# Novice teachers' struggles with connecting abstract algebra and school algebra through polynomial factorization

Hyungmi Cho, Miyeong Na, Kyungwon Lee, Oh Nam Kwon

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Distributed under a Creative Commons Attribution 4.0 International License
Novice Teachers' Struggles with Connecting Abstract Algebra and School Algebra through
Polynomial Factorization

Hyungmi Cho<fnref n="1" /> · Miyeong Na<fnref n="2" /> · Kyungwon Lee<fnref n="3" /> · Oh Nam Kwon<fnref n="4" />

Abstract

This study aims to explore how teachers apply their undergraduate mathematics knowledge in the context of school mathematics. Focusing on the unique factorization domains (UFDs) in undergraduate mathematics and polynomial factorization in school mathematics, we explored teacher knowledge of UFDs and how their understanding of factorization emerges in hypothetical teaching contexts. Through semi-structured interviews with eight novice teachers, we identified their approaches to number factors in polynomial factorization and classified their understanding of UFDs into four distinct groups. One group exhibited a clear knowledge of polynomial factorization from an undergraduate perspective, effectively bridging the gap between school and undergraduate mathematics in their factorization tasks. In contrast, the other groups showed limited ability to connect these two domains. Despite possessing similar content knowledge, the teachers' approaches to teaching polynomial factorization varied significantly. Notably, those who lacked a solid grasp of UFDs struggled to justify their teaching strategies consistently and concretely. The findings offer valuable insights for teacher educators aiming to

<footnote n="1">¹ Chuncheon National University of Education, Chuncheon, Korea, hmcho@cnue.ac.kr</footnote>

<footnote n="2">² Seoul Education Training Institute, Seoul, Korea, young005@sen.go.kr</footnote>

<footnote n="3">³ Department of Mathematics Education, College of Education, Seoul National University, Seoul, Korea, kyungwon.lee.snu@gmail.com</footnote>

<footnote n="4">⁴ Corresponding author. Department of Mathematics Education, College of Education, Seoul National University, Seoul, Korea, onkwon@snu.ac.kr</footnote>
improve the teaching of UFDs in abstract algebra, highlighting the need to link advanced
mathematical concepts with school mathematics in teacher education.

Keywords: polynomial factorization, unique factorization domain, abstract algebra, school algebra
Introduction

Bridging advanced mathematics courses with secondary mathematics content at the
undergraduate level is a critical issue in teacher education, particularly in the context of
preparing future teachers. Undergraduate mathematics in the domain of teaching knowledge
helps teachers understand school mathematics from an advanced standpoint. However, there is
no reason to expect an automatic transfer between the advanced knowledge gained at university
and the task of teaching mathematics at school. This raises a critical inquiry into the role of
undergraduate mathematics in the domain of teaching knowledge. It remains a matter of debate
whether preservice teachers' advanced mathematical experience plays a meaningful role in their
teaching.

Klein (2016) described the gaps between undergraduate mathematics and school
mathematics as two discontinuities. The first discontinuity involves the transition from school to
university, while the second discontinuity pertains to the shift from academic studies as a
preservice teacher (PST) to the professional role of a mathematics teacher. This paper focuses on
the second discontinuity.

To bridge the second discontinuity, Wasserman et al. (2017) proposed a model
emphasizing the transition between advanced mathematics and teaching practices. Core
mathematics courses should also be redesigned to help future teachers establish meaningful links
between the mathematics they study in college and the content they will teach in secondary
schools. Additionally, research has highlighted connections between school mathematics and
undergraduate mathematics in areas such as abstract algebra and real analysis. Some studies have
examined changes in the content and methods of undergraduate mathematics courses (Kwon, 2005; Rasmussen, 2001; Winsløw & Grønbæk, 2014). For example, Winsløw and Grønbæk (2014) designed a capstone course to enable preservice teachers to approach high school calculus more deeply and autonomously. Kwon (2005) reformed the differential equations course to incorporate a more innovative approach known as Realistic Mathematics Education.

As previously mentioned, many studies have proposed reflecting on undergraduate mathematics education and improving preservice teacher training methods to address these discontinuities. Another possible approach involves confronting the realities of the discontinuities teachers encounter in schools and exploring their perceptions of the role of advanced mathematical knowledge and its application in school mathematics. Despite ongoing discussions about undergraduate mathematics education, there remains a paucity of robust research on how teachers leverage advanced mathematical knowledge to facilitate student learning. While such research does exist, it remains limited in scope. Our objective in this study is to explore how teachers apply advanced mathematical knowledge in the context of secondary school mathematics instruction. Specifically, this research focuses on the concept of unique factorization domains (hereafter UFDs) in undergraduate mathematics and polynomial factorization in school mathematics. We aim to understand how UFDs can be connected to school mathematics through the responses of teachers who have learned UFDs. We examine novice teachers’ understanding of polynomial factorization, particularly those with substantial or recent advanced mathematical experience, to explore how they connect such advanced mathematical experience to school mathematics in the context of polynomial factorization. The research questions are as follows:

RQ1: How do novice teachers understand polynomial factorization from the perspective
of abstract algebra?

RQ2: How do novice teachers connect their knowledge of abstract algebra to school algebra in the context of polynomial factorization?

# Literature Review

## Advanced mathematical knowledge for teachers and their perceptions

Recent research has identified three prevalent perspectives on the importance of advanced mathematics in teacher education programs oriented toward practical classroom practices: (i) enhancing preservice teachers’ overall mathematical knowledge; (ii) deepening their understanding of specific secondary mathematics concepts; and (iii) strengthening their grasp of the disciplinary practices of mathematics, as the practices in advanced mathematics are considered analogous to those found in high school mathematics (e.g., Wasserman, 2017; Wasserman & Weber, 2017; Wasserman, 2018).

Despite widespread discussion on the necessity of advanced mathematics in teacher education programs, empirical studies indicate that both preservice and inservice teachers continue to struggle with appropriately applying undergraduate mathematics in their teaching practices (Zazkis & Leikin, 2010; Wasserman, 2018). Zazkis and Leikin (2010) reported a wide variation in teachers’ perceptions regarding the frequency with which they apply undergraduate mathematics in their teaching in everyday settings. While many mathematics teachers acknowledge the general value of academic mathematical content knowledge, the findings highlight significant difficulties in articulating specific problems and teaching contexts in which such knowledge has been applied to secondary mathematics teaching. Wasserman (2018)
investigated how undergraduate mathematics influences instructional practice by asking 14 secondary school mathematics teachers to discuss whether selected proofs could inform their teaching of secondary mathematics. Their analysis revealed that teachers' perspectives on using calculus in teaching secondary school mathematics aligned with the transport model, indicating that they primarily considered the potential benefits and limitations of directly incorporating such explanations in future classrooms.

## Abstract algebra for teachers’ mathematical knowledge for teaching

Within the framework of mathematical knowledge for teaching, horizon content knowledge can involve recognizing how school mathematics topics relate to more advanced mathematics (Ball et al., 2008). This form of knowledge refers to a teacher’s ability to perceive and articulate the connections between school mathematics and the broader, more abstract structures of advanced mathematical thinking. In this study, we consider abstract algebra as a form of advanced mathematics characterized by the study of highly abstract ideas such as algebraic structures. Abstract algebra is a proof-based course in which students are expected to actively engage in constructing and understanding theorems about algebraic structures.

The concept of structure is fundamental to algebra. Recognizing elements of a particular type and understanding how these elements behave and relate under defined operations are essential to both abstract algebra and school algebra. Wasserman (2018) proposed several pedagogical strategies and illustrative cases that connect abstract algebra and school algebra. Topics such as operations, functions, and factorization in abstract algebra can be meaningfully linked to secondary mathematics. Structural understanding, such as that required for number
systems or the relationships between sets, can play a central role in facilitating interconnected
learning across abstract and school algebra. Lee and Heid (2018) defined a structural perspective
as comprising three components. The first is the ability to recognize and identify elements of a
particular type (e.g., $a^6 - 1$ is both a difference of squares and a difference of cubes). The
second involves structural awareness, which refers to understanding the defining features of a set
of mathematical elements, along with the relationships and properties that connect the elements
within the set (e.g., recognizing that the properties of operations taught in school algebra reflect
the axioms of a field). The third component is the disposition to attend to and seek out structure
in doing mathematics (e.g., students studying calculus demonstrate this component when they
identify a Riemann sum structure while analyzing applications of calculus, such as determining
volumes of solids of revolution).

From an undergraduate mathematics perspective, polynomial factorization is a concept
for exploring the structural properties of sets. Accordingly, topics such as the conditions for
irreducibility of polynomials and the uniqueness of factorization are typically subjects of study.
In contrast, factorization in school algebra typically presents factorization as a procedural tool for
problem-solving rather than a concept for structural exploration. For example, students often use
factorization to solve equations in word problems quickly rather than to explore its underlying
structure. Factorization also supports computational efficiency, as seen when simplifying
expressions like $101^2 = (10 + 1)^2$, which highlights its utility in arithmetic contexts. Although
factorization constitutes important mathematical content, it is often viewed merely as procedural
knowledge necessary for solving equations. As a result, students tend to memorize algorithms as
inverse processes of expansion without necessarily understanding their underlying meaning (Lee
& Kim, 2002).
From an advanced perspective, polynomials exhibit a structural pattern analogous to
those of integers, and their factorization is closely linked to UFDs. Lee and Heid (2018)
explained the structural analogy between the factorization of polynomials and integers within an
integral domain. However, how teachers conceptualize and understand the uniqueness of the
factorization remains underexplored. Espinosa et al. (2024) investigated how preservice and in-
service teachers who had not yet formally learned about UFDs leveraged their existing
understanding of unique factorization to inform their teaching practices with the goal of making
explicit connections between abstract algebra and school algebra. Nonetheless, the specific
processes and challenges teachers encounter when attempting to draw on content from abstract
algebra have not been thoroughly examined.

Polynomial factorization in undergraduate mathematics and school mathematics

Prime factorization and polynomial factorization are structurally analogous. In the
context of polynomials, the Korean mathematics curriculum states that students should
"recognize that operations on numbers and prime factorization can be extended to polynomials"
when applying the achievement standards (Ministry of Education, 2015, 2022). Similarly, the
Common Core State Standards for Mathematics (CCSSM) emphasize: "Understand that
polynomials form a system analogous to the integers, namely, they are closed under the
operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials"
(National Governors Association Center for Best Practices & Council of Chief State School
Officers [NGA & CCSSO], 2010, p. 86). But what does it mean to recognize that prime
factorization can be extended to the factorization of polynomials?

For teachers to answer this question, there must be a meaningful connection between
their learning in undergraduate mathematics and the content of school mathematics. This connection particularly involves the ability to generalize from special cases (Baldinger, 2018).

When a polynomial is expressed as a product of factors, it can be interpreted structurally. Just as integers and polynomials behave similarly under multiplication and division, the concept of factorization supports a coherent understanding of the structure of the integral domain. The integral domain is a key mathematical structure for secondary teachers, as it provides a foundation for understanding the shared algebraic structures of integers and polynomials. Table 1 presents the parallel roles played by particular types of elements in the two structures—integers and polynomials (Lee & Heid, 2018).

<caption>Table 1</caption>

*The parallel structure of integers and polynomials* (Adapted with modification from Lee & Heid, 2018, p. 300)

<table><thead><tr><th>Within the set of integers...</th><th></th><th>Within the set of polynomials...</th></tr></thead><tbody><tr><td><strong>1 or -1</strong><br>1 and -1 are factors of every element in the integers.<br>NOTE: 1 is forced to be neither prime nor composite to conserve the simplicity of factorization or to avoid a trivial case for the prime factorization. If 1 were considered to be a nontrivial factor, there would be an infinite number of factorizations.</td><td>is parallel to</td><td><strong>Units</strong><br>A unit is an element that is a factor of every element in the set of polynomials.<br>NOTE: Unit is forced to be neither irreducible nor reducible to conserve the simplicity of factorization. If a unit were considered to be a nontrivial factor, there would be an infinite number of factorizations.</td></tr><tr><td><strong>Primes</strong><br>A prime number is defined as a positive integer (≠1) that has no factors except trivial ones within the set of integers.<br>If n is a prime number, whenever n is written as the product of two integers p and q (i.e., n=p·q), either p or q has to be 1 or -1.</td><td></td><td><strong>Irreducibles</strong><br>An irreducible polynomial is defined as a non-unit polynomial that has no factors except trivial ones within the set of polynomials.<br>If a polynomial f is irreducible, whenever f is written as the product of two polynomials p and q (i.e., f=p·q), either p or q has to be a unit.</td></tr><tr><td><strong>Prime factorization</strong><br>The act or result of decomposing a given integer into the product of primes.</td><td>is parallel to</td><td><strong>Complete factorization</strong><br>The act or result of decomposing a given polynomial into the product of irreducibles.</td></tr></tbody></table>
Irreducible elements in the integral domains of both integers and polynomials are defined as elements whose nontrivial factors consist only of units and the associates of the element itself. For example, in the set of polynomials over $Q$, $2(x - 1)$ is an associate of $(x - 1)$, and thus both $(x - 1)$ and $2(x - 1)$ are irreducible elements $Q[x]$. In contrast, in the set of polynomials over $Z$, $(x - 1)$ remains irreducible, but $2(x - 1)$ is not, since 2 is not a unit in $Z[x]$. Similarly, in the set of integers, 5 is irreducible because its only nontrivial factors are $\pm1$ and $\pm5$.

When discussing the unique factorization of a polynomial into irreducible polynomials, it is essential to clarify the meaning of ‘irreducible’ in order to fully understand the significance of its uniqueness. In abstract algebra, the definition of UFDs satisfies the following two conditions:

1. Every nonzero, non-unit element of $R$ is the product of irreducible elements.

2. This factorization is unique up to associates; that is if $p_1p_2\cdots p_r = q_1q_2\cdots q_s$ with each $p_i$ and $q_j$ irreducible, then $r = s$ and, after reordering and relabeling if necessary $p_i$ is an associate of $q_j$ for $i = 1,2,\cdots,r$ (Hungerford, 2014, p. 337)

To help students recognize the structural similarity between polynomial factorization and prime factorization of natural numbers, it is essential to explicitly include the concepts of ‘unit’ and ‘irreducible’, as illustrated in Table 1. While students typically encounter prime numbers and their unique factorization early in their mathematics classrooms and are often guided to understand this concept intuitively, they are also expected to extend these ideas to polynomials. However, this extension is often neither intuitive nor easy to explain. In particular, the concept of polynomials that act analogously to primes, namely, elements that cannot be factored further, is
not clearly defined in school mathematics. For instance, a polynomial $6x^2 + 12x - 18$ may be
factored as $(3x + 9)(2x - 2)$, $6(x + 3)(x - 1)$, or $2 \times 3 \times (x + 3) \times (x - 1)$, depending on
instructional goals or teaching context. There is often a lack of consensus among teachers
regarding which of these forms should be considered most appropriate. Moreover, students'
understanding of uniqueness in factorization tends to be superficial. They may simply recognize
that, just as a natural number can be expressed as a product of factors without grasping the
underlying algebraic structure that defines uniqueness in this context.

## Methods

Our study aims to investigate how novice teachers apply their undergraduate mathematics experiences, particularly those related to the UFDs, in teaching polynomial factorization.

Specifically, we examine whether the properties of algebraic structure described in the UFDs can be meaningfully applied to interpret polynomial factorization in school mathematics. We also explore the current state of such applications in classroom contexts.

## Participants

We selected novice teachers as our research participants, particularly newly graduated and early-career in-service teachers, based on the rationale that those with the most recent undergraduate mathematics experience are well-positioned to demonstrate how related undergraduate mathematics concepts are applied to understanding school mathematics concepts. Following Stansbury and Zimmerman's (2000) research, we defined novice teachers as those with less than five years' teaching experience. Eight novice teachers responded to public announcements, all of whom agreed to participate in this study.

The participants' teaching experience ranged from four months to two years and four
months. Seven participants had taught in high schools, while one had taught in a middle school. Before the interviews, we conducted a survey regarding the mathematics courses they had taken at university. All participants had completed more than 60 major credits in mathematics, including six required credits in abstract algebra. These abstract algebra courses covered topics such as groups, rings, fields, group actions, the Sylow theorem, and Galois theory. All participants had been introduced to the concept of UFDs; the relevant mathematical content referencing Fraleigh (2003) is included in the appendix.

All participants had passed the Korean secondary mathematics teacher employment test, which includes questions covering topics in abstract algebra. Therefore, it can be reasonably inferred that they had prior exposure to topics such as group theory, ring and field theory, integral domains, extension fields, and Galois theory. While this does not guarantee a deep conceptual understanding of all these areas, it suggests that participants were sufficiently exposed to advanced mathematical concepts relevant to the structural understanding of polynomial factorization in school mathematics. Each teacher was randomly assigned a number from one to eight to ensure their anonymity.

## Tasks and analysis

We developed two tasks for use in clinical interviews (see Figure 1). Task 1, which involves the factorization of $2x^2 + 4x - 6$, originated from a seminar conducted with graduate students in mathematics education. During the seminar, one participant who was also an inservice teacher posed a question regarding whether a specific factorization of a quadratic expression should be considered valid. The ensuing discussion revealed that teachers held differing views on what constitutes an appropriate or acceptable form of factorization in such cases. This issue has led some teachers to consciously avoid including such expressions in their
teaching materials. Accordingly, this task was identified as a suitable tool for examining how teachers connect their undergraduate mathematics knowledge to the teaching of school mathematics.

Although the tasks appear straightforward, they yield different factorizations across various polynomial rings when considered from the perspective of the UFDs. We anticipated that these tasks would prompt novice teachers to recall several abstract algebra concepts, including units, associates, reducibility, irreducibility, and UFDs. We also asked participants how they would teach these concepts in a school mathematics context in order to explore how they apply their undergraduate mathematics knowledge. Task 1 investigated how teachers applied their knowledge of UFDs to solve a polynomial factorization task. Task 2 examined the types of teacher knowledge revealed in a hypothetical teaching scenario, focusing on the connection between their undergraduate mathematics knowledge and the content they intended to teach.

**The following tasks are related to polynomial factorization.**

1. Considering the polynomial $2x^2 + 4x - 6$ as an element of the polynomial ring $\mathbb{Z}[x]$, where the coefficients are integers, factorize it. If we consider this polynomial as an element of the polynomial ring $\mathbb{R}[x]$ of which the coefficients are real numbers, does the result of the polynomial factorization change?

2. If a middle school student were to ask you how to factorize the polynomial, how would you explain it?

<caption>*Figure 1. Polynomial factorization tasks (Cho et al., 2017).*</caption>

We expected that one possible response would involve identifying contradictions present in school mathematics. Alternatively, we anticipated that teachers might offer consistent explanations in a teaching context by applying their undergraduate mathematics knowledge. For Task 1, factorization in $\mathbb{Z}[x]$ leads to $2(x - 1)(x + 3)$ as 2 is the greatest common factor of the terms $2x^2$, $4x$, and $-6$. In contrast, in $\mathbb{R}[x]$, the factor of 2 is not essential since constant multiples do not affect the equivalence of factorizations in this domain. We expected teachers to
recognize the uniqueness of factorization as it related to the algebraic structure of the domain.

For Task 2, due to the lack of consensus on this issue within the Korean curriculum, we anticipated that teachers' responses would vary. We thought some teachers might interpret the task as involving polynomials with integer coefficients in $Z[x]$, and therefore argue that the integer common factor should be extracted. In that case, $2(x - 1)(x + 3)$ would be considered the appropriate form of factorization.

From an educational perspective, we expected that as teachers engaged with Task 2, they would identify the mathematical reasoning underlying the controversies surrounding factorization results in school mathematics and recognize the basis for their own instructional decisions within this context. Providing such responses requires an understanding that the set of integers and certain polynomial rings are examples of UFDs. Although teachers may not explicitly introduce the concept of uniqueness of factorization to students, we contend that the nature of their mathematical explanations would differ depending on whether or not they understand the underlying concept. Furthermore, it is anticipated that teachers with a deep understanding of the principle of prime factorization would extend this understanding to polynomial factorization, emphasizing the structural analogies between integers and polynomial rings. In such cases, students would be encouraged to view polynomial factorization not merely as a procedural tool for finding roots but as a mathematical concept that reveals the deeper structural properties of algebraic systems.

We conducted semi-structured interviews using the tasks, with each session lasting less than one hour, and transcribed all responses. For the first question, teachers with similar responses were categorized based on their perception of whether the factorization results in $Z[x]$ and $R[x]$ were identical or different. For the second question, responses were classified
according to how teachers interpreted the results within the context of school mathematics, with particular attention to whether they maintained the mathematical justification provided in their response to the first question or adapted it to account for the teaching context. The analysis focused primarily on how teachers treated the number factor 2, which was closely examined in relation to their understanding of the uniqueness of factorization. Special attention was given to teachers who did not preserve their original mathematical justification. The results of polynomial factorization were coded based on the rationale provided within the teaching context. Two authors collaboratively developed and refined the codes through discussion until consensus was reached. To ensure the validity of the analysis, three authors cross-checked the categories derived from the interview data.

## Results

The categories of teachers' responses to the tasks are summarized in Table 2. This table indicates whether teachers incorporated knowledge of abstract algebra in their responses to Task 1 and whether they drew on both abstract algebra and school algebra in their responses to Task 2.

<table><caption>Table 2
Teachers' consideration of relevant algebraic knowledge in responses to the tasks</caption><thead><tr><th>Task</th><th>Relevant algebraic knowledge considered</th><th>Teachers</th></tr></thead><tbody><tr><td rowspan="2">Task 1</td><td>Relevant abstract algebra knowledge considered</td><td>Teacher 1, Teacher 2, Teacher 3</td></tr><tr><td>Relevant abstract algebra knowledge not considered</td><td>Teacher 4, Teacher 5, Teacher 6, Teacher 7, Teacher 8</td></tr><tr><td rowspan="3">Task 2</td><td>Both relevant abstract algebra and school algebra knowledge considered</td><td></td></tr><tr><td>Only relevant abstract algebra knowledge considered</td><td>Teacher 1</td></tr><tr><td>Only relevant school algebra knowledge considered</td><td>Teacher 2, Teacher 3, Teacher 5, Teacher 7, Teacher 8</td></tr></tbody></table>
<table><tr><td>Neither relevant knowledge considered</td><td>Teacher 4</td></tr></table>

In Task 1, only three teachers (Teacher 1, Teacher 2, and Teacher 3) referred to relevant knowledge of abstract algebra in their responses. In Task 2, all teachers except Teacher 4 incorporated either abstract algebra or school algebra knowledge. However, no teacher considered both relevant abstract algebra and school algebra simultaneously in their responses.

## Novice teachers' knowledge of polynomial factorization in abstract algebra

In Task 1, which involved factorizing the polynomial $2x^2 + 4x - 6$ in $\mathbb{Z}[x]$ and $\mathbb{R}[x]$, teachers' responses were categorized into four groups. They were classified based on how they interpreted factorization in each polynomial ring (see Table 3). Mathematically appropriate responses informed by abstract algebra were provided by three teachers (Teacher 1, Teacher 2, and Teacher 3). The analysis of these responses enables an examination of how these teachers explain the meaning of factorization and irreducible polynomials within polynomial rings, as well as their understanding of the uniqueness of factorization.

<table><caption>Table 3
*Factorization results depending on polynomial rings*</caption><thead><tr><th>Responses type</th><th>Teachers</th><th>Result in Z[x]</th><th>Result in R[x]</th></tr></thead><tbody><tr><td rowspan="2">Different</td><td>Teacher 1, Teacher 2,<br>Teacher 3</td><td>2(x - 1)(x + 3)</td><td>'2' does not matter</td></tr><tr><td>Teacher 4</td><td>(x - 1)(2x + 6)</td><td>2(x - 1)(x + 3)</td></tr><tr><td rowspan="2">Same</td><td>Teacher 5, Teacher 6,<br>Teacher 7</td><td>(x - 1)(2x + 6) or 2(x - 1)(x + 3)</td><td></td></tr><tr><td>Teacher 8</td><td>2(x - 1)(x + 3)</td><td></td></tr></tbody></table>
**Different factorizations in two polynomial rings.** In Task 1, four teachers (Teacher 1, Teacher 2, Teacher 3, and Teacher 4) responded that the factorization results in the two polynomial rings had to be different. Among them, three teachers (Teacher 1, Teacher 2, and Teacher 3) justified their responses by referring to the definition of UFDs. They argued that in $\mathbb{Z}[x]$, the polynomial should be factorized as $2(x-1)(x+3)$, whereas in $\mathbb{R}[x]$, extracting the constant 2 was unnecessary. Their explanations are provided below (underlined phrases indicate emphasis added by the authors).

Teacher 1: It is asking about the unique factorization in the polynomial ring. (...) 2 is not a unit when the coefficient is an integer. Therefore, 2 should be a factor.
(writing $2(x - 1)(x + 3)$ (...) Considering the coefficient to be a real number, 2 is a unit. ...) So, the result should be different.

Teacher 2: If we factorize in polynomials with the integers' coefficients (writing $2(x - 1)(x + 3)$), it would be this. However, in a ring with a real number coefficient, because "2" cannot be expressed as unique, the results of polynomial factorization can be different. (...) Because 2 is an irreducible prime in the integer ring, it doesn't matter. But it is not irreducible or prime over real number, the problem occurs; I think that's causing a problem.

Teacher 3: I think they have to be different. Because a polynomial with a real number coefficient is not expressed with one method. It has to be different. (...) When factorizing, it is reduced to its irreducible element. I do not remember the term. What was it? Anyway, "2" has an inverse in real numbers. Isn't it unit?

All three teachers considered whether 2 is *irreducible*, *unit*, or has an inverse in the given
polynomial ring. This suggests that they understood the definition of UFDs. In particular, Teacher 2 and Teacher 3 elaborated on the factorization of the polynomial across different contexts. For instance, when $6(x-1)(x+3)$ is factorized, 6 must be expressed as the product of 2 and 3, and then factorized as $2 \times 3 \times (x-1)(x+3)$ in $\mathbb{Z}[x]$.

Another teacher, Teacher 4, stated, “After factorization of polynomials in $\mathbb{Z}[x]$, it can be further factorized more in $\mathbb{R}[x]$,” and interpreted it as $(x-1)(2x+6)$ in $\mathbb{Z}[x]$, and to $2(x-1)(x+3)$ in $\mathbb{R}[x]$.

Teacher 4: If we factorize in a ring with real number coefficients, we originally factorize in a ring with the integer coefficients, and if we factorize in the real number coefficient, we can further decompose the integer. (...) I think that “2” should be extracted in real number polynomials. However, I didn’t remember.

Following Teacher 4’s response, for example, the polynomial $x^2 - 2$, which cannot be factorized in $\mathbb{Z}[x]$ can be factorized further to $(x + \sqrt{2})(x - \sqrt{2})$ in $\mathbb{R}[x]$. It seems that the constant “2” in the given polynomial was considered a factor in $\mathbb{R}[x]$. Thus, Teacher 4 responded that the given polynomial is factorized as $(x-1)(2x+6)$ in $\mathbb{Z}[x]$, and is further factorized as $2(x-1)(x+3)$ in $\mathbb{R}[x]$.

Teacher 4’s understanding of factorization in larger rings warrants closer attention. Teacher 4 appeared to believe that there more factors might exist in the larger ring (in this case, $\mathbb{R}[x]$), compared to the smaller ring (in this case, $\mathbb{Z}[x]$). While Teacher 4 seemed to know the concept of irreducible (non-constant) polynomials, they mistakenly treated constant factors as irreducible elements when factorizing in a larger domain.

Teacher 4: This is what is done in an integer ring, but I do not quite remember, and
when we factorize this over rational numbers, these elements need to be
irreducible elements, but I do not know whether it should be different or not.

Although Teacher 4's response reveals a limited understanding of polynomial factorization, this teacher recalled the significance of irreducible elements. We cannot conclude that this teacher lacks knowledge related to undergraduate mathematics in factorization. Even so, we can explain that Teacher 4 suggests only a vague recollection of what was learned at university, indicating reliance on partial memory of abstract algebra concepts.

To factorize the given polynomial, it is essential to recall the definitions of UFDs, the unique factorization theorem, invertible elements (or units), and irreducible elements. These concepts require a level of mathematical understanding beyond what they merely remember.

**Same factorization in two polynomial rings.** Three teachers (Teacher 5, Teacher 6, and Teacher 7) who stated that the two factorizations, $(x-1)(2x+6)$ and $2(x-1)(x+3)$, are equivalent explained that no distinction was necessary because both represent the product of two linear expressions. These teachers did not appear to relate the task to the definitions of UFDs or the unique factorization theorem. Furthermore, the treatment of numerical factors did not emerge as a significant concern.

In this case, it is unclear whether these teachers lacked the relevant knowledge or simply did not demonstrate it during the interview. However, their responses revealed an insufficient understanding of how to handle constant factors in factorization problems at the school level. They also did not demonstrate specialized mathematical content knowledge to relate undergraduate-level concepts, particularly those involving the UFDs, in a teaching context.

Teacher 6, for instance, interpreted the result as a product of two linear expressions. When asked whether the factor 2 needed to be extracted, the teacher misinterpreted the
factorization by focusing narrowly on the presence of factors rather than on the structural
properties of the polynomial ring.

Teacher 6: Now that I think about it, how far we need to factorize, if we do it like this, is it right? (...) The doubt I'm having is, for example, if there is a fourth-order polynomial and it is factored into four linear expressions, it's right to factorize it into two quadratic expressions. But I'm confused about whether we have to factorize it as far as we can or if we factorized it just once.

This teacher appeared to consider it acceptable to represent the result of polynomial factorization not as a product of irreducible polynomials but as a product of reducible ones. This suggests that the teacher understood factorization primarily as the multiplication of factors.

Lastly, Teacher 8, who stated that the factorization results in two polynomial rings were
the same and gave $2(x - 1)(x + 3)$ as the result, treated 2 as a factor and explained it as
follows:

Teacher 8: Here, in polynomial rings $Z[x]$ and $R[x]$, we get the same answer of $2(x + 3)(x - 1)$, I think. (...) the 2 can also be a factor in $Z[x]$. I think we can also consider the integer as a factor, and the result, which pulls the integer outside, can be considered a factorized one. (...) I remember that the factor of the polynomial corresponds with the divisor in (natural) numbers.

Teacher 8 considered 2 to be a factor in $Z[x]$. Since he gave the same factorization result, $2(x - 1)(x + 3)$ in both $Z[x]$ and $R[x]$, it can be inferred that he also regarded 2 as a factor in $R[x]$. However, while 2 is an irreducible constant polynomial in $Z[x]$, it is a unit in $R[x]$ and therefore cannot be considered irreducible in that ring. Teacher 8 appeared to generalize the concept of factorization by associating polynomial factorization with prime factorization. Since
2 is a prime factor in the context of integers, he may have extended this idea to $\mathbb{R}[x]$, leading to a misconception that 2 is a prime (i.e., an irreducible) element in that domain. Although he seemed to recognize a structural analogy between polynomial factorization and prime factorization, it was not possible to determine whether he fully understood the concept of irreducible polynomials within polynomial rings.

## Polynomial factorization in the teaching context of school algebra

Task 2, which asked teachers to explain to middle school students the extent to which a given polynomial should be factorized, was designed to examine how they draw on undergraduate mathematics knowledge of UFDs in an intended teaching context.

<caption>Table 4</caption>

Teachers' approaches to factorization in the context of school mathematics

<table><thead><tr><th>Responses to Task 2</th><th>Teachers</th></tr></thead><tbody><tr><td>Factor 2 not required to be extracted in school mathematics<br>→ Both (x - 1)(2x + 6) and 2(x - 1)(x + 3) are considered acceptable</td><td>Teacher 1, Teacher 2,<br>Teacher 4, Teacher 5</td></tr><tr><td>Factor 2 required to be extracted in school mathematics<br>→ Only 2(x - 1)(x + 3) is considered acceptable</td><td>Teacher 3, Teacher 6,<br>Teacher 7, Teacher 8</td></tr></tbody></table>

Teachers' responses to Task 2 were categorized into two groups: those who stated that the number factor 2 should be extracted during factorization and those who indicated that extracting 2 was unnecessary. It is not possible to determine which group of responses is mathematically appropriate as the domain of the polynomial in the task was not explicitly specified.

Notably, the classification of teachers based on their responses to Task 1 did not consistently align with their responses to Task 2. Among the three teachers (Teacher 1, Teacher 2,
and Teacher 3) who demonstrated robust mathematical knowledge in Task 1, different views emerged regarding the result of factorization in the context of school mathematics. Specifically, Teacher 3 responded differently from the others in the same group, suggesting that the appropriate form of polynomial factorization at the school level may differ from that at the undergraduate level. Similarly, among the three teachers (Teacher 5, Teacher 6, and Teacher 7) who had treated the two polynomial factorizations—$(x - 1)(2x + 6)$ and $2(x - 1)(x + 3) - as equivalent in Task 1, two teachers (Teacher 6 and Teacher 7) stated that teachers should teach students to extract number factor 2 in school mathematics. These responses are particularly noteworthy, as these teachers acknowledged both forms of factorization as mathematically acceptable, yet emphasized the importance of extracting 2 in teaching contexts. This discrepancy highlights a tension between their mathematical understanding and their instructional decision-making.

**Teachers who understand the definition of UFDs.** Three teachers (Teacher 1, Teacher 2, and Teacher 3) clearly understood the concept of UFDs and recognized that the result of polynomial factorization depends on polynomial rings. For example, Teacher 1, who stated that students do not need to extract number factor 2, appeared to interpret the polynomial ring relevant to the school algebra as $Q[x]$. Accordingly, from the perspective of undergraduate mathematics, extracting the factor 2 was deemed unnecessary.

Teacher 1: *In the mathematics curriculum, it says that students should factorize to the extent that the coefficient is a rational number. (...) I think you can say that you do not need to take out the (number factor) 2.*

Teacher 1 is a high school teacher. Several Korean high school mathematics textbooks state "When there is no special mention, factorize to the extent that the coefficient is a rational number."
number” (e.g., Lee et al., 2014). However, the domain of factorization is not explicitly specified in the Korean middle or high school mathematics curricula. It appears that Teacher 1 interpreted the domain of factorization in school mathematics based on the explanations provided in Korean high school textbooks. This suggests that he was aware of how the content of school mathematics relates to undergraduate mathematics, thereby demonstrating horizon content knowledge (HCK, see Jacobsen et al., 2012). On the other hand, Teacher 2 did not clearly explain the domain of factorization addressed at the middle school level.

Teacher 2: I do not think that students have to factorize uniquely, and because students' factorizations are done in order to solve equations, it is not important whether the 2 is pulled out or not, and since they can find the same answer when they do solve equations, I do not think it is an important question. Because they haven't learned the concepts for doing unique factorizations.

From his response that “I don’t think that students have to factorize uniquely”, it reflects that the teacher does not view polynomial factorization as a means of understanding the structural properties of a mathematical system. Instead, he regards it merely as a tool for finding roots. Although he was aware of the mathematically correct response, he asserted that there is no need to extract number factor 2 in school mathematics. Teacher 2 argued that polynomial factorization in school is primarily a procedural tool used to solve equations.

Teacher 3, who explained that the given polynomial in Task 1 had to be factorized differently in the two polynomial rings, did not clearly specify the domain of factorization addressed in school mathematics. Teacher 3 argued that students had to be taught to extract number factor 2, emphasizing that doing so would make the problem-solving process easier.

Teacher 3: When I teach, I teach that you take out the integer to factorize when you can
extract the integer. Because writing it like this (pointing at $2(x - 1)(x + 3)$),
*looks more organized* than writing it like this (pointing at $(2x - 2)(x + 3)$),
*and later when you solve the problem, students will find it easier*. So I teach
that this is the how to do it (pointing at $2(x - 1)(x + 3)$).

Teacher 1, Teacher 2, and Teacher 3 clearly explained factorization in two polynomial
rings by applying the definition of UFDs. While Teacher 1's explanations based on
undergraduate mathematics, Teacher 2 omitted the concept of uniqueness of polynomial
factorization and justified his claim by treating factorization as a procedural tool for solving
equations. On the other hand, Teacher 3 responded that the number factor 2 did have to be
extracted, based on its usefulness to problem solving. These three teachers identified
instructional issues that may influence student learning and evaluated the instructional
advantages and disadvantages of addressing the number factor 2 in polynomial factorization.

**Teachers who understand polynomial factorization as a product of two linear expressions.** Teachers 5, Teacher 6, and Teacher 7, who understand polynomial factorization as multiples of two linear expressions such as $(x - 1)(2x + 6)$ or $2(x - 1)(x + 3)$, but they can be separated into two groups based on their views. Teachers 6 and Teacher 7 stated that the number factor 2 should be considered in school algebra. Although they regarded both forms of factorization as mathematically acceptable, they believed that, in the school algebra, students should always be taught to extract the factor of 2.

While Teacher 5 and Teacher 6 did not consistently or clearly explain the basis for their
responses, Teacher 7 provided the following justification.

Teacher 7: School exams test how well a student has learned; therefore, he or she should be
taught to factor out integers. The fundamentals of factorization involve factoring
out “*m*” from “*ma* + *mb*.” I think *the reason why students do not take out (the number factor) 2 in the polynomial factorization are due to a lack of understanding of the rule given at school*.

In the Korean middle school mathematics curriculum, various forms of polynomial factorization, including $ma + mb = m(a + b)$ are presented as examples of the types of polynomials that students are required to factor. Based on this, Teacher 7 believed that students should be taught to factor out a common factor such as 2.

We analyzed that Teacher 7 intended for students to follow the rules explicitly presented in school algebra. In Task 1, Teacher 7 showed an understanding of polynomial factorization at the school mathematics level. However, his response did not provide evidence of knowledge that connects this understanding to concepts from undergraduate mathematics. He explained that the idea of a common factor is introduced in the section on polynomial factorization in Korean middle school textbooks. Based on this, he argued that students should extract the common factor 2 when factorizing. He also noted that students often make mistakes by ignoring the common factor 2, which indicates that he understood both the mathematical importance of common factors and the instructional sequence found in Korean mathematics textbooks.

## Discussion

There are subtle differences between the definitions of polynomial factorization in school and undergraduate mathematics. In particular, the concept of irreducibility is difficult to define and explain within the context of school mathematics. As a result, school mathematics remains inherently limited in its treatment of polynomial factorization. It seems to be assumed that
students will naturally extend the structural understanding of factorization from the prime factorization of natural numbers to the factorization of polynomials in school mathematics. However, our analysis suggests that this extension does not occur naturally. The concept of polynomials that behave like prime numbers and cannot be further factored remains underdeveloped in school mathematics. Consequently, while students often develop an intuitive understanding of the uniqueness of prime factorization, they tend to struggle with understanding the uniqueness of polynomial factorization. This difficulty arises because school mathematics does not provide a clear and explicit conceptual basis for understanding polynomials as concepts within algebraic structures, such as integral domains.

In the Korean mathematics curriculum, polynomial factorization plays a dual role. First, it is introduced as a conceptual parallel to prime factorization, allowing students to explore structural properties shared by integers and polynomials. This approach treats factorization as a mathematical idea in its own right. Second, it is treated as a procedural tool to solve equations by identifying the zeros of polynomials. This instrumental view emphasizes its utility in problem-solving. However, despite this dual role, school mathematics tends to emphasize the procedural aspect, with little attention paid to the structural meaning of polynomial factorization. As a result, teachers often do not guide students to extend their understanding from prime factorization to polynomial factorization in a meaningful way.

Structural understanding can play a central role in bridging school algebra and abstract algebra, supporting the development of deeper conceptual coherence across both domains (Wasserman, 2018). From the structural perspective proposed by Lee and Heid (2018), engaging meaningfully with polynomial factorization in school mathematics requires that teachers possess a solid understanding of advanced mathematical ideas. Teachers’ explanations are likely to differ
based on whether they grasp the structural nature of factorization. It is expected that teachers who understand the principle of prime factorization in depth will apply this knowledge to polynomial factorization, emphasizing the structural similarities between integers and polynomial rings. In such cases, students are more likely to perceive polynomial factorization not merely as a procedural tool for solving equations, but as a mathematical concept that reveals fundamental structural properties of mathematical systems.

However, this alone is insufficient. Our findings indicate that even teachers with a strong understanding of mathematical content did not naturally progress toward interpreting polynomial factorization in terms of its structural significance within school mathematics. A key issue lies in how undergraduate mathematics is taught to preservice teachers. Specifically, the disconnection between school and university mathematics is often overlooked, with the latter remaining isolated at a more abstract and formal level. Polynomial factorization can serve as a productive entry point for exploring this disconnection. It is necessary to identify which concepts from undergraduate mathematics align with those in school mathematics and how these relationships can be meaningfully articulated. This entails engaging in two parallel discussions: one concerning the fragility or incompleteness of mathematical content in school curricula from a higher-level perspective, and another concerning the pedagogical justifications for maintaining such simplifications. Teachers who have learned undergraduate mathematics may, by carefully reflecting on the concreteness and informality inherent in school mathematics, be able to gradually bridge the gap between school mathematics and undergraduate mathematics.

It is evident that teachers who engage in the practice of teaching school mathematics often encounter a disparity between their understanding of undergraduate mathematics and the curriculum they are expected to teach. This disparity can be attributed to several factors. First,
undergraduate mathematics is typically not taught in connection with school mathematics.

Second, there is a prevailing perception that the role of mathematics at the school level differs fundamentally from its role at the university level. Finally, knowledge of undergraduate mathematics may sometimes lead to misconceptions or misinterpretations when applied to school mathematics. As a result, contradictions can arise when addressing topics such as the uniqueness of factorization, which becomes difficult to meaningfully discuss within the constraints of school mathematics.

Reasoning about the properties of abstract algebraic structures can support teachers in developing robust understandings of the school algebra content they teach. Previous research has shown that such reasoning can help both students and teachers construct coherent and unified understandings of school mathematics (Lee & Heid, 2018; Zandieh et al., 2017). While we believe that discussions concerning the extension of the definitions of uniqueness and irreducible polynomials in school algebra can be postponed, our findings suggest that the disparity between undergraduate and school mathematics is not well understood, even by those who have studied undergraduate mathematics. This underscores the importance of encouraging teachers to identify, acknowledge, and reflect on this disparity. The structural disconnection between undergraduate and school mathematics makes it difficult for teachers to meaningfully bridge the two domains.

## Conclusion

This study investigated how teachers apply their knowledge of undergraduate mathematics in the context of school mathematics, with a particular focus on polynomial factorization. The analysis examined how the concept of factorization is influenced by the
underlying polynomial ring, especially through the lens of UFDs in undergraduate mathematics.
For instance, in $\mathbb{Z}[x]$, polynomials are factorized with attention to number factors other than 1
due to the presence of irreducible polynomials. In contrast, in $\mathbb{Q}[x]$ and $\mathbb{R}[x]$, where all
nonzero constants are units, such constants are often treated as interchangeable in factorization.
However, school mathematics rarely specifies the domain in which polynomial factorization
takes place, which may lead to misunderstandings about its meaning and applicability.

We conducted semi-structured interviews using tasks designed to prompt novice teachers
to recall and activate their understanding of UFDs and to explore how they apply this knowledge
in the context of school mathematics. Based on their responses, teachers were categorized
according to their understanding of UFDs, and their approaches to polynomial factorization in
school mathematics were analyzed. Our findings reveal a notable disparity in how teachers apply
their undergraduate mathematical knowledge to school-level instruction. Only three novice
teachers demonstrated a clear and accurate understanding of UFDs in solving factorization tasks.
Overall, novice teachers exhibited limited or incoherent connections between their undergraduate
mathematics education and school-level mathematical explanations. These findings reinforce the
observation by Zazkis and Leikin (2010) that it is often difficult for teachers to articulate the
relevance of advanced mathematics within specific teaching contexts. Wasserman (2018), in
describing undergraduate mathematics as a “transport model”, argued that teachers tend to use
advanced mathematics only when it can be applied directly. In our study, the perceived utility of
undergraduate mathematics was closely tied to teachers’ ability to directly explain concepts such
as UFDs and irreducibility within the school mathematics context. Furthermore, even among
teachers who demonstrated a relatively strong understanding of advanced mathematics, all but
Teacher 1 primarily relied on school-level algebraic reasoning when addressing school
mathematics tasks. This suggests that advanced mathematical knowledge is not automatically
transferred to teaching practice unless its relevance is explicitly recognized and pedagogically
integrated.

This underscores the importance of explicitly addressing the domain in which polynomial
factorization occurs within educational materials in order to bridge the gap between school and
undergraduate mathematics. Although teachers may possess mathematically correct knowledge,
the findings reveal a discrepancy between this knowledge and the content presented to students.
This highlights the need for teachers to effectively transfer their advanced mathematical
understanding into forms that are both appropriate and accessible for the school context.

Our analysis suggests that the weaknesses in school mathematics and the inconsistencies
in teachers' explanations arise from a lack of integration between school and undergraduate
mathematics. To address this, we propose enhancing both the curriculum and teacher education
programs to foster stronger connections between these two levels of mathematical knowledge. In
particular, greater emphasis should be placed on incorporating abstract concepts such as
irreducibility and the uniqueness of polynomial factorization into school mathematics
discussions. By doing so, we aim to reduce the disconnection and promote a more coherent
understanding of mathematics across educational levels.

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[Appendix]

Mathematical concepts in ring theory for polynomial factorization (Fraleigh, 2003)

<table><thead><tr><th>Concepts</th><th>Definitions</th></tr></thead><tbody><tr><td>A ring</td><td>A ring &lt;R,+,&gt; is a set R together with two binary operations + and -, which we call addition and multiplication, defined on R such that the following axioms are satisfied:<br>R1. &lt;R,+&gt; is an abelian group.<br>R2. Multiplication is associative.<br>R3. For all a, b, c ∈ R, the left distributive law, a &middot; (b + c) = (a &middot; b) + (a &middot; c) and the right distributive law (a + b) &middot; c = (a &middot; c) + (b &middot; c) hold.</td></tr><tr><td>A commutative ring</td><td>A ring in which the multiplication is commutative is a commutative ring.</td></tr><tr><td>A ring with unity</td><td>A ring with a multiplicative identity element is a ring with unity; the multiplicative identity element 1 is called “unity.”</td></tr><tr><td>An integral domain</td><td>An integral domain D is a commutative ring with unity 1 ≠ 0 and containing no divisors of 0.</td></tr><tr><td>An ideal</td><td>An additive subgroup N of a ring R satisfying the properties aN ⊆ N and Nb ⊆ N for all a, b ∈ R is an ideal.</td></tr><tr><td>Divisibility</td><td>Let R be a commutative ring with unity and let a, b ∈ R. If there exists c ∈ R such that b = ac, then a divides b (or a is a factor of b), denoted by a|b.</td></tr><tr><td>A unit of ring and associates in a ring</td><td>An element u of a commutative ring with unity R is a unit of R if u divides 1, that is, if u has a multiplicative inverse in R. Two elements a, b in R are associates in R if a = bu, where u is a unit in R.</td></tr><tr><td>An irreducible of an integral domain</td><td>A nonzero element p that is not a unit of an integral domain D is an irreducible of D if in every factorization p = ab in D has the property that either a or b is a unit.</td></tr><tr><td>A unique factorization domain</td><td>An integral domain D is a unique factorization domain (abbreviated UFD) if the following conditions are satisfied:<br>1. Every element of D that is neither 0 nor a unit can be factored into a product of a finite number of irreducibles.<br>2. If p₁...pₚ and q₁...qₚ are two factorizations of the same element of D into irreducibles, then r = s and the qᵢ can be renumbered so that pᵢ and qᵢ are associates.</td></tr><tr><td>A principal ideal domain</td><td>An integral domain D is a principal ideal domain (abbreviated PID) if every ideal in D is a principal ideal.</td></tr><tr><td>A prime</td><td>A nonzero nonunit element p of an integral domain D is a prime if, for all, a, b ∈ D, p|ab implies either p|a or p|b.</td></tr></tbody></table>