………………..
228
Appendix C: Letter of permission to the department
FACULTY OF EDUCATION
Mrs JJ Botha
Aldoel Building C04
Groenkloof Campus
[email protected]
Tel: 082 475 6096
15 March 2010
GAUTENG DEPARTMENT OF EDUCATION
Dear Sir/ Madam
Request from GDE for permission to do classroom observations and to conduct interviews
I am currently enrolled as a doctoral student at the University of Pretoria, where I am also a lecturer in the
Department of Science, Mathematics and Technology Education. The title of my proposed thesis is as
follows: The influence of Mathematical Literacy teachers’ knowledge, beliefs and attitudes on their
instructional practices. ML is a valuable subject and it is crucial to attain its purpose in our country by
addressing problems experienced by both teachers and learners. My research concerns the ML teacher’s
role in the classroom situation. It is important to determine who the ML teachers are, what knowledge they
have regarding the subject and what beliefs and attitudes they hold. Furthermore I want to explore and
interpret the influence of those elements on these ML teachers’ instructional practices. I hope, at the end of
my research, to be able to make a contribution to the improvement of pre-service training in order to perk up
ML teachers’ instructional practices.
In order to collect data for this project, I would like to observe and interview a purposive sample of
Mathematical Literacy teachers, preferably grade 11 teachers at approximately six schools in and around
Tshwane. Each teacher will be observed three times and interviewed twice. My observations will be
unobtrusive.
I therefore formally request your permission to observe and interview Mathematical Literacy teachers at
schools in and around Tshwane in the second term of this year. I trust that my request will meet with a
favourable response.
Yours faithfully
………………………………….
Researcher: Mrs JJ Botha
……………………………..
Date
………………………………….
………………………………
Supervisor: Dr G Stols
Date
______________________________________________________________________________________
I the undersigned, hereby grant consent to Mrs JJ Botha to conduct research for her PhD at schools
in and around Tshwane.
…………………………………
Departmental officer
……………………………..
Date
229
Appendix D: Ethical clearance certificate
230
Appendix E: Observation sheet for observing ML teachers’ lessons
OBSERVATION SHEET
(To be used for all three observations per teacher)
Name of school
Name of researcher
Mrs. J.J. Botha
Subject observed
Mathematical Literacy (ML)
Grade observed
Number of learners in class list (present in class)
Topic of the lesson
Name of teacher
Date of observation
Observation number
Table A and Table B are based on the different dimensions of teachers’ lessons. Use the indicators in Table B to complete
Table A.
Table C and Table D are based on the teachers’ pedagogical content knowledge (PCK) and beliefs. Use the indicators in
Table D to complete Table C.
231
Table A.
ASSESSING TEACHERS’ INSTRUCTIONAL PRACTICES THROUGH OBSERVATIONS
(Videotape lesson and make field notes during observations)
LESSON DIMENSIONS
COMMENTS (Support with examples)
Tasks
Modes of representation
Motivational strategies
Sequencing/difficulty level
Discourses
Teacher-learner interactions
Learner-learner interactions
Questioning
Learning environments
Social/intellectual climate
Modes of instruction/pacing
Administrative routines
Other
Mathematical content knowledge
Contextual knowledge
Evaluation scale: Table A: Description of the scale. 3 = commendable (strong presence of indicator); 2 = satisfactory (indicator is somewhat present); 1 =
needs attention (there is very little presence of indicator); N/O = not observed or not applicable.
232
Table B.
EVALUATING TEACHERS’ PCK AND BELIEFS THROUGH OBSERVATIONS AND INTERVIEWS
(Videotape lesson and make field notes during observations; audio-tape the interviews)
TEACHERS’ PCK
AND BELIEFS
COMMENTS (Support with examples)
PCK AND BELIEFS
Mathematical content
Knowledge
Content and learners
Content and teaching
Curriculum
BELIEFS
Nature of mathematics
Evaluation scale: Table B: Description of the scale. 3 = commendable (strong presence of indicator); 2 = satisfactory (indicator is somewhat present); 1 =
needs attention (there is very little presence of indicator); N/O = not observed or not applicable.
233
Appendix F: Interview schedule 1 (Prior to lessons 2 and 3)
INTERVIEW SCHEDULE 1
Semi-structured interview
GENERAL INFORMATION
Name of school
Name of researcher
Name of teacher
Date of interview
Teacher’s qualification
Level of Mathematics education
Number of years teaching Mathematics
Number of years teaching ML
Courses attended on teaching ML
Mrs. J.J. Botha
Based on the lesson that you are about to present and your preparation for the lesson, please answer
the following questions:
1.
What is the topic of the lesson you are going to present?
2.
a) What mathematical content do you predict the learners will understand?
b) Why do you think they will comprehend this content?
3.
a) What mathematical content do you predict the learners will not understand?
b) Why do you think they will not understand this content?
4.
a) Tell me about the context to which the mathematical content is applied in today’s lesson.
b) Is the context familiar or unfamiliar to the learners?
c) If unfamiliar, how do you plan to make it comprehensible to the learners?
5.
How did you plan to approach the lesson in order to bring the learners to understand the
content and context?
6.
a) Tell me about the task(s) you are going to give them.
b) In which way, in your opinion, will the learners approach these task(s)?
7.
What prior knowledge is needed by the learners to enable them to understand today’s new
work?
8.
What alternative or preconceptions do you believe the learners could have that may serve as
misconceptions?
234
ENDinterview)
OF INTERVIEW
Appendix G: Interview schedule 2 (Final
INTERVIEW SCHEDULE 2
Open-ended and semi-structured interview
GENERAL INFORMATION
Name of school
Name of researcher
Mrs. J.J. Botha
Name of teacher
Date of interview
This interview consists of three sections. The first section (Section A) is an open discussion based on
the lessons presented and focuses on the teacher’s demonstrated PCK and beliefs. The purpose is to
give the teachers the opportunity to reflect on their lessons and to identify justification for their
behaviour in the classroom. The second section (Section B) is a discussion according to a set of
predetermined questions on the teacher’s beliefs regarding the nature of mathematics as discipline, ML
as subject, the ML learners, the teaching of ML and the curriculum. The third section (Section C)
consists of questions regarding the NCS and CAPS and should be answered in writing.
SECTION A
Oral questions based on the observed lessons
Questions will be compiled once the observations have been done and will most probably vary from
teacher to teacher. The questions will be based on incidents where PCK was identified during the
lessons. Clips from the video recordings will be used as probes. Possible questions are the following:
1.
Tell me about your positive experiences regarding
a) the learners
b) your teaching of the lesson
2.
Tell me about your negative experiences regarding
a) the learners
b) your teaching of the lesson
3.
I noticed that you used … (lecturing, group work, discussion etc.) in today’s lesson.
Why did you choose this teaching strategy for the lesson?
235
SECTION B
Oral questions based on the teacher’s beliefs
The nature and value of mathematics and ML:
1.
How do you view mathematics as discipline?
2.
Complete the sentence: Mathematics is ……………
3.
How do you view ML as subject?
4.
What do you believe is the value of mathematics?
5.
What do you believe is the value of ML?
6.
What is your role as teacher in your ML classroom?
ML learners:
1.
Describe your Grade 11 ML learners in terms of their
a) mathematical abilities
b) motivation
2.
Give me a description/profile of your ML learners.
3.
How can you improve your learners’ appreciation of the subject ML?
4.
How can you improve your learners’ participation in the lesson?
5.
What is your belief about the way learners proficiently learn new work?
Teaching of ML:
1.
How do you feel about teaching ML?
2.
Describe the ideal ML classroom in terms of
a) instructional strategies used
b) discourse
c) learning environment
3.
How does this ideal classroom compare with your own class?
4.
What are your goals in teaching ML?
5.
To what extent is mathematical content knowledge a prerequisite to teach ML?
6.
a) Does the teaching approach of ML differ to that of Mathematics?
b) If you experience a difference, tell me about your experiences in teaching Mathematics versus
ML.
236
SECTION C
Written questions based on the teacher’s knowledge regarding the curriculum
1.
How does the Department of Education define Mathematical Literacy?
__________________________________________________________________________
__________________________________________________________________________
2.
What is the purpose of Mathematical Literacy according to the Department of Education?
__________________________________________________________________________
__________________________________________________________________________
3.
Which contexts does the Department of Education suggest you should use in teaching
Mathematical Literacy? ________________________________________________________
__________________________________________________________________________
4.
Write down what you know about the new Curriculum Assessment Policy Statement (CAPS)
for Mathematical Literacy. _____________________________________________________
__________________________________________________________________________
5.
a) Which topic are you currently teaching? _________________________________________
b) Name the instructional materials you use for your lessons on this topic.
__________________________________________________________________________
c) Comment on the availability and usefulness of the instructional materials. ______________
__________________________________________________________________________
6.
a) Which textbook(s) are you using? ______________________________________________
b) Which other material do you use? _____________________________________________
c) In your opinion, what are the strengths of these books and materials?
__________________________________________________________________________
d) In your opinion, what are the weaknesses of these books and materials?
__________________________________________________________________________
7.
Are you aware of the curriculum content being studied by your learners in other subjects that
integrate with Mathematics/ML? If yes, tell me about it.
__________________________________________________________________________
237
8.
a) Which departmental documents exist that you know of?
__________________________________________________________________________
__________________________________________________________________________
b) Which of these departmental documents do you find useful and valuable? ______________
__________________________________________________________________________
9.
What are the learning outcomes for Mathematical Literacy?
__________________________________________________________________________
__________________________________________________________________________
10.
A list of concepts and content to be covered in grade 10, 11 or 12 are provided per Learning
Outcome in the following table. Indicate in which grade the specific concept/content is
introduced. (Only complete the Learning Outcomes applicable to the observed lessons)
Table E: Concepts and content per learning outcome (DoE, 2003a; p. 38-42)
CONCEPTS AND CONTENT PER LEARNING OUTCOME
LEARNING OUTCOME 1
Cost price and selling price
Complex formulae
Currency fluctuations
Direct proportion
Financial and other indices
Fractions, decimals, percentages
Inverse proportion
Positive exponents and roots
Profit margins
Rate
Ratio
Ratio and proportion
Simple and compound growth
Simple formulae
Square roots and cube roots
Scientific notation
238
Grade
10
11
12
Taxation
The associative, commutative and distributive laws
LEARNING OUTCOME 2
Cartesian co-ordinate system
Compound growth
Formulae depicting relationships between variables
Graphs depicting the relationship between variables
Graphs showing the fluctuations of indices over time
Inverse proportion
Linear functions
Maximum and minimum points
Simple linear programming (design and planning problems)
Simple quadratic functions
Solution to linear, quadratic and simple exponential equations
Solution to two simultaneous linear equations
Rates of change (speed, distance, time)
Tables of values
LEARNING OUTCOME 3
Angles (0˚-360˚)
Basic transformation geometry, symmetry and tessellations
Circles
Compass directions
Conversion of measurements between different scales and systems
Conversion of units within the metric system
Floor plans
Location and position on grids
Measurement in 3D (angles included, 0˚-360˚)
Measurement of length, distance, volume, area, perimeter
Measurement of time (international time zones)
Polygons commonly encountered (triangles, squares, rectangles that
are
not squares,
parallelograms,
trapesiums,
regular
hexagons
Properties
of plane
figures and solids
in natural
and cultural
forms
Scale drawings
Scale models
Sine rule, cosine rule, area rule
239
Surface areas and volumes of right pyramids and right circular cones
and spheres
Surface area and volumes of right prisms and right circular cylinders
Theorem of Pythagoras
Trigonometric ratios: sin x, cos x, tan x
Views
LEARNING OUTCOME 4
Bivariate data
Compound events
Construction of questionnaires
Contingency tables
Cumulative frequencies
Histograms
Intuitively-placed lines of best fit
Line and broken-line graphs
Mean, median, mode
Ogives (cumulative frequency graphs)
Percentiles
Pie charts
Populations
Probability
Quartiles
Relative frequency
Scatter plots
Selection of a sample
Selection of samples and bias
Single and compound bar graphs
Standard deviation (interpretation only)
Tables recording data
Tally and frequency tables
Tree diagrams
Variance (interpretation only)
END OF INTERVIEW
240
Appendix H: List of research studies for Literature Control
AUTHOR AND
YEAR
Bansilal, S.
(2008)
Bansilal, S.,
Mkhwanazl, T. &
Mahlaboratoryela, P.
(2010)
Bowie, L.
(2009)
Fransman, J.S.
(2011)
TITLE OF ARTICLE
PARTICIPANTS
SOURCE
An exploration of teachers’ difficulties
with certain topics in Mathematical
Literacy
Mathematical Literacy teachers’
engagement with contexts related to
personal finance
In-service teachers
in ACE (ML)
programme
In-service teachers
in ACE (ML)
programme
Proceedings of AMESA
2008
Critical issues in school mathematics
and science: pathways to progress
Theory
Proceedings of an
Academy of Science of
South Africa Forum
Unpublished dissertation
for the degree Master of
Education
Exploring the practices of teachers in
mathematical literacy training
programmes in South Africa and
Canada
Frith, V.
A framework for understanding the
(2009)
quantitative literacy demands of higher
education
Frith, V.
How to make every graph a straight
(2010)
line (or not!)
Frith, V.
Towards understanding the
(2011)
quantitative literacy demands of a first
year medical curriculum
Geldenhuys, J.,
Grade 10 learners’ experience of
Kruger, C. & Moss, J. Mathematical Literacy
(2009)
Glover, H. & King,
The subject knowledge levels of some
L.
Mathematical Literacy teachers
(2009)
In-service teachers
in ACE (ML)
programme
Theory
Mathematics
University students
190 Grade 10
learners from three
types of schools
In-service teachers
in ACE (ML)
programme
241
Proceedings of
SAARMSTE 2010
APPLICABILITY
IP
Tasks: TSL
PCKB
MCK; PCK
Curr: C7
Discourse: DTL
Curr: C7
N/A
Learning
environment:
LEC, LESP
Journal: South African
Journal of Higher
Education
Learning and Teaching
Mathematics
African Journal of Health
Professions Education
Proceedings of
SAARMSTE 2009
Proceedings of
SAARMSTE 2009
MCK; PCK
(T); Curr: C1,
C3, C4, C5, C8
N/A
N/A
N/A
Learning
environment:
LESP
Curr: C1, C7
MCK
Govender, V.G.
(2008)
Govender, V.G.
(2011)
Govender, V.G.
(2011)
Graven, M. &
Venkat, H.
(2009)
Graven, M.
(2011)
Graven, M.
(2011)
Lessons learnt from the 2007 GMSA
foundation Mathematical Literacy
Olympiad
An investigation into learners’
approaches to solving problems in
mathematical literacy
University students’ experiences of a
mathematics service module:
Numerical Skills for Nursing
Mathematical Literacy
Mathematical Literacy in South Africa:
Increasing access and quality in
learners’ mathematical participation
both in and beyond the classroom
Creating new mathematical stories:
Exploring potential opportunities
within Maths Clubs
ML Olympiad
12 Grade 12
learners from 1
school
University students
Proceedings of AMESA
2008
Proceedings of AMESA
2011
Book: Chapter 4 in
Critical issues in
mathematics education
Theory
Book: Chapter 35 in
Mapping equity and
quality in mathematics
education
Proceedings of AMESA
2011
Hechter, J.
(2011)
Analysing and understanding teacher
development on a Mathematical
Literacy ACE course
In-service teachers
in ACE (ML)
programme
Unpublished thesis for
Master of Science
Hechter, J.
(2011)
Case studies of teacher development
on a mathematical literacy ACE
course
In-service teachers
in ACE (ML)
programme
Proceedings of AMESA
2011
Mthethwa, T.M.
(2009)
An analysis of Mathematical Literacy
curriculum documents:
cohesions, deviations and worries
Theory
Proceedings of AMESA
2009
242
Learning
environment:
LESP
Proceedings of
SAARMSTE 2011
Theory
Maths Clubs
N/A
Curr: C7
N/A
Tasks: TMS;
Learning
environment:
LEC; LESP
Curr: C7
Do not have the book
Discourse;
Learning
environment:
LEC;LESP
Tasks;
Discourse: DTL;
Learning
environment
Tasks: TMS;
TSL; Discourse:
DQ; Learning
environment:
LEC
Beliefs
Curr: C7
Question 4
N/A
Nel, B.
(2011)
Investigating the transformation of
teacher identity of participants in an
Advanced Certificate in Education in
Mathematical Literacy (Reskilling)
programme at a South African
University
The great Mugg and Bean mystery
In-service teachers
in ACE (ML)
programme
Proceedings from
SAARMSTE 2011
Theory
Journal: Learning and
Teaching Mathematics
North, M.
(2008)
North, M.
(2010)
Rughubar-Reddy, S.
(2010)
Progression in Mathematical Literacy
Theory
How mathematically literate are the
matriculants of 2008?
Beyond Numeracy: Values in the
Mathematical Literacy classroom
Grade 12
performance
5 Grade 10 learners
from 1 school
Proceedings of AMESA
2008
Proceedings from
AMESA 2010
Proceedings of
SAARMSTE 2010
Sidiropoulos, H.
(2008)
The implementation of a mandatory
mathematics curriculum in South
Africa: The case of mathematical
literacy
Two Grade 10 ML
teachers from 2
different schools
PhD thesis
Venkat, H.
(2008)
Senior certificate examinations for
mathematical literacy: findings from a
small study
Opening up spaces for learning:
Learners’ perceptions of
Mathematical Literacy in Grade 10
Grade 12 results
Journal: Learning and
Teaching Mathematics
All Grade 10 ML
learners in 1 school
Journal: Education as
Change
Critiquing the Mathematical Literacy
assessment taxonomy: Where is the
reasoning and the problem solving?
Theory
Journal: Pythagoras
North, M.
(2008)
Venkat, H. &
Graven, M.
2008
Venkat, H., Graven,
M., Lampen, E. &
Nalube, P.
(2009)
243
Beliefs
Question 4
Learning
environment:
LESP
Curr: C7
N/A
Learning
environment:
LEC
Learning
environment:
LESP
Curr: C1,C7,
C8
Beliefs
PCK (T/C)
Q.3
N/A
Tasks: TMS;
Discourse: DLL;
Learning
environment:
LEC,LESP,LEA
Learning
environment:
LESP
Problem solving
Venkat, H., Graven,
M., Lampen, E.,
Nalube, P. & Chitera,
N.
(2009)
Venkat, H.
(2010)
Vithal, R.
(2008)
Vithal, R.
(2008)
Zengela, C.
(2008)
‘Reasoning and reflecting’ in
Mathematical Literacy
Theory
Assessment tasks
Journal: Learning and
Teaching Mathematics
Tasks: TSL;
Discourse;
Scaffolding
Exploring the nature and coherence of
mathematical work in South African
Mathematical Literacy classrooms
Mathematical power as political power
– the politics of mathematics
education
Mathematical Literacy and
globalization
1 Grade 11 ML
teacher
Journal: Research in
Mathematics Education
Tasks: TMS;
Discourse
Theory
Book: Chapter in Critical
issues in mathematics
education
Book: Chapter 1 in
Internationalisation and
globalization in
mathematics and science
education
Learning and Teaching
Mathematics
Turning myself around – Experiences
of teaching Mathematical Literacy
Theory
1 Grade 12 teacher
244
Curr: C7
N/A
N/A
Curr: C1, C7
Appendix I: Analysis of discussions on Theme 1 and Theme 2
An analysis of discussions on Theme 1 and Theme 2 produced the following tables:
Table: Findings of my study listed according to a Teacher and Learner-centred approach
Teacher-centred
(Monty and Alice)
Learner-centred
(Denise and Elaine)
Instructional practices
Did not point out the value of mathematics to the
learners
Did not determine or appropriately use learners’
prior knowledge
Did not encourage learner participation and did
not require learners to explain their answers
Instead of providing scaffolding, either reexplained the work or solved the problem for
them
Insufficient knowledge of oral questioning in class
Created a formal atmosphere where focus was on
mastering the content
Used direct instruction as instructional strategy
Board work were incomplete and disorganised
Pointed out the value of mathematics to the
learners (Not Denise)
Lessons were build on learners’ prior knowledge
Involved learners through class discussions and
learners working on the board where learners
could also explain and/or demonstrate their work
Provided scaffolding to support learner
understanding
Asked various types of oral questions on different
levels (Not Denise)
Created a class atmosphere where learners were
comfortable and confident
Used class discussions and learners working on
the board as instructional strategies
Board and transparency work were organised and
no errors were made
PCK and beliefs
Superficial knowledge regarding learners. Believed
learners come to understanding by looking at
several examples and through much practice
Superficial knowledge regarding the teaching of
ML. Believed the teaching of ML is the same as
that of teaching Mathematics
Specific knowledge regarding learners. Believed
learners come to understanding by being involved
through sharing their ideas and where the teacher
build on their prior knowledge
Specific knowledge regarding the teaching of ML.
Believed ML teaching differs from teaching
Mathematics
245
Appendix J: Additional information verifying Question 1
I found that two of the four instructional practices of the ML teachers in my study can be described as
being exclusively teacher-centred, one teacher’s practice can be described as a combination of learnerand teacher centred, leaning more towards learner-centred, while the fourth teacher’s practice could be
described as exclusively learner-centred.
The practices of Monty and Alice
Monty and Alice’s instructional practices can be described as teacher-centred where they believed their
role as teachers was to transmit mathematical content, demonstrate procedures for solving problems,
and explain the process of solving sample problems. This finding is in accordance to the findings of
Artzt et al. (2008). From the observations and interviews prior to the observed lessons I realised that
their focus was on transmitting mathematical content and not on the needs of the learners to develop
conceptual understanding. Their practices are characterised by (according to the three lesson
dimensions):
•
Tasks: Not pointing out the value of mathematics so that the learners could appreciate the
mathematics learned; tasks being illogically sequenced; tasks being too easy or too difficult or
excessive; selecting tasks only from Level 1 of the ML Assessment Taxonomy;
•
Discourse: An absence of monitoring learners’ understanding; Instead of providing scaffolding,
solving the problems for the learners; expressing irritation with learners’ wrong answers; no
constructive learner-learner interaction; low level questioning with inappropriate wait times to
engage and challenge learners’ thinking;
•
Learning environment: Formal atmosphere where the focus was on mastering the content;
using direct instruction as instructional strategy; learners being passive recipients of
information.
There were differences between Monty and Alice’s practices: Alice’s practice was largely dysfunctional,
with inattentive learners and ineffective teaching. She did not connect the learners’ prior knowledge
with the new mathematical situation. As both Monty and Alice are novice teachers, a plausible
hypothesis seem to be the following: The difference between their practices could be attributed to the
fact that Alice had no formal mathematics education training, but Monty completed a BEd with
Mathematics and Methodology of Mathematics as major subjects. It is interesting to note that the
teacher-centred approach can serve as a mask for teachers who do not possess full knowledge of the
content, students and pedagogy (Artzt et al., 2008, p. 35). Compared to Franke et al.’s (2007) view of a
productive practice being a practice where the teacher creates ongoing opportunities for learning, the
246
practices of Monty can be described as somewhat unproductive, where Alice’s practice was
unproductive.
The practices of Denise and Elaine
Denise’s instructional practice can be described as a combination of learner- and teacher-centred,
leaning more towards being teacher-centred, while Elaine’s instructional practice can be characterised as
teacher-centred. Their purpose was that learners should develop both procedural and conceptual
understanding of the content. Using a learner-centred approach to teaching requires the teacher to
create opportunities for learners to come to understanding by being actively engaged with one another
and the problem solving process (Artzt, et al., 2008). Their practices are characterised by (according to
the three lesson dimensions):
•
Tasks: Lessons being built on learners’ prior knowledge; representations contributing to the
clarity of the lessons; tasks being logically sequenced and at a suitable level of difficulty;
•
Discourse: Encouraging learner participation; meaningful discourse between the teacher and the
learners; providing scaffolding to support learner understanding; recognising learners’
misunderstandings and misconceptions;
•
Learning environment: Having the ability to create learning environments that contributed to
proficient learning; having positive attitudes towards the subject and the learners; involving
learners through class discussions and learners working on the board; effective managing of
time to maximise learners involvement; board and overhead projector work being organised
and no errors were made.
There are some differences between the practices of Denise and Elaine. The following are
characteristics of only Elaine’s practice:
•
Tasks: Exploring contexts using mathematical content; pointing out the value of mathematics in
everyday-life situations to the learners; selecting tasks from Level 1-4 of the ML Assessment
Taxonomy;
•
Discourse: Having learners demonstrate and explain their answers; asking various types and
different levels of oral questions;
Elaine’s practice can therefore be described as a productive instructional practice as she created
ongoing opportunities for learning to occur (Franke et al., 2007) while Denise’s can be described as
somewhat productive.
247
Appendix K: Additional information verifying Question 2
The MCK of the four participants are described in the verification of question 2.
•
PCK and beliefs of two novice teachers
Knowledge and beliefs of ML learners: Monty and Alice believe that learners learn best by
receiving clear information transmitted by a knowledgeable teacher, a finding Artzt et al. (2008) also
found where teachers used a teacher-centred approach. They could not predict what content the
learners would and would not understand; how they would come to understanding; and what possible
misconceptions the learners might have.
Knowledge and beliefs of ML teaching: Once Alice introduced tasks that caused confusion for
her and the learners, she did not know how to adjust - a phenomenon that is according to Artzt et al.
(2008) typical of teachers in the initial phase of teaching. Monty and Alice could not predict the prior
knowledge that should have been present in the lesson for the learners to understand the new content
and could not choose appropriate instructional strategies to use in their teaching of ML. They
furthermore used examples too basic or too complex throughout the lesson presentations. They
believed the teaching of ML is different to the teaching of Mathematics and that group work and
discussions should be used in teaching ML.
Knowledge and beliefs of the ML curriculum: Monty and Alice had no knowledge of other
subjects integrating with ML, although they did have some knowledge about the definition, purpose
and learning outcomes of ML, but not of the various departmental documents. Most importantly they
taught content in the absence of contexts and did not adhere to the DoE’s (2008b) aim to develop in
learners [t]he ability to use basic mathematics to solve problems encountered in everyday life and in
work situations (p. 8), although they believe real-life scenarios should be used. They believe
mathematics as a constructivist discipline which is logical and that ML is valuable to learners.
According to Monty, ML is a unique subject, but Alice believes that ML is a lower level of
Mathematics.
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PCK and beliefs of two experienced teachers
Knowledge and beliefs of ML learners: Denise and Elaine have specific knowledge of learners’
prior knowledge, experiences and abilities. They could predict what learners would and would not
understand; how they would come to understanding; and what misconceptions learners have and
typical errors the learners make. They believe the learners should be active participants in their own
learning by explaining the work to each other in small groups.
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Knowledge and beliefs of ML teaching: Since they understand how learners learn mathematics,
they knew how to select appropriate instructional strategies and could adjust their teaching when
required. They predicted and integrated the prior knowledge needed to enable the learners to
understand the work and chose appropriate instructional strategies. They believed their role as teacher
is facilitating learners’ learning through selecting appropriate tasks and leading the discussions in class.
They furthermore believed that teachers should provide opportunities where learners can discover and
construct their own meaning through meaningful communication.
Knowledge and beliefs of the curriculum: Only Elaine knew about other subjects that integrate
with ML, and she knew the definition and learning outcomes. Denise and Elaine knew the purpose of
ML and were familiar with various departmental documents. Only Elaine taught the mathematical
content in context where all her tasks were based on applicable real-life scenarios (DoE, 2003a). Denise
taught content only although she believes a teacher should use contexts. Both these teachers believe
mathematics is a flexible and logical discipline and that ML is a unique subject and valuable to the
learners.
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