Research in Mathematics Education, 2014
Vol. 16, No. 1, 54–70, http://dx.doi.org/10.1080/14794802.2013.876157
Script writing in the mathematics classroom: imaginary conversations
on the structure of numbers
Rina Zazkisa* and Dov Zazkisb
Faculty of Education, Simon Fraser University, Burnaby, Canada; bGraduate School of
Education, Rutgers University, New Brunswick, NJ, USA
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a
Script writing by learners has been used as a valuable pedagogical strategy and a research
tool in several contexts. We adopted this strategy in the context of a mathematics course
for prospective teachers. Participants were presented with opposing viewpoints with
respect to a mathematical claim, and were asked to write a dialogue in which the
characters attempted to convince each other of their point of view. They had to imagine
and articulate fictional characters’ reasoning, as well as design a potential pedagogical
intervention. We outline what script writing revealed about the participants’ understanding of the structure of natural and rational numbers and of mathematical argumentation,
and discuss the affordances of this methodological tool in teacher education.
Keywords: script writing; rational numbers; prime numbers; prime factorisation
Mathematical knowledge and understandings of prospective teachers are recurrent themes
in mathematics education research. Investigating and detailing the nuances in this
knowledge can benefit from the use of innovative research tools. Our study uses one such
tool – script writing – in order to gain further insight into participants’ understanding of
the structure of natural and rational numbers.
Script writing
The use of the script writing method is rather novel in mathematics education research.
However, its roots can be traced to a Socratic dialogue, a genre of prose in which a ‘wise
man’ leads a discussion, often pointing to flaws in the thinking of his interlocutor. The
method echoes the style of Lakatos’s evocative Proofs and Refutations (1976), where
fictional interaction serves as a part of a learning progression. Recently, script writing by
learners has been implemented as an instructional tool, as well as a research tool for data
collection, in several studies.
For example, Gholamazad (2007) developed the “proof as dialogue” method.
Prospective elementary school teachers participating in her study were asked to clarify
statements of a given proof in elementary number theory by creating a dialogue, where
one character had difficulty understanding the proof and another attempted to explain
each claim. This method was amended and extended by Koichu and R. Zazkis (2013) and
D. Zazkis (2013) in their work with prospective secondary school teachers. In both
studies the participants had to identify problematic issues in the presented proofs, and
*Corresponding author. Email: [email protected]
© 2014 British Society for Research into Learning Mathematics
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clarify how these issues arise and how they might be resolved by creating a dialogue,
referred to as a proof-script. These scripts revealed participants’ personal understandings
of the mathematical concepts involved in the proofs as well as what they perceived as
potential difficulties for their imagined students.
Additionally, the ‘lesson play’ method was developed and used in teacher education.
In this method, participants are asked to write a script for an imaginary interaction
between a teacher-character and student-character(s) (R. Zazkis, Sinclair, & Liljedahl,
2013). The method is advocated as an effective approach in preparing for instruction and
also as a diagnostic tool for researchers and teacher educators.
In this study, we extend the script writing method by considering a disagreement
between characters with respect to a number property. To create a dialogue between the
characters the script writers have to assume each role, and present or rebut arguments
from the perspective of different characters.
The study
Participants and data
The prospective elementary school teachers who participated in this study were enrolled
in a ‘Mathematics for elementary school teachers’ course (n = 87). This is a 13-week
course, which includes four weekly hours of interactive lecture and is supported by a
‘workshop’, where participants are encouraged to meet for group work and may seek help
with the course content. The course includes a unit on number theory and another on
rational numbers both of which were explored for four weeks (on weeks #7–10 of the
course) prior to the data collection.
In the number theory unit the topics focused on divisibility, prime and composite
numbers, prime factorisation, and the Fundamental Theorem of Arithmetic. In particular,
the participants were introduced to the method of determining the number of factors of a
given natural number, based on its prime decomposition and the fundamental principle of
counting. That is to say, if N is a natural number and the prime decomposition of N is
a2
ak
pa1
1 ! p2 ! . . . ! pk , and if F is a factor of N, then each prime factor pi of N can
appear in F in a power s, where 0 ≤ s ≤ ai. Therefore the number of factors of N is equal
to (a1+1) (a2+1) … (ak+1). The participants practised this method in their homework,
and were successful in implementing it when it was explicitly requested by the task.
In the rational numbers unit the participants were introduced to the definition of rational
and irrational numbers. In particular, rational and irrational numbers were defined in terms
of the existence (or lack of existence) of a ratio representation. The resulting decimal
representation was discussed. The participants learned to convert repeating decimals to
fractions and to predict whether a given fraction will result in a terminating or a repeating
decimal representation. They practised converting repeating decimal representations into
common fractions, and were successful in performing this task in their homework.
Near the course’s conclusion (in week 11), about a week after the topics of natural
and rational numbers were completed, the participants were asked to provide a written
response to either Task 1 or Task 2 as one of the course assignments. They were asked to
complete the assignment within two weeks. Participants were encouraged to work in pairs
or in groups of three (group work was an integral part of the course). However, a few
opted to complete the task individually. We analysed 24 responses to Task 1 (composed
of 55 students: 3 individual, 14 pairs and 8 groups of 3) and 13 responses to Task 2
(composed of 32 students: 7 pairs and 6 groups of 3 students).
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The tasks
The participants were presented with two scenarios in which two characters have opposing
points of view with respect to a number property. Their assignment was to create a script of a
conversation, choosing one of the scenarios, in which each character attempts to convince
the other of the validity of their viewpoint. It was clarified that their personal knowledge of
mathematics involved in the tasks could be demonstrated through the claims of their
characters or in the accompanying commentary. The participants were further advised that
the characters in their scripts might expose incorrect justification or claims, or inappropriate
mathematical expressions, but that this should be acknowledged in the commentary. That is,
the commentary was meant to distinguish between the language and argumentation that the
script writers attributed to their characters and those that they themselves found appropriate.
Particular information on how the task would be assessed was neither provided nor sought.
The scenarios and specific instructions for completing the Tasks are presented below.
TASK 1
Bonnie and Clyde are discussing numbers and their factors. Bonnie claims that the
larger a number gets, the more factors it will have. Clyde disagrees.
Write a script for a conversation between these two characters that includes their
exchange of arguments as both sides are convinced they are right. Consider what
examples and what experiences could have led Bonnie to this conclusion. Consider
why Clyde would disagree. Consider what arguments and what examples they both
use to convince each other and what each one of them finds convincing.
Annotate your script analysing the arguments of your characters and their examples.
The scenario presented in Task 1 was adapted from Zazkis (1999). In that study the
erroneous claim of students – that larger numbers necessarily have more factors – was
analysed in terms of the “intuitive rules” framework suggested by Stavy and Tirosh
(1996). This claim was seen as an example of the “more of A – more of B” intuitive rule.
Indeed, if one were to choose a random n-digit number the expected number of factors for
that number would be lower than in a randomly chosen (n +1)-digit number. For example, the
expected number of factors for a 2-digit number is about 5 (450/89) and the expected number
of factors for a 3-digit number is about 7.3 (6580/899). However, there are infinitely many
examples of large numbers with a relatively small number of factors. In fact, for each natural
number n > 1, there is an infinite number of natural numbers that have exactly n factors.
TASK 2
Tom and Jerry are discussing rational and irrational numbers. Tom claims that 23/
43 is an irrational number, because his calculator shows 0.53488372 when 23 is
divided by 43, and there is no repeating pattern of digits. Jerry disagrees.
Write a script for a conversation between these two characters that includes their
exchange of arguments as both sides are convinced they are right. Consider what
examples and what experiences could have led Tom to his conclusion. Consider
why Jerry would disagree. Consider what arguments and what examples they both
use to convince each other and what each one of them finds convincing.
Annotate your script analysing the arguments of your characters and their examples.
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The scenario presented in Task 2 was adapted from Zazkis and Sirotic (2010). One of the
tasks used in that study invited participants to consider the number M = 53/83, for which
the calculator display showed 0.63855421687, and decide whether M was a rational or an
irrational number. They reported that over 30% of the participants either failed to
recognise the number represented as a common fraction as being rational or provided
incorrect justifications for their claim. The equivalence between the two descriptions of
rational numbers – one in terms of a ‘ratio’ and another in terms of repeating or
terminating decimal representation – was described by the authors as a ‘missing link’ in
their participants’ understanding of rational numbers.
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Data analysis
The main research question that drove this study was: “What is revealed by the script
writing method about the participants’ understanding of mathematics?” This overarching
research question was expanded upon when analysing the data from each of the two
tasks.
Different theoretical tools were used for analysing responses to each of the tasks. As
such, we report on each of the task separately, starting with the description of theoretical
considerations that guided our analyses. Both authors considered the scripts individually
and identified emerging themes. Both sets of themes were organised with respect to a
theoretical framework, and minor disagreements with regard to applying the frameworks
were resolved. Each author independently identified and compiled excerpts that
exemplify each theme. Then the authors jointly chose excerpts that were most
representative of an identified theme for inclusion in this paper.
Task 1: on numbers and factors
Theoretical considerations
In mathematics one counterexample is sufficient to conclude that a statement is not true in
general. However, this foundational norm in logic and in mathematics is not the one
practised in everyday reasoning. Researchers have observed students’ readiness to accept
conjectures following several confirming examples and students’ reluctance to abandon
conjectures when having disconfirming evidence (e.g., Harel & Sowder, 1998).
In non-mathematical situations one often weights evidence in order to confirm or
support an assertion. Fuzzy logic – as a response to the limitations of Aristotelian ‘crisp’
logic – was developed to capture decision making in situations of uncertainty and to
acknowledge a ‘grey area’ in which the truth-value of a statement is represented by a
number between 0 and 1. However, it was observed that students may have a tendency to
apply reasoning consistent with fuzzy logic to mathematical situations. For example, the
statement “even numbers are divisible by 4” was considered 50% true, as it is true for
every second even number (Zazkis, 1995).
In experiments with science students, Chinn and Brewer (1993) identified seven
responses to what they called “anomalous data.” The non-normative responses were:
ignoring, rejecting, or reinterpreting the data, excluding the data from the current theory,
and holding it in abeyance (not rejecting it, but not using it to modify the theory, either).
Only two responses followed scientific or mathematical norms: the first, making
peripheral changes to the currently-held theory; and the second, making substantial
changes. Science is epistemologically different from mathematics, particularly in regard
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to how single case counterexamples are treated. However, considering a counterexample
as a case of “anomalous data”, this framework was found applicable in analysing
participants’ responses to evidence in mathematical situations (Edwards & Zazkis, 2002).
Similarly, we find Chinn and Brewer’s (1993) framework to be useful for examining our
data; as such, it is utilised and extended in this study.
Our analysis focuses on participants’ ways of dealing with confirming and
disconfirming evidence, using the framework of Chinn and Brewer (1993) and ideas of
fuzzy logic. In particular, we expand our main research question (What is revealed by the
script writing method about the participants’ understanding of mathematics?) by
addressing the following: What arguments are presented in the scripts in support of
conflicting points of view? What kind of evidence is accepted as convincing?
Results and analysis (Task 1)
In examining the scripts and their annotations written by the participants, we focused on
several reappearing themes. Table 1 summarises the frequency of occurrence of various
themes.
Attention to concepts
Almost all the scripts (21 out of 24) devoted some attention to the mathematical concepts
involved in discussing Bonnie’s claim. Prime numbers were the focus of such attention as
most scripts included a definition of primes, and some reiterated the method for
determining whether a given number is prime. Consider the following excerpt from
S1.4(3)1:
Clyde: Your theory cannot be true for all the numbers. You have to take prime numbers into
consideration. Let’s go back to the definition of a prime number: “an integer that has no
integral factors but itself and 1”.
Bonnie: I’m not following you.
Clyde: Okay, in other words all prime numbers have only 2 factors, the number 1 and itself.
I’ll just name a few: 2, 3, 5, 7, 11, 13, 17, 19, 23. Prime numbers are the best examples for
you to realise that your theory is wrong.
Bonnie: Are you saying if we exclude prime numbers then my theory works?
Clyde: No dear, for each number you have to consider its factor tree.
Table 1. Frequency of themes in Task 1.
Theme
Attention to concepts
Use of “large” numbers in an argument
Recall of algorithm for determining the number of factors
Response to disconfirming evidence
Normative conclusion: rejection of claim
Theory amendment: primes as exceptions to the rule
Theory amendment: powers of primes as exceptions to the rule
Number of scripts
21
4
3
8
12
4
24
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As we discuss in the next section, the idea of ‘best examples’ featured here was often
interpreted as ‘the only examples’ or ‘exceptions’ to the general property. In the next
excerpt from S1.18(3), Clyde extends his argument by alluding to the infinity of primes:
Clyde: You didn’t take into account that there are prime numbers out there where their
factors only include themselves and 1, regardless of how large or small they are.
Bonnie: Prime numbers? I’ve heard of the term but I’m not sure what they are exactly.
Clyde: By definition, a prime number is a natural number greater than 1 that has no positive
divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is
called a composite number.
Bonnie: Okay, cool, that makes sense, but if the number keeps getting bigger, then it would
have more factors, wouldn’t it? For example, 4 has 3 factors (1, 2, 4) whereas 24 has 8
factors (1, 2, 3, 4, 6, 8, 12, 24).
Clyde: You have a point, but that only works to a certain extent. For some larger numbers,
there still is a possibility that they only have factors 1, and themselves, making them prime
numbers. It’s when they are divisible by another number that they are not prime.
Bonnie: Can you give me an example of a large number that would only have 2 factors?
[…after considering several examples of ‘large’ primes, 997 and 127]
Clyde: Clearly the point you made earlier is invalid because there are an infinite amount of
large prime numbers that only have two factors.
Other than explicit attention to primes in order to convince Bonnie, two additional themes
emerge in this dialogue: the convincing power of large numbers and the acknowledgement that “that only works to a certain extent”. Both themes are discussed in detail in the
next sections.
‘Factor’ is another concept that participants felt needed clarification. Some suspected
that Bonnie’s claim was based on a misunderstanding of this concept. Indeed, the word
factor is problematic for several reasons. First, it is used outside - the mathematical
register, and it is known that assigning everyday meaning to mathematical terms may lead
to errors (Durkin & Shire, 1991). Moreover, it is used in mathematics both as a noun and
a verb (as in factoring a2– b2 results in (a – b) (a + b)). The noun-factor is assigned its
meaning by the context. On the one hand, there is a number-theoretic definition, where
for natural numbers a and b, a is a factor of b if and only if there exists a natural number k
such as a × k = b. On the other hand, in the equation x × y = z, x and y are referred to as
factors, and z is their product, and there is no requirement for x, y and z to be integers.
This is the difference between a factor of a number as a relationship, and a factor in an
equation, as a role the number plays (Zazkis, 1998).
In the following excerpt from S1.2(2), the script writers show that Bonnie’s
understanding of the term factor is inconsistent with the expected number-theoretic
meaning, and Clyde attempts to correct this:
Bonnie: But Clyde, I thought we were dividing the cash into several bags, and with more
cash doesn’t that mean it can be divided into more bags?
Clyde: Sorry Bonnie but I don’t think you have your factoring rules straight
Bonnie: But I know for a fact that bigger numbers have more factors than small numbers. For
example, 38 has more factors than 32 because it has factors like 33.5, 34.25, 36.75, and 37.9.
Clyde: No Bonnie. I think you do not understand what a factor number is. A factor is a whole
number that is multiplied by another whole number to equal another whole number. A factor
can also be a number that divides into a multiple evenly. You do not include decimal
numbers, unless you want to be the one to carry all that chump change.
Bonnie: I understand now what a factor is, and I shouldn’t think of fractions as factors.
But wait!
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This excerpt was accompanied by the following commentary:
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Clyde is right because factors do not include decimal places. Factors can be numbers that
multiply evenly into another number. For example, 5 is a factor of 20 because 5 x 4 equals
20. Also, factors can be used to divide a larger multiple to get another factor of the larger
multiple. For example, 6 may be used to divide 30. When you do 30/6, you will get another
factor of 30, which is 5.
Though the language used in this commentary is imprecise, it shows that the script writer
wishes to employ the meaning of factor in number theory, but does not express the idea
rigorously.
Mathematical pedagogy was not the focus of the course, however, attention to
terminology reveals the participants’ conviction that familiarity with the concepts and
establishing a shared understanding of the meaning of particular terms is essential in
order to continue conversation. Such understanding is achieved when Bonnie acknowledges “I shouldn’t think of fractions as factors”, which leads her to focus on whole
number examples that confirm her claim.
Reconsidering terminology can be added to the list of responses to disconfirming
evidence identified by Chinn and Brewer (1993). As shown, disagreement among
characters can be attributed to a lack of common interpretation of concepts.
Who is right?
While no participant agreed with Bonnie (that larger numbers have more factors), there
were various degrees of disagreement. Only a third of the scripts (8 out of 24)
demonstrated a view in accord with mathematical convention by clearly rejecting
Bonnie’s claim. This is exemplified in S1.23(3):
Clyde: The answer to “True or False”: As a number gets bigger the more factors it will have
is False. It may sometimes have more factors, but to say that it always does would be
incorrect.
However, the verdict of ‘false’ to a statement that is ‘sometimes true’, or true in a large
number of cases, is inconsistent with everyday reasoning. As such, even when a
mathematically correct conclusion was drawn, some participants attempted to amend the
theory referring to a limited scope of applicability. Theory amendment is in accord with a
mathematical/scientific norm in response to disconfirming evidence (Chinn & Brewer,
1993), however, the amendment itself was usually incorrect. For example, the following
was included in the commentary in S1.12(2): “Large numbers do not always have more
factors. […] Her statement could be true for even numbers that are increasing but it is not
true for all numbers as a collective”.
A common tendency was to consider that Bonnie was “not totally wrong”, “not
completely right”, or “only partly correct”. This is consistent with intuitive application of
fuzzy logic to a mathematical situation. Further, prime numbers were the most notable
exceptions from the perceived ‘rule’, which is in accord with Chinn and Brewer’s (1993)
category of “excluding data from the currently held theory”, as we exemplify below.
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Prime numbers as exceptions
Prime numbers immediately falsify Bonnie’s initial claim. All script writers attended to
prime numbers, but this attention had different forms. Initial examples of small primes –
such as 5 has fewer factors than 4, or that 7 has fewer factors than 6 – were initially
treated by Bonnie in many of the scripts as an anomaly. Consider the following reaction
to disconfirming evidence from S1.16(1):
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Clyde: Exactly! Now haven’t we just shown that larger numbers don’t always have more
factors?
Bonnie: Damn you and your tricks! No, I refuse to give in, maybe you have just selected the
only two numbers that this general rule does not apply to. Maybe you chose an anomaly, the
only exception to the rule; it’s going to take more than just one counter example to
persuade me!
Providing evidence that supports the claim was the usual reaction to the initial
disconfirming evidence. However, as S1.16(1) wrote in his commentary: “Bonnie is
selecting only composite numbers, and that is her mistake, she seems to be forgetting that
there are more than just composite numbers.” This comment implicitly suggests that the
S1.16(1) may believe the statement to be correct for all composite numbers, that is, leaving
out the primes. Other script writers attributed this perception to their characters explicitly.
After considering several examples, the following conversation concludes the script
S1.7(2):
Bonnie: The larger the number, the more factors it has.
Clyde: True, unless it’s a PRIME NUMBER.
Bonnie: Why didn’t you tell me this rule before, it could have helped save so much time!
Clyde: I wasn’t sure myself either; I just didn’t want you to think you were right so I
denied it.
Moreover, in S1.20(3) below a similar idea is explicitly reiterated, after revisiting the
algorithm for determining the number of factors of a composite number. We note that
although the students were exposed to the algorithm for calculating the number of factors
of any given number, and were mostly successful in applying this knowledge on a test,
only three out of 24 scripts alluded to it:
Bonnie: This proves that I am right! That the larger the number, the more factors it will have.
Clyde: No! Actually this proves that these methods will work for composite numbers (large
or small) and not prime numbers.
Students’ tendency to reject evidence that is not in accord with their held beliefs was
noted in several studies (e.g., Chinn & Brewer, 1993; Edwards, 1997). Given that script
writers are prospective teachers, this tendency of their characters exemplifies their
awareness of such behaviour among students. However, in the assigned task the
participants were asked to acknowledge the erroneous claims of their characters in the
accompanying commentary. When the characters’ erroneous decisions were not noted, we
infer that they are likely to be in accord with the script writers’ personal views. Excluding
prime numbers from the generally accepted ‘rule’ was the most frequent conclusion
attributed to Bonnie’s character, and evidently accepted by writers of 12 out of 24 scripts
(See Table 1).
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Powers of primes as exceptions
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While prime numbers were the most frequently acknowledged exceptions, they were not
seen as the only exceptions to the rule. As indicated in Table 1, in four scripts the final
conclusion was to exclude powers of primes from the general assertion. The following
exchange takes place in S1.8(2) after several examples of prime numbers have been
considered:
Bonnie: Prime numbers are the exception to the rule. They do not behave like other numbers.
[…] The numbers that I am talking about when I say that the factors increase as the value of
the numbers increase, are any number other than a prime.
Clyde: Okay, so what about the squares of prime numbers. For example, the square of 7 is
49, so its only factors are 1, 7 and 49. That means that a smaller number, like 12, actually has
more factors than the larger number which is 49. I am confused.
Bonnie: Again, Clyde, we are looking at prime numbers in this situation. Any square of a
prime will only have three factors just like you said. The same thing happens when you find
the number of factors in the cube of a prime. But remember what I said before: prime
numbers are the exception to the rule.
Clyde: So what you are saying is that the number of factors increases with the value of a
number, unless the number you are looking at can be factored into the base of a prime
number. For example, 81 = 34 so it does not follow the trend that you are describing.
Bonnie: Yes! Clyde, I am really glad that you challenged me when I first suggested that the
larger a number is, the more factors it will have. I have also realised that this is not always
true. However, when leaving out numbers that can be factored into the base of a prime
number, like you said, the rule does hold true.
Here Bonnie acknowledges prime numbers as exceptions, but later she is invited to look
at squares of primes. As a result, the ‘exceptions’ to the rule are extended to include
powers of primes. Though the expression “numbers that can be factored into the base of a
prime number” used by both characters is colloquial, it is clear from the examples that
this phrase refers to numbers whose prime factorisation is a power of a single prime. This
is yet another example of theory amendment, a response that is consistent with a scientific
and mathematical norm (Chinn & Brewer, 1993), while the amendment itself is incorrect.
As the script writer does not comment on Bonnie’s conclusion – that “the rule does hold
true” once some numbers are excluded – we conclude that the participant shared this
belief. Other possible clusters of ‘exceptions’, such as the product of two large primes,
were not discussed in any of the scripts.
On the power of large numbers
In all the scripts one counterexample was insufficient in convincing Bonnie to abandon
her claim. This shows the awareness of script writers to possible robust beliefs held by
their potential students, beliefs that they themselves may also have possessed. The
following statement in a commentary from S1.22(1) summarises this phenomenon:
Bonnie insisted she was right until Clyde did more examples to prove her wrong. In order to
thoroughly prove that this theory is a reliable one (without just taking someone’s word for it),
one must test the theory multiple times with various numbers. After picking a few strategic
numbers, only a few examples are required before the trend can be seen that the size of the
number does not influence the number of factors.
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We mentioned the tendency to treat counterexamples as exceptions above, as in the case
of ‘large primes’. However, counterexamples that included ‘large’ composite numbers
that were close to each other had more convincing power than others. For example,
comparing the number of factors of 512 (having 10 factors) and 513 (having 8 factors),
or, in a different script, comparing the factors of 3800 and 3600 helped Bonnie reconsider
her position. The script writers demonstrated not only that several examples are essential,
but also that examples with large numbers are – using a notion introduced by Mason
(2006) – more ‘exemplary’, that is, are more likely to serve the intended purpose.
Attention to large numbers is an additional extension to the list of responses to
“anomalous data” identified by Chinn and Brewer (1993): rechecking the evidence with
additional and more convincing examples. Perhaps this extension is applicable to
mathematics settings more than science settings.
Task 2: on representing rational numbers
Theoretical considerations
The distinction between transparent and opaque representations originated with Lesh,
Behr, and Post (1987) and was further expanded by Zazkis and Gadowsky (2001). Lesh
et al. (1987) describe transparent representations as having no more and no less meaning
than the represented idea(s) or structure(s) and opaque representations as emphasising
some aspects of the ideas or structures and de-emphasising others. Zazkis and Gadowsky
(2001) shifted to discussing transparency and opacity with respect to particular number
properties. All representations can be considered either opaque or transparent with respect
to some number property. For example, representing the number 841 as 292 is transparent
with respect to the property of being a square number (perfect square). However, the
property that 841 leaves a remainder of 1 when divided by 7 is opaque in 292, but
transparent when this number is written as 7 × 120 + 1. In general, a representation is
considered transparent with respect to a certain property, if the property can be ‘seen’ by
considering the given representation.
The representation of the fraction 23/43 is transparent with respect to the definition of
a rational number, since a number that can be represented as a/b, where a and b are
integers and b ≠ 0, is rational. However, this representation is opaque with respect to the
property that rational numbers have either terminating or repeating decimal
representations.
The notion of transparency was used in framing our data analysis for Task 2. In
particular, we expand our main research question (What is revealed by the script writing
method about the participants’ understanding of mathematics?) by addressing the
following: In what ways do the script writers attend to the given transparent
representation? What evidence do they find convincing?
Results and analysis (Task 2)
All the script writers concluded that 23/43 was a rational number, however the reasons for
this conclusion varied. While this conclusion is immediate based on the transparent
representation of the number, this fact was explicitly featured only in three out of 13
scripts; in others it was either unnoticed (6) or deemed insufficient (4) (see Table 2).
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Table 2. Frequency of themes in Task 2.
Theme
Primary focus on ratio (transparency noted)
Ratio unnoticed
Ratio mentioned, but considered insufficient
Explicit calculation of a repeating cycle
“Stop at 43”
Number of scripts
3
6
4
6
3
13
Focusing on transparent representation as ratio
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In the following excerpt from S2.1(2), Jerry directs Tom’s attention to the fact that the
number in question is represented as a fraction:
Tom: The calculator tells me all I need to know! It’s a non-repeating and non-terminating
decimal for crying out loud!
Jerry: But what if the decimal repeats or terminates after the 2? If you had a bigger calculator
your answer would be different!
Tom: The calculator doesn’t lie! It’s irrational!
Jerry: You don’t even need a calculator to solve this problem! The answer lies in the question
itself.
Tom: What do you mean? You need the calculator to do everything!
Jerry: The problem gives the number to us in the form of a fraction and that’s part of what it
means to be a rational number.
Tom: Part of what it means…
Jerry: You do know what rational numbers are, don’t you Tom?
In the continuation of the dialogue definitions of rational and irrational numbers are
revisited. The transparency of representation here is the main focus; the script does not
contain any attempt to find a repeating decimal representation.
In four scripts the ‘ratio’ representation was acknowledged, but only after verifying
the digits of the decimal representation. For example, in S2.4(3) the following
conversation takes place after the repeating cycle was found via long division:
Tom: Oh yeah look on the 21st digit I have 5 again, followed by 3 and then 4.
Jerry: WOW!
Tom: So it does repeat!
Jerry: Ah ha! I knew I was right it must be rational.
Tom: Wait a minute, if it repeats why does that mean it’s rational? It’s still a really long
number that doesn’t terminate.
Jerry: Okay I see what you mean, because it is a really long number, you may think it would
be irrational. But remember the definition of a rational number is it can be either terminating
or repeating pattern! Let’s look at the definition again. […] I found on the internet was that a
rational number is any decimal that can be turned into a simple fraction.
Tom: Jerry, are you suggesting that 23/43 is a simple fraction?
Jerry: You bet I am Tom. Since we know that 23/43 is a simple fraction we already know that
it is a rational number!
Tom: So then all that work was a waste of time?
Jerry: Of course not, because you figured it out, and not just from the definition. Don’t you
love math?
Tom’s question of whether all that work (long division) was a waste of time is a
reasonable one. It could be the case that the script writer attempted to feature Jerry’s
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character as a nice tutor who was being careful not to devalue or belittle his partner
However, since no commentary to this effect is provided (and the task specifically
requested commentary to distinguish between the character’s claims and personal
dispositions) we believe that the script writer agreed with Jerry’s comment, which saw
value in figuring out the decimal expansion and not just relying on the definition.
Seeking transparency in decimal representation
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All the scripts mentioned that a rational number has either a terminating or repeating
decimal representation. Six scripts primarily focused on obtaining a repeating cycle in the
decimal expansion, as exemplified in the following excerpt from S2.8(3):
Tom: 23/43 is an irrational number! There’s no repeating pattern at all, you see?!
Jerry: Sometimes I think you’re too simpleminded, Tom. Just because that’s what appears on
your calculator doesn’t mean that the number is actually irrational!
Tom: Um, yeah it does? Irrational numbers are like pi, right? 3.141592654… see, no repeats
or an end in sight! They just go on and on forever, to infinity and beyond! And from what the
calculator shows, 23 divided by 43 doesn’t terminate or have any sort of pattern, just like pi!
Therefore, it must be irrational.
Jerry: Although, irrational numbers do have no definitive end or repetition pattern, you can’t
just say that 23/43 is irrational because you don’t see it repeat on the calculator! Because
guess what, smartie? Calculators’ displays are evil. They don’t actually show the entire
answer sometimes.
Tom: Yeah right, I highly doubt you’d be smarter than a calculator! I bet you can’t do the
same calculations a calculator can even if you were given tripled the time.
Jerry: Take 7/13 for example, sunshine! It’s actually 0.538461 repeating, but my calculator’s
showing 0.53846154! Which means the “4” is a number that was rounded up to make the
number look prettier. And you know, there could totally be a repeating pattern that appears
AFTER the stupid display cuts off!
Tom: Well, there is a possibility of that… [Tom grumbles] but I bet you can’t prove that!
As the script continues, the ‘proof’ is created by carrying out long division and pointing to a
repeating cycle. We note in this script two important issues. One is Tom’s total reliance on a
calculator’s display and the other is Jerry’s clever example of showing how a calculator
may conceal a repeating cycle in decimal representation by rounding the number. This
shows the script writer’s awareness of the potential calculator-dependence of her students,
as well as a nice method of demonstrating the limitation of a calculator’s display.
However, this script and five others (out of 13) do not pay any attention to the fact that
the number in question is presented as a ratio of two natural numbers. The fact that the
number is presented as the ratio 23:43 does not convince Tom that it is rational. Instead, the
explicit demonstration of a repeating decimal pattern does. To reiterate, the existence of a
repeating cycle of digits is accepted only after the specific digits in the cycle are calculated.
That is, the ratio representation is replaced with one that is transparent with respect to the
repeating cycle property. In S2.6(3) the participants included a cartoon with such a
calculation, shown in Figure 1. This was accompanied by the following commentary:
After deliberating within our group whether 23/43 is a rational or irrational number, we
decided that the only way to know for sure was to perform the long division and see if the
number terminated or if there was a repeating pattern. Since we know that the maximum
number of digits in the repeating pattern is (n – 1), which is the case of 23/43 is (43 – 1) =
42; there are a maximum of 42 digits in this repeating pattern. If, when we perform the long
division, the number does not terminate and there is no repeating pattern and we have
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reached 42 digits or more, we know that the number is irrational. If the number terminates or
reveals a repeating pattern within 42 digits, it is a rational number.
After carrying out the long division we discovered that 23/43 does in fact create a
repeating pattern, luckily for us the pattern begins to repeat after 21 digits, and not the
possible 42 digits. Thus, we have concluded that 23/43 is actually a rational number, and
Jerry is correct!
This commentary explicitly shows that the script writers, and not just the imagined
characters, see the necessity of determining the repeating cycle in order to conclude that
the number is rational. The commentary also echoes the formal knowledge that the
repeating cycle in this case has at most 42 digits. Nevertheless, the awareness of the
maximal number of digits in the repeating cycle does not seem to guarantee that such a
cycle exists, until explicitly calculated. Such a calculation assures transparency in
representing the repeating cycle:
534883720930232558139 534883720930232558139 53488372…
Surprisingly for us, the idea that one needs to calculate at least 43 digits – in order to
determine whether a repeating cycle exists – appeared in three different scripts. If no
repetition were observed within the first 43 digits the number would have been deemed
irrational.
Discussion
When scenario-based script writing is implemented in mathematics teacher education, the
goal is to explore further arguments from each side, to understand the origins of
erroneous claims and explore ways of helping students reconceive them.
On Bonnie and Clyde: the number of factors task
By creating a dialogue between Bonnie and Clyde, the participants revealed
their perceptions of the sources of the mistaken claim and their ideas of what the
characters may find convincing. While erroneous conclusions may have been
attributed to the characters, the included commentary pointed to participants’
personal understandings and biases. Although the given task is based on a falsifiable
mathematical statement, which can be disproved with a single counterexample, the
characters created by the participants treated the task as if it involved coordinating a
general tendency and the exceptions to that tendency. Very few included commentary
that clarified this point.
Primes are perhaps the most accessible counterexamples since they have a formal
name and are defined in terms of their factorisations. Actually, many participants created
characters who viewed primes as the only exceptions to the rule. Several script writers
expanded the set of exceptions to include powers of primes, that is, numbers of the form
pk. Only two participants were able to differentiate between what they called the
‘potential to have many factors’ and the actual general case.
The fact that participants used prime numbers as the first and most frequent
counterexample is in accord with prior research (Zazkis, 1999). However, the script
writing method revealed that primes are often mistakenly treated by the participants as
‘removable’ counterexamples. That is, the statement is treated as true if its scope of
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Research in Mathematics Education
Figure 1.
Seeking repeating pattern of digits in decimal representation of 23/43.
applicability is reduced. Reducing the scope of applicability, or, in other words, amending
a theory, is a normative response according to Chinn and Brewer (1993), however, as we
pointed out above, the amendment itself was erroneous.
The script writing method also suggested additional responses to disconfirming
evidence: (1) revisiting and renegotiating the use of the terms; and (2) rechecking the claims
with more convincing examples, in this case examples with relatively large numbers. These
extensions have pedagogical importance as claims that are conventionally considered incorrect
may have resulted from assigning unconventional meanings to the used terminology.
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Furthermore, while normatively one counterexample disproves a general claim, it is important
to recognise what kind of counterexample learners may find convincing.
On Tom and Jerry: the representing rational numbers task
Only three of the thirteen scripts focused specifically on the transparency of the ratio
representation. In these scripts, the conclusion that 23/43 is rational was based solely on
the fact that 23/43 is a ratio of two quantities. The other 12 scripts dealt with the decimal
expansion of 23/43 in some ways, and half of those did not address that 23/43 was
presented as a ratio at all.
The propensity for decimal representations over the fraction interpretation in the case
of rational numbers was noted in prior research (e.g., Zazkis & Sirotic, 2010). Our study
extends this finding by demonstrating participants’ need for transparency in decimal
representation, that is, the explicit computation of digits in the repeating cycle.
Interestingly, this need for transparency was so prevalent that decimal expansions of
23/43 were included even in scripts that explicitly mentioned the link between the ratio
and decimal interpretations.
Another interesting finding that emerged from the data is participants’ interpretation
of the maximal length of a cycle of repeating digits – a property that was discussed in
their course. The participants who mentioned that in division by 43 (n) the maximum
length of a cycle of repeating digits is 42 (n – 1) still treated the existence of the cycle as
something that needed to be tested. They interpreted this maximum as a bound on how
much they needed to test to find a cycle, and not as a guarantee that a cycle of length at
most 42 (n – 1) exists.
On script writing
As any method, script writing has its limitations. The scripts were produced as an
assignment for a mathematics course. It is unclear whether and how the scripts were
influenced by the participants’ experience during the course and their personal
interpretation of the course assignment, which could have been different from what was
intended by the researchers.
However, script writing has multiple advantages. For the script writers, it provides an
opportunity to imagine different characters as well as the interactions between these
characters. It provides opportunities to rethink and refine the interactions involved in the
scenario. As such, the scripts, together with the commentary, serve as a window for
researchers to investigate participants’ understanding of mathematics. In particular, when
script writers are prospective teachers, and the script situation involves a common
mistake, the exercise allows investigation of potential sources of error as well as
consideration of their remedy.
As demonstrated in this study, script writing provided insights into the participants’
understanding of mathematics, confirming and extending the findings of previous studies.
Our particular contribution can be seen on different arenas: Methodologically, we
extended a novel approach, script writing, by considering prospective teachers understandings of number structure. Implementing this approach provided some refinement to
the results of previous studies. In particular, considering Task 1, we described the special
role attributed to primes (removable counterexamples). Furthermore, we extended the list
of normative responses to disconfirming evidence, to include negotiation of terminology
Research in Mathematics Education
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and seeking ‘exemplary’ examples. Considering Task 2, we identified an unconventional
interpretation attributed to the length of a repeating cycle in considering whether a given
number is rational. The nuances in students’ understandings that were revealed by the
implementation of script writing in this study were not captured in prior research.
Although other methods may have revealed these subtleties, we suggest that script
writing provides a valuable approach for exploring students’ understanding of mathematical concepts and methods.
Note
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1. SA.B(C) is script #B written by C participants for Task A; in particular S1.4(3) is script #4
written by 3 participants responding to Task 1.
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