Evaluating data-analysis software: Exploring opportunities for
developing statistical thinking and reasoning.
Noleine Fitzallen and Natalie Brown
University of Tasmania
With the growing number of commercial graphing software packages it has been
recognised that there is a need to be able to critically evaluate software in terms of
their ability to address mathematical learning needs, and provide visualisation
representations. The ways in which these representations may be used to construct
mathematical knowledge and develop statistical thinking and reasoning is of
particular interest. This paper reports on the development of a theoretical
framework constructed from both the mathematical and technological
perspectives. It also discusses the application of the framework to a data-analysis
software tool, TinkerPlots Dynamic Data Exploration.
ICT and mathematics
The role of Information and Communication Technologies (ICT) in developing
mathematical thinking is highly valued by teachers and supported by research
(Fitzallen & Brown, 2006; Ward, 2003; Roschelle, Pea, Hoadley, Gordin, & Means,
2000). Furthermore, incorporating ICT into the curriculum is regarded as an essential
element of good teaching practice (National Council of Teachers of Mathematics
[NCTM], 2000; Australian Education Council [AEC], 1994). As a consequence there
has been an increase in the use of ICT in the mathematics classroom, providing
opportunities for students to develop their understanding of mathematical concepts
through problem-solving, critical thinking, communicating ideas, and reflecting on
their learning with ICT.
The utilisation of ICT in the mathematics classroom has the potential to influence
not only the teaching strategies employed by teachers but also what students learn,
and the way in which they learn (McGhee & Griffiths, 2004; NCTM, 2000; Roschelle
et al., 2000). In many cases, however, the majority of ICT use in the classroom
supplements established teaching practices and classroom activities (Goos, Galbraith,
Renshaw & Geiger, 2003; Reynolds, 2001). This is evident for example, when
calculators are used to speed up the calculation process, computing software games
are used to elicit improved automatic response to simple mathematics problems, and
computing software applications are used to allow students to publish neater work.
When only used to supplement existing teaching and learning practices, the influence
of ICT in terms of developing the desired higher-order thinking outcomes noted
above, is not realised (Ward, 2003), nor is the potential of ICT to influence the
learning experience or student thinking (Brooker, 2003; Roschelle et al., 2000). Goos
et al. go on to suggest:
… cognitive re-organisation occurs when learners’ interaction with the
technology as a new semiotic system qualitatively transforms their thinking;
for example, use of spreadsheets and graphing software can alter the
traditional privileging of algebraic over graphical or numerical reasoning.
Accordingly, learning becomes a process of appropriating tools that change
the ways in which individuals formulate and solve problems. (p.75)
Godwin and Sutherland (2004) contend that “the dynamic and symbolic nature of
computer environments can provoke students to generalise and make links between
1
their intuitive notions of mathematics and the more formal aspects of mathematical
knowledge” (p.132). Hoyles, Noss, and Kent (2004) suggest, together with Goos et al.
(2003), that the use of ICT can transform student thinking, extending beyond the
immediate concerns of the learning experience to an abstraction of the mathematical
concepts with student understandings being shaped by the technology used. However,
Hoyes et al. warns of the complexity that technology brings to learning experiences
and notes that it places many demands on students, both mathematical and
technological. Clearly this is a consideration for teachers when choosing technology
to support learning.
Guidelines to evaluate critically existing software and guide the development of
better software in the future have been advocated by Biehler (1997) and Shaughnessy
(2006). In addition, evaluation of software design drawing on relevant learning
theories has more recently been recognised by Goos and Cretchley (2003) particularly
with respect to developing higher order thinking. In the specific field of statistics
education, Shaughnessy has similarly recommended that research be conducted to
improve our understanding of the features of software that elicit and transform
students’ statistical thinking and reasoning. These calls indicate a continuing need to
evaluate software not only from a technological perspective but also from the point of
view of how it may contribute to developing statistical concepts for students.
ICT and statistics
In relation to statistics education, skills associated with the construction and
reading of graphs as well as data analysis is included in the curriculum for all the
years of compulsory schooling in Australia (AEC, 1994). Graphs are critical for data
representation, data reduction, and data analysis in statistical thinking and reasoning
(Shaughnessy, 2006). Traditionally, the teaching of statistics had an emphasis on
computations, formulas, and procedures such as constructing graphs. The inclusion of
technology in the learning experience has led to a shift towards the ability to interpret,
evaluate, and flexibly apply statistical ideas (Ben-Zvi, 2000). This has significant
pedagogical implications.
Teaching and learning of experiences in statistics education can be enhanced by
the application of software tools (Biehler, 1997). In particular, graphing software
provides the opportunity for students to create and change representations and “…this
may lead to exploration of important mathematical concepts” (Goos & Cretchley,
2003, p. 153). The features of graphing software enable students to move efficiently
from tabular and graphical representations to the visualisation of data. They also
allow students to design multiple graphical representations of the same data set
quickly and more accurately than with pen and paper (McGehee & Griffiths, 2004).
Konold and Higgins (2003) completed a literature review on how students reason
about data. They reported that students should begin with graphs in which they can
retrace each individual data value and should also have the opportunity to construct
graphs from primary data. Software with these characteristics is described as “bottomup software”. The contention is that this type of software may promote student
engagement that assists them to construct meaning from the data (Ben-Zvi, 2000).
This is clearly preferable to using software that produce accepted graphical
representations, such as histograms, at the click of a button, without providing the
opportunities to develop the relationship between the data and the graphical
representations (Konold & Higgins, 2003; Biehler, 1997).
Sivasubramaniam (2004) suggests that the software or computing device used to
display graphs takes over some of the cognitive processes employed to create
2
graphical representations, thereby allowing students to focus on the interpretation of
the graphs. He maintains the view that computing devices such as a graphing
calculator distributes the cognitive process of creating graphs differently than when
producing graphs with pen and paper. He describes this as the “effect of the
technology” because the graphing calculator determines the scale and performs the
mechanics of creating graphs. Sivasubramaniam does not, however, extend his
discussion into what features of the graphing software and subsequent graphical
representations contribute to students developing an understanding of the statistical
concepts explored, other than the ability of the technology to provide a concrete
model.
Evaluating software packages
The increased use of ICT identified by teachers for their application in graphing
activities has led to great increases in the availability of commercial software
packages. Drawing on research, critical examination of software is necessary to
determine how well they meet teaching and learning needs. This is not only in terms
of technology – how user friendly is the package? – but also in what ways the
software facilitates the learning of mathematical concepts. In the case of statistics
education, a wealth of literature exist describing key elements of best practice
teaching to develop statistical thinking and reasoning (Ben-Zvi & Garfield, 2004) and
in particular, graphing (Shaughnessy, 2006). This paper will detail the development of
a theoretical framework to evaluate software that incorporates those elements together
with the elements of software that provide meaningful visual representations (Kidman
& Nason, 2000), whilst also considering the capacity of software to influence
distributed cognition (Sivasubramaniam, 2004).
Model of graphing in an ICT environment
The following theoretical framework detailed in Table 1 (Fitzallen, 2006) was
developed by considering models of statistical thinking and reasoning in relation to
graphing and graph sense-making. A comprehensive consideration of the relevant
literature was undertaken to identify theoretical models of statistical thinking and
reasoning that were directly related to data analysis and in particular, graphing.
Models developed by Friel, Curcio and Bright (2001), Mooney (2002), and Moritz
(2004) were considered. Suggestions by Shaughnessy, Garfield & Greer (1996) for an
additional level to be added to Friel, Curcio and Bright’s levels of thinking and the
notion of transnumeration presented by Pfannkuch and Wild (2004) provided an
extensive view of the development of graphing and graph sense-making. It was
established from the literature review that although each of the models had elements
that were relevant, none of the models took into consideration the way in which
students learn in technological environments and how those environments may impact
on the construction of mathematical knowledge. The key elements extracted from
each of the models were grouped into four interconnected categories: Being creative
with data, Understanding data, Thinking about data, and Generic knowledge.
Another important feature that sets the Fitzallen (2006) model apart from the
others utilised in its development is that it recognises that there are some generic
understandings inherent in all levels of data analysis, graphing and graph sensemaking. In so doing, the model progresses from being a hierarchical model and aligns
more with the notion that learning experiences in relation to data analysis can be
accessed from multiple entry points (Moritz, 2004). Although the application of
technology in the data analysis process was considered, the Fitzallen model does not
3
take into consideration specific visual representations and how they may contribute to
the understanding of statistical concepts.
Understanding the dynamic nature of software and
technology environments
Thinking about data
Recognising the components of data and graphs
Understanding data
Speaking the language of graphs
Table 1: Model of graphing in an ICT environment
Category
Generic Knowledge
Descriptors of Category
Being creative with data
Reducing data to graphical representations
or statistical summaries.
Constructing different forms of graphs.
Translating verbal statements into graphs.
Making sense of data and graphs.
Understanding the relationship among
tables, graphs, and data.
Identifying the messages from the data.
Answering questions about the data.
Recognising appropriate use of different
forms of graphs.
Describing data from graphs.
Asking questions about the data.
Recognising the limitations of the data.
Interpreting data and making causal
inferences based on the data.
Looking for possible causes of variation.
Looking for relationships among variables
in the data.
Principles for analysing visual representations
Similar to the process undertaken by Fitzallen (2006), a review of research
literature in relation to analysing visual representations was conducted by Kidman &
Nason (2000). The review determined that visual representations can facilitate the
construction of mathematical knowledge in four ways:
i. visual representations can provide the source for mathematical analogs;
ii. visual representations can act as tools to facilitate problem solving;
iii. visual representations can facilitate mathematical meaning-making; and
iv. computer-based visual representations can act as tools for supporting and
reorganising thinking.
Kidman and Nason (2000) determined that existing frameworks did not take into
consideration research findings from the fields of semiotics and computer-based
visual representations as tools for supporting and reorganising thinking. The
integration of these factors with existing frameworks resulted in the development of
Principles for Analysing Visual Representations. The seven principles were applied
successfully by Kidman and Nason as a diagnostic tool to determine the effectiveness
of dynamic mathematical representations employed within Integrated Learning
Systems (computer-based learning activities).
Although the Principles for Analysing Visual Representations (Kidman & Nason,
2000) is a good framework for the general analysis of mathematical software, in order
to be considered for use with statistical software the principles need to be more
specific for evaluating the capacity of software to facilitate the development of
statistical concepts.
4
Aspects of Statistical Software
The limitations of presented frameworks – Fitzallen (2006) looking more broadly
at statistical thinking and reasoning without reference to the impact of visual
representations, and Kidman and Nason (2000) considering technology in relation to
mathematics in general, led to the melding of the two frameworks. The new
framework, Aspects of Statistical Software, is presented in Table 2. The relationship
between the literature and the elements of the framework are noted in the last column.
Table 2. Aspects of Statistical Software
No.
Aspects
1 The interface is accessible and features of the software are
easy to use.
 It is easy for students to enter data and understand
how it is represented by the software.
2 Software assists recall knowledge and data can be
represented in different and multiple forms.
 Students can access both graphical representations
and numerical data
3
4
5
6
Reference/s
Kidman & Nason (2000),
Principle 1
Kidman & Nason (2000),
Principles 1 & 5,
Fitzallen (2006), Being
creative with data,
Understanding data
Software facilitates the process of translating between
Kidman & Nason (2000),
mathematical expression and natural language.
Principle 6,
 Students can maintain the relationship between the Fitzallen (2006), Being
creative with data
context and the data of statistical investigations.
Software provides extended memory when organising and Kidman & Nason (2000),
reorganising data.
Principle 4,
 Students can display and manipulate data easily in Fitzallen (2006), Being
creative with data,
creative ways.
Understanding data,
Sivasubramaniam (2004),
Distributed cognition
Software provides physical/iconic environment that
Kidman & Nason (2000),
provides multiple entry points for abstraction of concepts. Principles 3 & 5,
Fitzallen (2006), Being
 Students can go between data, tables, and graphs,
creative with data,
developing an understanding of the relationship
Understanding data
between the different forms of data.
Visual representations used for both interpretative and
Kidman & Nason (2000),
expressive learning activities.
Principles 2 & 7,
Fitzallen (2006), Thinking
 Allows students to interpret data and make causal
about data
links between and within data. Students can ask
questions about the data based on the visual
representations.
The new framework (Table 2) progresses the work of Kidman and Nason (2000)
by specifically considering pedagogical content knowledge related to statistics
education. The work of Sivasubramaniam (2004) has also been considered. In
particular, the capacity of technology to reduce cognitive load by performing the
mechanics of creating graphs, thereby, allowing students to concentrate on
understanding the graphical representations. Although underlying understanding is a
key aim addressed by other frameworks, including the notion of distributed cognition
is a genuine ‘plus’ and allows for the focus to be on understanding. It should also be
recognised that each of the three sections of the Generic knowledge of the Fitzallen
(2006) model permeate through all aspects of the framework.
5
Evaluating software against the framework
The research project, from which this report is taken, utilises recently released
data-analysis software, TinkerPlots Dynamic Data Exploration (Konold & Miller,
2005). TinkerPlots provides educators with a graphing program that is designed
specifically for students in the middle years of schooling and gives students an
intuitive, informal set of operators to organise data flexibly to see patterns in them. Of
particular note, is the way in which explorations are performed without necessarily
having to learn first to make and interpret a standard set of graph types.
Early indications from research (Shaughnessy, 2006; Konold & Higgins, 2003;
Hammerman & Rubin, 2002) and anecdotal evidence (Konold & Miller, 2005)
support the conviction that TinkerPlots will be useful in developing students’
understanding of statistical concepts. It is, however, important to evaluate critically
TinkerPlots using relevant theories of statistical thinking and reasoning as well
drawing on frameworks that evaluate the visual representations afforded by
TinkerPlots. This will be of assistance when determining in what ways TinkerPlots
may assist students to construct mathematical knowledge.
Evaluation of TinkerPlots against the framework
Aspect 1. When initialised, TinkerPlots presents a clean and uncluttered interface.
The screen consists of a blank white page with a menu bar at the top of the screen. To
the novice user the blank screen is a little confronting as there are no visual clues as to
where to start. The user’s first reaction is to go to the menu with the expectation that it
will be a drop down menu similar to other software applications. This is not the case.
TinkerPlots utilises a drag and drop function to import data representations onto the
screen. As the user becomes familiar with the drag and drop function the manipulation
of graphs and data values becomes extremely easy.
To assist the user TinkerPlots has an extensive ‘Help’ section which is easy to
use. Additionally, the software has four tutorials demonstrating and explaining the
basic features of the software included within the program. These are in the form of
movies which only run for a few minutes each. The availability of the tutorials within
the program enables the user to access the tutorials at any time, to refresh the user’s
understanding of how to utilise the features of the software.
Aspect 2. TinkerPlots includes a stacked data card system in the plot window for
the organisation of case-based data. Information about the characteristics (variables)
of an individual case is presented on a single card and there is one card for each case
in a data set. Data entered into the data cards are automatically entered into a
spreadsheet and graphical representations can be constructed from both the data cards
and the spreadsheet. There is the option of entering the data directly into the
spreadsheet table. This facility will create the stacked data cards automatically.
Aspect 3. The interface of TinkerPlots allows students to create multiple graphical
presentations, import digital images, display tables of results, and add written
commentaries into the one file as they conduct statistical investigations. The context
of the learning experience is transferred to the software interface easily, allowing for
the user to construct an understanding of a statistical investigation by maintaining the
relationship of the context between the data and the graphical representations. Direct
access to the graphical representations and the ability to contribute written responses
to the interface facilitates the use of the language of graphs.
6
Aspect 4. Many features of TinkerPlots assist in distributing the cognitive load of
processing data between the user and the software, allowing the user to focus on the
interpretation of the data rather than the creation of the graphs (Sivasubramaniam,
2004). This is demonstrated when data points in a graphical representation in
TinkerPlots are manipulated by the drop and drag function in order to change the
scale of the graph and the graphical representation. Changes are quick and fluid, and
are viewed as animations. Additionally, the transformation of a dot plot to a histogram
is made easily by changing the shape of the icons representing the data from a dot to a
fused bar. Actions such as these change the scale and the form of the graph quickly.
Although the scale is determined initially by the software it can be changed by the
user to represent a different range.
Other functions such as mean, mode and median also allow the user to focus on
understanding the data when using TinkerPlots. As with most of the features in
TinkerPlots that are for interpreting data, they are accessed on the main menu bar. The
numerical calculations for these concepts of centrality are performed automatically by
the software and are represented on graphs by an icon and/or numerical value. This
allows students to explore what these concepts mean in terms of the data without
having to focus on the rules for determining the numerical value. This is also the case
when utilising the hat plot function (TinkerPlots version of a box plot)
Aspect 5. An outstanding feature of TinkerPlots is the way a set of data can be
explored in many ways. Graphs can be constructed by dragging and dropping the
names of the variables from either the data cards or the spreadsheet into a plot
window. Graphs can be constructed to explore the distribution of data as well as
covariation. Once a graph is constructed for one variable it can be changed to
represent a different variable by dragging the new variable name onto the axis of the
graph in the plot window.
Additionally, highlighting individual data points on a graph will automatically
display the data card associated with that particular data point on the top of the data
cards. It also highlights the data associated with that point in the spreadsheet. This can
potentially help the user to build an understanding of the relationship between the data
and the graphical representations.
Aspect 6. TinkerPlots provides for both interpretative and expressive learning
activities. Expression is the process of crafting visual representations to convey
meaning (Kidman & Nason, 2000). The program enables students to build meaning
from the data by being able to “create both simple and complex graphs by performing
actions such as sorting data into categories or ordering the information according to
the values of one of the variables” (Hammerman & Rubin, 2002). In relation to
interpretative activities, the box plot, mean, median, sorting and filtering of data
functions all contribute to the way the user can interpret data. An additional feature
that facilitates the interpretation of data is the way in which the software displays the
data in multiple categories within the same graph. This can lead to the interpretation
of the association of variables within and between data.
Conclusion
This paper has drawn on a model of statistical thinking and reasoning (Fitzallen,
2006), a framework for evaluating software (Kidman & Nason, 2000), and the notion
of distributed cognition in relation to technology (Sivasubramaniam, 2004) to develop
a framework for the evaluation of data analysis software. The framework, Aspects of
7
Statistical Software, was used to evaluate TinkerPlots Dynamic Data Exploration
(Konold & Miller, 2005). It was determined that TinkerPlots promotes the
opportunity for students to be involved in the data analysis process. This begins with
the input of the data into a spreadsheet or data card system, moves to the creation and
manipulation of graphical representations, and extends to the interpretation of graphs
through using box plots and other means of summarising data. It provides visual
representations that provide immediate feedback to students as the animations of
changes instigated are fluid and quick. Throughout the process the user is in control of
the functions and maintains the connection between the data and the graphical
representations. A student participating in a research project investigating students’
development of key statistical concepts - comparing distributions and comparing
samples – explained TinkerPlots in the following way:
It is sort of like a graphing sort of system that can be used to figure out that
you can make graphs out of and actually, it is pretty fun, too. It is sort of like,
… it’s got the same sort of things as any other sort of graphing thing but you
actually make the graph yourself instead of entering all the numbers in and
the computer does it for you. Bakker & Frederickson (2005, p. 88)
Clearly, TinkerPlots shows evidence of offering a dynamic learning environment
for students. It appears to cater effectively for middle school students and has obvious
advantages as an educational tool. It is, therefore, pertinent to explore the ways in
which students engage in the software and use it to analyse, display, and interpret
data. The research project from which this paper is taken will explore how
TinkerPlots contributes to and promotes the development of statistical concepts for
middle school students.
Acknowledgements: This research is funded by an APA(I) Scholarship associated
with an ARC Linkage Project LP0560543 and the industry partner, Department of
Education, Tasmania.
8
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