# Scriptwriting as a catalyst for linking undergraduate and school mathematics

Andrew Kercher, Rina Zazkis

## ► To cite this version:

Andrew Kercher, Rina Zazkis. Scriptwriting as a catalyst for linking undergraduate and school mathematics. Recherches en Didactique des Mathematiques, In press. hal-04733152

HAL Id: hal-04733152

https://hal.science/hal-04733152v1

Submitted on 11 Oct 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
SCRIPTWRITING AS A CATALYST FOR LINKING
UNDERGRADUATE AND SCHOOL MATHEMATICS

Andrew Kercher*, Rina Zazkis**

SCRIPTWRITING AS A CATALYST FOR LINKING
UNDERGRADUATE AND SCHOOL MATHEMATICS

**Abstract** – Scripting tasks are a powerful tool for both mathematics education researchers and teacher educators, in part because the resultant dialogues provide insight into the scriptwriters’ mathematical understanding and pedagogical inclinations. In this paper, we argue that scripting tasks used in mathematics education also provide an opportunity to deliver follow-up lessons that link undergraduate and school mathematics. These lessons facilitate mathematical connections by building directly on scriptwriters’ experiences, as captured in their dialogues, and in turn, enrich their teaching practice.

**Key words:** scripting tasks, mathematical knowledge for teaching, mathematics teacher educators

TITRE DE L'ARTICLE EN ESPAGNOL (STYLE RESUME TITRE)

Resumen – Texte espagnol du résumé (style resume texte)

Palabras-claves: en espagnol sans majuscule séparés par des virgules

RESUME (STYLE RESUME TITRE)

Texte français du résumé (style résumé texte)

Mots-Clés : en français sans majuscules séparés par des virgules.

<footnote>* Simon Fraser University; andrew_kercher@sfu.ca</footnote>

<footnote>** Simon Fraser University; zazkis@sfu.ca</footnote>
# INTRODUCTION

The potential of scripted dialogues for teaching and learning mathematics was realized as early as the fourth century BC, when Plato recorded Socrates in conversation with his students. Modern authors—see, for example, Lakatos’ *Proofs and Refutations* (1976) and Pólya’s *How to Solve it* (1945)—have continued to leverage dialogue as a didactical tool at the intersection of mathematics and pedagogy. More recently, contemporary mathematics education researchers have used dialogue with supporting commentary to foster discourse on learning and teaching (e.g., Zazkis & Koichu, 2015; Mason, 2018).

In the Socratic tradition, these exemplary dialogues tend to describe an interaction between a knowledgeable expert and a pupil. The expert characters often represent the viewpoint of the scriptwriters, and thus, their explanations are reflective of a comprehensive understanding of the topic at hand. Recent applications of scriptwriting to mathematics education research, however, have taken to analyzing dialogues written by student authors; in this way, the dialogues act as a tool for gaining insight into the scriptwriters’ developing ways of thinking about mathematics and pedagogy.

The emergence of these student-written scripts began with the *lesson play* (Zazkis et al., 2009; Zazkis et al., 2013). Originally introduced as a more robust form of lesson planning, lesson plays required teachers to first anticipate where in their lesson they would need to facilitate an important discussion with their class or attend to a particular question or misconception of a student. Then, they scripted this interaction in the form of a dialogue. Lesson plays are a means of instructional planning that simultaneously serve as an approximation of practice (as in Grossman et al., 2009), unlike the traditional method of organizing a future lesson’s planned activities in an outline.

*Scripting tasks* extend the use of student-written dialogues beyond lesson planning to more general mathematical settings. In a scripting task, scriptwriters are invited to continue a hypothetical mathematical dialogue that is already underway. The first lines of this existing dialogue are provided to the scriptwriters and introduce an unexpected question, observation, or disagreement about the mathematical setting. The scriptwriters then seek to bring the problematic situation to a satisfying resolution within the remaining dialogue.
Scripting tasks have an extensive and varied history as tools for
mathematics education and the research thereof (e.g., Bergman et
al., 2023; Brown, 2018; Buchbinder & Cook, 2018; Kontorovich
& Zazkis, 2016). Zazkis and Marmur (2021) review the benefits of
such tasks, as described in the literature, for three different
populations of interest. For researchers, scripting tasks are a "rich
data source that can be examined from various perspectives"
(Zazkis & Marmur, 2021, p. 85)—that is, they not only provide
researchers with insight into the scriptwriters' mathematical
understanding but also into their perception of what constitutes
effective pedagogy in practice. For students who compose the
scripts, crafting a dialogue facilitates the growth of mathematical
knowledge within an applicable pedagogical setting, and
conversely, the development of pedagogical experience in a
relevant mathematical context. We call attention to the fact that,
for both populations, scripting tasks occupy a unique space at the
intersection of mathematics and pedagogy.

The cross-disciplinary benefits of scripting tasks are especially
evident for the third population considered by Zazkis and Murmur
(2021): mathematics teacher educators. Instructors who are
responsible for the training and development of teachers can, like
mathematics education researchers, use scripting tasks to better
understand students’ mathematical and pedagogical content
knowledge. Teacher educators, however, then have the unique
opportunity to immediately leverage this insight to broaden and
advance the emerging competencies of teachers.

We submit that scripting tasks can also be leveraged in
mathematics teacher education in order to highlight connections
between secondary and undergraduate mathematics. In such tasks,
the teacher-character's dialogue within a prompt can point to a
connection to more advanced mathematics, but this is not essential.
Based on the completed scripts, a further connection to relevant
mathematical ideas can be brought to teachers' attention in follow-
up instruction, which may include explicit connections among
mathematical topics or concepts as well as ideas on how these can
be utilized pedagogically.

For example, Zazkis and Marmur (2018; 2021) describe the
ways in which they use scripting tasks as a springboard for future
lessons designed to make such connections. In the following
section we summarize how they accomplished this goal with a
scripting task based on secondary students' perception of the
function concept. We then explore how additional connections to teaching might appear in further study of the function concept.

After considering how the work of Zazkis and Marmur (2018; 2021) could be extended, the next sections introduce and extend other studies that feature scripting tasks. In these studies, Zazkis and Cook (2018) use scripting tasks in a mathematics classroom to probe undergraduate students' understanding of zero divisors; meanwhile, Kercher et al. (2023) features scripting in a mathematics education classroom as a way to record teachers' investigations of star polygons. In each case, we consider how follow-up lessons could build on the thinking exhibited in the scripted dialogue in order to highlight connections to secondary mathematics.

ON FUNCTIONS: CONNECTIONS FROM A SCRIPTING TASK

Zazkis and Marmur (2018; 2021) presented the prompt seen in Figure 1 to a group of secondary teachers in a professional development course. In addition to continuing the presented dialogue, the teachers were asked to explain the pedagogical choices made by the teacher-characters in their scripts and to elaborate on how their own understanding of the function concept might extend beyond what is evident from the dialogue.

<table><caption>Figure 1. - The prompt used by Zazkis & Marmur (2021, p. 87).</caption><tr><td>Teacher:</td><td>Consider the following table of values. What function can this describe?</td><td>x</td><td>y</td></tr><tr><td></td><td></td><td>1</td><td>3</td></tr><tr><td></td><td></td><td>2</td><td>6</td></tr><tr><td>Alex:</td><td>y = 3x</td><td>3</td><td>9</td></tr><tr><td>Teacher:</td><td>And why do you say so?</td><td>4</td><td>12</td></tr><tr><td>Alex:</td><td>Because you see numbers on the right are 3 times numbers on the left</td><td>5</td><td></td></tr><tr><td>Jamie:</td><td>I agree with Alex, but is this the only way?</td><td>6</td><td></td></tr><tr><td>Teacher:</td><td>...</td><td></td><td></td></tr></table>

Zazkis and Marmur (2018; 2021) considered the dialogues and supplementary explanations in order to identify two themes that characterized certain limitations of the scriptwriters' understanding of function. In response to each theme, Zazkis and Marmur (ibid.) planned and implemented a follow-up lesson that sought to address
these limitations while simultaneously presenting the teachers with
important mathematical connections.

**Theme 1: Expanding Teachers' Conceptions of Function**

Within the first theme, Zazkis and Marmur (2018; 2021) found evidence suggesting that the personal example spaces (see Sinclair et al., 2011) of the scriptwriters were mainly limited to functions continuous on the entire real line. Furthermore, certain interactions within the dialogue—such as when a teacher-character accepts a student’s claim that, for any function, “Each output can only have 1 input” (Zazkis & Marmur, 2021., p. 89)—indicated that some teachers also had incomplete or incorrect understanding of the definition of function. Taken together, these observations resulted in dialogues that explored only a small subset of possible functions that could be represented by the chart in Figure 1.

The follow-up activity associated with this theme first tasked the teachers, working in groups, to write a formal definition of function. Then, the groups were presented with a collection of definitions sourced from both mathematical history and modern textbooks. These definitions were supplemented by incorrect “definitions” that exhibited some of the explicit errors observed in the scripted dialogues. The teachers then sorted the collection of definitions into categories and compared them to their own definitions. Ultimately, the goal of this activity was to refine the teachers’ concept image and concept definition of function (in the sense of Tall & Vinner, 1981) to include a wider variety of functions and to more correctly align with the formal concept definition, respectively.

Achieving this goal highlighted an important connection between secondary and undergraduate mathematics. Several of the definitions presented to the teachers conceptualized functions as sets of ordered pairs. Such a definition is not likely to be appropriate for secondary mathematics classrooms; however, this advanced perspective helped draw attention to the existence of functions with discrete or disconnected domains. The historical definitions—in particular, the way they were seen to capture additional nuance over time with more precise language—emphasized the arbitrary nature of the relationship captured by a function. By responding to evidence from the dialogues that teachers held a view of function limited to continuous functions on the real numbers, Zazkis and Marmur (2018; 2021) grew teachers’
mathematical content knowledge with an appropriate follow-up lesson.

**Theme 2: Constructing a Polynomial**

The second theme was related to many teachers’ belief that only a linear function can contain the set of points described in Figure 1. However, the realization that the set of points itself already defines a function raised a question about the possibility of other nonlinear functions that contain the four colinear points given in the prompt. Zazkis and Marmur (2018; 2021) report on scripts wherein a teacher-character relied on a computer program to generate such a polynomial; purported to have found a higher-degree polynomial, which was revealed to be linear when simplified; and argued for the existence of such a polynomial without attempting to find a specific example. Zazkis and Marmur (ibid.) recognized in these responses an opportunity to broaden the scriptwriters mathematical understanding by introducing methods for constructing a family of nonlinear polynomials that all could contain points from the chart in Figure 1.

In the follow-up lesson that was designed to accomplish this goal, the teachers were initially asked to find a nonlinear polynomial $f(x)$ that had four distinct zeros. Then, they searched for a way to use $f$ as the foundation for a function that passed through the four colinear points from the table in Figure 1. They eventually created a linear combination of $f$ and the linear function $g(x) = 3x$ to yield the family of functions $h_k(x) = kf(x) + g(x)$. This process is visualized in Figure 2. Following this first activity, the class attended to an example, pulled directly from a script, of a third-degree polynomial that appeared to match the values given in the prompt. A discussion on the fundamental theorem of algebra (in particular, the fact that a third-degree polynomial has at most three real roots) explored how the teachers could use their knowledge of this advanced mathematical fact to recognize that the proposed function cannot include all the points in Figure 1. Thus, the conclusion could have been reached without conducting algebraic manipulations until the expression is recognized as a “linear function in disguise”.
<figure><img src="image_2.png" /><figcaption>**Figure 2.** - Constructing a family of nonlinear polynomials.</figcaption></figure>

Zazkis and Marmur (2021) argue that “the utility of advanced mathematical knowledge as a tool to instantly recognise student mistakes” (p. 96) is an important application of advanced mathematical knowledge to the work of teaching. The scripting task was a pivotal aspect of making this connection in that the motivating example of a “hidden” linear function was sourced directly from a dialogue. Furthermore, the challenge of finding a valid nonlinear polynomial to fit the required four points was directly relevant to the teachers because, as demonstrated in the scripts, they themselves had difficulty constructing such a function. In this way, the scripting task was a key aspect of the teachers’ mathematical learning.

Exploring Further Connections Involving Functions

Because of the ubiquity of the function concept in both secondary
and undergraduate mathematics, there are more opportunities to
make connections than just those presented by Zazkis and Marmur
(2018; 2021). In this section, we draw attention to other studies that
have engaged learners in meaningful conversation on the function
concept. Although these studies do not necessarily use scripting
tasks to prompt such conversations, we propose ways that scripting
tasks might naturally emerge from the existing mathematical
groundwork.

Mirin et al. (2021) present two different definitions for
function, each accepted as mathematically valid within the greater
mathematics community, which nevertheless result in conflicting answers when questions are asked about certain functions. The authors motivate their study by first presenting a short dialogue between two students who are in disagreement about whether a diagram similar to the one in Figure 3 represents a function. The disagreement centers around whether a function $f: X \to Y$ must act on every element of the domain $X$ or else if it is allowed to “ignore” some elements.

<figure><img src="image_3.png" /><figcaption>Figure 3. – Is the relationship pictured a function?</figcaption></figure>

The argument captured by this initial dialogue in Mirin et al. (2021) could be translated almost directly into an interesting prompt for a scripting task. Just as in Zazkis and Marmur (2021), a hypothetical follow-up activity could focus on the diversity of definitions for function. However, Mirin et al. (ibid.) also illustrate that such a task could invite teachers to question the implications for choosing one definition over another. For example, in Halmos’ (1960) definition of a function as a set of ordered pairs, the domain is the set that results from collecting all of the input values from these pairs and the codomain is the set that results from collecting all the output values; on the other hand, Bourbaki’s (1968) definition describes the domain and codomain first, then characterizes a function as some subset of the cross-product. Under these definitions, proponents of Bourbaki’s definition would recognize the relationship in Figure 3 as a function—whereas those who prefer Halmos’ definition would not. Whether the domain comes before or after the specification of the function relationship also has a direct effect on when two functions should be considered “the same”, and on what conditions are necessary for a function to have an inverse. In particular, does the condition for invertibility consist of injection and surjection or is injection alone sufficient?

This advanced perspective on the definition of function connects to school mathematics by illustrating how choice of definition can affect subsequent mathematical claims. Inverse functions, also taught in secondary mathematics, could be
presented as an immediate consequence of the way functions are
defined.

Functions and their inverses are also the topic of an
undergraduate calculus lesson described in Burroughs et al. (2023).
In that study, connections to teaching are presented in the form of
“applications to teaching” problems such as pictured in Figure 4.

Consider how Alex, Jordan, and Kelly found the inverse function of $f(x) = \frac{2}{3}x + 1$.

<table><thead><tr><th>Alex's Work</th><th>Jordan's Work</th><th>Kelly's Work</th></tr></thead><tbody><tr><td>y = 2/3 x + 1</td><td>f(x) = y</td><td>y = 2x + 1</td></tr><tr><td>x = 2/5 y + 1</td><td>f'(y) = 2/5 x + 1</td><td>y - 1 = 2x/3</td></tr><tr><td>y - 1 = 2/3 y</td><td>f(f'(y)) = 2/3 f''(y) + 1 = y</td><td>3(y - 1) = 2x</td></tr><tr><td>x - 1/3 = y</td><td>2/3 f'(y) = y - 1</td><td>3(y - 1)/2 = x</td></tr><tr><td>3/2 (x - 1) = y</td><td>3/2 f''(y) = 3/2 (y - 1)</td><td>B + f(x) = y &rarr; f''(x) + f'(x) = y &rarr; f''(y) &rarr; x = f'(y)</td></tr><tr><td></td><td>f(f'(y)) = 3/2 (y - 1)</td><td>50 | f''(y) = 3(y - 1)/2</td></tr></tbody></table>

Compare and contrast the key mathematical ideas used by Alex, Jordan, and Kelly to find the inverse function of $f(x) = \frac{2}{3}x + 1$. Make sure to identify which properties of inverse functions each student uses, if any.

<caption>**Figure 4.** – Student work illustrating how to find the inverse of a function, originally in Burroughs et al. (2023).</caption>

Again, questions of domain and codomain arise and are used to motivate teachers to address the common method of “switching x and y, then solving for y” often presented in secondary mathematics classrooms as a way to find an inverse of a function. Burroughs et al. (2023) argue that the limitations of this approach have repercussions in future classrooms—such as when obtaining the derivative of the inverse trigonometric functions in a calculus course. Highlighting these limitations acts as a connection between secondary and undergraduate mathematics for the benefit of prospective teachers taking an introductory calculus course.

The task presented in Figure 4 could be translated into a scripting task by including a discussion amongst the students as they compare their methods for finding an inverse of a function. This dialogue could motivate teachers to consider the role that the definition of an inverse function played in Jordan and Kelly’s work, and whether the three students in fact arrived at the same function. Like in previous studies, we note that this conversation would lay similar groundwork for a comparison of different definitions of function; however, the proposed scripting task might
also facilitate discussion about a teacher’s obligation to explain
why the common methods that they teach are mathematically valid.

EXTENDING SCRIPTING TASKS TO HIGHLIGHT CONNECTIONS

In this section, we follow Zazkis and Marmur’s (2018; 2021) example by proposing how existing scripting tasks from recent literature can serve as a foundation for subsequent lessons that highlight connections between secondary and undergraduate mathematics. We elaborate on these connections, their value to prospective and practicing teachers, and suggest directions for activities that might leverage the content of student-generated dialogues to make these connections meaningful.

Star Polygons

Kercher et al. (2023) introduced the *scripting journey* as a modification of the typical scripting task. In a scripting journey, the scriptwriters construct an entire dialogue, without any initial prompt, based on their own experiences engaging with a novel mathematical setting. Kercher et al. (ibid.) consider the scripting journey to be a practical method of collecting data on students’ activities during problem-solving tasks or mathematical investigations, supplementing other methods such as video recordings (e.g., Rott et al., 2021), problem-solving journals (Liljedahl, 2007), or portfolios (Gourdeau, 2019).

In Kercher et al. (2023), the participants’ scripting journeys were inspired by their investigations of star polygons and their properties. A star polygon is an equiangular and equilateral polygon that is typically self-intersecting (see Figure 5 for examples) and is represented using a Schläfli symbol (n, k). Geometrically, this notation represents *n* vertices equally spaced around the circumference of a circle wherein every vertex $p_i$ is connected to the vertex $p_{i+k}$ (or $p_{i+k-n}$, when $i + k > n$) by a line segment—in this sense, *k* dictates the distance around the circumference one must “skip” in order to connect two vertices. Algebraically, star polygons can be associated with the notion of cyclic subgroups of $\mathbb{Z}$ under addition modulo *n*. That is, for any number of “skips” $k \in \mathbb{Z}$, the subgroup $\langle k \rangle = \{ak \mid a \in \mathbb{Z}\}$ under addition modulo *n* is a star polygon. We note that, when $\text{GCD}(n, k) = 1$, *k* is a generator of the group $\mathbb{Z}_n$. In the geometric
interpretation, this corresponds to a unicursal connection joining
every vertex.

<figure><img src="image_4.png" /><figcaption>**Figure 5.** — A (5, 2) and (7, 3) star polygon.</figcaption></figure>

Prospective and practicing teachers (n = 27) who engaged with the investigation of star polygons in Kercher et al.'s (2023) study were not provided with the above definitions; instead, they engaged with a star polygon task. In this task, working in groups, the teachers generated and refined their own definition and characterized the values of *k* that would result in a star polygon for some given number of vertices *n*. Afterwards, they wrote a scripting journey inspired by their group's shared progress. These scripting journeys exhibited a variety of *advancing mathematical activities* (in the sense of Rasmussen et al., 2015; Rasmussen, et al., 2005). An analysis of the scripting journeys revealed that the teachers communicated their understanding through symbols; extrapolated and refined definitions from the examination of examples; executed algorithms, which they then attempted to generalize to different or more abstract settings; and made, tested, and justified conjectures. In particular, Kercher et al. (2023) note the interdependence of these *advancing mathematical activities*— for example, student-characters in the scripting journeys constructed multiple star polygon symbols according to an agreed-upon algorithm, and in comparing these symbols, proposed definitions or made conjectures.

Although the teachers in Kercher et al.'s (2023) study do not attend to star polygons using the language of group theory, some student-characters in the scripting journeys do provide explanations that informally capture ideas from abstract algebra. Furthermore, some scripting journeys feature explicit reference to important concepts from number theory. It is from these observations that we anticipate one way of making connections between secondary and undergraduate mathematics.
Connections between Mathematical Disciplines

Star polygons are clearly geometrical in nature, and in exploring their properties, teachers are likely to reconsider their understanding of related topics familiar to geometry curricula in secondary schools. To illustrate: consideration of the (5,1) star polygon calls into question whether regular polygons are in fact a subset of star polygons. This issue arose naturally in the scripting journeys of participants in Kercher et al. (2023). We propose that a follow-up lesson that builds on students’ observations would facilitate teachers in assimilating star polygons into their existing geometry schemas. This practice is in line with Skemp’s (1971) definition of understanding, and in this case, the understanding gained by teachers also allows them to connect school and undergraduate mathematics.

For example, if we accept regular polygons as a subset of star polygons, what familiar properties and theorems for regular polygons are then special cases of properties for star polygons—and what are the more general statements of these theorems? The sum of the interior angles of a regular polygon is often calculated in secondary schools using the formula $S_n = 180(n - 2)$. Is this formula also applicable to star polygons? To answer this, undergraduate students must first recontextualize what they understand to be an “interior angle” for something like a (5,2) star polygon (Figure 5). There is also a derivative relationship between the areas and perimeters of regular polygons (see Mamolo & Zazkis, 2012). Like the question of interior angles, deciding what to count as the perimeter and area of a self-intersecting shape is nontrivial. But once these terms have been clarified, does the same derivative relationship hold for star polygons?

Kercher et al. (2023) also note that they chose the star polygons task for their study in part because it allows for approaches that utilize multiple mathematical disciplines. For example, star polygons provide a geometric foundation for explorations in concepts from abstract algebra and number theory. One connection to the former subject has already been made in a preceding paragraph, but it can be extended. Consider the (7,3) and (7,4) star polygons; because $3 + 4 \equiv 0 \pmod 7$, these star polygons are the same shape. One can further justify this sameness by pointing to the cyclic subgroups (3) and (4) of $\mathbb{Z}$ under addition modulo 7 (i.e., the cyclic subgroups generated by 3 and 4, respectively) and noting that they are comprised of the same integers.

An interesting complication to this interpretation arises when the same shape appears on different numbers of vertices. For
example, Kercher et al. (2023) reported that two separate scripting journeys represented the (9,3) and (12,4) star polygons as equilateral triangles, respectively. Neither group individually explored whether these star polygons were in fact the same, but a class discussion making this question explicit could lead to a follow-up lesson on group isomorphisms. That is, the cyclic subgroups of $\mathbb{Z}$ corresponding to these star polygons are clearly different: they are $\{0,3,6\}$ under addition modulo 9 and $\{0,4,12\}$ under addition modulo 12, respectively. But because the star polygons are the same shape, there is still something intuitively “the same” about these groups. Formalizing this intuition creates a powerful connection between disciplines and prompts students to search for isomorphism and group structure in unexpected places.

With respect to number theory, a key element of many of the scripting journeys reported on in Kercher et al. (2023) was relative primacy. Student-characters were particularly interested in finding values of $k$ that were both less than and relatively prime to some given number of vertices $n$, and simultaneously, counting how many such $k$ exist. This led to scripting journeys that partially rediscovered Euler’s totient function, also called Euler’s phi function—a function which provides exactly such a count and is signified by $\phi$. The student-characters who began to recreate Euler’s phi function only explored cases in which $n$ could be decomposed into relatively few prime factors. An example of a meaningful task for a follow-up lesson would be to present teachers with a hypothetical student who wonders how many different values of $n$ exist such that $\frac{n}{100}$ is a proper fraction in lowest terms. This situation introduces an application for Euler’s phi function in the context of teaching, creating a practical connection to secondary mathematics. Deriving a version of the phi function that works for larger values (i.e., understanding how to calculate $\phi(100)$) could help teachers to grow the nascent ideas seen in the scripting journeys into general claims, engaging them in related questions of divisibility and combinatorics.

## A Formal Language for Informal Ideas

Multiple strands of mathematics education research promote pedagogies that utilize student’s ideas as the foundation for building mathematical theory—for example, realistic mathematics education (Gravemeijer & Doorman, 1999). For teachers who choose to employ one of these styles of instruction in their
classrooms, a key skill to develop is the ability to help students convert their informal intuitions about mathematical situations into the formal mathematical register (see Zazkis, 2000). This act of translation might manifest as a notational challenge, in which students must capture the nuances of spoken language with mathematical symbols. Additionally, we perceive a parallel challenge: helping students to align their potentially informal standards for argumentation and justification with the more rigorous standards of mathematical proof-writing.

Teaching students how to formalize ideas, create and use notation, and recognize the strengths or limitations of a justification are not typical elements of a mathematics curriculum—particularly in secondary schools. In this practical sense, it is not clear that these skills are “mathematics content”. We argue, however, that making connections between school mathematics and undergraduate mathematics transcends simply pointing to similarities between the mathematical content taught in both settings. Connections also include similarities in practice, an observation explored in more detail by Wasserman (2023a). Wasserman sought teachers’ input on which practices they associate with mathematics and which they associate with pedagogy, hoping ultimately to identify a selection of mathematical practices that are simultaneously effective pedagogical tools. Once these *pedagogical mathematical practices* have been identified, calling attention to their use in an undergraduate mathematics classroom creates a connection to school mathematics.

But more effective than simply calling attention to these practices, we argue, is providing teachers with opportunities to personally engage with and reflect on these practices in their own mathematical work—such as when investigating star polygons and writing an accompanying scripting journey. As Kercher et al. (2023) observed, the dialogues written by teachers in these scripting journeys featured extensive use of symbolic notation and argumentation strategies. We believe that a follow-up lesson designed specifically to call attention to these aspects of mathematical practice would be more likely to be received by teachers as valuable pedagogical practices in light of their own recent experiences.

For example, in Wasserman’s (2023a) study, one example of a pedagogical mathematical practice is using concrete examples to reason about a more general claim. This was a prominent strategy exemplified within the scripting journeys responding to Kercher et
al.'s (2023) star polygon task. An effective follow-up task could ask teachers to generate examples from secondary mathematics in which specific examples play an important role in helping students understand a more general concept. Discussion could later attend to situations in which concrete examples are (and importantly, are not) sufficient evidence to prove a general mathematical claim.

## Zero Divisors

Zazkis and Cook (2018) argued for integrating scripting tasks into a set of more conventional data collection strategies, such as conducting interviews, teaching experiments, and collecting students' written productions. The initial research of Cook (2014) identified students' difficulties with the Zero Product Property (ZPP), which states that, in an integral domain, if $ab = 0$ then $a = 0$ or $b = 0$. The illustrative cases included students' overgeneralizing the ZPP in a context where it is not applicable, such as when solving equations in a ring with zero divisors; conflating ZPP with its converse; and not taking advantage of the ZPP when an equation in $\mathbb{R}$ was presented in a factored form.

To further investigate students' understanding of the ZPP, Zazkis and Cook (2018) designed a scripting task, in which the authors presented a flawed proof that invoked the ZPP where it was not applicable (in a ring with zero divisors). They invited undergraduate students ($n = 17$) to respond to this proof in the form of a dialogue between a teacher and her students, as if the proof were presented in class by a student. The scriptwriters were asked to identify the possibility for misconceptions or incomplete understanding as their characters worked on the presented proof. The scripts confirmed the findings of prior research regarding students' difficulties with the ZPP. Even when the flaw in the presented proof was correctly identified, it was identified by a teacher-character responding to a student error. The authors considered this to be an instance of the scriptwriter's reflection on a personal learning experience. The results also highlighted nuances in students' understanding of rings, such as the tendency to infer existence of inverses from the existence of an identity element. The scripts also provided insights into students' ideas about the notion of mathematical proof.

We foresee a professional development session with teachers that capitalizes on the research results presented above and aims at establishing explicit connections between school mathematics and undergraduate mathematics. As reiterated in research literature,
such connections are not made by students without explicit
instructional techniques, so they should be purposefully developed
and articulated (Cuoco, 2001; Wasserman et al., 2017). Our claim
is that such a session may not only establish or strengthen the
connections, but also extend teachers' understanding of the school
mathematics they teach.

Attention to Basic Assumptions

According to Skemp (1971), “to understand something means to assimilate it into an appropriate schema” (p. 46). But how is it possible to understand better something that is already well understood? Zazkis (2008) contends that to understand something better means to assimilate it into a richer or more abstract schema.

Assimilating a known fact into a more advanced schema often
includes an explicit acknowledgement of the conventional
assumptions the fact carries and the possible generalizations it
suggests. For example, the statement "A whole number is divisible
by ten if and only if the last digit (i.e., the units digit) is zero" is
true under the implicit assumption that the number is represented
in base ten. This assumption limits the scope of applicability of this
divisibility test, but also invites variation and generalization: "a
number represented in base 5 (or in base B) is divisible by 5 (or by
B) if and only if the last digit (i.e., the units digit) is zero". So, a
familiar property becomes a special case of a general property
related to number representation.

We return here to the ZPP, which in school mathematics is
presented explicitly as a valuable strategy in solving quadratic
equations. That is, when a quadratic equation is written as a product
of two factors, a reasonable and often preferred approach towards
finding the solution set is to equate each factor to zero. This is an
instantiation of the ZPP. However, as is usual in school
mathematics, the basic assumption that limits the scope of this
property—in this case, that the mathematical setting is the real
numbers—is not mentioned explicitly. The same assumption is
made when it becomes necessary to introduce the multiplicative
inverse of a number, an idea often relevant to school mathematics
(e.g., when multiplying by a reciprocal). In R, every real number
y, other than zero, has a multiplicative inverse y' so that y × y' =
y' × y = 1. Zero is the only number that does not have
multiplicative inverse because no number multiplied by zero
results in 1. This lack of multiplicative inverse also serves as one
of the explanations for why division by zero is undefined.
However, both the ZPP and the idea of multiplicative inverses are
introduced well before students are aware of the field structure of the relevant number set.

Exposure to abstract algebra, and ring theory in particular, highlights the scope of applicability of the ZPP and the existence of multiplicative inverses. For example, in the ring $\mathbb{Z}_{12}$ – the set $\{0, 1, 2, \dots, 11\}$ with addition and multiplication modulo 12 – the product of 3 and 8 results in zero. This is a pair of zero divisors; other zero divisor pairs in this ring are 3 and 4, 2 and 6, and 4 and 9. Further, zero divisors never have multiplicative inverses. Returning to $\mathbb{Z}_{12}$, for example, there is no number that when multiplied (modulo 12) by 3 results in 1. On the other hand, in $\mathbb{Z}_{12}$, the number 5 is its own multiplicative inverse as $25 \equiv 1 \pmod{12}$. The same applies to the number 7, which when multiplied by itself results in $49 \equiv 1 \pmod{12}$. As such, the idea of rings (that are not integral domains) invites teachers to reconsider the properties of zero as related to the ZPP and its uniqueness in terms of the non-existence of multiplicative inverses.

We believe that a lesson that explicitly explores these issues is more valuable as a follow up to a scripting activity, similar to the one discussed by Zazkis and Cook (2018), as it helps teachers reflect and acknowledge their “met-befores” that may have overshadowed their newly learned abstract algebra. Álvarez et al. (2022) describe the structure of such a lesson, intended for use in an undergraduate abstract algebra course, that connects to secondary mathematics by asking teachers to reconsider basic algebraic assumptions that in fact are only true because $\mathbb{R}$ is a field. Álvarez et al. (2022) motivate this conversation by featuring hypothetical student work, such as when one student incorrectly applies the ZPP in an inappropriate setting but arrives at a correct solution (Figure 6).

<figure><img src="image_5.png" /><figcaption>Figure 6. – Applying the ZPP in $\mathbb{Z}_4$</figcaption></figure>

Conversations arising from considering this student's work can be extended to additional basic assumptions, such as geometry on a
plane or base ten positional representation, that are not
acknowledged explicitly unless breached.

The Many Faces of Multiplication

One important issue, which is implied but not highlighted in the
work of Cook (2014) and Zazkis and Cook (2018), relates to the
notion of multiplication and the multiplication symbol. In their
brief overview of the related background, Zazkis and Cook note
that "3 ·̇ 12 4 = 0". That is, the symbol "·̇ 12" is carefully used by
the authors to direct readers' attention to the fact that this
multiplication is carried out modulo 12.

However, a few lines later they write "in Z₁₂, 5 is invertible
because 5 · 5 = 1". Having established the applicable set in which
the multiplication is performed, they choose not to emphasize that
multiplication is performed modulo 12 and instead write the same
symbol as is usually used when multiplication is performed with
real numbers. This is a common trend in mathematics, where the
meaning of a symbol depends on the context in which the symbol
appears. That is, we call many operations "multiplication" and use
the symbol for "familiar multiplication", even when this familiar
symbol and familiar word invite interpretations that may not be
applicable in the given context.

We believe that this trend—which is an example of polysemy
of the word “multiplication” and its symbol—is a source of many
confusions and errors. Polysemy is a notion in linguistics that
refers to words with different but related meanings. This is to
distinguish from the notion of homonymy, which relates to words
with different meanings but the same pronunciation and often the
same spelling. While the meaning of homonyms is usually implied
by the context, for example, in distinguishing a deposit in a *bank*
from a steep *bank* of a river, polysemous words often appear in the
same context, such as a mathematics lesson. Zazkis (1998)
discussed the polysemy of the word quotient, which has different
(but related) interpretations in the context of division of whole
numbers and division of rational numbers. In considering 13
divided by 2, the whole number quotient is 6, but the quotient—as
the result of rational number division—is 6.5. Note that in both
cases, this division can be denoted by the same symbol: “13 ÷ 2 =
6.5” and “13 ÷ 2 = 6 remainder 1”.

Mamolo (2010) extended the notion of polysemy of words to
that of symbols. As in the example above, the same symbol is used
to indicate different kinds of division. Similarly, when we work in
group theory, we use “+” to indicate addition, whereas the specific
kind of addition may not be denoted explicitly. As a result,
beginners in group theory mistakenly consider the group $(\mathbb{Z}_3, +)$
to be a subgroup of $(\mathbb{Z}_6, +)$, focusing on the subset relationship
and not initially recognizing that the “+” symbol points to different
operations.

A famous quote from Henri Poincaré is that “Mathematics is
the art of giving the same name to different things”. As such, it is
the labor of the teacher (of undergraduate mathematics) to direct
learners’ attention to the different things called by the same name
and denoted by the same symbol, seeking to capitalize on the
differences and similarities. A possible task for teacher education
is to compare “regular” multiplication to multiplication in $\mathbb{Z}_{12}$ and
to identify similarities and differences. While the initial
interpretation may involve listing the same results (such as $2 \cdot 3 =$
6 and $2 \cdot_{12} 3 = 6$) and different results (such as $5 \cdot 5 = 25$, versus
$5 \cdot_{12} 5 = 1$), it should proceed to exploring relationships in the
related multiplication tables, properties of zero and 1, and the
existence of inverses. Such an exploration may also extend to
exploring different meanings of multiplication—for example,
when considered as a binary operation, multiplication exemplifies
a function of two variables (see further Wasserman, 2023b).

DISCUSSION AND CONCLUSION

We have demonstrated how engaging teachers with scriptwriting,
either to envision interactions with students on a mathematical
issue or to report on their mathematical investigations, can lead to
follow-up instruction that strengthens the links between
undergraduate and school mathematics. These links manifested in
different ways, each with different pedagogical affordances. For
example, Zazkis and Marmur (2021) demonstrated that knowledge
of the fundamental theorem of algebra could allow teachers to
quickly reason about an algebraically complex function suggested
by a student. In Kercher et al. (2023), we identified opportunities
for teachers to strengthen the mathematical practice of their
students by amplifying mathematically normative symbol use and
argumentation. Finally, in Zazkis and Cook (2018), knowledge of
zero divisors leads to discussions about hidden assumptions within
secondary mathematics. Each of these connections, and the
consequent implications for teaching, was catalyzed by a different
scripting task.
Obviously, the instructional activities described here can take
place unrelated to the scripting tasks. However, we note explicit
benefits when instruction follows scriptwriting. First, as a tool for
data collection, writing a script allows students to prioritize the
presentation and discussion of the most pivotal aspects of their
mathematical understanding. This affordance is especially evident
in scripting journeys, wherein the dialogue mirrors the most
impactful episodes of the students' authentic experiences. Second,
we suggest that the possibility to connect to a personal experience,
including personal stumbles when initially considering an
appropriate response to a student, makes the learning more
memorable. As a result, stronger mathematical knowledge informs
a more thoughtful pedagogical response, that in turn supports
students' mathematics.

For example, Zazkis and Kontorovich (2016) explored
reactions of teachers to a student’s question regarding the exponent
(-1). A student inquired whether this was the same symbol used to
indicate the inverse of a function *f*, denoted *f*⁻¹, and the reciprocal
of a number, as in 5⁻¹. The teachers were asked to write a script
that started by addressing the student’s question, “Have they ran
out or symbols, or what?” Most participants highlighted the idea
that the same symbol can be used for different purposes, and it is
important to distinguish whether it applies to a number or a
function. Only a few participants attempted to establish a
connection between the two uses of the exponent (-1), pointing to
the fact that in both cases it indicated “going backwards” or “doing
the opposite”. The follow-up instruction highlighted the
mathematical idea of inverse with respect to operation, comparing
the operation of multiplication of numbers to the operation of
composition of function. While such a perspective is informed by
a study of group theory, we claim that it can be accessible to
learners without any mention of groups. As such, we believe that
this experience equipped teachers with ways to respond to a
student’s curiosity regarding the exponent (-1), or possibly to
prompt such curiosity.

Recent research on scriptwriting outlined multiple benefits for
teachers, teacher educators and researchers. In particular, with the
assumption that the scripts reflect scriptwriters’ personal views,
teachers’ scripts provide researchers and teacher educators with
information about teachers’ perceptions of teaching. They also
offer a glimpse into scriptwriters’ mathematical knowledge and
their pedagogical aspirations. In this paper we have highlighted an
additional important benefit—scriptwriting by teachers serves as a
springboard and catalyst for a subsequent exploration of
connections between undergraduate and school mathematics.

BIBLIOGRAPHY

ÁLVAREZ, J. A., KERCHER, A., TURNER, K., ARNOLD, E. G., BURROUGHS, E. A., & FULTON, E. W. (2022). *Including school mathematics teaching applications in an undergraduate abstract algebra course*. Primus, 32(6), 685-703.

BERGMAN, A. M., GALLAGHER, K., & ZAZKIS, R. (2023). Prospective Teachers' Responses to Students' Dialogue on Fractions: Attribute Substitution and Heuristic Approaches. *Research in Mathematics Education*. 25(1), 85-104.
https://doi.org/10.1080/14794802.2021.2020155

BOURBAKI, N. (1968). *Elements of Mathematics: Theory of Sets*. Springer.

BROWN, S. (2018). Difficult dialogs about degenerate cases: A proof script study. *Journal of Mathematical Behavior*, 52, 61–76.

BUCHBINDER, O., & COOK, A. (2018). Examining the mathematical knowledge for teaching of proving in scenarios written by pre-service teachers. *Mathematics teachers engaging with representations of practice: A dynamically evolving field*, 131-154. https://doi.org/10.1007/978-3-319-70594-1_8

BURROUGHS, E. A., ARNOLD, E. G., ÁLVAREZ, J. A., KERCHER, A., TREMAINE, R., FULTON, E., & TURNER, K. (2023). Encountering ideas about teaching and learning mathematics in undergraduate mathematics courses. *ZDM Mathematics Education*, 1-11.
https://doi.org/10.1007/s11858-022-01454-3

COOK, J. P. (2014). The emergence of algebraic structure: Students come to understand units and zero-divisors. *International Journal of Mathematical Education in Science and Technology*, 45(3), 349–359.
https://doi.org/10.1080/0020739X.2013.837520

CUOCO, A. (2001). *Mathematics for teaching*. American Mathematical Society.

GOURDEAU, F. (2019). Problem solving as a subject and as a pedagogical approach, and the ongoing dialogue between mathematics and mathematics education. In P. Felmer, P. Liljedahl, & B. Koichu (Eds.), *Problem solving in mathematics instruction and teacher professional*
development (pp. 23-42). Springer. https://doi.org/10.1007/978-3-030-29215-7_2

GRAVEMEIJER, K., & DOORMAN, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. *Educational Studies in Mathematics*, 39(1), 111-129.

GROSSMAN, P., HAMMERNESS, K., & McDONALD, M. (2009). Redefining teaching, re-imagining teacher education. *Teachers and Teaching: theory and practice*, 15(2), 273-289.
https://doi.org/10.1080/13540600902875340

HALMOS, P.R. (1960). *Naive Set Theory*. D. Van Nostrand.

KERCHER, A., BERGMAN, A. M., & ZAZKIS, R. (2023). Counting the stars: advancing mathematical activity in a scripting journey. *Mathematical Thinking and Learning*, 1-20.
https://doi.org/10.1080/10986065.2023.2206510

KONTOROVICH, I., & ZAZKIS, R. (2016). Turn vs. shape: Teachers cope with incompatible perspectives on angle. *Educational Studies in Mathematics*, 93(2), 223-243. https://doi.org/10.1007/s10649-016-9699-2

LAKATOS, I. (1976). *Proofs and refutations*. Cambridge: Cambridge University Press.

LILJEDAHL, P. (2007). Persona-based journaling: Striving for authenticity in representing the problem-solving process. *International Journal of Science and Mathematics Education*, 5(4), 661-680.
https://doi.org/10.1007/s10763-007-9092-9

MAMOLO, A. (2010). Polysemy of symbols: Signs of ambiguity. *The Mathematics Enthusiast*, 7(2), 247-262. https://doi.org/10.54870/1551-3440.1186

MAMOLO, A., & ZAZKIS, R. (2012). Stuck on convention: a story of derivative relationships. *Educational Studies in Mathematics*, 81, 161-177. https://doi.org/10.1007/s10649-012-9391-0

MASON, J. (2018). Combining geometrical transformations: A mathematical narrative. *Scripting Approaches in Mathematics Education: Mathematical Dialogues in Research and Practice*, 21-51.
https://doi.org/10.1007/978-3-319-62692-5_2

MIRIN, A., MILNER, F. A., WASSERMAN, N., & WEBER, K. (2021). On two definitions of 'function'. For the Learning of Mathematics, 41(3), 22-24.

PÓLYA, G. (1945). *How to solve it*. Doubleday.
RASMUSSEN, C., WAWRO, M., & ZANDIEH, M. (2015). Examining individual and collective level mathematical progress. *Educational Studies in Mathematics, 88*(2), 259-281.
https://doi.org/10.1007/s10649-014-9583-x

RASMUSSEN, C., ZANDIEH, M., KING, K., & TEPPO, A. (2005). Advancing mathematical activity: A practice-oriented view of advanced mathematical thinking. *Mathematical Thinking and Learning, 7*(1), 51-73.
https://doi.org/10.1207/s15327833mtl0701_4

ROTT, B., SPECHT, B. & KNIPPING, C. (2021) A descriptive phase model of problem-solving processes. *ZDM, 53*(4), 737-752.
https://doi.org/10.1007/s11858-021-01244-3

SINCLAIR, N., WATSON, A., ZAZKIS, R., & MASON, J. (2011). The structuring of personal example spaces. *The Journal of Mathematical Behavior, 30*(4), 291-303.
https://doi.org/10.1016/j.jmathb.2011.04.001

SKEMP, R. (1971). *The Psychology of Learning Mathematics*. Penguin Books.

TALL, D., & VINNER, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. *Educational studies in mathematics, 12*(2), 151-169.
https://doi.org/10.1007/BF00305619

WASSERMAN, N.H. (2023a) Investigating a teacher-perspective on pedagogical mathematical practices: possibilities for using mathematical practice to develop pedagogy in mathematical coursework. *ZDM Mathematics Education*.
https://doi.org/10.1007/s11858-023-01468-5

WASSERMAN, N.H. (2023b). Mathematical challenge in connecting advanced and secondary mathematics: Recognizing binary operations as functions. In: Leikin, R. (eds) *Mathematical Challenges For All* (pp. 241-260). Springer.
https://doi.org/10.1007/978-3-031-18868-8_13

WASSERMAN, N. H., FUKAWA-CONNELLY, T., VILLANUEVA, M., MEJIA-RAMOS, J. P., & WEBER, K. (2017). Making real analysis relevant to secondary teachers: Building up from and stepping down to practice. *Primus, 27*(6), 559-578.
https://doi.org/10.1080/10511970.2016.1225874
ZAZKIS, D., & COOK, J. P. (2018). Interjecting Scripting Studies into a Mathematics Education Research Program: The Case of Zero-Divisors and the Zero-Product Property. *Scripting Approaches in Mathematics Education: Mathematical Dialogues in Research and Practice*, 205-228.
https://doi.org/10.1007/978-3-319-62692-5_10

ZAZKIS, R. (1998). Divisors and quotients: Acknowledging polysemy. *For the learning of mathematics, 18*(3), 27-30.

ZAZKIS, R. (2000). Using code-switching as a tool for learning mathematical language. *For the learning of mathematics, 20*(3), 38-43.

ZAZKIS, R. (2008). Examples as tools in mathematics teacher education. In D. Tirosh (Ed.) *Tools in Mathematics Teacher Education*. (in Handbook for Mathematics Teacher Education, Vol. 2, pp. 135-156). Rotterdam, Netherlands: Sense Publishers.

ZAZKIS, R., & KOICCHU, B. (2015). A fictional dialogue on infinitude of primes: Introducing virtual duoethnography. *Educational Studies in Mathematics*, 88, 163-181. https://doi.org/10.1007/s10649-014-9580-0

ZAZKIS, R., & KONTOROVICH, I. (2016). A curious case of superscript (-1): Prospective secondary mathematics teachers explain. *The Journal of Mathematical Behavior*, 43, 98-110.
https://doi.org/10.1016/j.jmathb.2016.07.001

ZAZKIS, R., LILJEDAHL, P. G., & SINCLAIR, N. (2009). Lesson plays: Planning teaching versus teaching planning. *For the Learning of Mathematics*, 29(1), 40-47.

ZAZKIS, R., & MARMUR, O. (2018). Scripting tasks as a springboard for extending prospective teachers' example spaces: A case of generating functions. *Canadian Journal of Science, Mathematics and Technology Education, 18*(4), 291-312. https://doi.org/10.1007/s42330-018-0019-y

ZAZKIS, R. & MARMUR, O. (2021). Pedagogical tasks towards extending mathematical knowledge: Notes on the work of teacher educators. In M. Goos, & K. Beswick (Eds.), *The learning and development of mathematics teacher educators: International perspectives and challenges*. Springer. https://doi.org/10.1007/978-3-030-62408-8_5

ZAZKIS, R., SINCLAIR, N., & LILJEDAHL, P. G. (2013). *Lesson play in mathematics education: A tool for research and professional development*. Springer. https://doi.org/10.1007/978-1-4614-3549-5