A coloured window on pre-service teachers’ conceptions of rational numbers

Abstract

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.

Full text

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A COLOURED WINDOW ON PRE-SERVICE TEACHERS' CONCEPTIONS OF
RATIONAL NUMBERS
Nathalie Sinclair, Michigan State University
Department of Mathematics
East Lansing, MI 48824
Voice: (517) 353.3833
E-mail: nathsinc@math.msu.edu
Peter Liljedahl, Simon Fraser University, Canada
Rina Zazkis, Simon Fraser University, Canada

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ABSTRACT: In undergraduate mathematics courses, pre-service elementary school teachers are
often faced with the task of re-learning some of the concepts they themselves struggled with in
their own schooling. This often involves different cognitive processes and psychological issues
than initial learning: pre-service teachers have had many more opportunities to construct
understandings and representations than initial learners, some of which may be more complex
and engrained; pre-service teachers are likely to have created deeply-held—and often
negative—beliefs and attitudes toward certain mathematical ideas and processes. In our recent
research, we found that pre-service teachers who used a particular computer-based microworld,
one emphasising visual representations of and experimental interactions with elementary
number theory concepts, overcame many cognitive and psychological difficulties reported in
the literature. In this study, we investigate the possibilities of using a similarly-designed
microworld that involves a set of rational number concepts. We describe the affordances of this
microworld, both in terms of pre-service teacher learning and research on pre-service teacher
learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-
service teachers. We also show how their interactions with this microworld provided many
with a new and aesthetically-rich set of visualisations and experiences.

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INTRODUCTION
In many undergraduate mathematics courses, pre-service elementary school teachers are faced
with the task of re-learning many of the concepts they themselves struggled with in their own
schooling. They are expected to develop a profound mathematical background in a subject that,
for many, makes them anxious and, for most, poses serious conceptual difficulties. It seems
clearly misguided to ask these pre-service elementary teachers to learn the concepts they will
have to teach—and which they themselves had difficulty learning—in the same way these
concepts have previously been encountered. First, from a psychological point of view: only
strong, positive mathematical experiences can help learners overcome existing fears and
anxieties (Goldin, 2000). Second, from a cognitive point of view: better understanding,
according to Skemp (1971) depends on the construction of richer, more multi-dimensional
schemas, in which previous understanding can be assimilated. As Piaget’s theory holds, the
construction of richer schemas can only occur through a process of dis-equilibration caused by a
perceived problem or unexpectedness and subsequent re-equilibration through assimilation
into existing schemas or, if necessary, reconstruction of new schemas.
In this paper, we discuss the re-learning of rational numbers (particularly those often
referred to as ‘fractions’ and ‘decimals’ in the context of school mathematics) by pre-service
elementary teachers. We present a computer-based microworld that has been found to provide
an unexpected and compelling way of re-presenting these frequently abhorred mathematical
objects, and to encourage and support mathematical exploration and problem solving (Sinclair,
2001). Our goal in using this microworld was to gain a window on pre-service teachers’
understanding of certain concepts involving rational numbers and to explore the benefits and
drawbacks of using such a microworld in pre-service courses.
THEORETICAL CONSIDERATIONS
Learning is about acquiring knowledge that can be a piece of information, a skill, an
understanding, or even a perspective. Re-learning for prospective elementary teachers is more

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than this. It is not simply adding knowledge to what has previously been incorporated into a
learner's repertoire. It is revisiting and reconstructing conceptions that may have been
developed long ago, and been affected by myriad—and often forgotten—experiences both in
and out of the mathematics classroom. It is often said that classroom teachers must develop a
better understanding of the kinds of mathematical conceptions that their students already hold,
before coming into the classroom. The same must be said of teacher educators, who are
frequently faced with the less malleable conceptions of pre-service teachers, who have
developed and reinforced their conceptions over a much broader set of everyday as well as
classroom experiences. Since prior structures have been constructed in learner's mind some time
ago, the reconstruction and reorganization processes involved in re-learning are more
challenging for the learner as well as for the instructor. The instructor needs windows on the
learners’ conceptions and learners need ways in which they can confront their own deeply
engrained conceptions.
Re-learning for pre-service elementary school teachers
Prospective teachers are often in a position of having to re-learn specific elements of their
mathematics. However, they are frequently content with what may be superficial
understandings: once they have formalised a procedure, it is difficult to re-visit the underlying
concepts for deeper understanding (Hiebert and Carpenter, 1992; Wilson and Goldenberg,
1998). Moreover, the existence of prior knowledge, which is robust while also being incomplete
and superficial, can result in significant difficulties. Markovits and Sowder (1990), for example,
reported that pre-service teachers participating in their research were more successful in
acquiring knowledge in a new mathematical content, than re-learning what has been perceived
as ‘familiar.'
One such ‘familiar’ topic is that of rational numbers. Through their own schooling,
prospective teachers learn many procedures associated with converting numbers from one
representation to another and operating on different types of representations. As such, many
feel that they know fractions or know decimals—these are ‘familiar’ entities to them—and are

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thus reluctant to re-visit some underlying concepts associated with rational numbers. Yet
student understanding of rational numbers is very complex, as the large body of research
literature studying it attests to (Behr, Wachsmuth, Post, and Lesh, 1984; Lamon, 1999). It is
therefore not surprising that, despite their reluctance, pre-service teachers do not have a good
grasp of rational numbers (Post, Harel, Behr, and Lesh, 1998).
One of the clearest findings of research on learning fractions and rational numbers is that of
the many different ways in which fractions can be understood (as a part of a whole, ratio,
quotient, and number), students who come to understand fractions as numbers have the
greatest success and flexibility (Lamon, 2001). Yet developing such an understanding has
proven to be difficult, in part because many of the helpful visual representations used in the
teaching of fractions tend to emphasise the part of a whole aspect of fractions, which is non-
trivial to reconcile with the idea of fraction as number.
A primary goal was therefore to give our research participants the opportunity to re-visit
fractions in a context that emphasises this ‘fraction as number’ way of understanding fractions,
and that makes it possible for them to work with fractions and their decimal representations
interchangeably. Given our previous research on various aspects of elementary number theory
(see Campbell and Zazkis, 2002), we saw this as an opportunity to bring to the fore some related
concepts such as the period of a repeating decimal fraction and the difference between rational
and irrational numbers. We wanted the pre-service teachers to encounter novel or unexpected
properties and relationships such as the period of a fraction in concepts they had perceived to
be entirely familiar both in order to increase the likelihood of prompting reconstruction and
reorganisation and to challenge the participants’ I-already-know-this complacency. Our
research participants were both short-term and long-term re-learners, as they had learned
rational numbers both long ago (in grade school) as well as quite recently (in a university-level
mathematics course).
To satisfy both our research aims—to understand better some of the participants
conceptions about certain aspects of rational numbers—and our teaching objectives, one author

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(Sinclair) designed a computer-based learning environment focusing on the domain of rational
numbers. We first outline our rationale for developing a rational numbers microworld. Then we
describe the design principles we followed, and conduct a mathematical analysis of the
concepts and behaviours supported by the microworld.
Affordances of computer-based learning environments
Many researchers have argued, along with Goldenberg (1989), that well-designed computer-
based learning environments can provide a scaffold for reasoning by fostering the development
and use of visual and experimental reasoning styles, which greatly complement the
traditionally taught symbolic deductive methods. For the learning of rational numbers, few
such computer-based environments exist, particularly in comparison to the number that exist to
support the learning of geometrical ideas, functions, probabilities and data analysis.
Pies, rectangles and number lines provide one type of visual representation frequently
encountered in the sub-domain of fractions. Independently from these visual tools, calculators
can support a certain range of numeric experimentation, particularly in terms of converting
fractions to decimals. However, each tool is often used in isolation: neither pies nor number
lines can be represented on calculators. In contrast to these isolated tools, there exists a class of
environments called “microworlds” which, as Edwards (1995) describes, “embody” or
“instantiate” some sub-domain of mathematics. The microworld is intended to be a mini-domain
of mathematics that essentially brings such tools together into a phenomenological whole. Thus
microworlds can be seen as specific forms of external representation of a subset of mathematical
ideas. The challenge for mathematics educators is to design microworlds that can offer new
external systems of representation that foster more effective learning and problem solving.
Noss and Hoyles (1996) argue that the computational objects of a microworld—its basic
building blocks—should maximise the chance to forge links with mathematical objects and
relationships. In the case of rational numbers, the computational objects should maximise the
learner’s chance to explore relationships among different types of numbers (rational, irrational,
terminating, repeating, etc.), as well as their underlying structure. In terms of design, Noss and

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Hoyles also argue that the development of a microworld should involve predicting where
student breakdowns might occur: breakdowns are incidents where learners’ anticipated
outcomes are not experienced. Microworlds that adhere to such design criteria provide the
learner with opportunities to build new meanings through new external representations. One
final design issue we wanted to bear in mind relates to the difficulties involved in incorporating
visualisation into mathematical activity. We recognise that without proper support, students are
either unwilling or unable to take advantage of potentially powerful visualisations (see
Eisenberg and Dreyfus, 1991). We also recognise that microworlds are essentially inert without
the animating presence of both the student and the task that invites investigation. The design of
our learning environment thus included the creation of the microworld as well as a set of
accompanying tasks that would encourage connections between analytic and visual modes of
reasoning (see Appendix 1). Many of these tasks involved the comparison or simultaneous use
of different denominators and numerators while others were intended to target particular
properties of certain denominators.
In the next section, we describe a computer microworld—the Colour Calculator—and draw
attention to some of its features, particularly as they compare to traditional paper-and-pencil
environments or handheld calculators. This will give a sense of the kinds of properties and
relationships that explorations with the Colour Calculator can emphasise, or make more
transparent. We then describe a study with a group of pre-service elementary teachers, who
used the Colour Calculator as part of their mathematics course. Finally, we describe the types of
meanings that different participants were able to construct, and the ways in which their
orientations (including attitudes and emotions, as well as richness of understanding) toward
fractions and decimals were affected.

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FRACTIONS AND DECIMALS WITH THE COLOUR CALCULATOR
Description of the Colour Calculator
The Colour Calculator is an internet-based
i
calculator that provides numerical results, but that
also offers its results in a colour-coded table. Conventional operations are provided, as shown
in Figure 1. Each digit of the result corresponds to one of ten distinctly coloured swatches
—reflected in a legend—in the table.
Figure 1. The Colour Calculator
The calculator operates at a maximum precision of 100 decimals digits, and thus each result
is simultaneously represented by a (long) decimal string and a table, or grid of colour swatches.
It is possible to change the dimension, or the width, of colour table to values between one and
thirty. Figure 2 shows the result of typing 1/7 into the calculator with the grid width set at ten,
thus generating the associated coloured table.
Figure 2. The Colour Calculator showing 1/7

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Using the button that controls the width of the table of colours, different table dimensions
can be selected, which results in different colour patterns. Figure 3 shows 1/7 displayed using a
table width of eighteen and seventeen, respectively. The table width of eighteen produces a
pattern we call “stripes” while the table width of seventeen produces a pattern we call
“diagonals.”
Figure 3. Different representations of 1/7 in the Colour Calculator
Of course, because of the way numbers are displayed in the Colour Calculator—only the
digits after the decimal point are represented in the coloured table—the division operation
produces the most interesting results, particularly when the rational quotient has a repeating
pattern. In fact, the Colour Calculator has the effect of reversing previously ‘transparent’ and
‘opaque’ (see Zazkis and Gadowsky, 2001) properties of rational numbers. For example, the
representation 1/7 makes transparent the number’s part-wholeness, while making more opaque
the fact that its decimal representation is non-terminating decimal. In contrast, the
representations of 1/7 shown above make transparent the number’s periodicity, while making
more opaque the fact that it corresponds to a simple fraction.
Faster than a pencil, bigger than a calculator, and more colourful than a textbook
We see the power of the Colour Calculator as an instructional tool as being rooted in three
important features that are not found either on the handheld calculator or in paper-and-pencil
environments: colour, speed, and size. The size feature relates to the number of digits that are
calculated and displayed (100 instead of the typical handheld calculator’s eight). With only
eight digits, the handheld calculator can barely display the repeating pattern found in a fraction

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with a denominator as little as seven (since its repeating pattern is of length six, the rounded
eighth digit can easily obfuscate the pattern). In contrast, the Colour Calculator easily handles
much larger units of repeat. This means that repeating decimals repeat transparently, and
therefore stand in greater contrast to non-repeating decimals. We hypothesised that the number
of digits displayed can help learners create a perception that infinitely repeating numbers really
do go on and on, and that non-repeating infinite decimals really do not repeat.
The speed feature relates to the Colour Calculator’s ability to compute and display quickly
the decimal expansion of a fraction; this contrasts especially with paper-and-pencil
environments where the process of conversion from fraction to decimal is frequently long,
tedious and error-prone. Speed thus allows learners to see a much wider range and greater
number of decimal expansions (it makes no distinction between ‘hard’ fractions like 3/41 and
‘easy’ fractions such as 5/6), and to test conjectures during exploration and problem solving
more quickly. In fact, it draws attention away from the procedure of conversion to the
properties and relationships that exist between common fractions and their decimal
representations. This should allow learners to work with the results of the conversion and to
treat these results as objects rather than processes.
Finally, the colour feature is responsible for translating strings of digits into a format where
patterns are more easily discernible. Since the colours are displayed within a manipulable grid,
they can be seen all-at-once (compare the tabular representation with the one-dimensional
array) and can be flexibly arranged to reveal certain patterns. (The grid width of eighteen in
Figure 3 arguably provides a more revealing pattern than the grid width of ten in Figure 2.)
Many of these patterns are recognisable and attractive (stripes, diagonals, and
checkerboards—see Appendix 1 for an illustration of these three types of patterns) and can thus
become motivational objects: Can you create a grid of stripes? Can you create an “all-red”
fraction? These kinds of question are qualitatively different than: Can you find a fraction that
has a repeating decimal representation? First, they involve creating a certain object, and, second,
they draw attention to the actual value of the length of the repeating decimal’s period.

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The three features of the Colour Calculator are intended to encourage and support
experimentation. Unlike other learning environments involving rational numbers, this one
makes it feasible to investigate the relationships that exist between, say, the denominator of a
fraction and the period of its decimal expansion. Since particularly interesting relationships
occur with denominators that are prime or ones that are of the form 10
n
- 1 (so 9, 99, 999, etc.),
the tasks we designed were oriented toward familarising the participants with those
relationships. In spite of its speed, size, and colour, the Colour Calculator can also provide an
exploratory environment well-suited to concepts previously explored in slower, smaller and
less colourful contexts. For instance, while the Colour Calculator is especially powerful for
displaying infinitely repeating decimals, it can also display terminating ones, and in fact,
provide a starker contrast between the two than is possible to achieve with other tools.
THE RESEARCH STUDY: A WINDOW ON RE-LEARNING
Setting
Participants in this study were pre-service elementary school teachers enrolled in a course
Principles of Mathematics for teachers, which is a core one-semester course in the teacher
certification program. The course is intended to deepen and extend students’ understanding of
topics underlying the elementary school mathematics curriculum. The Colour Calculator was
one of two “project”-options participants had to choose from as one of the requirements of this
course (the other option involved an investigation that was not computer based). Out of 90
students enrolled in the course, 42 chose to work with Colour Calculator. Out of these 42, seven
volunteered to participate in a clinical interview related to their experiences with this
microworld.
Participants’ work in the course included chapters on Fractions and Decimals (Musser,
Burger & Peterson, 2003). These chapters included the topics of decimal representation of
rational numbers, conversion between common and decimal representation of fractions,
classification of repeating decimals, representations of rational vs. irrational numbers, among

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others. These topics were completed shortly before the participants were introduced to the
Colour Calculator.
The participants were provided with written instructions about web access to the Colour
Calculator, a description of its commands and capabilities, and a list of suggestions for
mathematical explorations to be carried out before turning to the main assignment. These tasks
were intended to familiarise the participants with all the aspects of the Colour Calculator and
create an environment of experimentation and conjecturing. To support the participants,
computer lab hours were scheduled at different times on different days. Participation in the lab
was optional; nevertheless more than one-half of the participants (and all the interview
participants) chose to work, at least for part of the assignment, in this environment.
Our data consists of two main sources. The first source includes observations of the
participants’ work during the lab time, when they worked on series of tasks assigned to them
by the course instructor (see Appendix 1). During the allocated lab hours, the work of the
students was observed and supported where necessary. We noted frequently asked questions,
chosen routes for exploration, student’s conjectures, as well as their approaches toward testing
their conjectures. We also used these observations as a guideline for designing the interview
questions, which constitute our second source of data.
Our lab observations were focussed primarily on identifying instances in which the
participants engaged with concepts or ideas they had already encountered, either in their
university course or in the previous schooling. Thus, we initiated interactions with participants
who communicated various forms of recognition. For instance, if a participant noted a similarity
with something they had encountered in their university course or mentioned having heard of a
certain idea sometime during their pre-university schooling, we followed up with requests for
clarification. We also followed up with participants who expressed any form of surprise, in
order to probe the understandings they were bringing to their work with the Colour Calculator
and to discern how these understandings were being challenged.

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After the completion of the written assignment, clinical interviews were conducted with
seven volunteers from the group; they were audio-taped and transcribed. The interviews lasted
30-50 minutes; the Colour Calculator was available to the participants at all times during the
interview. These interviews had a semi-structured character, that is, the questions were
designed in advance (see Appendix 2), but the interviewer had the liberty to follow up with
prompts, include additional questions, or omit questions due to time considerations. In our
interviews, we posed a series of questions that were designed to assess various components of
the participants’ understanding and to further probe the kinds of cognitive, affect and aesthetic
representations that the participants had constructed in their interactions with the Colour
Calculator.
In addition to these two sources of data, we also drew supporting evidence from the
participants’ written work from their projects, particularly the descriptions of their visual
images of fractions and decimals elicited in the Final Task. We begin by reporting the
recognition instances—involving familiar though contextually different ideas for the
participants—observed first during the lab observations, and often re-visited during the
interviews. We then discuss the cognitive effects of the participants’ use of the Colour
Calculator on the problem-solving situations given to them during the interviews, followed by
some affective and aesthetic effects.
Lab observations: Enriching formal understandings
We had a total of 42 participants who spent on average three hours in the lab. Some of these
participants worked in pairs or groups at one computer while others worked individually,
though usually sharing findings and observations with their peers. In what follows, we have
chosen to describe several of the recognition instances that occurred most frequently in our
observations. When appropriate we have drawn on the interview data in order to further
illuminate or support our observations.

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Magic 9’s
During their University course, the participants had been exposed to the algorithm for
converting decimal numbers into fractions. They had completed an assignment in which they
had been asked to use this algorithm for a number of decimal numbers. Task 2 (see Appendix 1)
asked the students to investigate the several fractions whose denominators are of the form 10
n
–
1 (such as 9, 99, 999, etc.). We observed that the participants were very surprised to find that, for
example, 24/99 yields a repeating decimal with unit of repeat 24. However, they easily worked
through the questions in this task, generalising the “magic 9’s” pattern and noting some
“exceptions” such as 24/999. It is when they began encountering these exceptions that they
started to connect the phenomenon of the magic 9’s to the algorithm learned in class
(exemplified below), which, for several groups we interacted with, “explained” the
phenomenon.
Let n = 0. 238
___
Then 1000n = 238. 238
___
So 1000n – n = 999n= 238
And n = 238/999
One pair of participants remarked that their work with the Colour Calculator made them
realise what the algorithm was actually doing and how it was connected with fractions. Though
the algorithm always features nines (when the repetition starts immediately after the decimal
point; see the third and fourth steps above), those nines seem to remain somewhat opaque for
students, who focus on the algebraic manipulations involved. We observed many groups of
students writing out the algorithm on a piece of paper, accompanied by exclamations such as
“Oh! Now I see what’s going on.” Andrew in particular commented on how working with the
9’s in the Colour Calculator made “theories that were out there become related.” In an
interview, Blake also commented on the relationship between what he learned in class and what
he learned while working with the Colour Calculator: “I think that really like using the
numbers over 9, that really like related that, because it’s, you never really work backwards with

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that formula, or whatever, but in this case you are forced to go the other way, which really
makes it make sense. You know, it just gives you a better understanding of that, that it really
works.” This interplay between the “theory” learned in class and the “practise” was also
apparent in Leah’s interview: “Um, I think just better understanding, because I remember
before, Peter had mentioned something about, you know, there’s a very abstract theory for it,
but then seeing it and using it in practice really related it.
Perhaps students tend to view the algorithm as a process—or even a trick—that allows them
to turn decimal numbers into fractions rather than as a general relationship between fractions
and repeating decimals. We hypothesise that the patterned display of the Colour Calculator
helped that relationship become both more apparent and interesting for our participants. It
gave the participants an invitation to “play,” as Blake described later in an interview: “No, I
didn’t know that at all, because, well like I, you know, I kind of knew them but I just, I didn’t
relate the two, because I hadn’t seen it like that, we didn’t play with any numbers like that, so I
just didn’t make the connection.” And it gave them the opportunity to “see,” as Leah later
described: “[the instructor] talked about it in class and then yeah and then I did it on here just to
see what it looked like, that was weird. This was a more visual thing, it was, I mean it’s the
same thing, it’s just, just thinking about it more.” As we will discuss later, the participants also
gained an appreciation of how the “magic 9’s” phenomenon yielded a process that allowed
them to generate any repeating decimal they wanted, thus reversing the direction of the
algorithm.
Different but equal
As we observed the participants working in the lab, we were surprised by the number of times
we heard comments about the fact that the fraction they had inputted and the decimal string
that was being represented in the table were somehow one and the same. This realisation would
typically occur at the beginning of their sessions with the Colour Calculator.
Apparently, fractions and decimals look very different and it therefore requires some
cognitive work to see them as both representing the same number. Prior to using the Colour

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Calculator, the participants were perhaps more influenced by the form of these different
representations than by the quantities they denote. Typically, students learn algorithms for
converting one form into the other, but the assertion that ‘1/4 can be represented as 0.25’ is not
the same thing as ‘1/4 is equal to 0.25.’ Moreover, since decimal representations of fractions are
frequently rounded, they are actually only approximations (1/3 ≈ 0.33). This may obscure
students’ appreciation of the fact that fractions are numbers, and are thus equal to their decimal
representations—which students are more likely to perceive as numbers. In the Colour
Calculator, even though a repeating decimal is ultimately cut off, and thus is an approximation
of the fraction, the participants were able to interpret the patterned image of the repeating
decimal as an infinitely repeating object, and thus as equal to the fraction.
The beauty of the decimal system
When the participants were first working with the Colour Calculator, trying out different
fractions, they frequently expressed some form of disappointed when they had generated a
terminating decimal. This was often with fractions such as 1/4 or 3/10. We sometimes chided
them, asking “but don’t you know that 1/4 gives you 0.25?” We also reminded them of the
criterion they had seen in their course that checks whether or not a fraction terminates (can the
denominator of the reduced form be written as 2
n
5
m
?). Some of the participants insisted that
they did in fact know, but that “I just wanted to see” or “it’s so easy to just punch it into the
calculator without thinking.”
Although we suspect that many of these participants would have been able to determine
correctly that 1/4 and 3/10 terminate, we would argue that their purported knowledge was still
rather vulnerable. Moreover, we observed as at least three groups groped their way toward the
criterion by first focussing on whether the denominator was a factor of 10, 100, or 1000. This
may be due to the ease with which in fact they could experiment. By trying 3/8 and looking at
the decimal representation, it became obvious to them that there are three decimals and that 8 is
a factor of 1000. Two different groups of participants expressed surprise at this relationship
between the way decimal numbers are written (in the decimal system) and the denominators

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that generate terminating fractions. What role does the Colour Calculator play in prompting
this realisation? First, it simply allows students to test many fractions and quickly gain
feedback. Second, it is easier to distinguish a much larger class of terminating fractions than it is
in paper-and-pencil or regular calculator environments: a fraction as simple as 1/16 may not
appear as obviously terminating on a regular calculator. Third, we suggest that students need to
encounter similar ideas in a variety of settings. Their recollection of the algorithm likely
contributed to their developing experience with a large range of terminating decimals. And the
novelty of the setting may have motivated the participants to experiment rather than to assume
they have already “covered” the ideas involved. Many participants, such as Christine, were able
to admit that her experience helped her “re-establish things that I had forgotten about decimals
or about fractions.”
Interview findings: The transparencies of the Colour Calculator
In this section we report some of our findings from the semi-structured interviews. The
interview questions were designed to probe the participants’ understanding and familiarity
with terminating, infinitely repeating and infinitely non-repeating decimals.
Generating fractions from patterns of decimals
During the interviews, every single participant was immediately able to propose examples of
fractions that would generate certain patterns (see Question 5 in Appendix 2).
Figure 4: Colour Calculator pattern used in the interviews
When asked about the image shown in Figure 4, four of the participants responded as Andrew
does here, recalling specific fractions from his lab work (after he identifies the denominator
needed, he must identify the grid width, which he calls “the pattern,” that will produce the
diagonal pattern):

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Andrew: (pause) purple, (pause) 4, 2, it’s a pattern repeated, 6, so from previous
experience I know that 7 and 13 all have a repeating pattern of 6, um I’m going
to go with (pause), I’m going to try 7. . .
Interviewer: Okay, so 1/7. . .
Andrew: (pause) purple, red, orange, that’s the one I want, and we set the pattern at 11.
Interviewer: 11, how did you know 11?
Andrew: Um, yeah, okay, because you’ve got the repeating pattern of 6 and then you
basically yank the last one and start the next line, or would you bump it up or.
The other three interviewees relied on the “magic 9’s” and proceeded as Darren did below:
Darren: I’m just going to see if there’s a pattern of repeat, I think there is a pattern of
repeat, so, (pause) okay so the pattern of repeat is 6, the digits are . . .
Interviewer: It doesn’t matter if you don’t get exactly the colours, as long as you get the
pattern. So you got here what, 1,4,2,8,5,7, okay. . .
Darren: Yeah, so I have a repeat of 142857, now I’m going to, the denominator will be
rather large, it will be about 99,000. . .
Interviewer: Okay, try it.
Darren: Yeah it’ll be, okay I’ll try that, (pause) . . .
Interviewer: 142857 and how many 9's are you going to put. . .
Darren: I’m going to put in, I’m going to put in six 9's.
With those who used Darren’s method, the interviewer asked whether there was a smaller
fraction that would work. One participant was able to identify immediately a fraction with a
denominator of seven while the other two proceeded by simplifying the 9’s fraction they had
generated. During the interview, with the prompting of the interviewer, these two participants
were able to see that sevenths, for example, had a period length of 6 because they could be
written as fractions over 999,999. Nicole remarked that “13 was a 6 pattern, so it would
obviously go into 999,999 some amount of times, I guess there we go, that’s why it works. So
there, I just learned something new, because of the colour calculator.”

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19
Figure 5: Colour Calculator pattern used in the interviews
For the third image shown in Figure 5, every participant used the 9’s method and found that
226/999 generated the correct pattern. And for fourth image (see Figure 6(a)), one participant
attempted to use the 9’s method, after having correctly determined the period length of 46
ii
,
while the others were able to identify immediately 47 as the correct denominator. In addition to
being able to call upon ‘basic’ fractions such as 1/3, 1/4, 6/10, 2/5, the participants knew what
fractions having denominators such as 7, 13, 17, or 47 would look like (one participant
commented that “Well I know, I know I can now say well 7 is actually a routine example with a
length of 6”), and they could describe some relationships between the denominators and the
decimal expansions (particularly for prime number denominators and denominators of the
form 10
n
-1). This kind of relational understanding— which differs from the participants’
existing algorithmic way of understanding—might account for Danielle’s claim “I now have a
better understanding of how certain fractions create different types of decimals, such as finite or
infinitely repeating.” She may not have had a better understanding of how to convert fractions
to their decimal representations, but she had a larger set of concrete (and colourful) examples
corresponding to different types of decimal representations.
It is interesting to note that students are frequently more comfortable with ‘nice’ fractions
that terminate than with fractions that repeat infinitely. Students’ typically biased exposure to
these ‘nice’ terminating decimal representations of fractions may mislead them into believing
that fractions having terminating decimal representations are the most common types of
fractions. Referring to the frequency of occurrences of non-terminating fraction relative to the
terminating ones, Blake noted that he now thought that terminating fractions were “weird

A coloured window on teachers’ conceptions…
20
compared to what we've been doing with this program.” An increased exposure to non-‘nice’
fractions acquainted Blake with the preponderance of non-terminating numbers. His comment
reveals the way in which students’ typical experiences with rational numbers may bias them
towards thinking that terminating numbers are more frequent than non-terminating ones. It
may be that just having access to a larger set of concrete examples of non-terminating decimals
can rectify this bias.
Focus on denominators: repeating versus non-repeating
In addition to gaining more familiarity with the difference between terminating and repeating
decimals, participants worked with several examples of irrational numbers in the lab, that is,
numbers that have infinite non-repeating decimal representations. These types of numbers are
frequently difficult to discern with handheld calculators, which only display eight digits.
Christine explained how calculators had left her confused about the status of many decimal
representation: “I think it helps a lot that is shows so many numbers, like on a calculator you
would just see, like if you were to take 1/47, it would look like it's an irrational number.” The
Colour Calculator helped the participants realise that fractions such as 1/47 can have very long
periods (in this case, 46), yet still repeat. Christine knew the definition of irrational numbers but
had been unable to reconcile this definition with the results shown on regular calculators.
However, by building up a strong image of many, many repeating decimals, she could more
distinctly grasp the notion of a non-repeating infinite decimal. This became very clear in the
interviews when the participants were asked to explain how certain coloured patterns had been
created with the Colour Calculator. Even an image such as in Figure 6(a) posed no problems, for
the participants knew that some fractions could have long periods. In this excerpt, Christine
shows that she simply needs to find whether the first sequence of digits (or colours) repeats
again:
Interviewer: Okay, how could the Colour Calculator produce this image?
Christine: Umm first I’d find some colours that repeats, so the starting goes pink, yellow,
blue, red, so I need to find that sequence somewhere, what’s yellow, blue, red,

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21
so it would repeat, 1,2,3,4,5,6,7,8,9, 1,2,3,4,5, so 45, 46 times, so the denominator
would have to be 47, because that’s a prime, so it’ll give 46 repeaters.
Instead of restricting themselves to the first digits in the sequence as Christine does, most
interviewed participants were drawn to a certain local pattern that occurred twice in the image,
thus recognising that a repeating pattern in the decimal representation will create a certain
pattern within the table of colours. For example, the diagonally-arranged powder blue squares
pop out for Andrew, and enable him to assert that there is indeed a pattern, and that its length
is 46.
Andrew: This is an awfully long pattern.
Interviewer: What do you see. . .
Andrew: Well I see those two blue guys right there. They’re repeating further
down there and I’m sure there is a pattern and then once you get these
blues here you can count 9, uh, 18, 27, 36, this is an awfully long pattern,
36, oh 45, 46, so it’s probably something over 47.
(a) 3/47on the Colour Calculator
(b) √2 on the Colour Calculator
Figure 6: Two outputs from the Colour Calculator set at grid width 9
With the image in Figure 6(b), most of the participants reasoned, like Leah below, that they
could only deduce that either the period was greater than the number of colours shown, or the
pattern represented an irrational number.
Leah: I’m just trying to see if I can see a pattern at all. Yellow here, and like
there’s one, that yellow there. No, it doesn’t repeat, you never get these
two guys, these three guys here, it looks like irrational number.
Interviewer: You say it looks like an irrational number, why?
Leah: It looks like, because I can’t find any pattern, I mean it could just be that
this is, the pattern goes from here to here.
Leah’s interview shows that she has clearly understood what a non-repeating decimal will
look like, and has even grasped the limitations of the Colour Calculator’s output. Darren placed

A coloured window on teachers’ conceptions…
22
somewhat more trust in the Colour Calculator in his description of π: “I realize that it can’t be a
repeating decimal and so, I can say, I could go into the colour calculator and it can show me
what pi really looks like as a colour.”
Some special denominators
We now consider a typically less familiar relationship than the ones described above. Students
frequently learn how to convert a fraction to a decimal using long division, but this process
does not make explicit the role played by either the numerator or the denominator in
determining a fraction’s decimal expansion. Through their use of the Colour Calculator, we
observed that several participants were just beginning to realise that the denominator of a
fraction plays an important part in determining the decimal expansion. This realisation has
familiar as well as novel components, and Nicole describes them both: “Regarding fractions, my
visual image depends a lot upon the denominator, which plays a large factor in determining
what type of decimal representation [terminating or repeating] will occur as well as how long
the period length will be.” The participants were previously aware that different fractions
produced different decimals but the specific role of the denominator—which determines the
length of the period—became more pronounced, as Leah points out: “I didn't know anything
about like denominators having anything to do with the length of the periods and stuff.”
Not only does the denominator play a large part in determining the pattern of a decimal
expansion, but there is also a relationship between the denominator and the period length.
Kyle’s articulation of this property—which is not completely accurate—is typical of the
understanding achieved by participants: “specific denominators will consistently have a certain
period length no matter what the numerator is.” This understanding is elaborated upon by
Tracy, who sums up the role of both parts of the fraction: “The numerator is responsible for
determining which numbers are in the decimal but the denominator dictates the period length
and therefore is responsible for the structure of the decimal.”

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23
Focus on numerators
Another property perceived as novel—though discussed and practised in one of the homework
assignments—that the participants encountered relates to the relationships between different
fractions with the same denominator. The participants had completed a homework assignment
the previous week that involved looking at the decimal numbers for family of sevenths (1/7,
2/7, 3/7, 4/7, 5/7, 6/7). They were also asked to do this with the Colour Calculator. We had
thought this would be trivial for them, a mere repetition of their homework. Yet, Andrew
reported some of the differences: “That was surprising to me, you don't even see that when you
do it on the calculator and your calculator has a tendency to round for you, so you never see
that 1/7 has same numbers as 2/7.” Granted, this property—that each member of the family
has the same sequence of numbers, though starting at a different term in the sequence—lays
outside the realm of material usually covered in school mathematics, but Andrew explained
how it added to his previously flat understanding of infinite decimals:
I don't think I would have known that because my previous experience with any decimal, I
mean in high school they just said, that's a never-ending decimals, 1/7 is a never ending
decimal. That's what we were told. Just put three dots beside it and don't worry about it.
Well, no more little dots for students of mine, no way.
The use of the “three dots” ended up obscuring many relationships for Andrew. By knowing
more about the family of sevenths, including the length of its period and the relationship
between the numerators and the actual numbers in the period, the fraction 1/7 gained some
personality.
How were the participants’ orientations towards fractions and decimals affected?
As the participants began working with the Colour Calculator in the lab, they were engaged
and explorative. Many responded with surprise and even delight upon creating their first
patterns of stripes and diagonals. Both in their written assignments and during their interviews,
the students had opportunities to describe qualitatively their experiences working with the

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24
Colour Calculator. They were explicitly invited to do so in the Final Task of the assignment and
in the last interview question. Two major themes emerged: vivifying properties and
relationships encountered more formally in the classroom and developing understanding
through experimentation and visualisation.
Vivifying properties and relationships
Colour seemed to play a role in vivifying the participants’ understanding of various properties
and relationships. It is difficult to tell whether she could have described 1/3 prior to using the
Colour Calculator, but Kimberley wrote in her project that a “denominator 3 will always give
me a block of mono-colour in my mind.” She adds “I'll forever see fractions and decimals in
colour.” In a slightly different vein, Darren talks about how the colour-coded decimal
representation helps make clear distinctions that might otherwise be vague: “Fractions in their
decimal representations (with colour) give a better understanding of the relationship between
fractions and decimals. Thus 3/99 and 3/9 seem similar but the results are very different.”
Some participants described and wrote about the way in which the Colour Calculator
helped bring to life concepts they had encountered in the classroom. Christine talked about the
Colour Calculator gave her shortcuts to doing things: “Like with this whole program, the
questions we would try to figure them out how we learned them in a lecture, or just how we
would figure them out, and then we'd realise, there's a shortcut to it.” Once again, Christine is
describing how the Colour Calculator helped make certain ideas more accessible, more flexible
to work with. In her statement, Kelly alludes not only to the vivification of the logical
procedures encountered in the classroom, but also to the role that the actual tasks played in her
experience: “ I have been able to gain a whole new perspective of fractions and decimals. These
tasks forced us to look for logical procedures as opposed to stumbling upon the correct answers.
Looking to create stripes, diagonals, and checkerboard patterns on the grid allowed me to
search for an understanding rather than being handed the answers.”

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25
Part of the process of vivification can include developing emotional or aesthetic responses to
mathematical ideas. Aimee clearly was able to do both using the Colour Calculator, as she
reports in her project:
The repeating fractions have a flow that I find comforting. As a child I disliked fractions that
did not terminate, but now I see them in a light of beauty. I find that the decimals which
terminate sad. They are unable to touch the fingertips of forever, like the repeating ones can.
Though they did not make statements such as Aimee’s, we noticed that many participants used
aesthetically-charged words both in the lab and in their projects to describe the patterns they
created in the Colour Calculator. Some of them even developed strong preferences for certain
types of patterns over others, such as stripes over diagonals; these preferences emerged when
they were asked for the best way of displaying different fractions.
Developing understanding through visualisation and experimentation
As Kelly’s comment above states, the assignment tasks encouraged experimentation. In fact, we
propose that both the use of colour representation and the possibility of experimentation were
crucial to the participants’ experiences with the Colour Calculator. With regard to
experimentation, Danielle notes that “with this program you can try several things and just play
around with fractions and decimals until you figure out patterns and simpler ways of finding
what you want.” Because of this, some of the participants were able to engage in exploration
and problem solving, which for Brad, “helped me justify some unknown curiosities.”
With regard to the visual representation afforded by the Colour Calculator, Kevin wrote
about the increase in pattern possibilities now available: “patterns can be seen as you move left
to right, up and down, and even left and up, or right and down. Patterns are more visible and
meaningful in the sense than when written down on paper in the standard 0.blah blah blah blah
blah blah way.” A couple participants talked about the way in which the colourful patterns
attracted and held their attention, in a way that numbers might not. For example, Kyle noted
that “without colours to represent numbers, patterns are much more difficult to discern, and
can impact highly on an individual's ability to focus. Similarly, Kelly explained “the colours

A coloured window on teachers’ conceptions…
26
were important in not only clarification purposes, but were also important in keeping me
attentive.” The colour also seems to make patterns more accessible than they are on handheld
calculators, as Dianna insisted: “Being able to see the numbers represented as colour helps the
patterns to become more pronounced for me. Normally, on the regular calculator, you cannot
see that there are sets of repeating numbers—they usually just look like a jumble with no rhyme
or reason.” Lauren agreed: “It is easier to see patterns of colours than patterns of numbers.”
CONCLUDING REMARKS
Frequently, pre-service elementary school teachers do not have a deep understanding of the
mathematical topics related to the school curriculum. However, they do possess a superficial
understanding, a certain familiarity with the mathematical concepts in question, and thus need
to embark on a process of re-learning. But re-learning does not necessarily come from re-
teaching, a process that can be contaminated by existing understandings, inaccurate or
fragmented as they may be. A construction metaphor is illustrative: when an old house isn't
solid, it is at times preferable to flatten it and construct a new one on the same lot, rather than
strengthen, restore, and renovate the old structure. In teacher education, we do not have this
option: we must work with a learner’s existing structure, and we must reengage the learners in
the process.
This is not an easy task in that students tend to shy away from reengaging with a concept
that they already have some familiarity with, often overestimating their abilities and
understandings of the concept in question. Furthermore, if the concepts are at all unpleasant, as
‘fractions’ and ‘decimals’ have a tendency to be for learners, then reengagement is even more
challenging. In the specific case of pre-service elementary school teachers, deepening prior
understandings may require new, or more sophisticated approaches than simply building new
understandings does. In the context of education and relearning, such sophistication takes the
form of experiences and challenges that provoke learners to re-examine and enrich their
understandings of the mathematical concepts previously encountered.

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In this study we have shown how pre-service teachers’ interactions with a web-based colour
calculator both enriched their experiences with rational numbers and challenged their
understandings of several properties related to fractions and their decimal representations. The
novelty and aesthetic appeal of the Colour Calculator—in synchronisation with the tasks we
chose—helped the participants overcome their reluctance to reengage with properties and
relationships associated with these concepts. Through colour, speed, and size of display,
previously opaque qualities such as repetition and length of the period became transparent, and
thus available for exploration and problem solving. More importantly perhaps, our research
participants had the opportunity to engage with representations of fractions in decimal form as
objects, rather than as final steps in a procedure, and to construct vivid, memorable, and positive
images of important mathematical ideas.
REFERENCES
Behr, M., Wachsmuth, I., Post, T., and Lesh, R. (1984). Order and equivalence of rational
numbers: A clinical teaching experiment. Journal for Research in Mathematics Education 15, 323-41.
Campbell, S., & Zazkis, R. (Eds.) (2002). Learning and teaching number theory: Research in
cognition and instruction. Journal of Mathematical Behavior Monograph. Westport, CT: Ablex
Publishing.
Edwards, L. (1995). Microworlds as representations. In A. A. diSessa, C. Hoyles, and R. Noss
(Eds.), Computers and Exploratory Learning, 127-154. Berlin/Heidelberg: Springer-Verlag.
Eisenberg, T., and Dreyfus, T. (1991). On the reluctance to visualize in mathematics. In W.
Zimmerman and S. Cunningham (Eds.), Visualization in Teaching and Learning Mathematics (Vol.
MAA Notes Series, pp. 25-37). Washington, DC: MAA Press.
Goldenberg, P. (1989). Seeing beauty in mathematics: Using fractal geometry to build a spirit
of mathematical inquiry. Journal of Mathematical Behavior, 8, 169-204.
Goldin, G. (2000). Affective pathways and representation in mathematical problem solving.
Mathematical Thinking and Learning, 2(3), 209-220.

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Hiebert, J. and Carpenter, T. (1992). Learning and teaching with understanding. In D. A.
Grouws (Ed.), Handbook of research on mathematics teaching and learning (65-97). New York:
Macmillan.
Lamon, S. (2001). Presenting and representing: from fractions to rational numbers. In A.
Cuoco and F. Curcio (Eds.), The roles of representation in school mathematics. Reston, VA: National
Council of Teachers of Mathematics.
Lamon, S. (1999). Teaching fractions and ratios for understanding: Essential content knowledge and
instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum Associates.
Markovits, Z. & Sowder, J. (1990). Students' understanding of the relationship between
fractions and decimals. Focus on Learning Problems in Mathematics, 13(1), 3-11.
Musser, G., Burger, W., and Peterson, B. (2003). Mathematics for elementary teachers: A
contemporary approach, 6
th
edition. New York: John Wiley & Sons, Inc.
Noss, R., and Hoyles, C. (1996). Windows on mathematical meanings. Dordrecht, The
Netherlands: Kluwer Academic Publishers.
Post, T., Harel, G., Behr, M., and Lesh, R. (1988). Intermediate teachers’ knowledge of
rational number concepts. In E. Fennema and T. Romberg (Eds.), Rational numbers: An
integration of research (107-130). Hillsdale, NJ: Erlbaum.
Sinclair, N. (2001). The aesthetic is relevant. For the learning of mathematics, 21(1), 25-33.
Skemp, R. (1971). The psychology of learning mathematics. XX: Penguin Books.
Steffe, L. P. & Nesher, P. (1996). Theories of mathematical learning. Mahwah, NJ: Lawrence
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Wilson, M. and Goldenberg, M. (1998). Some conceptions are difficult to change: One
middle school mathematics teacher’s struggle. Journal of Mathematics Teacher Education, 1, 269-
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Zazkis, R., & Gadowsky, K. (2001). Attending to transparent features of opaque
representations of natural numbers. In A. Cuoco (Ed.), The role of representations in learning
mathematics. Reston, VA: National Council of Teachers of Mathematics.

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APPENDIX 1
Written response tasks
Task 1. Stripes, Diagonals and Checkerboards
Stripes
Left Diagonals
Right Diagonals
Checkerboard
1. Can you find three different ways to create stripes?
2. Can you find three different ways to create diagonals? What makes them right or left?
3. Can you find three different ways to create a grid with only one colour in it?
4. Can you find three different ways of creating a checkerboard pattern? What about a “real”
checkerboard with a grid width of 8?
Task 2. Magic number 9.
1. Try the following fractions: 2/9, 5/9, 7/9. What do you notice about these fractions? Can
you explain why the denominator of 9 has this effect?
2. Try the following fractions: 23/99, 48/99, 73/99. What do you notice about these fractions?
Can you explain why the denominator of 99 has this effect? Can you create a grid that starts
blue, orange, and repeats?
3. Predict what will happen if you display the fractions 123/99 and 145/999. Test your
conjectures and explain what you observe.
4. Can you make a grid of blue and yellow repeating without using the digits 2 or 6 in your
fraction?
Task 3. The period of primes.
a) Describe how you would set the width of the grid to best illustrate the period of each of the
following fractions: 3/11, 1/29, 1/59.

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30
b) Display the fractions 1/7, 2/7, 3/7, 4/7, 5/7, and 6/7. What do you notice?
c) For denominator ≤ 10, the longest period length is 6. What is the longest length of period
you can find for denominator ≤ 50? Describe how you looked for/found it.
d) Predict the length of period of the following fractions: 1/13, 1/17, 1/19. Can you predict the
length of period of any fraction whose denominator is a prime number?
Final Task. Write a paragraph that describes the visual image you have of various fractions and
decimals.
APPENDIX 2
Interview Questions
1. Can you get the fraction 11/13 into stripes?
2. How would you find a fraction that produces a table of stripes when the grid width is 9?
3. The grid only displays 100 decimals. Could you tell what the colour of the 103
rd
cell would
be? What about the 175
th
cell?
4. We are looking at 14/99. Can you predict what will happen if we look at 140/990? Can you
find another way of obtaining the same sequence of colours?
5. How could you make a table of colours that looks like each of these?
6. Can you use the Colour Calculator to show why π is not equal to 22/7?
7. What have you found surprising or helpful or interesting in your experience with the
Colour Calculator?

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31
Footnotes
i
It can be accessed on the web at: http://hydra.educ.queensu.ca/maths/. The design of this
calculator was inspired by work at the Centre for Experimental and Constructive Mathematics
(CECM) at Simon Fraser University, where techniques are being developed to employ the
natural visual capacities of human perception to search for complex relationships and patterns
in numerical distributions. Though the CECM’s use of visual calculators is aimed primarily at
looking for the fundamental underlying structures of mathematical objects such as sequences of
polynomials and continued fraction expansions, author Sinclair created a modified and
simplified calculator that would be more suitable for the exploration of simpler mathematical
objects.
ii
This would require inputting a fraction whose denominator is a 46-digit number, a computation
that the Colour Calculator cannot handle.