Journal of Mathematical Behavior 32 (2013) 364–376
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The Journal of Mathematical Behavior
journal homepage: www.elsevier.com/locate/jmathb
Decoding a proof of Fermat’s Little Theorem via script writing
Boris Koichu a,∗ , Rina Zazkis b
a
b
Technion – Israel Institute of Technology, Israel
Simon Fraser University, Canada
a r t i c l e
i n f o
Keywords:
Proof as dialog
Students’ difficulties with proving
Indirect proof
Numerical examples
Number theory
Equivalence by modulus
Fermat’s Little Theorem
a b s t r a c t
Twelve participants were asked to decode – that is, interpret and make sense of – a given
proof of Fermat’s Little Theorem, and present it in a form of a script for a dialog between
two characters of their choice. Our analysis of these scripts focuses on issues that the participants identified as ‘problematic’ in the proof and on how these issues were addressed.
Affordances and limitations of this dialogic method of presenting proofs are exposed, by
means of analyzing how the students’ correct, partial or incorrect understanding of the elements of the proof are reflected in the dialogs. The difficulties identified by the participants
are discussed in relation to past research on undergraduate students’ difficulties in proving
and in understanding number theory concepts.
© 2013 Elsevier Inc. All rights reserved.
1. Introduction
The extensive professional literature on mathematical proof and proving tells us that virtually any aspect of understanding
and producing mathematical proofs is a stumbling block for learners. The particular difficulties that have been identified for
undergraduate students include: unawareness of the social norms developed within the mathematical community regarding
proof and underdeveloped conceptions of proof (e.g., Harel & Sowder, 1998; Knuth & Elliott, 1997); the lack of knowledge
of mathematical theorems, definitions and concepts needed for proof understanding and production (e.g., Edwards & Ward,
2004; Knapp, 2005); the lack of strategic knowledge needed in order to produce a proof (e.g., Weber, 2001); poor ability to
unpack the structural logic of a given proof (e.g., Selden & Selden, 1995); misuse of examples (e.g., Ruthven & Coe, 1994);
inadequate use of mathematical language (Edwards & Ward, 2004; Epp, 2003; Moore, 1994), among others (cf. Harel &
Sowder, 2007; Knapp, 2005; Weber, 2003, for comprehensive reviews).
As a rule, students’ difficulties with constructing and understanding proofs are exposed by means of documenting and
interpreting their (often poor) performance when coping with various proving tasks. This research approach implies that
students’ understanding of proofs and their difficulties are mainly examined from an expert point of view. A complementary
approach – inquiring what students themselves see as issues of difficulty – is still underrepresented in research on proof
and proving. As such, the goals of our study were to inquire what students themselves perceive as problematic issues in a
given non-trivial proof in number theory, to compare these with the expert view, and to describe how students cope with
the identified difficulties.
∗ Corresponding author. Tel.: +972 48293895; fax: +972 48293895.
E-mail address: bkoichu@technion.ac.il (B. Koichu).
0732-3123/$ – see front matter © 2013 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.jmathb.2013.04.001
B. Koichu, R. Zazkis / Journal of Mathematical Behavior 32 (2013) 364–376
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2. Theoretical underpinnings
Our study is influenced by the idea of writing a fictional script of interaction as a part of a learning process. The roots
of this method are inspired by the style of Lakatos (1976) evocative Proofs and Refutations and can be traced to a Socratic
dialog, a genre of prose in which a ‘wise man’ leads a discussion, often pointing to flaws in thinking of his interlocutor.
Recently, the use of narrative and dialog representations of teaching has been featured in mathematics education research
(e.g., Herbst & Chazan, 2011). Our focus, however, is on dialogs and scripts created by research participants, who are teachers
and learners of mathematics. In particular, we refer to the dialogical approach for presenting proofs (Gholamazad, 2006,
2007), and to composing imaginary instructional interactions (Chen, 2012; Crespo, Oslund, & Parks, 2011; Herbst, Chazan,
Chen, Chieu, & Weiss, 2011; Zazkis, Liljedahl, & Sinclair, 2009; Zazkis, Sinclair, & Liljedahl, 2009; Zazkis, Sinclair, & Liljedahl,
2013). To reiterate, we focus on imaginary interactions created by participants, rater than those created by researchers and
reflected upon by participants.
Presenting proofs in form of a dialog is a method used by Gholamazad (2006, 2007) in her work with prospective elementary school teachers. This type of proof presentation consists of a script of a dialog between characters who ask and
answer questions about different steps in a proof. The method was implemented both in asking participants to create a proof
in a form of a dialog and to write a script for a dialog based on a given proof. Gholamazad suggested that this method of
presentation provided insights into the students’ cognitive obstacles when creating and interpreting proofs.
Gholamazad developed the idea of engaging students in writing and interpreting proofs in the form of a dialog based on
Sfard’s (2001, 2008) communicational framework, which conceptualizes thinking as a form of communication, specifically,
as “individualized version of interpersonal communication” (p. 81, italics in the original). That is, thinking resembles a conversation between two people in a form of asking and answering questions. A request for a student to present a proof in
a form of a dialog makes such personal thinking salient. As such, in assigning the Task for our participants (see Fig. 2) we
expected to learn about their explanations of concepts and justifications of claims presented in the given proof that may not
be apparent in a ‘standard’ form of presenting a proof.
Further, the idea of learning-via-scripting was implemented in mathematics teacher education in different contexts.
Zazkis and her colleagues (Zazkis, Liljedahl, et al., 2009; Zazkis, Sinclair, et al., 2009; Zazkis et al., 2013) introduced a ‘lesson
play’ as a novel construct in research and teachers’ professional development in mathematics education. Using the theatrical
meaning of the word ‘play’, lesson play refers to a lesson or part of a lesson written by a teacher or a prospective teacher
in a script form, featuring imagined interactions between a teacher and her students. This approach was developed to
complement the traditional ‘lesson plan’.
In a similar approach, Crespo et al. (2011) engaged prospective teachers in creating a hypothetical classroom dialog based
on a given prompt. Taking advantage of technology-enhanced environment, Chen (2012) analyzed imagined classroom
interactions created by prospective teachers using comics-based representations with lesson sketching software.
Zazkis and her colleagues (Zazkis, Liljedahl, et al., 2009; Zazkis et al., 2013) argued that asking prospective teachers to
think about their future teaching in terms of fictional interactions draws their attention to how the mathematics of their
students can be developed. They described the affordances of this approach both in teacher education and in research. In
teacher education it provided a valuable tool for engaging prospective teachers in considering particular students’ mistakes
or difficulties, presented in prompts that serve as a starting point for the play. In research it provided a window on how
prospective teachers envision addressing students’ difficulties, both mathematically and pedagogically. In particular, the
prospective teachers’ personal understanding of the mathematics involved became apparent in their attempts to guide
students’ solutions. As such, we wondered what mathematical understandings would surface when students decode – that
is, interpret, and make sense of – proofs through script-writing.
3. Our study
In light of the above considerations, our study addresses two interrelated research questions:
• What problematic issues, that is, points of potential difficulty in understanding the proof, do students identify in the given
proof and how do they deal with these issues when decoding the proof into a script? In particular, which issues are treated
as central?
• What is revealed from the dialog method of interpreting a proof about participants’ understanding of particular concepts in
number theory that appear in the given proof? In particular, how are students’ correct, partial or incorrect understandings
of the number theory concepts reflected in their scripts?
3.1. Participants
Twelve students participated in our study. Two of them were graduate students in mathematics education holding
undergraduate degrees in mathematics; the other 10 were working toward completion of a teaching certificate for secondary mathematics. An extensive mathematical background – an undergraduate degree in mathematics or in mathematics
education – is required for teaching certification at the location of the study. Therefore, all the participants had broad and rigorous exposure to undergraduate mathematics, having completed at least eight upper-division courses, including a course in
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Fig. 1. Fermat’s Little Theorem and its proof.
Adapted from Wikipedia.
Number Theory. In particular, in the courses of Discrete Mathematics and Number Theory, the participants studied rigorously
the topics of congruence and modular arithmetic. Further, in most of their undergraduate mathematics courses participants
were exposed to ‘traditional’ mathematics teaching, and as such had considerable experience in studying formal proofs of
theorems and writing their own deductive proofs. Consequently, it is reasonable to conclude that they have accepted, at
least at its face value, the institutionalized in the mathematical community view of deductive proofs.
The study was conducted in the context of an elective mathematics education course entitled “Proofs and proving.” The
course was taught by the second author. The objectives of the course were two-fold: to introduce or re-introduce participants
to several “classical” theorems and discuss their various proofs (such as irrationality of the square root of 2, infinity of primes
and Euler’s formula), and to expose them to mathematics education research in the area of proofs. Specifically, an essential
part of the course was devoted to reading and discussing contemporary research on proof and proving. In addition, the
students were systematically engaged in proving and reflecting on their own proving experiences. The following Task was
part of a final assignment in the course and was completed by students individually.
3.2. The Task
The students were given the theorem and its proof, as presented in Fig. 1.
Based on the substantial mathematical background of the participants, we expected that they were familiar with all the
concepts and symbols presented in the proof. The fact that the theorem is Fermat’s Little Theorem was not announced, but
we assumed that at least some of the students would recognize the theorem from their prior studies. The Task is presented
in Fig. 2.
In short, the participants were asked to consider the given proof, to identify points of potential difficulty – referred to
as ‘problematic points’ – in understanding this proof, and to write a script for a dialog between two characters of their
choice, in which these difficulties are addressed (see Fig. 2 for the detail of the Task). The Task provided participants with
an opportunity to define both the interlocutors and the issues that they discuss. This flexibility of the task allowed for the
analysis of participants’ choices and their ways of addressing mathematical concepts and claims involved in the proof.
Create a dialogue that introduces and explains the attached theorem [see Figure 1] and its
proof. Highlight the problematic points in the proof with questions and answers. In your
submission:
-
Describe the characters in your dialogue.
-
Write a paragraph on what you believe is a “problematic point” (or several points) in
the understanding of the theorem/statement or its proof for a learner.
-
Write a dialogue that shows how you address this hypothetical problem (THIS IS
THE MAIN PART OF THE TASK)
-
You may add a commentary to several lines in the dialogue that you created,
explaining your choices of questions and answers, in connection to the characters,
which may not be obvious for the reader.
Fig. 2. The Task.
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Table 1
Overview of problematic issues identified in the dialogs.
Tova
Omer
Ron
Iris
Hugo
Ivan
Linda
Helen
Anna
Gaps in the flow of the proof
A potential learner may not understand parts of the proof because they are presented too briefly and should be unfolded
Rearrangement of the list of reminders
PI, m
PI
PI
pi
PI
pi, m
PI
pi, m
Multiplication in equivalence relation
pi
pi
pi
pi
pi, m
pi
Division in equivalence relation
pi, m
pi
pi, m
pi
pi, m
pi
The presumed lack of preliminary knowledge
A potential learner may not understand the theorem or its proof because s/he does not possess necessary knowledge
pi, m
PI
pi
PI
pi
pi, m
pi, m
pi, m
The concept of equivalence relation or its
symbol a ≡ b (modp)
pi
pi
pi
PI
pi
Basic concepts (natural numbers, prime
pi
numbers, co-prime numbers, factor,
factorial)
Amira
pi
pi
Viktor
Eliot
PI
pi
pi
pi, m
pi, m
PI
pi
pi
pi
“pi” (stands for “problematic issue”): an issue is explicitly identified as problematic in the student’s comments or dealt with in the dialog; “PI” – the central
problematic issue in the dialog; “m” (stands for “mistake”): a corresponding problematic issue is treated incompletely or incorrectly in the dialog.
4. Results and analysis
We present below our analysis of the proof that guided the initial stage of our data analysis. We then explain the data
driven framework that guided our subsequent data analysis.
4.1. Analysis of the proof
Among a variety of proofs of Fermat’s Little Theorem we chose this particular proof because it had certain gaps which
required decoding. Specifically, we, as well as the anonymous Wikipedia writers of the proof, deemed two points particularly
problematic, as they required further justification1 :
(1) Rearrangement of the remainders: Why does reducing the numbers 0, a, 2a, 3a, . . ., a(p − 1) by p results in the
rearrangement of the list of remainders 0, 1, 2, 3, . . ., p − 1?
(2) Division in a congruence statement (or cancellation property): Why does the equivalence remain when both sides of a
congruence by modulo statement are divided by (p − 1)!?
The first point can be decoded, for instance, by assuming that there are two numbers on the list 0, a, 2a, 3a, . . ., a(p − 1),
which give the same remainder when divided by p, and concluding that this assumption is wrong as it contradicts the
conditions that p is a prime number and that a and p are co-primes. Decoding the second point requires recalling the fact
that dividing both sides of a congruence by a number does not always preserve the equivalence (e.g., 12 ≡ 16 (mod 2) is true,
but the division of both sides by 4 – 3 ≡ 4 (mod 2) – results in a false statement). In the given proof, the division is possible
because it is given that p is a prime number, and thus (p − 1) ! and p are co-primes.
4.2. Framework for data analysis
Based on our analysis of the proof and the two ‘problematic points’ defined a priori, we were initially interested to see
how these issues (rearrangement of reminders and division in a congruence statement) were addressed in the scripts and
whether the participants identified them as ‘problematic’. However, the data provided a wider variety of problematic issues
to be considered.
We examined, first independently and then together, each student’s work for problematic issues, which were explicitly
identified as such in the students’ comments or dealt with in the script. We considered a problematic issue to be “dealt with
in the script” if there was an excerpt, in which the dialogue’s characters explicitly addressed the issue with questions and
answers. Then we distinguished between two main kinds of problematic issues identified by the students, for which the data
seemed to have provided rich and solid evidence: (1) gaps in the flow of the proof, and (2) the presumed lack of preliminary
knowledge (see Table 1).
Gaps in the flow of the proof included the two issues identified in our a priori analysis, rearrangement of reminders and
division in a congruence statement, as well as an additional issue that have emerged from the data, multiplication in a
congruence statement. By presumed lack of preliminary knowledge – this issue also emerged from the data – we considered
the participants’ attention to descriptions and exemplifications of number-theoretic concepts appearing in the proof, such
as prime or co-prime numbers, or the basic idea of congruence relationship.
1
See the second proof at http://en.wikipedia.org/wiki/Proofs of Fermat%27s little theorem.
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Further, a problematic issue was considered to be a ‘central problematic issue’ if it was explicitly identified by a participant
as such, or if it took significantly more space than discussions of the other issues (e.g., a particular issue, which was treated in
about 30 lines of the script whereas the other issues – in 5–10 lines each, was identified as central). Such issues are denoted
in bold letters PI in Table 1; up to two central issues were identified in each dialog.
At the next stage of data analysis, we isolated the excerpts, in which the students’ incomplete or incorrect knowledge was
manifested. An excerpt was taken as such a manifestation if a mathematically incorrect or incomplete statement was treated
by both the dialogue’s characters as a true one. In Table 1, such excerpts are juxtaposed with the corresponding problematic
issues and denoted as “m” (for “mistake”). Table 1 provides an overview of the findings and serves as an organizational
framework for presenting examples discussed in detail in the rest of this section.
5. Gaps in the flow of the proof
5.1. Focus on “rearrangement of the remainders”
Ten out of 12 students treated “rearrangement of the remainders” issue as problematic, and for six of the 12 this issue
was treated as a central one (refer to PI in Table 1).
These six students elaborated on it by means of constructing a proof by contradiction. All invested effort in elaborating
on a statement, which falsification would be sufficient in order to complete the proof, and only then presented the technical
part of the proof. An example below is taken from the script by Hugo,2 who composed a dialog between a teacher-character
and a ‘smart’ undergraduate student-character. Note that the last question in the excerpt below is a prelude to the proof by
contradiction, and it is attributed by Hugo to Student.
Teacher3 :
Teacher:
Student:
Teacher:
Student:
Teacher:
Student:
Teacher:
Student:
Teacher:
Student:
Teacher:
Student:
So let’s look at the numbers 0, 1, 2, . . ., p − 1. Now let’s multiply them by a. What do we get?
It is clear. We will get 0, a, 2a, . . ., (p − 1)a.
Excellent. Now let’s divide all these numbers by p, and look at the remainders in these divisions.
But you cannot know what we will get! a and p are just general numbers, I cannot know the remainder of say, 5a when
divided by p.
Right, it is impossible to know what a particular remainder will be. But it is possible to know what all the remainders will be.
I don’t understand. What do you mean?
I mean that when we divide all the numbers by p and look at the remainders, we get exactly the same remainders as we had
before, namely, 0, 1, 2, . . ., p − 1.
Do you mean that the remainder of 0 is 0, the remainder of a when divided by p is 1, the remainder of 2a when divided by p is
2 and so on?
No. It can happen that the remainder of a when divided by p is p − 1, or 7, or any other number. But when we look at all the
remainders, we find out that we have exactly all the remainders between 1 and p − 1, even if they are in the order different
from the original one.
I think I understand. You mean that there is exactly one [the emphasis by Hugo] number among the numbers between 0, a,
2a, . . ., (p − 1)a that when divided by p, gives 1, and then another number will give 2 when divided by 2, and that, in fact, every
number gives a new remainder when divided by p?
Exactly! This is what I mean.
Just a second. But one of the number is 0, and 0 divided by any number gives the remainder 0!
Right, and indeed 0 is the only number in our group of numbers for which we know its remainder exactly. But the rest of the
numbers give all additional remainders that can be.
It was a lot of remains. . . remainders. . .But I understood. That is, I understood what you claimed, but I really don’t understand
why it is true! Why it cannot be that there are two numbers that give the same remainders when divided by p?
The above excerpt shows how the teacher-character is leading the student-character in helping him understand the
proof. It further shows that Hugo was sensitive to a subtle, but important point: how can one say something about the group
of the remainders without knowing exactly what each remainder is?4
Leron (1985) suggested that the core cognitive difficulty of indirect proofs for learners is the lack of an object that can be
directly constructed and mentally manipulated, and that other non-trivial proofs “pivot around an act of construction – a
construction of a new mathematical object (a number, a function, a point, a line, a set, a partition of a set, etc.)” (p. 323). The
appeals to special cases in the above excerpt seem to reflect Hugo’s own attempts to decode the proof by interspersing the
dialog with mathematical objects to manipulate. This phenomenon of decoding the “rearrangement of the remainders” part
of the proof by introducing and discussing mathematical objects that can be directly constructed and manipulated within
the script can also be seen in the continuation of Hugo’s script. It was also evident in five additional student scripts; the
objects constructed and manipulated by the characters in a dialog included numerical examples and algebraic expressions
that correspond to different ways of constructing the remainders.
2
All the names are pseudonyms.
All the excerpts were translated by the authors (who are fluent both in English and in the language of instruction) preserving the authentic nature of
interaction.
4
Two of the participants, Ivan and Anna, attributed to their characters a wrong claim—that numbers 0, a, 2a, . . ., (p − 1)a give, when divided by p, the
remainders 0, 1, 2, . . ., (p − 1), correspondingly (see ‘m’ on Table 1).
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For instance, the characters in Ron’s script – two undergraduate students named Mike and Matt – consider the case of
a = 4 and p = 7 when trying to understand why the claim about rearrangement of the remainders is true. At some stage, Mike
and Matt consider the remainder in division by 7 – 0, 1, 2, 3, 4, 5, and 6 – multiply them by 4 – 0, 4, 8, 12, 16, 20, and 24 – and
compute the remainders in division by 7 of this list, getting – 0, 4, 1, 5, 2, 6, and 3, respectively, which is a rearrangement of
the initial list. They accept this result as an empirical verification of the claim, but are eager to understand the mechanism
behind it. Apparently, it is Ron’s advanced mathematical background that leads his characters to seek formal justification
and not to stop at the empirical evidence.
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
As a matter of fact, we need to ask a question why for prime numbers we get all the numbers from 0 to 6, that is, from 0 to p − 1,
though in a different order. Because the proof relies on this.
Good question. I remember that the lecturer advised to think about congruencies in terms of a clock. For instance, if we talk about
modulo 7, so it is a clock with 7 digits, from 0 to 6.
Yes, I remember this too.
And, in essence, what you did was to jump by fours.
It means that if we start from [the remainder] 0, and jump by fours, we should come back to 0 after exactly 7 jumps, and we get
the rest of the numbers [from 0 to 6] on the way.
Why?
Let’s assume that it comes back to 0 after less than 7 jumps.
Oh! A proof by contradiction.
Yes. So what would it say?
Let’s assume that it comes back after exactly 2 jumps. So it would say that 7 is divisible by 4, but it is impossible.
In the above excerpt, as well as in the next 30 lines of the dialog, Ron’s characters gradually develop the proof by
contradiction, by mentally manipulating the clock, making computations, and then by introducing algebraic notation, which
turns a generic example a = 4, p = 7 into a general proof. In terms of Leron (1985), Ron attempted to create a mental reality,
which made a proof by contradiction similar to a proof by construction.
An interesting, though not fully developed, attempt to bypass a proof by contradiction by considering how the remainders
are created appears in Ivan’s script.
Teacher:
Student:
Teacher:
Student:
Teacher:
Student:
Teacher:
Now let’s take the list of all possible remainders and multiply by a. We will get 0, a, 2a, . . ., (p − 1)a. If we divide each number by p,
what will we get?
I don’t know.
We will get a sort of displacement of the remainders.
Why?
Because when we create [italicized by the authors of the paper] a new positive integer, its greatest common factor with a prime
number p would be 1, as we take a, which is co-prime to p, and multiply it by a number between 1 and (p − 1), which are also
co-primes [to a]. Therefore, the remainder still will be a number between 1, 2, 3, 4, . . ., and (p − 1).
OK, I understand why we still get a remainder between 1 and p − 1. But why is it a displacement?
Let’s say that division of a by p gives the remainder k. So division of 2a by p gives the remainder 2k, on condition that 2k < p, and if
2k > p then, in fact, the remainder will be 2k − p. The remainder is moved and does not remain at the same place.
After this explanation, Teacher just turns to the next step of the proof, even without asking Student whether or not he
understands. It is evident, however, that Ivan tried to construct the remainders and show that they ‘move’, rather than to
prove that something about them is impossible. However, the condition 2k > p, on which the explanation of displacement is
based, suggests that the remainder (2k) is larger than the divisor (p), which is inconsistent with the definition of reminder
(2k). Note also that the main mathematical ideas are attributed by Ivan to a teacher-character in the above dialog, and the
role of a student-character is mainly to ask questions and indicate whether he understands the answers. Noticeably, the
tone of the dialog is more didactic than in the dialog by Ron.
Iris also wrote a dialog of a didactical tone, in which the main insights came from a teacher-character. The studentcharacter, however, is more active than the student from Ivan’s dialog. In particular, he asks a question related to the logic
of proof by contradiction, which highlights the difficulty with this proof method in a different fashion.
Teacher:
Student:
Teacher:
Student:
Teacher:
Student:
Teacher:
Student:
Now let’s check that there are no two identical remainders.
How would it help us?
If there are p remainders, and there are no two equal ones, it would say that all the remainders are different.
That is, this is what we wanted, since if they are different, we will have all of them.
Exactly. Let’s prove it by showing that the situation, in which there are two equal remainders, is impossible.
And would this be enough? If there are no two, possibly, there are three, or four?
It is not possible. If there are three, there are also two. We just choose two among three.
I see. If we prove that there are no two equal [remainders], we also prove that there no more than two.
Iris was the only participant who incorporated an excerpt devoted to the structure of a proof by contradiction into her
script. To our surprise, which was informed by the literature about students’ difficulties with the logic of indirect proofs (e.g.,
Brown, 2012; Leron, 1985; Tall, 1979; Tall et al., 2012), the rest of the participants did not consider, at least not explicitly,
the logical structure of proof by contradiction as a possible source of difficulty.
5.2. Focus on the properties of multiplication and division in equivalence relations
The ‘rearrangement of the remainders’ step of the proof is needed in order to justify the equivalence
a × 2a × 3a × . . . (p − 1)a ≡ 1 ×2 × 3 × . . . (p − 1) (mod p). Seven out of 12 students elaborated on why this congruence holds
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in their scripts, and 6 of them presented mathematically accurate explanations (see Table 1). All of them had their characters
explain why it is possible to multiply congruencies with the same modulus. Jane and John, two student-characters in Omer’s
dialog, treated the issue as follows:
Now they multiply all the numbers from the two lists [i.e., 1, 2, 3, . . ., p − 1 and a, 2a, 3a, . . ., (p − 1)a], and claim that the
products are congruent by modulo p.
That is, if one multiplies congruencies by modulo p, it is possible to multiply the remainders of the numbers when divided by
p instead of multiplying the numbers themselves, and the products are equal modulo p.
Just a second, let me think about it. Assume that a · b is d modulo p, or it is p · k + d, and a is e modulo p, that is, p · l + e, and b is f
modulo p, that is p · m + f. So, on one hand, a · b is d modulo p, and on the other hand it is (p · l + e) · (p · m + f), which is
p2 lm + plf + epm + ef or p(plm + lf + em) + ef, that is, e · f modulo p. In brief, it is d ≡ ef (mod p), that is, the remainder of a product is
congruent to the product of remainders modulo p.
We’ve shown this for the multiplication of two, but it is OK. We can take the product of two, and multiply it by the third, so it
will again be a product of two, so it will work, and then to multiply by forth so on, until we get the entire product.
There is a more elegant way of doing so. It is called induction. . .
John:
Jane:
John:
Jane:
John:
It is apparent from the above dialog that Omer possesses a sophisticated understanding of how modular multiplication is
different from regular multiplication. Consequently, we were surprised to read in the continuation of his script the following
conversation:
John:
Jane:
John:
Jane:
John:
Jane:
John:
Now they only change the order of the multiplications in the left side of the congruence [he refers to
a × 2a × 3a × . . . (p − 1)a ≡ 1 ×2 × 3 × . . . (p − 1) (mod p)].
Is this permitted? Does it work with modulo?
Yes, clearly, commutativity and associativity properties of multiplication do not disappear by modulo.
Yes, Tamar [the imaginary teacher of the characters] mentioned it a year ago.
OK, so ap−1 (p − 1) ! ≡ (p − 1) ! (mod p), and now we divide two sides by (p − 1) !, and we get ap−1 ≡ 1 (mod p), and then we multiply
both sides by a, and get ap ≡ a (mod p), which is what we wanted to prove. Walaa! Next task!
Just a second, not that fast! Who said that also in the arithmetic of congruencies it is permitted to divide or multiply both sides by
the same number? Have we proven this?
Well, it is the same principle. But you’re right, it is not trivial. Let’s write “It is known that also in the arithmetic of congruencies it
is permitted to divide and multiply two sides by the same number.”
The same mistake – the claim that both sides of a modular equivalence can be divided by the same number with no
additional specifications – appeared in 4 additional scripts.5 Apparently, the inevitable analogy between operations with
regular equations and operations with modular arithmetic equations is responsible for these students’ confusion. The analogy
is particularly salient in the following excerpt taken from Iris’ script.
Teacher:
Student:
Teacher:
Let’s divide both sides by (p − 1)! and get ap−1 ≡ 1(mod p).
Is it allowed to divide like this?
Yes. p is a prime number, so it is different from 1, therefore, (p − 1) is different from 0 and so it is possible to divide by it.
As we see, the teacher-character argues that the division is possible just because it is not division by 0. Thus, she acts
as if the same justification that applies to ‘regular’ algebraic and arithmetic expressions also applies in the modular case.
The (problematic) role of the analogy between familiar algebraic equations and modular equivalencies is further discussed
below.
6. The presumed lack of preliminary knowledge
6.1. Focus on the meaning of equivalence relation
The formal mathematical definition of congruence, introduced by Gauss in his 1801 work Disquisitiones Arithmeticae,
states the following:
For a, b ∈ N c ≡ b (mod m) if and only if m divides |c − b|.
In other words, natural numbers c and b are said to be congruent modulo m if they have the same remainder in division by
m. In particular, with respect to the statement of the theorem discussed here, ap and a have the same remainder in division
by p.
However, in the common usage of congruence, what appears on the right hand side of the equivalence statement is the
remainder in division of the left hand side by the modulus. That is, while statements (1), (2) and (3) below are all correct
according to the definition, (1) is the one that is usually used when working with congruence classes of integers.
(1) 13 ≡ 3 (mod 5)
(2) 3 ≡ 13 (mod 5)
(3) 13 ≡ 8 (mod 5)
5
On positive side, five (out of nine, see Table 1) students correctly treated the division issue by referring to that the division by (p − 1)! is possible because
(p − 1)! is a co-prime with p. No one elaborated on this property by proving it.
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371
This is likely what leads to a rather common view that the right hand side of the congruence statement indicates the
remainder. In this section we exemplify and discuss this confusion.
The characters in Amira’s script are Impatient (professor) and Clueless (student). Her dialog starts with Impatient interpreting for Clueless the statement of the theorem:
Impatient: The statement says that for 2 relatively prime numbers a and p, where p is prime and a is natural, the remainder in division of ap by p is a.
We found similar misinterpretations of congruence in five scripts. For example, in the following excerpt from Ivan’s dialog
between a teacher and a student, the teacher starts with the observation that it is important to understand the meaning
of all the concepts and operations in the theorem. What is attended to, following the student-character’s question, is the
meaning of the term ‘mod’ in general, not in particular relation to the statement of the theorem.
Student:
Teacher:
Student:
Teacher:
I have never seen the word ‘mod’, what does it mean?
Modulo means the remainder in division of whole numbers. For example, 7 modulo 6 equals 1 because the remainder in
division [of 7 by 6] is 1. How much is 22 modulo 5?
If we divide 22 by 5, we get 4 and remainder 2, therefore modulo it is 2.
Do you have any other questions?
While the notion of ‘mod’ is attended to, the symbol for the congruence remains unattended. This may not be surprising if we
note the expression “7 modulo 6 equals 1”. That is, if congruence is viewed and interpreted as equality, then no explanation
is deemed necessary. We find further evidence in this dialog that the teacher-character describes equivalence as equality
(referring to the expression a × 2a × 3a × . . . (p − 1)a ≡ 1 ×2 × 3 × . . . (p − 1) (mod p).)
Teacher:
Asker:
Researcher:
Because we know we have the same remainders on both sides, so the product of modulo on both sides, we get two sides of the
equations are equal to one another.
In the next example taken from Anna’s script the researcher-character explains the asker-character the congruence
relationship, rather than attending to the meaning of “mod.”
What is the meaning of ap ≡ a(mod p)?
It means that when ap is divided by p the remainder is a.
Actually, this claim about the remainder holds true only if a is smaller than p. For cases where a is larger than p, the remainder
in division of ap by p (or anything else by p) should be smaller than p (by the definition of a remainder), as such it cannot be
a. Consider a simple example of a = 3 and p = 2. The remainder in division of 32 by 2 is 1, and not 3.
However, while some students misinterpreted the meaning of congruence, Omer attends to this common misinterpretation of congruence statements in a conversation between two students, Jane and John. This dialog suggests a possible source
for the confusion and acknowledges the correct interpretation.
Jane:
John:
Jane:
John:
Jane:
John:
Jane:
John:
Jane:
John:
Jane:
John:
Jane:
John:
Jane:
John:
Jane:
So we have a prime number p and a natural number a, so that their GCD is 1, and we need to prove that ap = a(mod p).
That is, if we divide ap by p, do we get remainder a?
Modulo, like in Pascal, we learned this.
Wait a second, if we divide some number by p, the remainder can be 0 – if it is divisible – or 1, or 2 and so on, until p − 1.
So?
So why do they say that the remainder will always be a? The remainder has to be smaller than p, and the theorem does not
require that a < p!
Wow, you are right. So this isn’t like the mod function in Pascal?
Probably not. Let’s take an example. Let us choose something where a is greater than p, for example a = 5, p = 3.
Let us check, 5 is natural, 3 is prime and their GCD is 1, OK.
53 = 125, and if we divide by 3, . . . so 123 is divisible, so we get remainder 2.
And not 5!
Clearly it’s not. We said that if we choose a > p then it cannot be equal.
But 5 divided by 3 also leaves the remainder of 2.
Good, good. So that is what it says. When it is written ap ≡ a(mod p), it means that if each one of them is divided by p we get
the same remainder.
Indeed. That is why it is written “equal” with three lines, which is in fact “equivalent” for all, not just some specific value. That
is, if we look at the numbers modulo p, it is the same number.
What do you mean by the same number?
Like in our example, 125 and 5 leave the same remainder when divided by 3.
John starts by considering a as the remainder in division of ap by p, but then he notices that this cannot be the case, as
there is no requirement for a to be smaller than p. Considering a particular example the characters cooperatively draw an
appropriate conclusion: ap and a have the same remainder in division by p.
The misinterpretation of congruence relations is rather common. The research of Smith (2002) also pointed to such
misconceptions. She noted that undergraduate students enrolled in a Number Theory course, “tended to view the relation of
congruence operationally rather than relationally, similarly to the way that children tend to view the relation of equality” (p.
259). Smith elaborated that “Having an operational view of the congruence a ≡ b (mod n) means interpreting the statement
in terms of a completed operation, with the number on the right as the ‘answer’. Having a relational view of congruence
means interpreting the statement as a relationship between two equivalent quantities” (p. 260). In particular, Smith noted
that students expressed preference for reducing the integer on the right side of a congruence to the smallest positive integer
smaller than the modulus. Furthermore, when the participants in Smith’s study were asked in an interview to explain
the meaning of the statement a ≡ b (mod n), five out of six students gave the following interpretation: “a divided by n has a
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remainder of b.” This is despite the fact that three appropriate equivalent definitions were provided by the professor teaching
their course.
While this particular inappropriate interpretation may not interfere with the understanding of the theorem and its proof,
it could be troublesome in other contexts. In our informal survey with a similar group of 15 participants, about half indicated
that statements such as 37 ≡ 12 (mod 5) or 2 ≡ 7 (mod 5) were false, and that there is only one correct way to fill in the blanks
in 18 ≡ (mod 5)
These responses are reminiscent of the extensive research literature on young children treating the equality sign as an
instruction to find a solution, rather than an indication of equivalence (e.g., Behr, Erlwanger, & Nicholos, 1980; Booth, 1988;
Kieran, 1981; Matthews, Rittle-Johnson, McEldoon, & Taylor, 2012). While the resemblance between the misconceptions
in both cases can be explained by an inappropriate analogy, Omer’s dialog offers another possible reason: the influence
of programming experience. The command mod refers to a remainder in Pascal, as well as in several other programming
languages, spreadsheet applications, some graphing calculators and mathematical programs. In Maple, for example, when a
user enters 17 mod 3, the computer interprets the command as “find the remainder in division of 17 by 3” and outputs 2. In
computer science mod is a function of two variables, at times denoted as mod (a,b). In congruence statements in mathematics
mod is followed by a number which is the modulus, that is, the natural number that divides the difference of two other natural
numbers.
6.2. Focus on the basic concepts
In some dialogs we find extended attention to clarifying all the concepts that appear in the theorem. While we agree with
the view that understanding of the underlying concepts is essential, we believe that at the stage of dealing with the given
theorem most of the concepts should not be problematic for a learner. The following is an excerpt from the script created
by Amira.
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
Impatient:
Clueless:
The statement says that for 2 co-prime numbers a and p, where p is prime and a is natural, the remainder in division of ap by p
is a.
Wait a second, what are co-prime numbers?
This is when their greatest common divisor is 1.
And what is a common divisor?
This is some whole number, which divides the two numbers and gives whole quotients.
Can you give an example?
Yes indeed, 3 for example is the greatest common divisor of 3 and 6.
Why is this true?
Because 3 divides 3, and 3 divides 6, and there is no number greater that 3 that does this.
I get it. But I do not understand why 1 is permitted, since it is clear that 1 divides all the numbers.
The number 1 appears when there is no other number that divides both numbers together, like 5 and 6. Can you give me a
number that divides both 5 and 6, other than 1?
Obviously no. Only 1 divides both.
That is why they are co-prime.
Now I understand.
If you understand, can you give me an example for 2 co-prime numbers?
Sure. 7 and 9, right?
Excellent!
Now, what is a prime number? And don’t worry, I will not ask about natural numbers because I already know what they are.
What a relief! (the professor laughs) Now for the primes. A prime number is a whole number that has two divisors different
from one another, one and itself.
What? Can I get an example please?
It is clear that you would want an example. For example, 5. 5 is a whole number and its only divisors are 1 and 5, that is, 1 and
itself, and it is clear that these two are different.
So, from what I understood, 1 is not a prime, because 1 and itself divide 1, but they are not distinct.
Very true. You already understand well.
So, how do prime numbers help us here?
Finally, we come to the proof. So here you have a natural number a and a prime number p, and they are co-primes, OK?
OK, no problem.
So the statement says that if we divide ap by p we get a remainder of a.
Wow, just a small issue before we proceed, what is a remainder?
The next 10 lines of the dialog clarify and exemplify division with remainder. Only then the dialog proceeds to the
proof itself. When all the concepts are clarified, the lines of the proof are presented with minimal explanation. In fact, this
script includes an additional clarification, by Impatient, of the meaning of factorial. However, the issues that most of the
participants as well as we considered as problematic are simply restated without additional explanation. This corresponds
to Amira’s stated belief that complete understanding of all the concepts in the theorem paves the way for understanding the
proof. In her submission she explicitly stated, “Mathematics resembles a building; if the foundation isn’t solid, that is, if we
skip the small detail, the building may collapse”.
We noted that those students, who were less successful in the course in general, devoted in their dialogs unnecessary
extended attention to details that could be considered ‘trivial’, or taken for granted at the expected level of mathematical
sophistication. They then passed quickly through the statements that required clarification. Such extended attention to
B. Koichu, R. Zazkis / Journal of Mathematical Behavior 32 (2013) 364–376
373
particular concepts appears to us as a ‘shield’ that protects Amira from exposing her personal difficulty in understanding
and explaining the ‘real’ problematic sections of the proof.
7. Additional evidence: the use of numerical examples
As shown above, one general strategy students used to decode the proof was to incorporate in their scripts mathematical
objects that can be directly constructed and manipulated. However, we note the relative lack of numerical examples of such
objects. We provide theorized reasons for this at the end of this section. The students’ use of numerical examples attracted
our attention as we consider it closely related to our first research question, that is, how do students deal with the identified
issues in their proof-scripts?
Numerical examples often provide helpful guidance in understanding and following proofs in general (e.g., Hazzan &
Zazkis, 2003; Leron, 1985) and in number theory in particular (Rowland, 2002). In order to understand (or understand
it better) a particular idea numerical examples can be useful both pedagogically and mathematically. With respect to the
particular theorem and our analysis of its particular proof, examples can be used (a) for verifying the statement of the theorem,
(b) for exploring the need for particular requirements, and (c) for testing the rearrangement of remainders mentioned in the
proof, that is, what happens to remainders in division by p when a, 2a, 3a, . . ., (p − 1)a are divided by p. As such, the value
of examples in examining a particular proof is both mathematical (exploring and confirming personal understanding) and
pedagogical (assisting understanding of others). However, only five out of 12 students attended to (a), only one attended to
(b), and only two attended to (c).
Further, numerical examples were used in five scripts to exemplify the meaning of ‘modulus’ in an equivalence statement,
either correctly (see for example Omer’s dialog above, which also attends to (a)) or incorrectly (see for example excerpts from
Ivan, Anna or Amira’s scripts above). Also, as demonstrated above in the script written by Amira, numerical examples were
used to exemplify or check familiarity with prerequisite basic concepts, such as prime numbers, remainders or common
divisors. Three dialogs had no numerical examples at all.
The numerical example for rearrangement of remainders (c) provided by Ron is exemplified in a previous section. Of
note is that the characters in his dialog, Matt and Mike, initially choose to check the rearrangement of remainders for a = 4,
p = 6, and then for a = 5, p = 6; they were initially confused with the results, until realized that p stood for a prime number.
Attending to (b) above, the necessity of p being a prime number is exemplified in the following excerpt from Ron’s script.
After checking the rearrangement of the sequence of remainders, the following conversation takes place:
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
Mike:
Matt:
What are we supposed to get?
That ap−1 ≡ 1 (mod p). So let us check. Is 45 ≡ 1 (mod 6)?
This will be a big number. . . (takes out a calculator)
1024, not that big. And modulo 6, I’m dividing this by 6, this is 170 point something, so 170 times 6, and subtract this from
1024, we get 4 !?! How can this be???
One moment, let us check for a = 5. I bet you this will work.
55 this is 3125. Divide by 6 and get 520 point something. . .multiply this by 6 and subtract from 3125 and we get 5?! What is
happening? There is a mistake in the expression ap−1 ≡ 1 (mod p).
Moment, let us read the theorem. There it says ap ≡ a (mod p).
Let us check this one. 46 is. . . 4096. Divided by 6 is. . . 682 point something, multiply by 6, subtract from 4096, it is. . . 4. This
works.
And what about 5?
56 is. . . 15,625, divided by 6 gives. . . 2604 point something, multiplied by 6, subtract from 15,625, it comes to. . . 1. What?
What is going on here? This can’t be. I am checking again. [he repeats the calculation and gets the same answer]
It gets complicated.
This instructor gave us a false theorem. Here is a counterexample. For a = 5 and p = 6 the theorem isn’t true. Isn’t this so? One
counterexample is enough to refute the theorem, right Mike?
In general, yes.
So this theorem isn’t true. It’s simply not true.
I don’t think the theorem isn’t true. You know, this instructor didn’t come up with the theorem. It has existed for at least 200
years. It can’t be false.
So what is going on here? You explain this to me. Let me check this for 3.
Maybe we didn’t understand something about the theorem. (he reads the statement of the theorem for the first time since the
beginning of the work). Wow, we’re @&$% [this string of symbols appears in the original text]! Look what is written here!
What?
Here, read!
for prime number p. . .
In what follows students read the requirements for a and p and exemplify the statement of the theorem with appropriately
chosen numbers. They also verify with numbers the idea of rearrangement of remainders.
The students’ initial choice of numbers ignored the requirement that p should be a prime. As such, Matt rushed to the
conclusion that the theorem was not true. However, the belief that the instructor would not give an incorrect statement
directed students to attend carefully to the particular requirement of the theorem. The particular problematics identified by
Ron relate to the observation that rather often students attempt to prove a statement without paying sufficient attention to
the specified constraints of that statement. However, in order to fully understand the necessity of particular requirements
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it is important to see what happens when these requirements are not satisfied (cf. Koichu, 2008, for the teaching strategy
based on the same idea).
The relative lack of numerical examples in the dialogs is in discord with a variety of prior studies reporting that undergraduate students and mathematics teachers gave undue weight to numerical evidence, at times in lieu of a proof (e.g., Knuth,
2002; Smith, 2006). In fact, some of the best students, who provided complete and correct explanations (e.g., Hugo), did not
use numerical examples to clarify the proof. The reason for this could be in the mathematical background of the participants
or in their interpretation of the didactical contract. It may be the case that on a higher cognitive level of understanding the
proof particular numerical examples are not essential. As such, if particular numerical examples were not considered to confirm a personal understanding of a script-writer, s/he did not use them in explaining the proof to the interlocutor. It may also
be the case that students who studied a wide variety of mathematics courses and moved beyond empirical proof schemes
(Harel & Sowder, 1998) have accepted the notion that examples do not suffice and that examples can be misleading. This
may be why these students decided to avoid the important explanatory power of examples, which can be a useful didactical
tool.
8. Discussion
Within a wide variety of research in mathematics education that attended to undergraduate students’ ability to handle
proofs, the tasks presented to students requested them to produce proofs (e.g., Smith, 2006) or to evaluate given proofs (e.g.,
Selden & Selden, 2003). The task of interpreting a given correct proof in the form of a script for a dialog is a relatively novel
approach that provides several methodological advantages.
As mentioned, this method was previously carried out by Gholamazad (2006, 2007), out only with prospective elementary school teachers and only on very simple proofs. Such proofs were drawn from introductory number theory and were
occasionally denoted as “one line proofs” (e.g., if a divides b, and b divides c, then a divides c) (Gholamazad, Liljedahl, & Zazkis,
2003). The main difficulties that surfaced when prospective elementary school teachers interpreted or presented proofs in a
form of a script for a dialog related to students’ ability to communicate ideas mathematically, such as choosing appropriate
symbolic representation, manipulating it correctly and interpreting the results. Additional difficulties were related to the
participants’ insufficient understanding of the idea of a mathematical proof and what situations require it.
Given the extended mathematical background of the participants in our study neither the idea of a proof nor symbolic
notation presented a difficulty. Nevertheless, we wondered what is revealed when students with extended backgrounds in
mathematics interpret a proof in the form of a dialog, what issues they chose to pause on and explain, how mathematical
issues are treated in these explanations, and what is taken as shared understanding or assumed knowledge. The chosen
method provided insights into these questions.
With respect to identifying problematic issues, ten out of 12 students noted that the rearrangement of remainders required
elaboration, and 9 students noted this with respect to division of a congruence statement. Nevertheless, three out of ten in the
former case and four out of nine in the latter case did not provide a correct explanation for this issue. We discussed particular
mistakes or omissions in previous sections. In particular, we noted limited interpretation of the congruence relation and
unjustified application of some operations in congruencies. As such, our study contributed to the research on students’
understanding of number theory concepts. Of note is that prior research related to learning and understanding number
theory focused primarily on the introductory concepts in this area, such as divisibility and prime decomposition (e.g., Zazkis
& Campbell, 2006; Zazkis & Liljedahl, 2004).
However, how students deal with issues that we identified as problematic was not unexpected. What was unforseen,
at least initially, is that students treated as ‘problematic’ a variety of issues that related to possible lack of preliminary
knowledge. Of course, one cannot understand the proof if it involves unfamiliar concepts. Clearly, the concept of congruence,
or equivalence with respect to modulus, is in the heart of this proof. It is interesting to note that the three students who made
a mistake in explaining the rearrangement of remainders were among those five who assumed a common misinterpretation
of the equivalence relation, even though there is no logical connection between the two mistakes. This implies that requesting
students to interpret details of proof in a form of a script for a dialog may serve as an appropriate instrument for assessing
students’ understanding. This method is applicable in undergraduate education and in teacher education and it can provide
both researchers and instructors with more information than a proof constructed by a student in ‘standard form’.
Yet, congruence was not the only prerequisite concept that was attended to and explained in the dialogs. Some students’
devoted extended attention to prerequisite concepts such as prime or co-prime numbers, factors, and factorial, and further,
considered these as central problematics. There are two possible explanations for such choices. One, a student may believe
that familiarity with concepts and symbols is all that is needed in order to understand a proof that uses these concepts
and symbols. Two, since the task asked to identify a “problematic point”, rather than “the most problematic point”, it is
reasonable to believe that students chose to focus on issues they could handle, rather than to identify a difficulty that they
themselves were unsure how to explain. This suggestion is further supported by an interesting regularity in the participants’
choices of the characters for their dialogs. Specifically, all six participants, who treated “rearrangement of the remainders” as
a central issue, attributed the main insights to (rather smart) student-characters. In contrast, two participants, who treated
the presumed lack of knowledge of the basic concepts as the only central issue, wrote the dialogs between a teacher-character
and a student-character, who needs to be straightforwardly instructed. These choices are informative with respect to the
participants’ strategies of how to cope with the task in an optimal ways. In particular, we note that focus on prerequisite
B. Koichu, R. Zazkis / Journal of Mathematical Behavior 32 (2013) 364–376
375
knowledge, which was motivated for the readers of the dialog as a need of one of the characters, may have served as a shield
for a script writer, that is, as something that protects from exposing personal incompetence or, in other words, creates a
feeling of self-efficacy in coping with the task.
8.1. On affordances, limitations and future research
We conclude that the task of working through a proof and presenting it in the form of a dialog proved to be fruitful on
several accounts: it provided a window into students’ abilities to handle identified difficulties; it exposed misconceptions
as well as personal strengths. Our contribution can be seen on several arenas: methodological innovation in task design and
implementation, further insight on understanding proofs by students with strong mathematical backgrounds, and extension
of research on understanding particular concepts in number theory.
A possibility to choose a focus of the dialog and decide on time and space allocation of various issues can be considered
both as an affordance and a limitation of the method. The issue of affordance is clear as the dialog provides an opportunity
of explaining what is not apparent in the dry formalism of mathematical proofs. However, it also provides an opportunity
to avoid ‘real problematics’ by directing the focus of attention to other issues.
Further, the dialogs provided students with an avenue to demonstrate personal creative and pedagogical skills, as evident
in their choice of characters, playfulness and even mathematical humor. As we focused primarily on mathematical ideas, the
above mentioned matters have not received sufficient attention in the current study and created an opportunity for further
research. For example, the choice of characters and the roles they play, and the influence of this choice on the content
of the dialogs – this theme was mentioned, but not developed in our analysis – may provide a fruitful avenue for further
research. Consider for example Amira’s choice of Clueless as her character, which reminds of Simplicio from Galileo’s “Dialog
Concerning the Two Chief World Systems”.
Research in mathematics education acknowledges contextual and cultural issues in writing proofs (e.g., Weber, 2003).
These issues could be explored further in decoding/interpreting proofs in a form of a script for a dialog. In particular,
what proof schemes of the participants and their characters are exposed in the dialogs? How are the scripts influenced by a
particular context in which they were composed? The omissions of the current study are avenues for imminent explorations.
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