The Development of Graph Understanding in the Mathematics Curriculum

The Development of Graph Understanding in the Mathematics Curriculum
April 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Report for the NSW Department of Education and Training
Jane Watson and Noleine Fitzallen
Acknowledgements
The preparation of this report was supported by funding from the New South Wales Department of Education and Training.
The Development of Graph Understanding in the Mathematics Curriculum.
© State of New South Wales through the Department of Education and Training, 2010. This work may be freely
reproduced and distributed for personal, educational or government purposes. Permission must be received from the
Department for all other uses.
ISBN 9780731386871
SCIS 1458886
The Development of Graph Understanding
in the Mathematics Curriculum
Contents
1. Introduction
5
2. History and other background
7
2.1 Preliminary comments: Describing graphs
7
2.1.1 Inconsistency of terminology
8
2.1.2 Defining “bar-like” representations
10
2.2 Historical developments in graphing
12
2.2.1 Spatial organisation for data analysis
13
2.2.2 Discrete quantitative comparisons
13
2.2.3 Continuous distributions
14
2.2.4 Multivariate distributions and correlation
14
2.2.5 20th Century and future developments
15
2.3 The history of statistical graphing in the school curriculum
16
2.4 Current graphing curricula
19
2.4.1 Description of curricula
20
2.4.2 Comparison of curricula
24
3. Graphing and technology
26
4. Development of graph creation and interpretation
37
4.1 Development based on individual tasks
37
4.2 Development based on surveys
41
4.3 Development based on student interviews
48
4.4 Development suggested by the two Rasch analyses
56
4.5 Building a general developmental model for graph creation
57
4.5.1 The first cycle: The concept of graph
58
4.5.2 A second cycle: The ability to create or choose appropriate
graphs when more than one attribute is involved 59
4.5.3 A second cycle: The ability to create or choose appropriate
graphs for large data sets
60
4.5.4 The third cycle: Informal decision-making for graphs
61
4.6 Building a general developmental model for graph interpretation
62
5. Implications
64
5.1 The NSW context: Working Mathematically
64
5.2 Sequencing learning and the curriculum
65
References
© State of New South Wales through the NSW
Department of Education and Training, 2010
69
Contents
3
The Development of Graph Understanding
in the Mathematics Curriculum
4
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
1. Introduction
Graphs play a highly significant role across the mathematics curriculum, providing
visual means of presenting information that may be held for example in a functional
relationship or a data set. The visual representations provide numerical, pictorial,
and statistical information by combining symbols, points, lines, a coordinate system,
numbers, shading and colour (Tufte, 1983), with the aim of conveying information
quickly and efficiently. This report focuses on the application of graphs for portraying
data, and their potential as instruments for reasoning about quantitative information.
Whereas the conventions for creating graphs in the algebraic side of mathematics in
various coordinate systems appear fixed, in statistics there is much more flexibility and
lack of agreement on what the conventions should be. This may be due to the more
recent emergence of the field of statistics and in the next 50 years perhaps more rigid
conventions will emerge. It also may be that graphing in statistics has needed to be
more flexible because the stories held within data sets are so varied. Authors such as
Tufte (1983, 1990, 1997) and Wainer (1997) demonstrate the very large differences in
representations of data that have been used to tell the stories of various kinds of data.
Most of the graphs they show are not included in the school mathematics curriculum,
probably because of the influence of the rest of the curriculum in requiring tight
conventions and because there would be too many “types” to cover reasonably in an
already crowded curriculum. This rigidness has encouraged various graph types to be
introduced at particular stages of the curriculum, only to be superseded by other graph
types introduced later on. It is somewhat unfortunate for example that the pictograph,
which is included in the mathematics curriculum for the early years of schooling, has
traditionally been forgotten, whereas it is often used in media and older students
need to be able to analyse such forms critically, particularly where “area” is involved in
representing quantity.
One of the difficulties with being too rigid in prescribing graphical conventions for
statistical data across the school years is that it may have the effect of stifling students’
creativity in thinking of ways to tell the stories in their data sets. Research such as that
of Moritz (2006) demonstrates many successful but unconventional attempts by quite
young students to show association between variables. To suggest to teachers that
students cannot display stories of association until they have been taught the rules
for creating a scatterplot, would be very unfortunate. Allowing creativity within bounds,
however, is likely to put stress on the assessment of graphing skills. “Correct” answers
for items such as “draw a graph to show that for children, as they get older they grow
taller” and “draw a scatterplot for the following data set of children’s ages and heights”
are unlikely to look similar to each other. The second will display the procedural skills
whereas the first is likely to determine if students understand how to connect the two
variables and create a representation.
Another difficulty arises when students use software applications to create graphs.
Software applications are designed to enable students to visualise data to promote
sense-making from the arrangement of information in space. At their best, they provide
dynamic interactive structures that can be manipulated easily, reducing the burden
of graph creation on students. Some graphing software packages, however, are not
interactive and apply rigid graphing conventions, often producing graphs that do
not make sense and are not useful. An emphasis on applying graphical conventions
enforced by the software may not only limit the way in which students use graphing
© State of New South Wales through the NSW
Department of Education and Training, 2010
Introduction
5
The Development of Graph Understanding
in the Mathematics Curriculum
software to be creative but also limit the way it can be used to influence students’
thinking and understanding. The selection of graphing software to enact the curriculum
should be based on its ability to provide the best learning environment for achieving the
outcomes of developing students’ thinking about data as well as graph creation.
How to build creativity of graph construction into the curriculum given the constraints
noted is a great challenge. Curricula need to be constructed and implemented carefully
and writing realistic assessment items (plus having the resources to mark them) is
not easy. Teachers also need to have enough appreciation of tasks undertaken in
the classroom so that they can recognise appropriate uses of a variety of graphical
representations in order to guide both adaptations of created forms and movement
toward conventions. There is no doubt that conventions are needed but students need to
appreciate why particular graphs do the appropriate job of telling the story in the data.
The underlying themes that contribute to this report are:
• the history of graphing, both outside and inside the school curriculum;
• the 21st century graphing technology, including what it does and does not
offer;
• research on the development of student understanding;
• recognition of the close and critical relationship of graph creation and graph
interpretation; and
• the implications of graph understanding for Working Mathematically
interpreted as facilitating decision-making.
These themes contribute to the recommendations for implementation of a 21st century
graphing curriculum.
6
Introduction
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
2. History and other background
2.1 Preliminary comments: Describing graphs
Tufte’s three books, The Visual Display of Quantitative Information (1983), Envisioning
Information (1990), and Visual Explanations: Images and Quantities, Evidence and
Narrative (1997), provide a commentary on the different types of visual images
developed since the 1600s. They show a variety of representations that are often
complex and multi-dimensional, requiring well developed analytical and critical thinking
skills to interpret. They also illustrate the way in which powerful imagery has been used
to communicate with audiences.
A recurring theme throughout Tufte’s three books is the need for graphing practices to
be committed to finding, telling, and showing the truth about the data. He emphasised
the importance of visual representations being able to sum up and convey information
as well as stimulate ideas. He also recognised that the construction of data displays is
as much about reasoning about statistical evidence as it is about displaying statistical
information effectively. The effectiveness, however, depends heavily on the ability of
the user to understand the imagery used and the conventions applied. Playfair (1805)
recognised the potential complexity of graphs and cautioned:
Opposite to each Chart are descriptions and explanations. The reader
will find, five minutes attention to the principle on which they are
constructed, a saving of much labour and time; but, without that trifling
attention, he may as well look at a blank sheet of paper as at one of the
Charts. (p. xvi)
With this in mind it seems appropriate to develop an understanding of the way graphs
are structured to appreciate the way in which they communicate information. Although
there are many types of graphs they are all made up of the same basic-level constituents
(Kosslyn, 1989). Kosslyn suggests a schema for the analysis of graphs that can be
used to communicate information clearly and concisely. The elements include the
“background,” the “framework,” the “specifier,” and the “labels.” Fig. 2.1 illustrates the
basic-level constituents of a typical graph.
Figure 2.1. The basic-level constituent parts of a graph (Kosslyn, 1989, p. 188).
The background is the pattern over which the other component parts of a graph are
presented. In most instances the background is blank as is not necessary to include
a pattern or picture. The pattern of a background such as a photograph can assist in
© State of New South Wales through the NSW
Department of Education and Training, 2010
History and other background
7
The Development of Graph Understanding
in the Mathematics Curriculum
conveying the information of the graph but when too detailed, may interfere with the
ability to read the graph.
The framework extends to the edges of the graph and its function is to organise the
graph as a meaningful whole. Some graphs may have an inner framework which is
nested within the outer framework. An inner framework is a structure (e.g., a grid) that
maps points on the outer framework to other parts of the display.
The specifier conveys specific information about the framework by mapping parts of the
framework to other parts of the framework. The specifier may be a point, line, or bar and
is often based on a pair of values.
The labels of a graph are an interpretation of a line or region. They may be letters, words,
or pictures that provide information about the framework or the specifier.
To analyse graphs it is necessary to understand the interrelated connections among
the constituents of a graph. The connections foster the interpretation of graphs on
three levels (Kosslyn, 1989). First the individual elements and their organisation can
be described. Second, understanding of the display can be determined by looking
at the relations between the elements of the graph. Third, the analysis can extend
to interpretation of the symbols and lines that goes beyond the literal reading of the
information. The interpretation of the meaning of the graph is conveyed by the way in
which the information is organised.
Influenced by the work of Kosslyn, Curcio (1989) considered school students’
interpretation of graphs from three perspectives, reading data directly, reading
“between” the data, and reading “beyond” the data. Shaughnessy (2007) added to this
by suggesting the further need to read “behind” the data. These phrases reflect to some
extent the increasing demands of the levels suggested by Kosslyn (1989). The work of
Curcio in turn influenced other developmental models as is discussed in Section 4.
2.1.1 Inconsistency of terminology
One of the frustrating aspects of studying the history of graphing, research on graphing
and suggestions for specific graphs to be included in the school curriculum is the lack
of consistency in nomenclature and definition. Sometimes the same name is applied to
slightly different representations and sometimes a particular graph has several names.
The bar graph or bar chart is one that has various definitions, sometimes representing a
total frequency of a category or individual values of data, and at other times displaying
internal frequencies or percentages with respect to the category. The box plot or boxand-whisker plot is plagued with distinctions about how long its whiskers ought to be.
For younger students the whiskers extend to the extreme values in the range of the data
but later some definitions insist that the whiskers should be 1.5 times the length of the
interquartile range, with values further out marked individually as outliers. Moore and
McCabe (1989) also suggest that at times for large data sets it may be appropriate to
use the 10th and 90th percentiles as the ends of the whiskers.
Perhaps the most confusing nomenclature or lack of it is associated with a graph that
plots measurement values, e.g., temperature, for each case of a variable, e.g., various
dates throughout the year. This temperature-date graph is the example presented in
the National Council of Teachers of Mathematics’ (NCTM) Standards (1989, p. 55) with
no name to accompany it (see Fig. 2.2). Chick, Pfannkuch, and Watson (2005, p. 87)
8
History and other background
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
presented a graph of similar appearance for fast food consumption of 16 students,
again without naming it. Whether such a graph tells the story of the relationship between
two variables depends on the data. When looking at correlation, Konold (2002) said
such a graph was made up of “case-value” bars, creating a case-value graph. The term
value bar graph is used in the software TinkerPlots and the bars can be displayed either
horizontally or vertically. Choosing the type of icon for the representation makes it
possible to switch between value bars and dots that can create a stacked dot plot. Cobb
(1999) suggested that moving from horizontal value bars to a stacked dot plot assisted
students to make the connection between the magnitude of the measurement and
the scale upon which the dots were plotted. This discussion is further developed in the
following section with examples.
Figure 2.2. Graph of temperature values (NCTM, 1989, p. 55).
The confusion of naming and usage for such “value” graphs is a focus here because it is
the kind of graph often created spontaneously by less-experienced but creative students
(e.g., Chick & Watson, 2001; Pfannkuch & Rubick, 2002). Whether the graph is a step to
a uni-variate distribution or a scatterplot of two variables, it is important to acknowledge
its existence and potential for helping students tell the story in a data set. Associated
with the creation of various types of value graphs is the idea of transnumeration.
Wild and Pfannkuch (1999) introduced the term transnumeration for the process of
“changing representations to engender understanding” (p. 227). They included three
aspects: (i) capturing measures from the real world, (ii) reorganising and calculating
with the data, and (iii) communicating the data through some representation. Although
all three aspects are highly relevant to students doing genuine statistical investigations,
the second and third aspects are the most closely related to graphs and graphing.
The second is relevant because it is through the different arrangements of data (e.g.,
combining across different characteristics or choosing to display frequencies rather
than values) that different representations are created. Success is likely to depend on
knowing what types of representation are useful and having a range of techniques for
transforming data into forms conducive to such representations (Chick, 2003).
Another source of confusion is associated with “line plot” or “dot plot” or “stacked dot
plot,” all of which refer to a scaled (usually) horizontal line with dots indicating all values
in the data set. Particularly useful with smaller data sets, such plots display shape –
gaps, clumping, skewness, and spread. Stacked dot plot is the phrase used in this report
due to the confusion of “line plot” with “line graph,” the latter being a phrase used for
© State of New South Wales through the NSW
Department of Education and Training, 2010
History and other background
9
The Development of Graph Understanding
in the Mathematics Curriculum
graphs usually displaying sequential data (e.g., hourly-temperatures over a day) where
the dots for the data values are connected with straight lines.
2.1.2 Defining “bar-like” representations
The difference between a case-value plot and a bar chart, which may superficially look
the same, is that each “bar” for a case-value plot represents an individual data point,
whereas a bar chart collects together like data values and reports their total frequency.
The time this can be most confusing is when those data in the set themselves are
frequency or count values, for example “how many books each child in the class has
read.” For a class of 12 children the following might be the data:
Mary
2
Anne
4
George
4
Barb
4
Tom
3
Jerry
0
Dan
2
Laura
3
Carol
4
Fred
2
Ken
1
Pat
1
The case-value plot could look like the plot in Fig. 2.3.
Figure 2.3. Case-value plot of number of books read by students.
This plot puts the names in alphabetical order. The plot might also be arranged in
numerical order, say from least number of books read to most as in Fig. 2.4.
Figure 2.4. Case-value plot ordered by number of books read by students.
For young children this type of plot is likely to aid the transnumeration into a bar chart. A
bar chart of these data represents the number of data points (students) who have read
each number of books. Hence looking across the data there are five possible values the
10
History and other background
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
data take: “0,” “1,” “2,” “3,” and “4.” The frequencies for each value are determined
by the number of students who read that number of books, as in Fig. 2.5. In this
situation using words rather than numbers on the horizontal axis may be helpful to ease
confusion.
Figure 2.5. Bar chart of frequency of number of books read by students.
Sometimes the suggested transition from a case-value plot (Fig. 2.4) to the bar chart
(Fig. 2.5) is to stack the names of the students in a column to create the bars in the bar
chart. The names are associated with the number of books read not the actual books
themselves (e.g., Fig. 2.6).
Figure 2.6. Stacked data to create a bar chart.
The transnumeration of measurement data, e.g., height, from a case-value plot is in
many cases likely to create a stacked dot plot rather than a bar graph (Fig. 2.7) so
perhaps there is likely to be less confusion of the two forms (see also Konold & Higgins,
2003).
© State of New South Wales through the NSW
Department of Education and Training, 2010
History and other background
11
The Development of Graph Understanding
in the Mathematics Curriculum
Figure 2.7. Case-value plot with corresponding stacked dot plot.
2.2 Historical developments in graphing
Quantitative graphics have been used since ancient times to represent information.
They have their origins in mapping but have evolved over time to become important data
analysis tools. The Egyptians used a coordinate system to show the location of points in
real space c3200BC and used graphical representations to show the area of shapes,
including squares, trapeziums, triangles and circles c1500BC (Beniger & Robyn, 1978).
From that time until the 1600s, for the most part, graphical representations were used
for mapping, and recording the orbits of planets over time. It was not until the late 18th
and early 19th centuries that the use of graphs and charts for data displays became
accepted practice (Fienberg, 1979). Since that time the development of graphic methods
has depended on advances in technology, data collection and statistical theory (Friendly,
2007).
In a brief overview of the history of quantitative graphics in statistics, Beniger and
Robyn (1978) describe four stages that correspond to successive historical periods that
began in the early 1600s. During these periods developments were made that came
about as a result of major graphical problems that preoccupied scientists and data
analysts at the time. The four stages are: spatial organization for data analysis, discrete
quantitative comparisons, continuous distributions, and multivariate distributions
and correlation. Beniger and Robyn also include another section that introduces the
innovations developed in the 20th century. Collectively, the four stages and the 20th
century innovations describe progressive developments that have influenced the way
in which data are represented and analysed today. Although the developments were
12
History and other background
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
introduced in successive historical periods, the ideas introduced in earlier periods were
not superseded by successive developments. Elements of each stage were carried over
and incorporated into the next.
2.2.1 Spatial organisation for data analysis
Technological innovations in the form of automatic measuring devices invented in
the 17th and 18th centuries made it possible to collect and record large sets of data.
Measuring devices such as the air and water thermometer, weather-clock, pendulum
clock, and mercury thermometer were used to develop scientific instruments that
were capable of making multiple measurements. As a result, new ways of organising
and analysing data were necessary to handle the large collections of data. Generally,
the automatic recording devices produced moving line graphs that represented data
collected over a period of time using a coordinate system. At the time, it was common
for the data to be translated from this graphical form into tabular form for analysis. The
graphical form was considered a means of recording data and the potential for it to
be used for analysis did not occur until later. The coordinate system of Cartesian plots
reintroduced in mathematics by Descartes in 1637 did not become an important tool for
data analysis until the 1830s (Beniger & Robyn, 1978).
2.2.2 Discrete quantitative comparisons
The combination of visual imagery and statistical data to create graphical
representations of information other than scientific data was instigated by Playfair
in 1786. He replaced tables of numbers with visual representations, creating the
opportunity to use pictures and graphics to reason about quantitative information (Tufte,
1983). The graphs Playfair produced were very complex, often displaying multivariate
data on the same graphic. Although others preceded Playfair in using graphics to display
data, he extended their use to the areas of economics and finance, making the use of
statistical graphics popular for general interest information (Wainer & Velleman, 2001).
One of Playfair’s first innovations was the bar chart. This representation was used to
display categorical data. At the time, he was very cautious about the effectiveness of
bar charts and apologised for their lack of detail as they were not related to a particular
duration of time, as with time-series graphs (Funkhouser, 1937). Ironically, bar charts
have become a universal language and it is the simplicity of the representation that has
made them so useful.
During the 1700s scientists and economists developed ways of gathering large amounts
of information about populations and social activities, such as trade data. Although it
was commonplace to organise the data into tables for analysis, it was time consuming
and difficult. To address this difficulty, Playfair developed the circle graph (pie chart) to
allow for the visualisation of data. He recognised that people were able to make direct
comparisons of proportion and magnitude of shapes intuitively by eye, stating: “it is the
best and easiest method of conveying a distinct idea” (Playfair, 1801, p. 4). He also went
on to elaborate about the compelling nature of graphical representations.
It is different with a chart, as the eye cannot look on familiar forms
without involuntarily as it were comparing their magnitudes. So that
what in the usual mode was attended with some difficulty, becomes not
only easy, but as it were unavoidable. (p. 6)
© State of New South Wales through the NSW
Department of Education and Training, 2010
History and other background
13
The Development of Graph Understanding
in the Mathematics Curriculum
Playfair applied these principles in a graph displaying the area, population, and
revenue of European countries, with circles representing the area of countries and
lines representing population and tax data. The areas of the circles were proportional
to the areas of the countries, allowing for direct comparison. Some of the circles were
segmented and coloured, displaying the data as parts of the whole and differentiating
the parts by colour. The resulting graphic is a pie chart whose main purpose is to display
the relationship of a part to the whole (Spence, 2005).
2.2.3 Continuous distributions
In 1821, J. B. J. Fourier was the first to apply graphical analysis to population statistics,
furthering the development of vital statistics. In order to show the number of inhabitants
of Paris per 10,000 in 1817 who were of a given age or over, he developed the
cumulative frequency distribution. He began with a bar chart representing the age
groupings, and then placed the bars one atop each other for a particular age range
(Beniger & Robyn, 1978). This was repeated for other age ranges at regular intervals to
produce a graph. Fourier analysed the cumulative frequency distribution to determine
geometrically “the mean duration and probable duration of life, the mean age of
a population and the stability of life” (Funkhouser, 1937, p. 296). The cumulative
frequency distribution was named an “ogive” by Galton in 1875 (Beniger & Robyn,
1978). Cumulative frequency is used to construct box plots, a semigraphical innovation
designed by Tukey in 1977.
Another innovation developed from the bar chart was the histogram. In 1833, A. M.
Guerry produced histograms by arranging ordered categories for continuous data
(Beniger & Robyn, 1978). He used columns of equal width to represent the frequency for
each class at equal intervals of the data. A frequency polygon was obtained by joining
the midpoints of the class intervals. The broken line formed begins and ends on the
horizontal axis, resulting in an irregular polygon. The frequency curve was the smoothed
curve derived from the frequency polygon (Funkhouser, 1937). The word “histogram”
was first used by Karl Pearson (1895) in Contributions to the Mathematical Theory of
Evolution – II. Pearson recorded data collected in a histogram and compared the graph
with a corresponding theoretical skewed frequency curve. Adolphe Quetelet furthered
the development of the graphics of continuous distributions by applying the theory of
probabilities to graphical methods (Funkhouser, 1937). In 1846 he recorded the results
of sampling from urns as symmetrical histograms, and then showed the limiting “curve
of possibility.” This was developed further and later called the normal curve (Friendly,
2009).
2.2.4 Multivariate distributions and correlation
During the mid-19th century the data related to vital statistics became more complex
and involved interrelationships among more than two variables. Contour maps and
stereograms were developed to accommodate this increase in complexity as they
provided two dimensional representations of multivariate distributions and correlations.
The use of contour maps included the display of population density in a geographic
region and stereograms were used to represent the density of a population by age
groupings for a particular region. When the relationship between two variables is
examined, the data are referred to as being bivariate.
14
History and other background
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Bivariate data provide information about two variables that are not necessarily
dependent on each other. A scatterplot is a graphical technique used to display bivariate
data: paired measurements of two quantitative variables. It is a useful exploratory
method for identifying clusters of points and outliers in a distribution of bivariate data. It
assists in the identification of the relationship and dependence between the variables,
and variation from those (Cleveland, 1993).
2.2.5 20th Century and future developments
After a period of time in the early 1900s when formal statistical analysis of data was
favoured over graphical analysis, the importance of the visualisation of data for graphical
analysis regained prominence. This can be attributed to three main developments. First,
Tukey (1977) introduced the concept of exploratory data analysis, second Bertin (1981)
developed a theory of graphics, and third technological innovations allowed for the
computer processing of statistical data (Friendly, 2009).
Exploratory data analysis (EDA) is an informal, robust, and graphical approach to
data analysis that focuses on the appearance of graphs to provide insights about the
data rather than making formal inferences from statistical calculations. It is based
on graphical representations of data with a few added quantitative techniques. EDA
is a paradigm that is flexible and allows data analysis to be repetitive, iterative, and
creative (Tukey, 1977). Tukey suggests that data analysis is not just about getting the
right answer to a question. It is also about what influences the answer, what questions
are asked and the way in which they are asked. These notions are in opposition with
confirmatory data analysis (CDA). CDA employs principles and procedures that look
at a sample and what it can tell us about the larger population. It is then assessed to
determine the precision with which the inference from sample to population is made.
Tukey (1977) suggests that exploratory data analysis should be taught along with the
techniques of confirmatory data analysis. He states: “We need to teach exploratory as
an attitude, as well as some helpful techniques, and we probably need to teach it before
confirmatory” (Tukey, 1980, p. 25). He refers to exploratory data analysis as:
It is an attitude, AND
A flexibility, AND
Some graph paper (or transparencies, or both).
The graph paper – and transparencies are there, not as a technique,
but rather as a recognition that the picture-examining eye is the best
finder we have of the wholly unanticipated. (Tukey, 1980, p. 24)
Tukey (1977) designed semigraphical displays that provided a visual representation
of the data as well as statistical information. These included the stem-and-leaf display
and the box-and-whisker plot. The stem-and-leaf display is an alternative to tallying
values into frequency distributions. It displays a distribution of a variable with numbers
themselves. In overall appearance the display resembles a horizontal histogram
(Emerson & Hoaglin, 1983). The distribution of two data sets can be compared when
displayed as a back-to-back stem-and-leaf plot. Another useful display for comparing
multiple data sets is the box-and-whisker plot (Emerson & Strenio, 1983; Feinberg,
1979). The box-and-whisker plot can be determined from cumulative frequency and is
directly related to the ogive representation.
© State of New South Wales through the NSW
Department of Education and Training, 2010
History and other background
15
The Development of Graph Understanding
in the Mathematics Curriculum
The second development of the 20th century is attributed to the work of Bertin (1983).
In 1967 he published his theory of information visualisation in French, Semiologie
Graphique. Bertin’s interest in this area started when he identified that graphical
representations produced in scientific publications were not understood. The theory
he developed focused on the interpretation of the visual and perceptual elements of
graphics. His work made a distinction between how the qualitative and quantitative
elements imparted meaning (Card, Mackinlay, & Shneiderman, 1999). Tufte (1983) also
developed a theory of data graphics that emphasised the maximisation of the density of
information and the minimisation of extraneous information he termed “chart junk.”
Advances in computer technologies have had and will continue to have a significant
impact on the analysis of data and the visual displays used to represent the data.
Graphing software and associated technologies provide an alternative to hand-drawn
graphics, embellishment of older graphical types, analysis of large multivariate data
sets, and representation of multivariate data sets in two or three dimensions (Beniger &
Robyn, 1978). Graphing software such as TinkerPlots: Dynamic Data Exploration (Konold
& Miller, 2005), are evidence of the way in which interactive and digital technologies are
intersecting with the theories of graphics and the philosophy of EDA. In the future, the
application of computer technologies to data analysis has the potential to produce new
and innovative data representations.
2.3 The history of statistical graphing in the school curriculum
This section considers graphs related to portraying data, not related to functions and
relationships elsewhere in the mathematics curriculum. In fact early arithmetic and
algebra books (e.g., Hall & Knight, 1885; Pendlebury, 1896) had no graphing at all in
them. There were no early books on statistics for schools but by the 1930s it is possible
to see what kinds of graphs were being used in applications of statistics. Boddington
(1936) in writing about statistical applications to commerce devoted two chapters to “the
graphic method.” The first and simplest form of graph introduced was what today would
be called an ordered case-value plot (for 80 items) (cf. Sections 2.1.1, 2.1.2), including
the lower quartile, median, and upper quartile. This was followed by a frequency
polygon, a normal frequency curve, an ogive for cumulative frequency, “historigrams,”
as well as other “bar” and “block” graphs. He then described various diagrams as
“pictograms,” including segmented bar diagrams, comparison diagrams based on areas
of squares and circles, as well as an adaptation to show three variables. Each of the
representations is elementary enough to be included in the school curriculum.
In 1980 the NCTM in the US proposed an Agenda for Action in Mathematics for the
1980s including eight recommendations. In 1983 the Council’s Yearbook, The Agenda
in Action (Shufelt, 1983), reported on the recommendations with articles reporting
“actions” related to them. Although none of the recommendations specifically mentioned
statistics, Recommendation 2 concerning basic skills being more than computational
ability, provided the opportunity to tie statistics to the quantitative needs of an
information society. Swift (1983) did this and included scatterplots and stem-and-leaf
plots. Recommendation 6 called for “more mathematics” and Noether (1983) suggested
applying mathematical concepts to lines of fit for scatterplots. This was a very humble
beginning in gaining a place for statistics and graphing in the mathematics curriculum.
Also in 1980, the Schools Council published its report on Teaching Statistics 11-16
(Holmes, 1980), which presented comprehensive coverage of the state of statistics
16
History and other background
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
education in the UK, the needs and proposals for schools. Graphs and graphing were an
innovative part of the suggested lessons.
In the meantime, the NCTM had acknowledged the potential of statistics and probability
by publishing its 1981 Yearbook (Shulte, 1981) on the topic. Among the contributions,
Maher (1981) introduced side-by-side and back-to-back histograms, back-to-back stemand-leaf plots, and box plots, all shown to be useful for examining classroom data.
The 1980s also brought a growing recognition that statistics deserved a place in the
mathematics curriculum and the Quantitative Literacy series produced by the NCTM and
the American Statistical Association (ASA) provided realistic contexts and straightforward
explanations for meaningful applications. The graphical forms introduced in this series
(Landwehr & Watkins, 1986) were exactly those that came to inhabit the curricula that
followed: stacked dot plots (line plots), stem-and-leaf-plots, box plots, scatterplots, and
time series plots.
By the time of the publishing of the NCTM’s Curriculum and Evaluation Standards for
School Mathematics (1989), statistics and probability were accepted as warranting
inclusion among the standards. For the K-4 Standard, the graphing form displayed,
although not given a name, was again equivalent to a value bar plot (cf. Section 2.1.1).
For the 5-8 Standard for Statistics, line graphs and box-and-whisker plots were the
examples used with an emphasis on interpreting the information provided in them. The
graphical display in conjunction with the 9-12 Standard for Statistics was a scatterplot
with a regression line. In the Probability Standard at the 9-12 level the standard normal
distribution was displayed, linked to the actual data scale and showing a shaded region
of probability. A stem-and-leaf plot was used to display simulation outcomes but it was
not stated that this is the level for the stem-and-leaf plot to be introduced. In a document
with the scope of the NCTM Standards, it would be impossible to suggest all the
appropriate graphical types for each of the three levels. Subsequent research has shown
that it may be preferable to introduce box-and-whisker plots later (e.g., Bakker, 2004)
and students can understand stem-and-leaf plots before grade 9.
Following the NCTM’s 1989 statement and a similar document from the Department
of Education and Science and the Welsh Office (1989) in the UK, various individuals
and groups made suggestions for appropriate graphical forms for various ages.
Rangecroft (1991a) gave examples of representations suitable for the first Key Stage of
the UK curriculum for children up to age 7. She included a progression of pictographs
and mapping diagrams to tell stories of relationships, and included several types of
“temporary” graphs made with concrete materials or with the children themselves,
as well as “block” graphs, ending with bar charts. Rangecroft considered bar charts
quite abstract and not the appropriate starting point for young children’s graphing
experiences. Many of her suggestions, for example, related to tallying and pictograph
symbols representing more than one value, are likely to be the type of representation
created independently by students without instruction if given initial starting points.
Rangecroft (1991b) went on to suggest a progression for the secondary years that again
paralleled the UK curriculum, moving from bar charts to include stem-and-leaf plots,
scaled bar charts, scaled strip graphs, stick graphs, line graphs, box-and-whisker plots,
pie charts, scatterplots, frequency polygons, cumulative frequency curves, histograms,
graphs with non-linear scales, special distributions, and plots on special paper (e.g., star
diagrams). Rangecroft was particularly concerned to emphasise the importance of scale
in graph creation and this is confirmed by other research showing students’ difficulties
© State of New South Wales through the NSW
Department of Education and Training, 2010
History and other background
17
The Development of Graph Understanding
in the Mathematics Curriculum
in this area (e.g., Moritz & Watson, 1997). According to Rangecroft’s progression no new
graph types were introduced between ages 7 and 12.
At about this time the Center for Statistical Education of the American Statistical
Association put out its own guidelines, Teaching Statistics: Guidelines for Elementary
through High School (Burrill, Scheaffer, & Rowe, 1994). It is interesting that the 10
principles in the document only mentioned graphing explicitly in Number 5: “The
exploration of and the experimentation with simple counting and graphing techniques
should precede formal algorithms and formulas” (p. 5). Throughout the 28 activities
suggested for grades K to 12, the usual representations are suggested: pictographs, bar
graphs, stem-and-leaf plots, stacked dot plots (line plots), and histograms. “Broken-line
graphs” were the only new graph suggested in the activities. It is significant to note that
the ASA guidelines are not built around graphing and various graph types but on the
types of experiences students should have and where graphs are tools for assisting in
the overall process to reaching a conclusion based on data.
Australia’s A National Statement on Mathematics for Australian Schools (Australian
Education Council, 1991) followed the United States (US) and UK models, including
Chance and Data as one of five content strands. In the Statement itself, no different
types of graphs from those already mentioned were introduced but several salient
comments were made about the process of creating and using graphs.
Whether the ‘graphs’ are concrete, pictorial or more symbolic, the need
to use a common baseline when comparing frequencies or measures is
a fundamental notion which is not at all obvious to young children, and
which develops gradually as they gain experience in using graphs to
compare occurrences. (Band A, p. 165)
Graphs should not be regarded as an end in themselves; rather
they should serve purposes which are clear to children. As the
children perceive the need for increasingly sophisticated forms of
data representation, the teacher can assist them by introducing new
methods of representation. Little is likely to be achieved by providing a
collection of data (found in a text) and having children practise drawing
graphs types in isolation. (Band B, p. 168)
Many interpretations of data are based on summary statistics, such as
measures of central tendency, variability and association, and graphs
such as line plots [stacked dot plots], histograms, stem-and-leaf plots,
box plots, scatter plots, and lines of best fit. Students should be able
to interpret these various representations, understand the conditions
under which their use is appropriate, and compare and select from
different possible representations of the same data. (Band C, p. 173)
The second iteration of the NCTM’s suggested curriculum, Principles and Standards for
School Mathematics (2000), had similar structure to the 1989 document but divided the
years of schooling into four sections rather than three, extending the earliest downward
to include Pre-K-2 and changing the title related to statistics to Data Analysis and
Probability. The representations shown for Pre-K-2 included placing counters in bowls,
creating horizontal [value] bar graphs for data for children in the class, and creating a
stacked dot plot to display the frequency for each “group” of the data. An innovation in
this document was the introduction of a misleading display to raise students’ awareness
18
History and other background
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
of potential difficulties of interpretation at an early age. At the 3-5 level, spreadsheets of
data were introduced but no advice was given on using the graphing software likely to
be available with spreadsheets. The translation of data from tallies in tables to stacked
dot plots was shown along with recommendations for comparing two different plots.
Bar graphs, although not named, were shown for recording frequencies for several
categories of two attributes. The focus on comparing and contrasting data sets was
significant because it was intended to create greater interest in students and motivate
them to use their graphing skills productively to answer meaningful questions. The Data
Analysis and Probability Standard for grade 6-8 included a relative-frequency histogram,
box-and-whisker plots and a scatterplot, whereas the standard for grade 9-12, extended
the usage of these graphs combined with other analysis methods such as line-fitting to a
scatterplot and using a [value] frequency plot to represent the distribution of simulations
for creating random samples. By this level the graphs were seen very much as tools to
be used as part of more comprehensive analyses.
It is interesting to return to the introductory paragraph in this section and the summary
of the graphs suggested for application to commerce in 1936 (Boddington, 1936). Of
Boddington’s representations, the main omission in curriculum documents is to any
reference to graphs where area represents magnitude. The exception to this might be
considered to be the Pre-K-2 reference to a misleading representation by the NCTM
(2000). As critical statistical literacy is now part of the curriculum and the use of such
graphs based on area in the media is often misleading, it might have been expected that
more explicit mention might have occurred. Certainly such examples are consistent with
expectations of the 9-12 NCTM Standard to “evaluate published reports.”
The most comprehensive recent document on statistics at the school level is the
Guidelines for Assessment and Instruction in Statistics Education (GAISE) framework
(Franklin et al., 2007) from the ASA, suggesting curriculum guidelines from pre-K to
grade 12. Complementary to the NCTM’s Standards (2000), it provides much more
detail across the school years and again graphs are seen as the tools in the data
analysis stage of statistical problem solving. As such they feature prominently in
the report, with nomenclature that may become the common usage: picture graph,
bar graph, dot plot, time plot. Presenting the framework in three levels (A, B, and C),
Franklin et al. introduce the graphical forms in levels A and B, applying them in more
sophisticated settings in level C. Misleading graphs based on area representations are
featured, completing the link back to Boddington (1936). The only one of Boddington’s
graphs not featured in the GAISE report is the cumulative frequency graph or ogive. It
must be expected that the GAISE report will guide curriculum development in statistics
education for the next decade, at least in the US.
2.4 Current graphing curricula
In this section current curriculum documents from government schools in New South
Wales (NSW), Western Australia (WA), Tasmania, and the draft Australian Mathematics
Curriculum released for consultation in March 2010 (Australian Curriculum, Assessment
and Reporting Authority [ACARA], 2010), are examined to determine how graph creation
and graph interpretation are incorporated into K-10 syllabuses. These curriculum
documents are considered in relation to the progression of development of graphing
proposed by Rangecroft (1991a, 1991b), summarised in the previous section.
© State of New South Wales through the NSW
Department of Education and Training, 2010
History and other background
19
The Development of Graph Understanding
in the Mathematics Curriculum
2.4.1 Description of curricula
The Tasmanian Mathematics Curriculum (Department of Education Tasmania [DoET],
2007) is divided into five standards, with each standard equating approximately to two
years of the compulsory years of schooling. The Data component of the curriculum
that includes graph creation and graph interpretation is organised within the Chance
and Data strand. The development of these notions across the curriculum is seen
as complementary, in that most references to graph creation are followed by graph
interpretation outcome statements.
In Standards 1−3 the emphasis for graph creation is on the introduction of graph types
such as pictographs and bar charts. It is not until Standard 4 that more complex graphs
are introduced. Like Rangecroft’s (1991a, 1991b) model, scatterplots and pie charts
are introduced in the first two years of high school. Standard 5 does not refer to the
introduction of any new graph types but focuses on extending the use of graph types
encountered previously.
Graph interpretation is developed progressively across all the standards of the
Tasmanian Mathematics Curriculum (DoET, 2007). Collecting, organising, representing,
summarising, and describing variation in data are the emphases for graph interpretation
in the primary years (Standard 1−3), with Standard 4 focusing on using measures of
centre to represent data sets. In Standard 5 graph interpretation shifts towards the
application of higher order thinking skills, with an emphasis on analysing, interpreting,
justifying, and drawing conclusions from the data.
The NSW Mathematics Curriculum for years K−10 is divided into five stages, Early
Stage 1 and Stage 1, Stage 2, Stage 3, Stage 4, and Stage 5 consisting of sub-stages
5.1, 5.2, and 5.3 (Board of Studies New South Wales [BoSNSW], 2002a, 2002b). The
curriculum includes a Data strand with a substrand of Data for Early Stage 1−Stage 3,
and substrands of Data Representation, Data Representation and Analysis, and Data
Analysis and Evaluation for Stages 4−5. In the early years of schooling graph creation
starts with the use of concrete materials and pictures to construct different ways of
displaying data. In Stage 2 data are represented abstractly in graphs constructed on
grid paper. Vertical and horizontal column graphs are introduced during this stage, with
the addition of divided bar graphs in Stage 3. In both Stage 2 and 3 the emphases are
on using axes with marked scales to construct graphs and naming the components of
graphs. In Stage 4 histograms and frequency polygons are introduced, whereas box-andwhisker plots are introduced during Stage 5.2.
In Stages 1−3 of the NSW Mathematics Curriculum (BoSNSW, 2002a) graph
interpretation is limited to reading and interpreting different graph types. It is not until
Stage 4 that graph interpretation extends to using data to make predictions and the
application of measures of centre to analyse data. During Stage 5.2 the nature of graphs
in terms of skewedness and the shape of distributions are used to describe graphs.
The WA Mathematics Curriculum (Department of Education and Training Western
Australia [DoETWA], 2007) has a Chance and Data strand, with sub-strands of Collect
and process data, Summarise and represent data, and Interpret data. Like other
curricula, the WA curriculum starts using objects as data and creating graphs from
physical models. In contrast to other curricula it does not specify learning about many
different graph types but does introduce new ideas about graphing evenly throughout
the nine stages of the curriculum. Initially, for graph creation, univariate data are
20
History and other background
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
represented in picture and column graphs. In Stage 4 and 5 frequency graphs and
scatterplots are used. This is followed by the introduction of histograms and cumulative
frequency in Stage 6. During Stages 7 and 8 the focus shifts to calculating trend
lines and quantifying association. As in graph creation, graph interpretation in the WA
curriculum is developed evenly across the curriculum. It is specifically about using
graphical representations to compare data sets, make inferences from samples for the
population, and comment on predictions made.
The Australian Mathematics Curriculum (ACARA, 2010) consists of three main
organisers: Number and Algebra, Statistics and Probability, and Measurement and
Geometry. The curriculum covers the compulsory years of schooling from Kindergarten
to Year 10, with an additional section for Year 10 Advanced. In the Statistics and
Probability strand, the topics relevant to graph creation and graph interpretation include
data representation, data investigations, data interpretation, summary statistics,
data measures, and bivariate data. Not all the topics are covered every year, with
the emphasis for each year varying. The content related to graph creation and graph
interpretation for each year level is summarised in Tables 2.1 and 2.2.
Table 2.1. Summary of the Statistics and Probability strand of the Australian Mathematics Curriculum
relevant to graph creation and graph interpretation – Years K-6.
Year
Graph creation
Graph interpretation
Kindergarten
Pictographs
Determining the mode from a graph
Year 1
Pictographs, bar charts
Comparing information in data categories
Making connections between different
representations of the same data –
tables, graphs, and lists
Year 2
Pictographs, bar charts,
column graphs
Understanding information stays the
same even though the representation may
change
Explaining how information can be
extracted from graphical representations
Year 3
Pictographs, column
graphs, and dot plots
Other graphs from
prepared baselines
as well as student
generated graphs
Pictographs and dot
plots involving manyto-one ratios between
symbols and data points
Understanding the purpose and
usefulness of different data
representations
Understanding the importance of scale
and equally spaced intervals on an axis
Comparing different student-generated
data representations
Scale
© State of New South Wales through the NSW
Department of Education and Training, 2010
History and other background
21
The Development of Graph Understanding
in the Mathematics Curriculum
Year 4
Year 5
Pictographs, column
graphs and dot plots
using hand drawn
and computer based
methods
Comparing different student generated
data representations
Presents results from
investigations, including
use of ICT, to illustrate
best how the data
answer the question
being investigated
Exploring bivariate data collected over
time
Pie charts
Interpreting data representations in the
media where there is a one-to-many
correspondence between data symbols
and data points
Interpreting data representations and
drawing conclusions
Justifying the choice of data
representation used to display results
from investigations
Identifying the mode and median on dot
plots
Using and comparing effectiveness of
different data representations to interpret
data
Considering if data representations
provide an unbiased view
Making decisions from data
representations
Year 6
Stem and leaf plots,
pie charts and other
simple representations
including the use of
technology
Understanding the proportional nature of
pie charts
Using ordered stem and leaf plots to
determine the median and mode of data
Identifying misleading representations
Investigating data representations in the
media and critiquing claims made
Interpreting the messages conveyed in
data representations in the media
Identifying potentially misleading data
representations in the media
Understanding variation in measurements
22
History and other background
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Table 2.2. Summary of the Statistics and Probability strand of the Australian Mathematics
Curriculum relevant to graph creation and graph interpretation – Years 7-10A.
Year
Graph creation
Graph interpretation
Year 7
Ordered stem and leaf
plots, dot plots, scatter
plots, back-to-back stem
plots,
Understanding that summarising data
using measures of centre and spread can
be used to compare data
parallel dot plots, line
graphs, column graphs
Calculating mean, mode, median and
range from graphs
Identifying outliers
Comparing data sets and showing how
outliers may affect the comparison
Locating measures of centre on graphs
and connecting them to real life contexts
Using graphical representations to
compare univariate data
Collecting bivariate data to explore the
relationship between variables
Suggesting questions that can be
answered by bivariate data, providing the
answers, and justifying the reasoning
Identifying patterns in bivariate data to
suggest a relationship between variables
Year 8
Construct graphs,
including frequency
column graphs with and
without technology
Interpreting spread and measures of
centre resulting from natural variation in
nature
Using sample properties, such as large
gaps on a graph, to predict properties of a
population
Using graphs to identify the modal
category
Describing the shape and spread of data
representations
Year 9
Scatterplots, time
series plots and using
technology to sort, graph
and summarise data as
well as report results
Identifying and describing trends in data
from graphs
Line graphs
Displaying, identifying and describing
relationships among data from
scatterplots
© State of New South Wales through the NSW
Department of Education and Training, 2010
Analysing data and making conclusions
based on data representations
History and other background
23
The Development of Graph Understanding
in the Mathematics Curriculum
Year 10
Box plots, parallel box
plots
Comparing visually and numerically the
centre and spread of data sets
Judging the spread of data visible on dot
plots
Determining whether one data set is more
or less spread than another
Choosing among graphs and measures
of centre and spread to suit data analysis
purposes
Suggesting alternative models for
representing data
Year 10
Advanced
Scatterplots, graphs
showing linear
relationships in bivariate
data
Interpreting the slope and intercept of the
least squares line
Distinguishing between interpolation and
extrapolation when using least squares
lines to make predictions
Describing the relationship between two
numerical values
In relation to the discussion of terminology in Section 2.1.2, the elaboration for column
graph in Year 2 is a case-value plot whereas for Year 3, the elaboration for a column
graph is a frequency plot using children’s names as counters. In Year 7 again column
graph represent case values (lengths) over time and the term “frequency column graph,”
does not occur until Year 8. A striking omission is the term “histogram.” Its presence
and importance across the history of statistics (cf. Section 2.2) and the school statistics
curriculum (cf. Section 2.3), as well as current usage (e.g., Shaughnessy, Chance &
Kranendonk, 2009), would suggest this may lead to recommendations for its inclusion
at Year 9 or 10.
2.4.2 Comparison of curricula
The WA (DoETWA, 2007) and Tasmanian (DoET, 2007) curricula are similar as they are
quite descriptive about the ways in which data and graphs could be used to compare
groups, describe association, make predictions, and interpret data within a context. The
detail provided for graph interpretation is quite specific in both documents, providing
vital information about how data can be used to make inferences and inform decisions.
The WA curriculum (DoETWA, 2007) is less crowded than the other curricula examined,
staging the introduction of new graph types to coincide with the complexity of the data
collected. Column and picture graphs are used to represent univariate data and when
bivariate data are introduced they are represented in scatterplots. The interpretation
of graphs also aligns with the graph creation concepts. When compared to the WA
curriculum the NSW curriculum (BoSNSW, 2002a) introduces more graph types but
does not include how graphs and data could be used to make inferences. Statements in
the Data strand are limited to “read and interpret graphs” with supporting information
on how to use data situated in the Working Mathematically Strand of the curriculum.
24
History and other background
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
The Tasmanian curriculum (DoET, 2007) includes more of the conventional graph types
than the other curricula but introduces most of them at the beginning of high school
as described by Rangecroft (1991b). The advantage of this is that the choice of graph
representation available is extensive, potentially providing the opportunity to use the
graph type most appropriate for the data collected and the questions explored.
The Australian Curriculum model (ACARA, 2010) develops the notions of graph
interpretation progressively across the curriculum and stages the introduction of
different graph types across most of the years of schooling similar to the other curricula.
It is, however, more general in its descriptions as it often refers to the analysis and
interpretation of data without being explicit about how to use graphs for these purposes.
It has an emphasis on students developing graphing and data analysis skills when
conducting investigations but does not extend the application of these skills to make
informed decisions and make conclusions based on the context of the investigation. One
aspect of the Australian curriculum not mentioned in any detail in the other curricula is
the incorporation of students using secondary data sources, particularly from the media.
The need for students to understand how the media and other sources use graphical
representations to convey messages cannot be underestimated in today’s society of
expanding electronic and digital communication mediums.
In most of the curricula examined in this section there is a lack of recognition that
students need to learn about and understand the specific characteristics of different
graph types. Kosslyn (1989) suggests that an understanding of the constituent parts
of graphs and their relationships with each other is vital to students’ understanding of
graphs and their ability to communicate clearly with graphs. This becomes extremely
important when students work with different data representations. The story told by
the data representations used and the way in which they are displayed can impact on
students’ ability to interpret the messages within the data.
Another aspect noticeable for its absence across the curricula is the opportunity for
students to develop an understanding of variation. In most of the curricula variation is
expressed in terms of “spread” but they do not extend the application of this concept
more broadly. The Australian Mathematics Curriculum (ACARA, 2010) does refer to
students using the spread of data and the measures of centre to compare data sets in
the later years of schooling, as do some of the other curricula, but does not incorporate
explicitly these notions with graph interpretation. In Year 6, for example, under the
heading of “Variation,” there are elaborations related to measurement variation and
collecting repeated measurements but an opportunity is lost to link this to the related
observations in graphical representations. For the most part the curricula align the
development of variation with the application of box plots.
All of the curricula provide the opportunity for students to use a variety of graph types
and stage the introduction across most of the years of schooling. With some curricula
there is a density of new graph types introduced at the beginning of high school whereas
others spread the introduction more evenly. This spacing of material probably places
fewer demands on students to learn a large amount of new information at the same
time but it needs to be recognised more explicitly that graphs introduced in the early
years should be used continually even after more complex and sophisticated graphical
representations are introduced. The emphasis needs to be on using and applying the
graphical representation that is most appropriate for the data collected, the questions to
be answered, and the context of the problem under investigation.
© State of New South Wales through the NSW
Department of Education and Training, 2010
History and other background
25
The Development of Graph Understanding
in the Mathematics Curriculum
3. Graphing and technology
The issue of the use of technology in relation to the prescribing of graphing skills in the
mathematics curriculum is a vexing one. Similar to elsewhere in the curriculum where
there is debate about what algebra skills students need to have when a CAS calculator
can complete the required procedures, there are many software packages that can
create graphs for students. In both cases those who go on to study mathematics
or statistics at tertiary level, or enter careers in these fields, will undoubtedly use
technology to perform basic procedures, just as nearly everyone today uses the fourfunction calculator in their mobile phone. What is important in all of these areas is
that the student understands the principle behind the process being carried out, be it
multiplication or plotting an association of two numerical variables. Hence it appears
that to the present time all curriculum documents indicate types of graphs that students
should understand (and presumably be able create) before moving on to use technology
to create the graphs. This is fine as far as it goes but there are two complications. One is
that some software packages, such as Excel, readily produce graphs that are not in the
curriculum; some of these are colourful and attractive and students use them when they
are inappropriate for the story in the data. The other complexity is that some software
packages, such as TinkerPlots (Konold & Miller, 2005), do provide the opportunity for
students to be creative, similar to how they might be without software, but in ways that
are not specified by the curriculum.
It is the view of the authors of this report that students should have the option of
creating graphs with or without technology (even rulers might once have been described
as technology), with the understanding that they can explain what they have created and
the conclusions they can draw from the representation. Having said this, it is necessary
to discuss what is possible with a software package such as TinkerPlots. In contrast
to Excel, from the Microsoft suite of computer applications, which produces “finished”
products for those who specify the variables correctly, TinkerPlots allows students to
begin with the data randomly arranged in a plot window. An example is shown in Fig.
3.1. Variables can be placed (with a “drag and drop” feature) on either the horizontal or
vertical axis. Students can explore the possibilities of different representations, changing
and adapting as necessary.
Figure 3.1. Initial random arrangement of data icons for a data set with 57 values.
26
Graphing and technology
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
One particularly useful representation created in TinkerPlots is the “Hat plot.” Hat plots
divide a numeric attribute into three sections that look somewhat like a hat. There is
a central “crown” and, on either side of the crown, a “brim.” The crown represents the
middle 50% of the data and the brims represent the lower 25% and the upper 25% of
the data. The brims extend out to the minimum and maximum values of the attribute
(Konold & Miller, 2005). The characteristics of the hat plot are presented in Fig. 3.2.
Figure 3.2. Characteristics of hat plots created in TinkerPlots.
The following examples are provided using TinkerPlots to illustrate two aspects of
graphing in relation to the school curriculum: first, the fact that the software can produce
most of the types of graphs likely to be specified in any mathematics curriculum and
second, the ability of the software to create, under the instructions of the user, other
imaginative presentations that tell stories well, without the constraints of curriculum
conventions. Examples of graphs created by students are used to show the variety of
representations they can produce. Then other examples are provided in order to show
the similarity of plots produced by the software to the graphs expected by the curriculum
to be produced by hand.
The variation in choosing representations occurs for students using TinkerPlots in the
same way it does if students are given the opportunity to create them with paper and
pencil. An example of this is shown by grade 7 students asked to decide which class had
done better at a spelling test (scored out of 9 with higher scores being better) (Watson &
Donne, 2009). Data were presented for 36 students in the Pink class and 21 students in
the Black class. The graphs in Fig. 3.3 show the display of data that had been presented
to students in an earlier study but the later students only had the data in a TinkerPlots
file.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Graphing and technology
27
The Development of Graph Understanding
in the Mathematics Curriculum
Figure 3.3. Comparing groups task.
Because TinkerPlots initially displays values in bins when attributes are introduced, one
student based her decision that the Black class had done better on the left hand plot in
Fig. 3.4, saying that Black did better because most of them were in [5-9] and only four in
[0-4]; the 25 for Pink in [5-9] was only because there were more in the Pink class. When
prompted for detail, the student separated the data as in the right hand plot in Fig. 3.4,
and said it confirmed her view that Black did better because it had less kids but more in
the higher scores than the lower ones.
Figure 3.4. Determining which class did better with bins then two stacked dot plots.
A second student coloured the data by class, created bins to count the number in each
class, then created one stacked dot plot and observed that there were more Black on
the higher numbers and more Pink on the lower ones. The plots are shown in Fig. 3.5.
28
Graphing and technology
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Figure 3.5. Determining which class did better with bins then one stacked dot plot coloured by class.
A third student created the plot in Fig. 3.6, which is more conventional but stacked
against the vertical axis rather than the horizontal. The graph also includes Hat plots
showing the middle 50% of the data in each group. The hat helped some students, for
example one stating that Black was better because 75% of its values were “up” in the
higher scores.
Figure 3.6. Determining which class did better with two stacked dot plots and hats.
In another activity the same students were given data on various attributes for 16
students, including age and weight. Asked to explore the data set some students
considered the association of these two attributes. The four plots in Fig. 3.7 show the
variety of representations that these students used to claim that generally the older
students in the data set weighed more.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Graphing and technology
29
The Development of Graph Understanding
in the Mathematics Curriculum
Figure 3.7. Different representations to show the association of Weight and Age.
In another study of grade 7 students in a classroom learning environment (Watson,
2008), students collected data on their resting and active (after jumping rope) heart
rates. Following a class discussion on expectations, in the first lesson students were
free to explore the data in TinkerPlots. As is typical some students were very interested
in the rates of the students in the class and finding themselves in particular. Three
students created the plots in Fig. 3.8. Only the top plot has ordered the data from
lowest to highest rate and these are shown in bins rather than on a scaled plot as in the
middle plot. Each value is labelled and it appears that perhaps both resting and exercise
values were entered for “peter.” The bottom pair of plots shows case-value plots for the
students.
30
Graphing and technology
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Figure 3.8. Different representations for students’ names and reaction times.
In the second lesson the teacher reviewed understanding of percentage and students
were introduced to Hat plots. The plot shown in Fig. 3.9 shows one student’s plots,
having created two plots with the same scale in order to compare the resting and
exercise heart rates (note that this was not done in the two case-value plots in Fig. 3.8).
The student who created these plots discussed the range of the crown of the hat and
the entire range, noting that the resting values were “all in together” and in the exercise
they were “all spread out.” The overlap of the data sets made the student want the
classmates with the lowest exercise values to “do it again,” not believing the data were
correct.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Graphing and technology
31
The Development of Graph Understanding
in the Mathematics Curriculum
Figure 3.9. Stacked dot plots and hats to compare resting and exercise heart rates.
TinkerPlots does not produce stem-and-leaf plots but the stacked dot plot using bin
widths of the same size as the stem-and-leaf intervals holds the same information
and since it can be positioned on either axis, provides either a direct or transformed
representation. This is shown for a data set in Van de Walle (2007, p. 461), which
includes test scores for two classes (Mrs. Day and Mrs. Knight). Fig. 3.10 shows
the possibilities for displaying one of the classes (Mrs. Day), whereas Fig. 3.11 is a
representation based on the horizontal axis.
Basic horizontal stacked dot plot
Basic horizontal stacked dot plot
with labels
Basic horizontal stacked dot plot
ordered with labels
Figure 3.10. TinkerPlots versions of stem-and-leaf plots.
32
Graphing and technology
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Figure 3.11. Stacked dot plot with bins similar to stem-and-leaf intervals.
TinkerPlots cannot do back-to-back plots, as would be used with stem-and-leaf plots but
Fig. 3.12 shows how a similar comparison can take place either with the scale vertical,
as in a stem-and-leaf plot, or horizontal.
Figure 3.12. TinkerPlots comparisons similar to back-to-back stem-and-leaf plots.
The Hat plot, as described earlier in this section, is related to the Box plot, as a simpler
representation that can introduce the ideas of density and spread in data distributions
(Watson, Fitzallen, Wilson, & Creed, 2008). In Fig. 3.13 the data for all winners of the
Melbourne Cup are displayed by the Weight attribute, that is how many kg the horse
had to carry in the race as a handicap. The stacked dot plot with a hat on the left is for
all horses from 1861 to 2009, whereas the plot on the right is separated into 30 year
intervals with hats included to help consider the change over time. The hat on the left
shows that the middle 50% of the weights lie between 48 and 55 kg and the overall
range is from 33.5 to 66 kg. In particular there appears to be less variation overall in
the weights carried over time, with the range decreasing in each interval and the last 30
year’s middle 50% being 50.9 to 55.4 kg. For students, having the data visible at the
same time as the hats helps them to appreciate the relationship between the two.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Graphing and technology
33
The Development of Graph Understanding
in the Mathematics Curriculum
Figure 3.13. Use of the hat plot to monitor the change in distribution of Weight over time.
In Fig. 3.14 the same data are displayed but the axes are swapped and box plots are
used. In TinkerPlots it is possible to hide the data icons and have them reappear.
The plot on the left is the type of display of box plots that is often used, for example
in reporting statewide student outcomes for schools. With no data present, this
representation is much more difficult for students to interpret. With TinkerPlots it is
possible to switch back and forth to seeing the data icons. It is also possible to revert to
the representations in Fig. 3.13.
Figure 3.14. Use of the box plot to monitor the change in distribution of Weight over time.
TinkerPlots does not provide lines of best fit but it is possible to draw lines over data
displays using the mouse as a pencil. The software, Fathom, which extends TinkerPlots
for the senior years, provides this facility. Other representations that are possible with
TinkerPlots are bar charts, segmented bar charts, histograms, line graphs, and pie
charts (circle graphs). Examples are given in Fig. 3.15.
34
Graphing and technology
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Bar chart of Sex of Melbourne Cup winners
Bar chart of Sex segmented by Colour for Melbourne Cup
winners
Histogram of Winning Margin for Melbourne Cup winners
Line graph of Weight carried over the years for Melbourne
Cup winners
Pie chart for Sex of Melbourne Cup winners
Pie charts for Gender segmented by Sex for Melbourne
Cup winners
Figure 3.15. Various types of graphs available in TinkerPlots.
TinkerPlots also displays means, medians, modes, and midranges, with labels (or
reference lines). The plot in Fig. 3.16 contains the dates on 100 randomly collected
Australian 10-cent pieces (in 2007). In the plot the mean is 1995.1, the median is
© State of New South Wales through the NSW
Department of Education and Training, 2010
Graphing and technology
35
The Development of Graph Understanding
in the Mathematics Curriculum
2000.5, the mode is 2005, and the midrange is 1987.5. This data set was chosen to
show the spread in the four values for the skewed distribution.
Figure 3.16. Plot displaying midrange, mean, median, and mode (left to right).
The purpose of this section has been to illustrate the potential use of software that
allows students to create and manipulate graphs in order to tell the stories in data sets.
It is not necessary, and in many cases it is not realistic, to expect students to know the
appropriate graph for the data set before they start to produce it. After introductory work
understanding the elements of graph creation, students have the opportunity to explore
potential representations rather than just have a category from which to choose a fixed
form.
36
Graphing and technology
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
4. Development of graph creation and interpretation
Although researchers in statistics education have shown considerable interest in
students’ development of understanding in relation to graphs and graphing, none have
specifically focused on particular types of graphs and the order in which they might be
introduced in the school curriculum. In fact the main contribution to suggestions for
ordering in the curriculum comes from historical curriculum documents based on advice
from tertiary statisticians, observed practice, and the historical order of introduction
(mainly throughout the 20th century).
From a theoretical point of view statisticians generally build up single-variable
techniques (e.g., z-tests) and then move on to more than one variable (e.g., ANOVAs),
based on the complexity of the mathematics required for the tests. What research with
school students has found is that for small and moderate sized data sets, students
can handle, and find interesting, dealing with two variables (e.g., comparing attributes
for boys and girls, or comparing height and armspan) at a younger age than might be
anticipated. Although they cannot make formal statistical inferences, they can compare
groups and make informal inferences about underlying “populations.” Hence it is not
necessary to know everything about graphing large data sets for a single attribute before
moving on to considering more than one attribute.
Research into school students’ understanding of graphing generally has employed one
of three methods. In one it has been based on analysis of a single, perhaps multi-step,
task, where student development is observed and modelled in relation to cognitive
functioning for that particular graphical context. The analysis may be based on student
surveys or interviews. Second several interview tasks have been combined to suggest
a broader picture of statistical understanding, with graphing being a component of the
analysis. Third many shorter tasks have been combined in surveys, with data analysis,
such as Rasch (1980) techniques, used to provide a hierarchy on a variable such as
“statistical literacy.” Graphing tasks are often among the items used in the surveys but
do not constitute the total set.
The following three sections focus on these three aspects of research into school
students’ understanding of graphing: first, some examples of individual tasks; second,
outcomes based on surveys; and third, developmental models based on interviews.
Finally a short section ties together the Rasch analyses from the survey study and the
interview study. Within the sub-sections, the discussion distinguishes between the tasks
of graph creation and graph interpretation. Given the historical development of graphical
representations for various types of data it is clear that for some people the creation of
graphs was important to tell their stories. Creators had to be aware that others would
need to interpret their graphs in order to recreate the stories behind them. The question
of how well the creators do at their job is seen when the interpreters are tested on the
meaning of graphs they are given.
4.1 Development based on individual tasks
Research that is based on students’ creation of graphs must give students freedom to
create and hence cannot be specifically tied to graphical forms prescribed in curriculum
documents. Idiosyncratic graphs may or may not tell the story of a data set as well as a
form from the formal curriculum. The stages of development displayed are related to how
the students use their representations to tell the stories in the data. In doing so however,
higher levels are usually associated with more conventional representations.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
37
The Development of Graph Understanding
in the Mathematics Curriculum
In studying students’ ability to create a concrete pictograph of how many books a small
number of children had read, Watson and Moritz (2001) provided concrete materials in
the form of cards with drawings of children and drawings of books for students who were
interviewed face to face (grades 3 to 9). The four levels of representation they observed
are shown in the photos in Fig. 4.1. The representations labelled Prestructural at Level 1
associated books with children but not in a manner that they could be counted (unless
physically picked up and manipulated). At Level 2, labelled Unistructural, students
spread books out around the children in a way that could be counted, whereas at Level
3, Multistructural, they arranged the cards in a more conventional grid-like fashion.
At the Relational level, Level 4, the data were ordered (on the left) or an additional
feature was added (on the right, where an extra “book from the library” is shown in the
representation).
Level 1 (Prestructural/Idiosyncratic)
Level 2 (Unistructural)
Level 3 (Multistructural)
Level 4 (Relational)
Figure 4.1. Development of pictograph representation (from Watson & Moritz, 2001, Figure 3, p. 58).
38
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
In the time series context of describing the maximum daily temperature over a year
Watson and Kelly (2005) asked students in interview settings (grades 3 to 9) to create
such a graph for Hobart, Tasmania where the average maximum daily temperature for
the year was 17oC. The representations shown in Fig. 4.2 show a similar progression
to that related to the pictographs in Fig. 4.1 but in a more data intensive context with
two attributes, time and temperature. Again the graphs contained more decipherable
information and conventional formatting as the levels increased.
Level 1 (Prestructural)
Level 2 (Unistructural)
Level 3 (Multistructural)
Level 4 (Relational)
Figure 4.2. Development of graphical representation involving time series
(from Watson & Kelly, 2005, Fig. 2, p. 259; Fig. 3, p. 259; Fig. 4, p. 260; Fig. 5, p. 261).
In a chance setting where interviewed students (grades 3 to 9) were asked to imagine
the outcomes of randomly drawing 10 lollies from a container of 100, 50 of which
were red, and counting the number of reds, Kelly and Watson (2002) again observed
four levels of development. These ranged from a story-telling imaginative picture (Level
1), through a representation more like a time series, to a frequency tally. The Level
2 representations recognised frequency but not centre, whereas the Level 3 plots
recognised variation about the centre in a horizontal trial-order (like a time series) bar
representation. The Level 4 plot was a precursor to a frequency distribution. These are
shown in Fig. 4.3.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
39
The Development of Graph Understanding
in the Mathematics Curriculum
Level 1 (Prestructural)
Level 2 (Unistructural)
Level 3 (Multistructural)
Level 4 (Relational)
Figure 4.3. Development of graphical representation involving chance outcomes
(from Kelly & Watson, 2002, Fig. 2, p. 370; Fig. 3, p. 370, Fig. 4, p. 371; Fig. 5, p. 371).
These three sequences of graphical representations suggest features in the
development of graph creation, taking into account the meaning of the attributes, the
way to show the data, and the variation present. They appear to show an increasingly
more understandable way of representing the data and their stories. The second two
tasks are combined with others later in exploring a more general hierarchy related to
variation and expectation (cf. Section 4.4).
In studying students’ ability to interpret, rather than create graphs, Watson and Kelly
(2003) used survey items with larger numbers of students. One item employed a
pictograph showing, with icons for girls and boys, how 27 children came to school one
day, with the possibilities Bus, Car, Walk, Train and Bike. There were between 5 and 9
icons for each mode of transport except Train, which had none. The item is shown later
in Fig. 4.5. Interpretation and prediction questions included:
Would the graph look the same everyday? (Why or why not?)
A new student came to school by car. Is the new student a boy or a girl?
(How do you know?)
What does the row with the Train tell about how the children get to
school?
Tom is not at school today. How do you think he will get to school
tomorrow? (Why?)
The responses ranged from idiosyncratic, such as, “Yes [the graph would look the same]
because their legs would hurt;” to looking at patterns in the pictograph, such as, “[The
new student is a] boy, because there is a pattern” and balancing, such as, “[The new
student is a] boy, it could make 14 of both in the class.” Using frequencies presented in
40
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
the graph provided higher level responses, e.g., “[Tom will come by] bus, more people
catch the bus.” The highest level responses, however, included acknowledgement of
uncertainty in the interpretation or prediction: “No [the graph would not be the same],
sometimes people get sick.” “[The new student] is probably a girl because more girls
get a car to school.” The questions in this task were devised to allow students the
opportunity to display appreciation of variation and uncertainty in predictions. Not all
tasks encourage such recognition.
In asking an interpretation question about a pie chart that did not add up to 100%, again
in a survey setting, Watson (1997, 2006) worded the question as “Is there anything
unusual about it?” rather than “What is the error?” in order to measure students’
memory for the conventions associated with pie charts. Some responses at the lowest
level, took the unusual aspect to be visual, such as “It’s cut into all different shapes”
or “The black part.” At the next level, students took into account relevant features
but missed the main point: “Other [a category] is bigger than the rest.” The highest
level response reflected the error in the graph in several ways: “Where it has Other, it
says 61.2% and the percentage of that section on the pie is less than 50%” and “The
percentages add up to 128.5. They should equal 100!!”
These types of questions are useful in delving into specific aspects of students’ ability to
interpret graphs in particular contexts. Combining them requires more complex analysis
and this is considered later (cf. Section 4.5).
4.2 Development based on surveys
Much of the developmental research into students’ interaction with graphical concepts
has focussed on evidence of progression with respect to specific tasks (e.g., Moritz,
2000; Watson, 2000; Watson Collis, Callingham, & Moritz, 1995; Watson & Kelly,
2005). The work of Watson and Callingham (2003), however, used some items based
on graphing in their study related to defining a statistical literacy construct. The study
was based on Partial Credit Rasch modelling (Masters, 1982), which ranks task-steps
for rubrics associated with levels of observed performance for each question. The
task-steps hence make it possible to suggest ability levels descriptively in six stages.
The hierarchical development in understanding of creating and interpreting graphs
as extracted from this study and that of Watson, Callingham, and Kelly (2007) is
summarised in the following section. The focus of this section is graphing based on the
survey items used in the statistical literacy study (Watson & Callingham, 2003), one
of which related to graph creation and six of which related to graph interpretation. The
overall statistical literacy construct, based on surveys that included the graphing tasks,
is summarised in Table 4.1.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
41
The Development of Graph Understanding
in the Mathematics Curriculum
Table 4.1. Statistical Literacy Construct (from Watson & Callingham, 2003)
Level
Brief characterisation of step levels of tasks
6. Critical
Mathematical
Task-steps at this level demand critical, questioning engagement
with context, using proportional reasoning particularly in media or
chance contexts, showing appreciation of the need for uncertainty
in making predictions, and interpreting subtle aspects of language.
5. Critical
Task-steps require critical, questioning engagement in familiar
and unfamiliar contexts that do not involve proportional reasoning,
but which do involve appropriate use of terminology, qualitative
interpretation of chance, and appreciation of variation.
4. Consistent
Non-critical
Task-steps require appropriate but non-critical engagement with
context, multiple aspects of terminology usage, appreciation of
variation in chance settings only, and statistical skills associated
with the mean, simple probabilities, and graph characteristics.
3. Inconsistent
Task-steps at this level, often in supportive formats, expect selective
engagement with context, appropriate recognition of conclusions
but without justification, and qualitative rather than quantitative use
of statistical ideas.
2. Informal
Task-steps require only colloquial or informal engagement with
context often reflecting intuitive non-statistical beliefs, single
elements of complex terminology and settings, and basic one-step
straightforward table, graph, and chance calculations.
1. Idiosyncratic
Task-steps at this level suggest idiosyncratic engagement with
context, tautological use of terminology, and basic mathematical
skills associated with one-to-one counting and reading cell values in
tables.
Among the survey items was one that asked students to draw a graph showing an
association described in a newspaper article about car usage and heart deaths.
This item is shown in Fig. 4.4. Responses that created appropriate bivariate or
series comparison graphs appeared at the Critical level (5) of the hierarchy, whereas
incomplete representations that showed a trend or a double comparison appeared
at the Consistent Non-Critical level (4), and basic labelled graphs or single-value
comparison graphs as responses were at the Inconsistent level (3).
Family car is killing us, says Tasmanian researcher
Twenty years of research has convinced Mr Robinson that motoring is a health hazard.
Mr Robinson has graphs which show quite dramatically an almost perfect relationship
between the increase in heart deaths and the increase in use of motor vehicles.
Similar relationships are shown to exist between lung cancer, leukaemia, stroke and
diabetes.
Draw and label a sketch of what one of Mr. Robinson’s graphs might look like.
Figure 4.4. Graph creation item from the statistical literacy survey.
42
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Six survey items with problem contexts associated with some aspect of graph
interpretation provided 38 task-steps in the statistical literacy hierarchy. The items
are presented in Fig. 4.5. The items and their task-steps are then listed in Table 4.2
in relation to the hierarchical levels summarised in Table 4.1. The 38 task-steps were
distributed relatively evenly across the six levels of the hierarchy. Reading “up” the
right side of Table 4.2 gives an impression of the increasing complexity of skills and
understanding associated with graph creation and interpretation, although mostly the
latter.
TRV. How children get to school one day
Number of students
TRV1. How many children walk to school?
TRV2. How many more children come by bus than by car?
TRV3. Would the graph look the same everyday? Why or why not?
TRV4. A new student came to school by car. Is the new student a boy or a girl? How
do you know?
TRV5. What does the row with the Train tell about how the children get to school?
TRV6. Tom is not at school today. How do you think he will get to school tomorrow?
Why?
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
43
The Development of Graph Understanding
in the Mathematics Curriculum
SP. A class did 50 spins of the above spinner many times and the results for the
number of times it landed on the shaded part are recorded below.
SP6.What is the lowest value?
SP7. What is the highest value?
SP8. What is the range?
SP9. What is the mode?
SP10. How would you describe the shape of the graph?
SP11.Imagine that three other classes produced graphs for the spinner. In some
cases, the results were just made up without actually doing the experiment.
a) Do you think class A’s results are made up or really from the experiment?
__Made up
__Real from experiment
Explain why you think this.
b) Do you think class B’s results are made up or really from the experiment?
__Made up
__Real from experiment
Explain why you think this.
44
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
c) Do you think class C’s results are made up or really from the experiment?
__Made up
__Real from experiment
Explain why you think this.
BT1A (finding error) and BT1B (noting variation).
These graphs were part of a newspaper story reporting on boating deaths in
Tasmania. Comment on any unusual features of the graphs.
M2PI. Explain the meaning of this pie chart. Is there anything unusual about it?
TWN. A class of students recorded the number of years their families had lived in
their town. Here are two graphs that students drew to tell the story.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
45
The Development of Graph Understanding
in the Mathematics Curriculum
TWN1. What can you tell by looking at Graph 1?
TWN2. What can you tell by looking at Graph 2?
TWN3. Which of these graphs tells the story better? - Why?
M9C. Suppose the standard rate is $1.00
for 1 minute. You have already talked for
30 minutes. How much would the next 10
minutes cost?
M9D. How much did the first 30 minutes
of the phone call cost?
M9. The longer your overseas call, the
cheaper the rate.
Figure 4.5. Graph interpretation items from the statistical literacy survey.
Table 4.2. Student performance on the statistical literacy survey items in relation to the
hierarchical levels of the Statistical Literacy Construct1
Level
Task-step
Description of student performance
6. Critical
Mathematical
TRV4.3, TRV5.2,
TRV6.5
used subtle language that evidenced an
appreciation of uncertainty in making
predictions from pictographs
TWN1.3, TWN2.3
made high level, comprehensive summaries
in context about the information presented
in dot plots
M9C.2, M9D.2
worked out the costs of telephone calls from
the complex graph
BT1A.2
identified the error in the boat deaths graph
SP9
identified the mode/s of a dot plot
M2PI.2
noted the percentage error in the pie chart
BT1B.2, BT1B.3
identified variation features in the boat
deaths graph
5. Critical
46
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
4. Consistent
Non-critical
3. Inconsistent
2. Informal
1. Idiosyncratic
SP7, SP8
identified the highest value and the range of
a dot plot
SP10
acknowledged variation in describing the
shape of the dot plot
SP11.2
chose appropriately the real and made-up
dot plots
TRV4.2, TRV6.3
provided arguments for predictions for the
pictograph that were based on balance or
majority in the graph
TWN3.3
identified the scaled dot plot as better than
the non-scaled plot
M2PI.1
focussed on peripheral issues rather than
critical ones when commenting on the
unusual features of the pie chart
BT1B.1
little focus on variation for the graph of boat
deaths
TWN1.2, TWN2.1,
TWN3.2
read data values only in commenting on
stacked dot plots or could make only one
summary statement from the plot
BT1A.1
made specific comments about elements of
the graph, some of which were incorrect
TRV4.1, TRV6.2
used patterns to provide reasons for
questions about the pictograph
SP6
identified the lowest value in a dot plot
TWN1.1
made comments based only on reading
data from a dot plot
TWN3.1
gave personal preference (non-data based)
for which of the two graphs was better
M9C.1, M9D.1
made incorrect calculations of costs for
telephone calls based on a graphical
representation
TRV1, TRV2
read values from a pictograph
TRV3
recognised the presence of variation in
potential changes in the travel graph from
day to day
TRV5.1, TRV6.1
gave personal non-data-based
interpretations of the pictograph
SP11.1
identified correctly one of the three dot plot
scenarios as real or made up
The presence of a decimal value indicates that the item is associated with a coding rubric with
more than a correct/incorrect response, i.e., was associated with partial credit.
1
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
47
The Development of Graph Understanding
in the Mathematics Curriculum
This section has focused on graphing items used in larger surveys that also considered
other aspects of statistical literacy, for example, chance, sampling, averages, and
inference. The fact that the increasingly complex demands of the graphing items mesh
at every level with the demands of the other items indicates the interrelationships of
the concepts in the statistics component of the school curriculum. Graphing cannot
be seen as something separate but must be integrated with the other content at all
levels as the expectations for students grow from idiosyncratic engagement to critical
mathematical ability. The importance of context throughout in reaching the critical
thinking levels foreshadows not only the need to acknowledge the importance of a
Working Mathematically component in the curriculum but also the need to expand the
influence and impact of graphing across into other curriculum areas.
4.3 Development based on student interviews
The earliest comprehensive research on the developmental aspects of statistical
thinking centred on graphing (or data representation) is that of Jones, Thornton, Langrall,
Mooney, Perry, and Putt (2000) based on the Biggs and Collis (1982, 1991) cognitive
development model (SOLO) and interviews with 20 children, four in each of grades 1 to
5. Mooney (2002) then extended the work with the model based on interviews with a
total of 12 students in grades 6 to 8. Although reporting under the phrase “statistical
thinking,” in fact the research was not concerned with issues of data collection and
sampling or the part played by chance in statistical analyses. Jones et al. and Mooney
considered four levels of performance for each of four components of statistical thinking,
each component dependent to some extent on graphical representations. The first
component, Describing data displays, reflected the work of Curcio (1987, 1989) in being
able to read information from graphical displays, being aware of graphing conventions,
and recognising different displays of the same data as such. The second, Organizing
and reducing data, included being able to group and order data, aware of potentially
losing information, representing typicality, and representing spread. Although these
processes could be carried out without graphical representations, for example with
lists, tables or formulas, the tasks used with children all employed some type of visual
graphical display. The third component, Representing data, was based on constructing
representations and visual displays, including the elemental conventions for such
constructions. Given the constraints of their design and tasks, this did not allow for
tremendous freedom or creativity on the part of students in that they were presented
with frameworks (axes) to add information to or asked to present data currently in a
graph in a “different” way. The fourth component, Analysing and interpreting data, was
again based on the presentation of graphical displays and reflected Curcio’s (1989)
reading “between” and “beyond” the data, expecting students to recognize patterns,
trends, and missing elements in order to make “inferences” and predictions. Some of
the questions incorporated in their tasks were difficult to distinguish between being
related to Describing data and being related to Analysing and Interpreting data. The
significant aspect of the research of Jones et al. and Mooney for this report is that
graphical representation is central to all four components of statistical thinking as
considered in the research. In comparing this conception of statistical thinking with that
say of Holmes (1980), the Holmes’ views are more broadly based including the asking of
questions and understanding appropriate methods of collecting of data, before reaching
a point where display of data is likely to take place, and later in the process appreciating
the necessity to deal with uncertainty using basic concepts of probability.
48
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
For each of the four Jones et al. (2000) components of statistical thinking, four levels
of development were hypothesised based on the work of Biggs and Collis (1982, 1991)
and trialled, adapted, and confirmed based on the student interviews. The descriptive
labels attached to the levels were “Idiosyncratic,” “Transitional,” “Quantitative,”
and “Analytical,” parallel to the SOLO model levels of Prestructural, Unistructural,
Multistructural and Relational, in the stage of development called the Concrete
Symbolic Mode, the stage relevant to children during the school years. The Jones et
al. descriptors are closely related to the context of statistical thinking but the details
reflect the structural nature of the SOLO model. “Idiosyncratic” responses are often
subjective or irrelevant to the content of tasks set, missing the point, which fits with the
Prestructural description of not employing any of the elements relevant to the task set.
“Transitional” responses generally recognize the meaning of the task set but can only
make limited progress toward achieving it, for example providing a partial reading or
interpretation of data from a graph, appreciating the nature of typicality but not providing
a measure, and identifying some aspects required to extend a partially completed graph.
These responses often focus on a single relevant feature of the task and correspond
reasonably with the Unistructural level of the SOLO model. “Quantitative” responses
pick up more than one relevant aspect of task set, often in a procedural fashion, in
order to meet basic requirements, which correspond to the SOLO Multistructural level
where several elements of the task are employed in sequence to reach a conclusion.
Whether this is considered sufficient depends on the nature of the task. For Jones et
al., Quantitative responses generally read and compare data in graphs correctly, order
data, appreciate centres at a surface level, can follow provided models in completing
graphs, and can make multiple responses in interpreting data. “Analytical” responses
correspond to Relational SOLO responses in that all of the elements provided in the
task are combined to provide coherent and comprehensive solutions to tasks. For
each of the four components of statistical thinking Analytical implies satisfying the
task requirements in a statistically appropriate fashion, for example, showing complete
understanding of the link between data and displays, understanding measures of centre
and spread (e.g., range), constructing valid graphs, and making valid “inferences” from
data representations. Only one of Jones et al.’s students could approach the Analytical
level. Similarly Mooney’s (2002) older students were generally unable to surpass the
Quantitative level on any of the four components of statistical thinking as described by
the team of researchers.
The work of Jones and his team (2000) highlights the importance of the graph as the
instrument, tool, and foundation of much of the statistical learning that takes place
at the school level. Putting aside issues of initial question-setting, data collection, and
understanding chance through formal probability, the concrete, visual aspects of graphs
make them an appropriate vehicle for developing statistical concepts. Their work also
indicates the breadth of opportunity to answer questions provided by graphing, beyond
the basic construction of graphs to display data.
Reading (2002) produced a profile of statistical understanding based more broadly
on Holmes (1980) five basic areas of statistics: Data collection, Data tabulation and
representation, Data reduction, Probability, and Interpretation and inference. Her model
of development, using an adapted SOLO model, consisted of nine levels: three in the
Ikonic mode and two cycles of three levels in the Concrete Symbolic Mode. With respect
to Data tabulation and representation, the summary of the performance in the Ikonic
mode was “only mention title or the axis labels (variables) involved” (p. 2, Fig. 1); for the
first Concrete Symbolic cycle, “interact with the data in non-statistical terms – describe
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
49
The Development of Graph Understanding
in the Mathematics Curriculum
individual data points” (p. 2, Fig. 1); and for the second Concrete Symbolic Cycle,
“interact with the data in more statistical terms – describe features of behaviour of the
data” (p. 2, Fig. 1). For Interpretation and inference, the three descriptions began with
“describe patterns but prediction impossible” (p. 2, Fig. 1) for the Ikonic mode, and then
increased to “describe patterns but predict using personal experience and not the data”
(p. 2, Fig. 1) for the Concrete Symbolic Cycle 1 and “predict using pattern descriptions
– justification makes use of the data” (p. 2, Fig. 1) for the second cycle. These were the
two components dependent on graphing and the hierarchical description runs parallel
to that of Jones et al. (2000). In suggesting grade levels associated with the progression
in her hierarchical model, Reading found that Data tabulation and representation was
the component of statistical thinking that students progressed through most rapidly
across the high school years. They appeared to start at lower levels on this component
compared to the other components but progressed to the second Concrete Symbolic
cycle by grade 12. For Interpretation and prediction, the students progressed through
the first Concrete Symbolic cycle by grade 7 but then did not enter the second cycle
until grade 12. Perhaps this outcome reflects a focus on graphing skills across the high
school years but little recognition of using graphs in statistical decision-making during
this time in the state where her data were collected (NSW).
In making recommendations for teaching statistics at the high school level, Reading
and Pegg (1995) suggested tasks for interpreting and analysing data with respect
to the “story” told in the graphs presented and basing data reduction on graphical
representations. Again the tasks are related to graph interpretation rather than creation.
In the interview study of Watson, Callingham, and Kelly (2007), six comprehensive
settings provided the opportunity to explore student understanding of expectation
and variation in statistics based on Rasch analysis (1980) for 73 students in grades
3 to 9. Of the 11 tasks defined in the six settings, one task with four task-steps asked
for the creation of a graph, two tasks with 9 task-steps required only interaction with,
explanation of or interpretation of a graph presented in the task, whereas one task
with four task-steps asked for both graph creation and interpretation. The overall
analysis suggested six levels of a single variable. In the context of the interviews
the Idiosyncratic level (1) displayed little or no appreciation of either expectation or
variation, the Informal level (2) reflected primitive or single aspects of expectation
and/or variation and no interaction of the two, the Inconsistent level (3) displayed
acknowledgement of expectation and variation, often with support, but few links
between them, and the Consistent level (4) showed appreciation of both expectation
and variation with the beginning of acknowledged interaction between them. Given the
contexts of tasks used for the interviews, the highest two levels of the hierarchy were
related to the ability to apply proportional reasoning for either one or two variables.
Level 5 was termed Distributional, displaying established links between variation and
proportional expectation in a single setting (e.g., with a single variable), whereas Level
6 was called Comparative Distributional, where responses displayed established links
between expectation and variation in comparative settings (e.g., with two variables) with
proportional reasoning.
The task that requested a graph to be created (LGR) was based on a scenario of drawing
10 lollies “randomly” from a container of 100 with 50 red. Initial questions based on
expectations were part of another task. For the graphing scenario, students were asked
to imagine performing this process 40 times (with replacement each time) and to
draw a graph of the outcomes (cf. Fig. 4.3). Two of the tasks (CGV, CGX) were based on
50
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
interpreting pairs of bar charts to distinguish which represented a class that had done
better on a spelling test – one task considered expectation and the other variation based
on the interpretation of the graphs. The fourth task (WGR) asked for both the creation
of a graph to describe Hobart’s maximum daily temperature over a year (cf. Fig. 4.2)
and interpretation of three other graphs. The four tasks used in the interview related to
graphs are shown in Fig. 4.6. The tasks and their levels in relation to the hierarchy of
understanding of expectation and variation are summarised in Table 4.3.
LGR. Lollies Scenario: [Suppose you were given a container with 100 lollies in it. 50
are red, 20 are yellow, and 30 are green. You pull out 10 lollies. How many reds do
you expect? Would you get this many every time? What would surprise you?...]
Suppose that 40 students pulled out 10 lollies from the container, wrote down the
number of reds, put them back, mixed them up.
(a) Can you show what the number of reds look like in this case? [Blank space]
(b) Now use the graph below to show what the number of reds might look like for
the 40 students. [2 labelled axes with no data]
CGV/CGX. Two schools are comparing some classes to see which is better at
spelling.
a) Number of People
5
4
2
3
2
2
2
1
2
BLUE
1
2
Number Correct
Number of People
5
4
3
2
1
RED
1
2
Number Correct
3
3
3
3
4
4
5
6
7
8
9
5
6
6
6
6
7
7
7
7
8
9
Now look at the scores of all students in each class, and then decide. Did the two
classes score equally well, or did one of the classes score better? Explain how you
decided.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
51
The Development of Graph Understanding
in the Mathematics Curriculum
b) Number of People
5
4
3
2
1
YELLOW
1
2
Number Correct
Number of People
5
4
3
2
1
BROWN
1
2
Number Correct
3
3
3
4
4
4
5
5
5
5
5
5
6
6
6
7
8
9
4
4
4
5
5
5
5
6
6
6
7
7
8
9
Did the two classes score equally well, or did one of the classes score better?
Explain how you decided.
c) Number of People
7
6
5
4
3
2
1
2
PINK
1
2
Number Correct
52
3
3
3
3
3
Development of graph creation and interpretation
4
4
4
4
4
4
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
8
8
8
8
© State of New South Wales through the NSW
Department of Education and Training, 2010
9
9
The Development of Graph Understanding
in the Mathematics Curriculum
Number of People
7
6
5
4
3
2
1
2
BLACK
1
2
Number Correct
3
3
4
4
4
5
5
5
5
6
6
6
6
6
7
7
7
7
7
7
7
8
8
8
8
8
9
9
Again look at the scores of all students in each class, and then decide. Did the two
classes score equally well, or did one of the classes score better? Explain how you
decided.
WGR. 1. Some students watched the news every night for a year, and recorded
the daily maximum temperature in Hobart. They found that the average maximum
temperature in Hobart was 17º C.
g) How would you describe the temperature for Hobart over a year in a graph?
a)
b)
Figure 4.6. Interview tasks related to graphing.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
53
The Development of Graph Understanding
in the Mathematics Curriculum
Table 4.3. Student performance on the interview items in relation to the hierarchical levels
of the construct related to developing understanding of variation and expectation1
Level
Task-step
Description of student performance
6. Comparative
Distributional
CGV.4
integrated, compared, and contracted multiple
features with a global focus
CGX.5
integrated all available information from visual
comparisons and calculation of means to support
a response in comparing groups of unequal size
LGR.4
created a frequency graph at “5” with appropriate
symmetric variation
CGX.4
used either single visual comparisons
appropriately in comparing groups of unequal
size, or multiple-step visual comparisons or
numerical calculations (using the mean) in
sequence on a proportional basis to compare
groups
WGR.4
combined ability to draw a graph with relevant
features of yearly change and appropriate
interpretation of other graphs
LGR.3
created logical time-series graph with values
around “5” or a frequency graph centred on “5,”
noting change
CGV.3
considered multiple columns of graph in
sequence
CGX.3
used all information for simple group comparisons
but appropriate conclusions restricted to groups
of equal size
LGR.2
created time-series-like graphs for 40 draws;
showed data with variation or data with a centre
CGV.2
considered single columns used terms like
“more” with no justification (Parts b and c)
WGR.3
produced one of two responses: graph created
with change but no trend and appropriate
interpretation of one other graph or inability to
produce more than an informal graph or labelled
axes but appropriate interpretation of other
graphs
5. Distributional
4. Consistent
3. Inconsistent
54
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
2. Informal
1. Idiosyncratic
LGR.1
wrote single numbers or lists, drew pictures, or
sketched graphs without context
CGV.1
considered single columns and used terms like
“more” with no justification (Parts b or c)
CGX.2
used multi-step comparisons or numerical
calculations in sequence for absolute values for
simple equal-sized group comparisons
WGR.2
produced graph showing change but no trend
and focused on single features of other graphs
presented
CGX.1
compared single features in equal-sized groups to
determine which class had done better
WGR.1
produced axes only for created graph and
misinterpreted the others presented
The presence of a decimal value indicates that the item is associated with a coding rubric with
more than a correct/incorrect response, i.e., was associated with partial credit.
1
The highest two levels of the analysis of expectation and variation from the student
interviews correspond closely with the ability to carry out calculational aspects of the
statistics curriculum, the mean and standard deviation. Understanding of the standard
deviation was not part of the student survey but average was, in terms of both mean
and median. Two tasks (CGV.4, CGX.5) on these topics appeared at Level 6 for their
most difficult task-step, indicating a relationship between understanding the measures
of central tendency and the intuition to appreciate the underlying concept of variation
applied in the context of comparative graphical representations. Other task steps were
distributed downward throughout all levels reflecting less appreciation of expectation
and variation as observed (and interpreted) in the graphs.
The four levels of graph creation and interpretation for Hobart’s maximum temperature
over a year range across Idiosyncratic (WGR.1), Informal (WGR.2), Inconsistent
(WGR.3), and Distributional (WGR.4). For the chance task based on drawing 10 lollies
randomly from a container of 100, the four levels of graph creation shown by tasks in
Fig. 3.7 correspond to Informal (LGR.1), Inconsistent (LGR.2), Consistent (LGR.3), and
Distributional (LGR.4). The relative placement of the task-steps indicates that for these
students and tasks, creating the representation of Hobart’s maximum temperature over
a year was slightly easier than creating the distribution of the chance outcomes from
random draws of lollies. In relation to the other tasks in the analysis the two tasks were
considered to be based on a “single” variable and their placement in the hierarchy is
reasonable in relation to the more difficult task of comparing the two groups of spelling
scores when numbers in the groups are unequal.
Again, as was noted with the survey items in the previous section, the tasks that involve
graphing are spread across all levels of the variable documenting students’ increasing
understanding of variation and expectation within the statistics curriculum. This is the
case in terms of both creating and interpreting graphs. By acknowledging the importance
of linking understanding to visual representations, a solid foundation can be developed
for the introduction of theoretical statistics for some students.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
55
The Development of Graph Understanding
in the Mathematics Curriculum
4.4 Development suggested by the two Rasch analyses
Combining the information in Tables 4.2 and 4.3, the summary presented in Table 4.4
is meant to give an overall feeling for how the progression of graphical understanding
develops across the years of schooling from roughly grade 3 to grade 10. The range of
years observed for any of the stages is quite large and it is not appropriate to label them
with specific years. The summary only represents evidence from the tasks used in the
studies reported here. Tasks, such as, “please draw a pie graph,” were not employed
in the research. The use of more open, flexible tasks was intended to allow students
to display their intuitions in telling stories about data in any way they felt appropriate.
Hence responses were judged on their success in conveying the meaning in the story,
not just meeting graphing conventions. The insight obtained from the research, however,
informs the suggestions made for a developmentally friendly progression of introducing
various graph types across the years. The research also points to the importance of
context in devising a curriculum including graphing, as well as links to understanding of
variation and expectation as intuitive ideas from an early age.
Table 4.4. Summary of proposed development of graph understanding (from two Rasch analyses)
Level
Graph Creation
Graph Interpretation
6. Critical
mathematical/
Comparative
Distributional
[No tasks at this level.]
Speculation: Can represent
expectation and variation in
graphs produced to compare two
groups (e.g., one categorical and
one numerical variable)
Includes variation and
expectation in representing a
single distribution; can create
a graph appropriately showing
the association of two numerical
variables
Successful in representing
individual variables involved in
association; represents centre,
where appropriate, intuitively
without explanation
With support, may begin to focus
on centre, when creating graphs,
but unlikely to display trends
Considers expectation (means)
and variation (shape) when
comparing groups of unequal
size; uses subtle language of
uncertainty and mathematical
skills in assessing graphs
Recognises some critical
features of graphs (for example
in the media), including the
need to account for unequal
sized groups when comparing
two groups
Recognises pattern and/or
majority in assessing graphs;
willing to suggest reasons
(but usually inappropriate) for
unusual features of graphs
Moving between just reading
individual data to summarising
several aspects in single
dimension graphs (like dot plots)
Beginning to appreciate features
of graphs (like minimum value)
and can read data from dot plots
5. Critical/
Distributional
4. Consistent
3. Inconsistent
2. Informal
1. Idiosyncratic
56
Willing to try out a range
of mostly inappropriate
representations; starting to
appreciate the existence of
variation
Can create a picture to tell a
story or draw axes
Development of graph creation and interpretation
Ability to read information from
pictographs
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Research such as that summarised here documents what students can do when given
opportunity, i.e., to create or interpret a graph. Moving up the hierarchy, the activities
they are observed to do become more complex. When there are aspects that students
struggle to comprehend or perform inadequately, these are signposts for teachers to be
aware of, in order to plan activities to assist in the transition to appropriate responses.
In thinking of a teaching sequence, teachers would never exclude looking for trends
for example (something students struggle with at the Inconsistent level) but might
produce an inadequate representation in order to challenge students to think of ways
of improving it. At every grade level teachers can encourage their students to be critical
thinkers. The traits observed at the highest level need to be developed over a long period
of time and should not be “discovered” only in later years.
In suggesting a structure for teaching graph creation in the next section, the purpose is
to make the reader aware of the elements involved and how they are combined to create
meaningful representations of data and their attributes. At the same time, based on the
research summarised here, teachers should be aware of the gaps to look for and be
prepared to fill them to enable students to reach higher levels in their understanding.
4.5 Building a general developmental model for graph creation
The focus of this section is on graph creation, ending with its potential to contribute to
the larger scene of data analysis. The following section considers graph interpretation.
In this developmental context the situation for graphing is similar to that for average.
There is a basic hierarchical process of building up the concept of average as a number
that is representative of a data set in several possible ways. The concept is more
than a formula, which is one step in the development. Once the concept of average
is consolidated it can then be applied in further developmental hierarchies to solve
problems, for example weighted average problems (Watson & Moritz, 2000) or in
comparing two groups (Watson & Moritz, 1999). If the process is thought of as cyclical
based for example on the work of Biggs and Collis (1982, 1991) and Pegg (2002a,
2002b) then a first cycle develops the concept of average and a second cycle (or more)
applies it in statistical contexts; some of these contexts may involve graphs. Fig. 4.7
shows that the development of the concept of average is based on four basic elements,
which are first used singularly in describing average and then combined in pairs before
constructing an integrated concept.
The concept of average
Relational
stage
Chooses measure to represent data appropriately
Multistructural
stage
Links two elements, e.g., Combining procedures with elements
Unistructural
stage
Colloquial usage – “okay”
Elements
Middle
Most
Balance
Representation
Figure 4.7. Developmental sequence for Average.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
57
The Development of Graph Understanding
in the Mathematics Curriculum
In another regard, however, the situation for graphing is different and more complex
than that for average. There may be three or four basic measures for average but there
are many more types of graphs relevant to the school curriculum. Also the procedure for
finding an average does not change if there are a small or large number of data points
(it just may become more tedious). The procedures associated with creating graphs,
however, change if large data sets are involved or if the variables involved are numerical
(e.g., measurements) or categorical (e.g., gender). It is hence suggested here that after
an initial developmental cycle that produces the concept of a graph, there are two
parallel cycles of development building on the basic graphing concept: one is based on
the ability to create and/or choose appropriate graphs for large data sets, and the other
is based on the ability to create and or choose appropriate graphs when more than one
attribute needs to be displayed at the same time. As examples, the first may require
development of understanding of the histogram, whereas the second may require
development of understanding of the scatterplot.
Once these understandings are developed the consolidated concepts are available to be
combined with the concept of average to embark on a yet higher level cycle (probably in
the formal mode of the Biggs and Collis model), the decision-making processes that lead
to informal statistical inference. Another cycle or perhaps more than one, building the
mathematical procedures and understanding, is needed to achieve application to formal
statistical inference, but speculation on this is beyond the scope of this report. The
following three subsections focus on the first cycle building the concept of graph and the
two “second” cycles that follow it for larger data sets or more attributes.
4.5.1 The first cycle: The concept of graph
The basic elements required to develop the concept of graph are attribute – something
of interest that is measured or counted; data – the existence or collection of information
about the attribute for different cases; variation – the acknowledgement of difference
or change among the data/cases; and scale – the necessity to keep track of the data
in a fashion that makes fair comparisons possible. The idea that variation is absolutely
fundamental to statistics (Moore, 1990) makes it one of the critical elements of
developing the concept of graph. Students are unlikely to have sophisticated ideas about
variation but the expectation that data for attributes will vary/change is essential to the
process of graph creation. The fact that the word variable is synonymous with attribute
reflects this understanding but variable and variation are not exactly the same – both
are needed in the process and using the term “attribute” assists to make the distinction.
Although context is not mentioned explicitly in the model, it is assumed that the context
represented by the attribute and the data are understood.
At the first stage in the process of developing the concept of graph, students are likely
to comprehend one or more of the four elements but not be able to link them together
meaningfully. For example they may recognise variation in the characteristics of the
students in the classroom but not be able to measure or represent it; they may be able
to count accurately but not represent numbers to compare; they may see data as “dots”
but not be able to connect them with the attributes they represent. Following the Biggs
and Collis (1982, 1991) model this stage is labelled unistructural (e.g., Watson & Moritz,
2001). At the next stage students can link two or more of the elements, perhaps creating
simple pictographs and discussing difference or discussing how the data tell about the
attribute. This stage is labelled multistructural. At the stage where the four elements are
connected meaningfully, called relational, students can create a meaningful picture/
58
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
diagram with appropriate scale and tell the story of the variation in the data and what
it means for the attribute being displayed. With this concept of a graph as a pictorial
representation of the elements, the student is ready to move on to a second cycle and
extend the concept in two directions. This developmental structure is shown in Fig. 4.8.
The Concept of Graph
Relational
stage
Combines all elements to create a representative graph
Multistructural
stage
Links 2 or 3 elements to create basic graphs
Unistructural
stage
Uses single elements unlinked
Elements
Attribute
Data
Variation
Scale
Figure 4.8. Developmental sequence for the basic Concept of Graph.
Having reached this point it does not seem reasonable to suggest that one of the
next two cycles should precede the other. Conceptually they are parallel. In fact from
considering curriculum documents there appears to be a mix of graph types of the two
sorts across the years. The potential motivational value of adding more attributes to
the mix, however is now acknowledged (e.g., NCTM, 2000) and since it can often be
managed with small data sets, it is likely that dealing with two attributes occurs earlier.
4.5.2 A second cycle: The ability to create or choose appropriate graphs when more
than one attribute is involved
The basic elements needed for the development of graph creation for more than one
attribute are: the consolidated graph concept from the first cycle; different types of
attributes – categorical, numerical, discrete, continuous; two-dimensional scaling; and
measurement of two or more attributes on a single case unit. These are the elements
that are combined to create the types of graphs that tell stories about two (or more)
attributes: split stacked dot plots, time series graphs, line graphs, scatterplots. Time
deserves special mention in this context because often events are documented over
time. Time may also be implicit in “before” and “after” measurements, as well as
being the focus of one of the measurements, such as heart rate or “winning” time.
A hierarchical sequence can be imagined to create each variation on the theme of
displaying the relationship between two attributes and the aim at the relational level is to
understand the range of possible graphs and where they are appropriate for application
to tell a story. At the unistructural stage students struggle to go beyond their basic
concept of graphing and incorporate the other elements, perhaps confusing the different
types of attributes or how to plot two values on different axes to create a single point in
a scatterplot. At the multistructural level of this cycle students create some or all of the
representations for the various possible pairs of attributes. Having built a repertoire of
graph types at this level, students may find it difficult to use each. As noted, however,
at the relational level, students can recognise among the graph types they understand,
the one appropriate for particular associations between attributes and apply it to tell the
story of the data. Fig. 4.9 summarises this progression.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
59
The Development of Graph Understanding
in the Mathematics Curriculum
The Concept of Graph for Multiple Attributes
Relational
stage
Chooses and or creates appropriate graph for attributes and
explains their application
Multistructural
stage
Creates graphs from elements: split dot plots, time series, line
graph, scatterplots
Unistructural
stage
Builds elements, cannot combine into complete graphs
Elements
The Concept of
Graph
Types of
attributes
2-D scaling
Relationship
of two
attributes to
single case
Figure 4.9. Developmental sequence for the Concept of Graph for Multiple Attributes.
4.5.3 A second cycle: The ability to create or choose appropriate graphs for large
data sets
The basic elements for the development of graph creation for large data sets are: the
consolidated graph concept from the first cycle; percentage understanding; the fivenumber summary; using area to represent frequency; equal intervals with respect to
scale. The elements for this cycle appear somewhat more procedural than for the other
second cycle. It is, however, essential that students understand the procedures they
bring to creating graphs and when to use them. At the unistructural stage they struggle
to remember which procedure is appropriate when and are likely to confuse the use of
percentage and frequency. At the multistructural level they are able to create histograms,
cumulative frequency graphs and ogives, frequency polygons, box plots, and pie charts
for large data sets. They may not yet have the ability to select the appropriate graph for
a particular data set and attribute. This ability comes at the relational stage with the
integrated understanding of the various representations. Fig. 4.10 summarises this
progression.
The Concept of Graph for Large Data Sets
Relational
stage
Chooses and or creates appropriate graph for attributes and
explains their application
Multistructural
stage
Creates graphs from elements: histograms, cumulative frequency
graphs, ogives, frequency polygons, box plots, pie charts
Unistructural
stage
Builds elements, cannot combine into complete graphs
Elements
The
Concept of
Graph
Percentage
5-number
summary
Area
representing
frequency
Figure 4.10. Developmental sequence for the Concept of Graph for Large Data Sets.
60
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
Equal
interval
grouping
The Development of Graph Understanding
in the Mathematics Curriculum
4.5.4 The third cycle: Informal decision-making for graphs
The third cycle takes the appropriate creation and choosing of graphs one step further
and applies the graphs to decision-making, potentially in the form of informal inference.
The elements are the two advanced graphing understandings related to multiple
attributes and to large data sets, the concept of average – required to consider central
tendency in graphs; and the concept of variation – revisited although an element
of the original concept of graph due to its significance in contrasting with average.
At the unistructural stage students are likely to struggle for example to combine the
comparison of multiple attributes with presence of large data sets, or to appreciate the
link between central tendency and variation. Combining pairs of the elements is likely
to provide partial answers to statistical questions at the multistructural level but for
most statistical enquiries all elements will need to be integrated for completely justified
conclusions to be reached. The overall developmental model is summarised in Fig. 4.11.
Informal* Decision Making for Graphs
Relational
stage
Combines all elements as required by the question to reach an
informal conclusion
Multistructural
stage
Combines two or more elements to reach partial informal
conclusions for questions about a data set
Unistructural
stage
Appreciates elements in isolation, has difficulty combining
Elements
Concept
of
Variation
Concept of Graph for
Multiple Attributes
Concept of
Graph for Large
Data Sets
Concept of Graph
Concept
of
Average
*The term informal is used here to distinguish the hierarchy from one that would involve formal statistical t ests.
Figure 4.11. Development sequence for Informal Decision Making for Graphs.
The important aspect of teaching graph creation is to be certain that all of the elements
are made known to students as the work is begun. Since it is likely that it will take some
time before all students are able to integrate the elements, repetition will be needed
when each new type of graph is introduced. The need to understand the element,
attribute, means that the creation of graphs should not be considered without a context.
Data are needed to create a graph and they must be “measuring” some attribute. The
concept of graph is built over time from seeing many different examples in different
contexts. It is building this repertoire that makes it possible to choose and use graphs
when given data handling questions.
Young children for example may be able to create partial pictographs, for example
without scale and this may be adequate as a starting point given their ability to
understand basic scaling. They may be able to consolidate the ideas attribute, data and
variation, to which they can add scale later.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
61
The Development of Graph Understanding
in the Mathematics Curriculum
4.6 Building a general developmental model for graph interpretation
In considering the hierarchy of observed development of students’ understanding in
relation to graphing in Table 4.4, the type of graph interpretation that is the goal of the
mathematics curriculum only really appears at the top two (critical) levels of the variable.
This seems reasonable since it is necessary to have a consolidated concept of what a
graph is before one can interpret its meaning and perhaps question its validity. It is the
basic concept of graph that is built up in the first four stages of the hierarchy.
It is possible then, building on the proposed model in the previous section, to imagine
graph interpretation as another “second” cycle after the first of building the concept
of graph. Depending on the complexity of the graph and the context it represents,
however, the interpretation process may be built after another second cycle, say where
understanding of graphs of more than one attribute is consolidated. Hence a cycle
related to graph interpretation may be considered as flexible, or moveable, having as
one of its basic elements the type of graph that is represented. Another element that
is foreshadowed in the earlier cycles for graph creation is that of context. At the level
of graph interpretation, however, the need for understanding of context is broader.
It is not only necessary to understand what the data actually measure in terms of
the attributes but also necessary to have understanding of what are reasonable and
meaningful boundaries and shapes, and whether there are other factors, such as
biased sampling, impinging on what is shown in the graph. A third element required
for graph interpretation is a questioning attitude. This can be seen in critical thinking,
in appreciating uncertainty, and in sometimes looking outside of the graph for further
information. These are combined with the elements of variation and average implying
that graph interpretation can become quite involved and complex. This is reflected to
some extent in the descriptions from the Australian Mathematics Curriculum (ACARA,
2010) listed in Table 2.2.
In a sense then, graph interpretation is one of the “third” cycles alluded to in Fig. 4.11.
Graph interpretation almost inevitably leads to decision-making and represents the
successful application of the consolidated concept of graph relevant to the context
presented in a given task. It is likely that success at graph interpretation will reflect
Curcio’s (1989) and Shaughnessy’s (2007) characteristics of reading behind and beyond
the data. How sophisticated the interpretation is depends on the initial task and its
associated graph. Fig. 4.12 presents a possible developmental sequence. The number of
elements employed will depend on the complexity of the graph and task.
Graph Interpretation
Relational
stage
Ability to question or draw implications from the graph by combining
understanding of the elements present
Multistructural
stage
Consolidating the message in the graph based on the elements
present
Unistructural
stage
Appreciation of the single elements as they appear in the graph
presented
Elements
Concept
of Graph
(Basic or
Advanced)
Concept of
Variation
Concept
of Average
Context
Figure 4.12. Developmental sequence for Graph Interpretation.
62
Development of graph creation and interpretation
© State of New South Wales through the NSW
Department of Education and Training, 2010
Critical
Questioning
Attitude
The Development of Graph Understanding
in the Mathematics Curriculum
Do students have to be able to create a certain type of graph before they can interpret
a similar one? Probably not, as long as they have a general appreciation of the graph
concept and its elements. It is well known that most school leavers will have more
opportunity to interpret graphs than to create them. Graphs are very common in
advertising and popular sporting contexts. Hence developing the overall concept of
graph with flexibility in terms of the elements is very important even if the ability to
create complex representations may not be complete. What is important as well, is
to have the ability to interpret, and hence not use inappropriately, the fancy graphs
presented by spreadsheets.
An application of a developmental structure to a practical guide for graph interpretation
has been suggested by Kemp and Lake (2001; Lake & Kemp, 2001), again based
on the SOLO model (Biggs & Collis, 1982). The Five Step Framework for Interpreting
Tables and Graphs provides a generic template for teachers to use when assisting
students to develop strategies for interpreting graphical and tabular data. In relation to
graphs, it can be used by teachers to construct questions for students to answer when
developing graph interpretation skills, particularly for graphs presented in the media.
The Five Step Framework recognises explicitly the importance of students developing
an understanding of the variation within and among data. Originally developed for
tertiary students, Kemp has extended the application of the framework to primary and
secondary contexts. Lake and Kemp (2001) provide details of the 5-step process for
graphs and these are summarised in Table 4.5. The parallel of Steps 2 to 4 with the
unistructural, multistructural, and relation levels of the SOLO model discussed earlier is
evident.
Table 4.5. Developmental teaching model for interpreting graphs (adapted from Lake & Kemp, 2001)
Step
Examples of Characteristics
1. Getting organised
Find the topic, scope, and definitions from
the title, labels, footnotes on the graph.
2. WHAT do the numbers mean? Looking
at individual points
Consider individual values that stand
out. Make generalisations but avoid
premature closer.
3. HOW do they change? Looking at
trends
Use several pieces of information to
consider local variation and trends in the
data.
4. WHERE are the differences? Looking
at relationships
Consider data as a whole, interrelating
the information. Seek consistency and
use simple statistics.
5. WHY do they change? Looking at
meaning
Use and interrelate all information and
test against theories or hypothetical
situations.
© State of New South Wales through the NSW
Department of Education and Training, 2010
Development of graph creation and interpretation
63
The Development of Graph Understanding
in the Mathematics Curriculum
5. Implications
5.1 The NSW context: Working Mathematically
The Working Mathematically strand of the NSW Mathematics Syllabus (BoSNSW, 2002a,
2002b) consists of five interrelated processes – Questioning, Applying Strategies,
Communicating, Reasoning, and Reflecting. These are noted with some examples
for every stage of every content area and appear from introductory material to be
considered undefined terms. In creating an annotated bibliography of sources for this
project, the processes from Working Mathematically were used repeatedly, with all
five considered a part of the research or teaching focus of many of the sources. They
appear so intrinsically entwined that placing an excessively strong emphasis on one
process to the exclusion of others is likely to lead to learning that is limited in its scope
and application. Concentrating learning on graph creation, for example, to facilitate the
development of Communication and Applying Strategies skills may be successful for
those two but not develop Reflection. A holistic approach to learning about data and
associated graphs hence is required to achieve comprehensive learning outcomes that
encompass all five aspects of Working Mathematically. This can be achieved using an
inquiry-based approach to learning. Inquiry-based learning encompasses investigations
that require students to answer questions set in meaningful contexts. Students work
through learning activities that include gathering information, making decisions about
the best way to display and transform the information, deciding the validity of the
information, looking for relationships in the information, and making connections
between what they have learnt from the information and their existing knowledge. As
these are also the fundamental skills and processes that are developed through graph
creation and graph interpretation, embedding all components together as a fundamental
part of the mathematics curriculum has the potential to satisfy all aspects of Working
Mathematically.
As students progress through the years of schooling they will build up their knowledge of
graph types. This process is cumulative, giving the students a tool kit of representations
that they can select from when given the freedom to do so. When students make the
decisions about which representation to use to answer a particular question about the
data, they engage in critical thinking, drawing on their understanding of the graph type
as well as their graph interpretation skills. Having a selection of graph types to choose
from also allows them to be creative and explore various representations of the same
data, with and without the use of technology. Engaging in the learning process in this
way also allows for the development of critical thinking, thereby honing their Working
Mathematically skills.
The situation exists today where technology is available in the classroom when students
begin school. Young students have access to and are able to use graphing software
to create accurate graphs. The task of creating a graph by hand can be laborious and
prone to error and inaccuracy, particularly when specific graph conventions are to be
adhered to. Further, as drawing graphs by hand is very time consuming, the creation
of multiple graphs and looking at the same data using different graphs types may not
occur. Graphing software provides the freedom to create multiple graphs and various
graph types to explore how data can inform decisions, influence opinions, and support
hypotheses. This is not to say students should not create graphs by hand. There is great
value in being able to draw graphs as well as create graphs by manipulating data in the
64
Implications
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
form of data cards and pictures. Technology provides the additional opportunity to learn
how to work mathematically using data without the burden of physical graph creation.
It is important to remember that the thinking about and interpretation of graphs
produced by software, or by hand for that matter, cannot occur if students do not
understand the representations applied. Therefore, learning about different graphs
types and the subsequent creation of them must go hand-in-hand with developing an
understanding of what they mean and what they can be used for. To address this issue,
also keeping in mind the comments made in Section 4.6, any curriculum must be
flexible enough to allow for the development of graph creation in coordination with graph
interpretation. At times, when new graph types are introduced the learning emphasis will
be on graph creation and to a lesser extent graph interpretation. At other times, when
students are proficient at creating particular graph types the emphasis will be on making
sense of the data and making inferences, from their own graphs and others. Overlaying
student learning about data representations and using data is the way students develop
an understanding of not only the underlying mathematics but also the way in which it
can be used. Surely this encompasses what Working Mathematically is about.
5.2 Sequencing learning and the curriculum
The implication of considering students’ development of understanding is not that
teachers introduce graphing and graphing types in incomplete stages but that they
are aware of the elements that they need to make explicit at the start and constantly
reinforce how the elements are linked in order to create a meaningful representation.
It is not just a matter of drawing lines with rulers and plotting numbers in a procedural
fashion but of thinking about each element and how it is linked to the others. At the end
of any activity it is essential for the teacher to reinforce the relationships and how they
tell the story in the data.
As an example in relation to Fig. 4.8 on the Concept of Graph consider the introduction
of a stem-and-leaf plot. An appropriate attribute might be heart rate, which could be
discussed with students in terms of exercise or rest. The associated data might be
measured on the students and the link to the attribute can be made in terms of a
decision about whether the rate recorded is “per 15 seconds,” “per 30 seconds,” or “per
minute.” The recorded data can then be discussed in terms of place value, focussing on
the tens and units digits. This information can then be linked to the scale to determine
end points (top and bottom) and intermediate tens values. As data are recorded in the
plot, variation is noted as the leaves extend further for some tens values than others.
Labelling the plot reminds students of the attribute and they can write a summary about
the story present in the data they have plotted.
To use the same data to introduce a stacked dot plot instead of a stem-and-leaf plot
would only change the discussion of creating the scale on a baseline, determining the
end points and markers in between them. The implications for discussing variation
would be similar as data values were plotted.
For the next cycle of introducing another attribute (cf. Fig. 4.9), one of the elements
assumed is the basic stem-and leaf or stacked dot plot. Teachers need to be certain
students have this element consolidated. It is then important to introduce the new
attribute, carefully distinguishing it from the first; for example it might be another heart
rate, say after exercise if the first measurements were taken at rest. Or the second
© State of New South Wales through the NSW
Department of Education and Training, 2010
Implications
65
The Development of Graph Understanding
in the Mathematics Curriculum
attribute might be the categorical attribute, gender. In either situation it needs to be
made clear that for each case (e.g., student) on which measurements are made, there
are now two pieces of information, either active and rest heart rates, or heart rate and
gender. For the stem-and-leaf plot scenario, the adaptation and perhaps extension of the
central scale is straightforward but students need to be aware of the meaning attached
to the leaves that appear on each side of the stem. In the two examples suggested
the meaning is somewhat different. For the resting and active heart rates, a student is
represented by two leaves, one on each side of the stem. For heart rate and gender, a
student’s numerical value only occurs once, the second attribute being determined by
the side of the stem the leaf is on. The link to the attributes is made explicit through the
labelling of the plot.
Similarly, for the stacked dot plot, adding a second heart rate may suggest the
introduction of a second scaled horizontal baseline, labelled with the name of the
attribute, perhaps longer than the first but arranged so common values are seen
in relation to each other. In this situation a student will be represented by a dot on
each plot. As well, however, a scaled vertical axis could be introduced to intersect the
horizontal line, providing the opportunity to represent each student with only one dot
but a dot that conveys two pieces of information, a heart rate from each labelled axis,
thereby constructing a scatterplot. These two possible representations show the relative
power of the scatterplot to show trends based on individual values rather than group
values (as is the case in two stacked dot plots). Further than this, however, having
both representations in their repertoire allows students to make choices about how
they tell the stories they find in data sets. Depending on the variation in the data, one
representation might be better or simpler at getting the message across than the other.
Considering the attributes heart rate and gender, again two stacked dot plots with linked
scales (one for each gender) would be used and this time a student’s data value would
be represented by only one dot, placed on the plot for the appropriate gender.
These examples show how complex the process of introducing even relatively simple
graphical forms can be. Once the elements are made clear and reinforced, however,
students have power to create meaningful representations to tell the stories in the data
sets.
The order in which graph types are introduced is likely to reflect the complexity
associated understanding the basic elements as other related aspects of mathematics
are developing. Conservation of number and counting are clearly prerequisites for work
with graphing, although early graphs can be used for reinforcement of counting and
comparison skills. Categorical attributes, such as colours or fruit types, are likely to be
understood before measurement attributes, for example, measuring with rulers. The
related complexity of the scales for axes suggests an ordering such as categories (e.g.,
red, blue, green, …), followed by whole number markings for “stacking dots,” then scales
with markings in fractions or decimals between whole number values. Later interval
scales associated with histograms may be quite variable and hence more complex.
The Australian Mathematics Curriculum (ACARA, 2010) aligns the introduction of graphs
with the specific years, or sometimes a series of years. Once a graph type is introduced
it should be available in successive years depending on the requirements of a particular
investigation and its associated data set. What is missing from the curriculum, as
noted in Section 2.4.1, is a clarification on terminology associated with “column”
graphs and the case-value plots discussed in this report. It is also possible to argue
66
Implications
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
that stem-and-leaf plots could be introduced earlier than Year 6, as soon as students
have a consolidated understanding of place value. Work with pie charts depends
on the understanding of the proportional representation possible using areas of the
segments of the circle. The actual construction of pie charts is relatively simple and
does not depend on calculations involving central angles and 360o in the early years.
Overall, the sequence of introduction of graph types appears reasonable, although in
some instances they may be accessible in earlier years. If the earlier types of graphs
are consolidated well, then it seems reasonable to leave box plots until Year 10, as
students are more likely to have the proportional reasoning understanding necessary to
appreciate the density representations they provide. Hat plots (cf. Section 3) would be a
useful intermediate step as suggested by Watson et al. (2008).
In relation to graph interpretation, again it is useful to be aware of the potential levels of
development that students are likely to experience and to help them make connections
necessary to reach higher levels. Kemp’s work (cf. Section 4.6) provides a workable
model that can actually be given to students to assist them to get the most out of graphs
they wish to interpret. Links back to the suggestions of Kosslyn (1989) are also very
relevant in terms of graph interpretation. He first focuses on the individual elements or
constituent parts of the graph (cf. Fig. 2.1) and then moves to the relationships shown
among the elements. He finally suggests extending the interpretation to the symbols
and lines that go beyond a literal reading of the graph. Taking this suggestion to include
the context of the graph fits well with the framework of Lake and Kemp (2001) and the
recognition of student’s developmental needs (Rangecroft, 1991a).
The history of graphs and graphing (cf. Section 2.2) is instructive in thinking about the
future of graph creation and interpretation in the school curriculum. Across several
centuries, advances in scientific technology provided more types of data, which in
turn forced innovation in graph creation. Some of these graph types have come into
common usage and others have become quaint historical oddities. Similar scientific
or social science innovations in data collection today continue to require new software
tools to provide visual representations to tell the stories in the data. In terms of the
school curriculum, Wall and Benson (2009) capture the dilemma in their title, “So
many graphs, so little time,” for an article showing some of the latest innovations in
graphing. The dilemma for the school curriculum also includes the potential complexity
of representations of limited application that could become a distraction from the basic
understanding that will be applicable, with variation, across many contexts. As students
are unlikely to have the state-of-the-art software tools available to them for graph
creation, the importance of graph interpretation grows, with the further potential of
cross-curriculum links.
The extracts from the Australian Mathematics Curriculum (ACARA, 2010) in relation
to graph interpretation outlined in Tables 2.1 and 2.2 demonstrate the importance of
graph interpretation across all years K−10. Some statements relate, for example, to
averages, which could be thought of a separate procedural part of the Statistics and
Probability curriculum. The view taken here is that relating averages explicitly to visual
representations is essential to make complete and meaningful interpretations of the
messages from graphs. Hence the elaborations that include graph interpretation serve
two important purposes: they reinforce graph creation at every level and they link to
wider goals of statistical literacy. The latter is important not only for the Mathematics
Curriculum but also more widely for the entire Australian Curriculum, as statistical
literacy is a critical element of “Numeracy,” which is seen in the Shape Statement
© State of New South Wales through the NSW
Department of Education and Training, 2010
Implications
67
The Development of Graph Understanding
in the Mathematics Curriculum
(National Curriculum Board, 2009), as an essential skill, along with Literacy (p. 7).
The Shape Statement further says, “Numeracy knowledge, skills and understanding
need to be used and developed in all learning areas” (p. 12). The door is now open
for much more cross-curriculum engagement with both graph creation and graph
interpretation. Having built a firm foundation on creating and then interpreting graphs in
the Mathematics Curriculum, there is great opportunity to illustrate their value as part of
Numeracy across the curriculum.
68
Implications
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
References
Australian Curriculum, Assessment and Reporting Authority. (2010). Draft Consultation
1.0 Mathematics. Canberra: Author. Retrieved March 20, 2010 from http://www.
australiancurriculum.edu.au/Home
Australian Education Council. (1991). A national statement on mathematics for
Australian schools. Melbourne: Author.
Bakker, A. (2004). Design research in statistics education: On symbolizing and computer
tools. Utrecht, The Netherlands: CD-ß Press, Center for Science and Mathematics
Education.
Beniger, J.R., & Robyn, D.L. (1978). Quantitative graphics in statistics. The American
Statistician, 32(1), 1-11.
Bertin, J. (1981). Graphics and graphic information-processing. Translated by W.J. Berg
and P.S. Berlin: De Gruyter.
Bertin, J. (1983). Semiology of graphics: Diagrams, networks, maps. (W.J. Berg, Trans.).
Madison, WI: University of Wisconsin Press. (Original work published 1967)
Biggs, J.B., & Collis, K.F. (1982). Evaluating the quality of learning: The SOLO taxonomy.
New York: Academic Press.
Biggs, J.B., & Collis, K.F. (1991). Multimodal learning and the quality of intelligent
behaviour. In H.A.H. Rowe (Ed.), Intelligence: Reconceptualization and
measurement (pp. 57-76). Hillsdale, NJ: Lawrence Erlbaum.
Board of Studies New South Wales. (2002a). Mathematics K-6 syllabus. Sydney:Author.
Board of Studies New South Wales. (2002b). Mathematics years 7-10 syllabus.
Sydney:Author.
Boddington, A.L. (1936). Statistics and their application to commerce. (7th ed.). London:
Sir Isaac Pitman & sons, Ltd.
Burrill, G., Scheaffer, R., & Rowe, K.B. (Eds.). (1994). Teaching statistics: Guidelines for
elementary through high school. Palo Alto, CA: Dale Seymour Publications and the
Center for Statistical Education American Statistical Association.
Card, S.K., Mackinlay, J., & Shneiderman, B. (Eds.) (1999). Readings in information
visualization: Using vision to think. San Francisco, CA: Morgan Kaufmann
Publishers
Chick, H. (2003). Tools for transnumeration: Early stages in the art of data
representation. In L. Bragg, C. Campbell, G. Herbert, & J. Mousley (Eds.),
Mathematics education research: Innovation, networking, opportunity (Proceedings
of the 26th annual conference of the Mathematics Education Research Group of
Australasia, Geelong, pp. 207-214). Sydney, NSW: MERGA.
Chick, H.L., & Watson, J.M. (2001). Data representation and interpretation by primary
school students working in groups. Mathematics Education Research Journal, 13,
91-111.
Chick, H.L., Pfannkuch, M., & Watson, J.M. (2005). Transnumerative thinking: Finding
and telling stories with data. Curriculum Matters, 1, 87-108.
© State of New South Wales through the NSW
Department of Education and Training, 2010
References
69
The Development of Graph Understanding
in the Mathematics Curriculum
Cleveland, W.S. (1993). The visualisation of data. Summit, NJ: Hobart Press.
Cobb, P. (1999). Individual and collective mathematical development: The case for
statistical data analysis. Mathematical Thinking and Learning, 1, 5-43.
Curcio, F.R. (1987). Comprehension of mathematical relationships expressed in graphs.
Journal for Research in Mathematics Education, 18, 382-393.
Curcio, F.R. (1989). Developing graph comprehension. Reston, VA: National Council of
Teachers of Mathematics.
Department of Education Tasmania. (2007). The Tasmania curriculum: Mathematicsnumeracy. K-10 syllabus and support materials. Hobart: Author.
Department of Education and Science and the Welsh Office. (1989). Mathematics in the
National Curriculum. London: HMSO.
Department of Education and Training Western Australia. (2007). K-10 overview
- Mathematics - Chance and data. Retrieved January 23, 2010 from http://k10syllabus.det.wa.edu.au/content/syllabus-documents/k-10-overviews/k-10overview-mathematics
Emerson, J.D., & Hoaglin, D.C. (1983). Stem-and-leaf displays. In D. C. Hoaglin, F.
Mosteller, & J. W. Tukey (Eds.), Understanding robust and exploratory data analysis
(pp. 7-32). New York, NY: John Wiley & Sons, Inc.
Emerson, J.D., & Strenio, J. (1983). Boxplots and batch comparison. In D. C. Hoaglin, F.
Mosteller, & J. W. Tukey (Eds.), Understanding robust and exploratory data analysis
(pp. 58-96). New York, NY: John Wiley & Sons, Inc.
Fienberg, S. E. (1979). Graphical methods in statistics. The American Statistician, 33(4),
165-178.
Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M., & Scheaffer, R.
(2007). Guidelines for assessment and instruction in statistics education (GAISE)
report: A preK-12 curriculum framework. Alexandria, VA: American Statistical
Association. Retrieved July 3, 2009 from http://www.amstat.org/education/gaise/
Friendly, M. (2007). A brief history of data visualization. In C. Chen, W. Hardle, & A. Unwin
(Eds.), Handbook of Computational Statistics: Data Visualization, Vol III, Heidelberg:
Springer. Retrieved November 18, 2009, from: http://www.math.yorku.ca/SCS/
Gallery/milestone/
Friendly, M. (2009). Milestones in the history of thematic cartography, statistical
graphics, and data visualization. Retrieved November 18, 2009, from: http://www.
math.yorku.ca/SCS/Gallery/milestone/
Funkhouser, H.G. (1937). Historical development of the graphical representations of
statistical data. Osiris, 3, 269-404.
Hall, H.S., & Knight, S.R. (1885). Elementary algebra for schools. London: MacMillan and
Co.
Holmes, P. (1980). Teaching statistics 11-16. Slough, UK: Schools Council and Foulsham
Educational.
Jones, G.A., Thornton, C.A., Langrall, C.W., Mooney, E.S., Perry, B., & Putt, I.J. (2000). A
framework for characterizing children’s statistical thinking. Mathematical Thinking
and Learning, 2, 269-307.
70
References
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Kelly, B.A., & Watson, J.M. (2002). Variation in a chance sampling setting: The lollies
task. In B. Barton, K.C. Irwin, M. Pfannkuch, & M.O.J. Thomas (Eds.), Mathematics
Education in the South Pacific (Proceedings of the 26th Annual Conference of the
Mathematics Education Research Group of Australasia, Auckland, Vol. 2, pp. 366373). Sydney, NSW: MERGA.
Kemp, M., & Lake, D. (2001). Choosing and using tables. Australian Mathematics
Teacher, 57(3), 40-44.
Konold, C. (2002). Alternatives to scatterplots. In B. Phillips (Ed.), Proceedings of the
Sixth International Conference on Teaching Statistics: Developing a Statistically
Literate Society, Cape Town, South Africa. [CDrom] Voorburg, The Netherlands:
International Statistical Institute.
Konold, C., & Higgins, T.L. (2003). Reasoning about data. In J. Kilpatrick, W.G. Martin,
& D. Schifter (Eds.), A research companion to Principles and Standards for School
Mathematics (pp. 193-215). Reston, VA: NCTM.
Konold, C., & Miller, C.D. (2005). Tinkerplots: Dynamic data exploration. [Computer
software] Emeryville, CA: Key Curriculum Press.
Kosslyn, S.M. (1989). Understanding charts and graphs. Applied Cognitive Psychology, 3,
185-226.
Lake, D., & Kemp, M. (2001). Choosing and using graphs. Australian Mathematics
Teacher, 57(3), 7-12.
Landwehr, J.M., & Watkins, A.E. (1986). Exploring data: Quantitative literacy series. Palo
Alto, CA: Dale Seymour.
Maher, C.A. (1981). Simple graphical techniques for examining data generated by
classroom activities. In A.P. Shulte (Ed.), Teaching statistics and probability. 1981
yearbook (pp. 109-117). Reston, VA: National Council of Teachers of Mathematics.
Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149174.
Mooney, E.S. (2002). A framework for characterizing middle school students’ statistical
thinking. Mathematical Thinking and Learning, 4(1), 23-63.
Moore, D.S. (1990). Uncertainty. In L.A. Steen (Ed.), On the shoulders of giants: New
approaches to numeracy (pp. 95-137). Washington, DC: National Academy Press.
Moore, D.S., & McCabe, G.P. (1989). Introduction to the practice of statistics. New York:
W.H. Freeman and Co.
Moritz, J.B. (2000). Graphical representations of statistical associations by upper
primary students. In J. Bana & A. Chapman (Eds.), Mathematics Education Beyond
2000 (Proceedings of the 23rd Annual Conference of the Mathematics Education
Research Group of Australasia, Vol. 2, pp. 440-447). Perth, WA: MERGA.
Moritz, J.B. (2006). Developing students’ understandings and representations of
statistical covariation. Unpublished doctoral thesis. Hobart: University of Tasmania.
Retrieved December 12, 2009 from http://eprints.utas.edu.au/1473/
Moritz, J.B., & Watson, J.M. (1997). Graphs: Communication lines to students? In F.
Biddulph & K. Carr (Eds.), People in mathematics education (Proceedings of
the 20th Annual Conference of the Mathematics Education Research Group of
Australasia, Vol. 2, pp. 344-351). Waikato: MERGA.
© State of New South Wales through the NSW
Department of Education and Training, 2010
References
71
The Development of Graph Understanding
in the Mathematics Curriculum
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation
standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for
school mathematics. Reston, VA: Author.
National Curriculum Board. (2009). The shape of the Australian Curriculum. Barton, ACT:
Commonwealth of Australia.
Noether, G.E. (1983). Statistics in the curriculum for the 1980s. In G. Shufelt (Ed.), The
agenda in action. 1983 yearbook (pp. 195-199). Reston, VA: National Council of
Teachers of Mathematics.
Pearson, K. (1895). Contributions to the mathematical theory of evolution – II. Skew
variation in homogeneous material. Philosophical Transactions of the Royal Society
of London. A, 186, 343-414.
Pegg, J.E. (2002a). Assessment in mathematics: A developmental approach. In J.M.
Royer (Ed.), Mathematical cognition (pp. 227-259). Greenwich, CT: Information Age
Publishing.
Pegg, J.E. (2002b). Fundamental cycles of cognitive growth. In A. Cockburn & E. Nardi
(Eds.), Proceedings of the 26th Conference of the International Group for the
Psychology of Mathematics Education (Vol. 4, pp. 41-48). Norwich, UK: University of
East Anglia.
Pendlebury, C. (1896). Arithmetic. (9th ed.). London: George Bell and Sons.
Pfannkuch, M., & Rubick, A. (2002). An exploration of students’ statistical thinking with
given data. Statistics Education Research Journal, 1(2), 4-21.
Playfair, W. (1801). The statistical breviary; Skewing on a principle entirely new, the
resources of every state and kingdom in Europe. London. Retrieved November 13,
2009, from http://books.google.com
Playfair, W. (1805). An inquiry into the permanent causes of the decline and fall of
powerful and wealthy nations. London. Retrieved November 13, 2009, from http://
books.google.com
Rangecroft, M. (1991a). Graphwork – Developing a progression. Part 1 – The early
stages. Teaching Statistics, 13(2), 44-46.
Rangecroft, M. (1991b). Graphwork – Developing a progression. Part 2 – A diversity of
graphs. Teaching Statistics, 13(3), 90-92.
Rasch, G. (1980). Probabilistic models for some intelligence and attainment tests.
Chicago: University of Chicago Press (Original work published 1960).
Reading, C. (2002). Profile for statistical understanding. In B. Phillips (Ed.), Proceedings
of the Sixth International Conference on Teaching Statistics: Developing a
Statistically Literate Society, Cape Town, South Africa. Voorburg, The Netherlands:
International Statistical Institute.
Reading, C., & Pegg, J. (1995). Teaching statistics: Background and implications. In L.
Grimison & J. Pegg (Eds.), Teaching secondary school mathematics: Theory into
practice (pp. 140-163). Sydney: Harcourt Brace.
72
References
© State of New South Wales through the NSW
Department of Education and Training, 2010
The Development of Graph Understanding
in the Mathematics Curriculum
Shaughnessy, J.M. (2007). Research on statistics learning and reasoning. In F.K. Lester,
Jr. (Ed.), Second handbook on research on mathematics teaching and learning (pp.
957-1009). Charlotte, NC: Information Age Publishing.
Shaughnessy, J.M., Chance, B., & Kranendonk, H. (2009). Focus on high school
mathematics: Reasoning and sense making in statistics and probability. Reston,
VA: National Council of Teachers of Mathematics.
Shufelt, G. (Ed.). (1983). The agenda in action. 1983 yearbook. Reston, VA: National
Council of Teachers of Mathematics.
Shulte, A.P. (1981). Teaching statistics and probability. 1981 yearbook. Reston, VA:
National Council of Teachers of Mathematics.
Spence, I. (2005). No humble pie: The origins and usage of a statistical chart. Journal of
Educational and Behavioral Statistics, 30(4), 353-368.
Swift, J. (1983). A statistics course to lighten the information overload. In G. Shufelt (Ed.),
The agenda in action. 1983 yearbook (pp. 110-119). Reston, VA: National Council
of Teachers of Mathematics.
Tufte, E.R. (1983). The visual display of quantitative information. Cheshire, CT: Graphics
Press.
Tufte, E.R. (1990). Envisioning information. Cheshire, CT: Graphics Press.
Tufte, E.R. (1997). Visual explanations: Images and quantities, evidence and narrative.
Cheshire, CT: Graphics Press.
Tukey, J.W. (1977). Exploratory data analysis. Reading, Mass: Addison-Wesley Publishing
Company.
Tukey, J.W. (1980). We need both exploratory and confirmatory. The American
Statistician 34(1), 23-25.
Van de Walle, J.A. (2007). Elementary and middle school mathematics: Teaching
developmentally. Boston: Pearson.
Wainer, H. (1997). Visual revelations: Graphical tales of fate and deception from
Napoleon Bonaparte to Ross Perot. New York: Copernicus.
Wainer, H., & Velleman, P.F. (2001). Statistical graphics: Mapping the pathways of
science. Annual Review of Psychology, 52, 305-335.
Wall, J.J., & Benson, C.C. (2009). So many graphs, so little time. Mathematics Teaching
in the Middle School, 15, 82-91.
Watson, J.M. (1997). Assessing statistical literacy using the media. In I. Gal & J.B.
Garfield (Eds.), The assessment challenge in statistics education (pp. 107-121).
Amsterdam: IOS Press and The International Statistical Institute.
Watson, J.M. (2000). Statistics in context. Mathematics Teacher, 93, 54-58.
Watson, J.M. (2006). Statistical literacy at school: Growth and goals. Mahwah, NJ:
Lawrence Erlbaum.
Watson, J.M. (2008). Exploring beginning inference with novice grade 7 students.
Statistics Education Research Journal, 7(2), 59-82.
Watson, J.M., & Callingham, R.A. (2003). Statistical literacy: A complex hierarchical
construct. Statistics Education Research Journal, 2(2), 3-46.
© State of New South Wales through the NSW
Department of Education and Training, 2010
References
73
The Development of Graph Understanding
in the Mathematics Curriculum
Watson, J.M., Callingham, R.A., & Kelly, B.A. (2007). Students’ appreciation of
expectation and variation as a foundation for statistical understanding.
Mathematical Thinking and Learning, 9, 83-130.
Watson, J.M., Collis, K.F., Callingham, R.A., & Moritz, J.B. (1995). A model for assessing
higher order thinking in statistics. Educational Research and Evaluation, 1, 247275.
Watson, J.M., & Donne, J. (2009). TinkerPlots as a research tool to explore student
understanding. Technology Innovations in Statistics Education, 3 (1), 1-35.
Retrieved December 20, 2009 from http://repositories.cdlib.org/uclastat/cts/tise/
vol3/iss1/art1
Watson, J.M., Fitzallen, N.E., Wilson, K.G., & Creed, J.F. (2008). The representational
value of hats. Mathematics Teaching in the Middle School, 14, 4-10.
Watson, J.M., & Kelly, B.A. (2003). Inference from a pictograph: Statistical literacy in
action. In L. Bragg, C. Campbell, G. Herbert, & J. Mousley (Eds.), Mathematics
education research: Innovation, networking, opportunity (Proceedings of the 26th
Annual Conference of the Mathematics Education Research Group of Australasia,
Geelong, pp. 720-727). Sydney, NSW: MERGA.
Watson, J.M., & Kelly, B.A. (2005). The winds are variable: Student intuitions about
variation. School Science and Mathematics, 105, 252-269.
Watson, J.M., & Moritz, J.B. (1999). The beginning of statistical inference: Comparing two
data sets. Educational Studies in Mathematics, 37, 145-168.
Watson, J.M., & Moritz, J.B. (2000). The longitudinal development of understanding of
average. Mathematical Thinking and Learning, 2(1&2), 11-50.
Watson, J.M., & Moritz, J.B. (2001). Development of reasoning associated with
pictographs: Representing, interpreting, and predicting. Educational Studies in
Mathematics, 48, 47-81.
Wild, C., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International
Statistical Review, 67(3), 223-248
74
References
© State of New South Wales through the NSW
Department of Education and Training, 2010
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement