Including School Mathematics Teaching Applications in an Undergraduate Abstract Algebra Course

2.1. Choosing Content Connected to Teaching

Each lesson focuses on mathematical concepts that readily link to secondary school mathematics topics, deepening the understanding of these topics for prospective secondary mathematics teachers. Simultaneously, each lesson addresses concepts that are not only valuable to prospective teachers but to all undergraduates, regardless of their background or degree plan.

To create connections to teaching, we examined undergraduate mathematics courses and identified topics either directly or indirectly related to school mathematics curricula and practices. Although multiple connections between abstract algebra and school mathematics can be identified, we focus on two such topics to explore how instructors use the lessons in their classrooms and whether the lessons influenced undergraduates' mathematical understanding.

The first lesson, Groups of Transformations, acclimates undergraduates to the properties of groups by prompting them to investigate sets of matrices that represent rotations and reflections of vectors in ${\mathbb{R}}^2$. Matrices are often presented in high school mathematics classrooms as arbitrary objects that calculators use to solve unwieldy systems of equations; on the other hand, rigid motions are used to establish the congruence of shapes but are not treated as especially formal mathematical objects. We simultaneously address these shortcomings by demonstrating that certain rigid motions can be represented as linear transformations on ${\mathbb{R}}^2$ via matrices. This lesson addresses the Common Core State Standards for Mathematics [Citation15] (CCSSM) related to matrix operations and vector multiplication with matrices as well as transformations in the plane. The sets of these matrix transformations, once explored for group structure, are used to interpret a high school geometry problem through the lens of abstract algebra.

In Solving Equations in ${\mathbb{Z}}_n$, undergraduates attempt to find solutions to linear equations and roots of quadratic equations in ${\mathbb{Z}}_n$. Solving equations is a significant component of high school algebra. This lesson highlights the properties of $\mathbb{R}$ often taken for granted, such as the zero product property and the existence of multiplicative inverses for every non-zero element, by exploring the same properties in alternative settings. In doing so, we connect to CCSSM [Citation15] related to solving linear and quadratic equations.

A full lesson includes detailed instructor notes in the form of an annotated lesson plan, a pre-activity assignment for undergraduates to complete some time prior to a class activity, the class activity itself, annotated solution documents for each activity, and a selection of suggested homework and assessment questions. We discuss the mathematical content of each lesson in more detail in Section 4.1.

2.2. Making Explicit Connections to School Mathematics

The primary difference between our lessons and traditional materials lies in our deliberate effort to connect advanced mathematics to school mathematics. To guide the design of the activities and lessons, we formulated five types of connections (see Table 1) based upon the Mathematical Knowledge for Teaching framework [Citation4]. These five connections are not necessarily mutually exclusive. For example, Thuy's problem (Figure 1) could align with each of Content Knowledge, School Student Thinking, and Guiding School Student Thinking. In Álvarez et al. [Citation1], we further describe how we designed activities that embed our five connections to teaching.

2.3. Fostering Undergraduate Student Engagement

Research suggests that active learning in mathematics classes increases student performance on exams regardless of class size, course type, or course level [Citation9]. On this basis, our lessons promote a classroom environment based on active learning. Included in lesson materials are suggestions for facilitating collaborative work and student-led discussions. Such suggestions are examples of annotations, discussed in Section 2.4.

2.4. Including Annotated Lesson Plans

Because our lessons provide a novel twist to common topics in undergraduate mathematics courses, an essential feature in the design of the lessons was to create annotated lesson plans. An annotated lesson plan differs from typical notes college instructors create for themselves because they include annotations. Annotations are carefully crafted textual asides that include pedagogical suggestions, reasons for including these suggestions, and explicit connections to the mathematics for teaching [Citation13]. Annotations also help instructors adapt their presentation of the lesson by providing them with anticipated undergraduate responses, optional discussion prompts, and indication of where particular emphasis or additional explanation may be needed. In writing our annotations, we focused on capturing and recording the implicit knowledge that instructors use in their classrooms.

The annotated solution documents (Figure 2) that supplement our activities also include the most important annotations for lesson implementation, creating a compact set of notes which can readily be used as a teaching aid. For example, either labeled or in a different font color we see: Sample Student Responses, which also appear in the annotated lesson plans when they are especially unexpected or could influence the direction of classroom discussions; a Facilitation Suggestion for when and how an instructor might begin a classroom discussion; and a Connection to Teaching annotation, so that instructors can make these connections at the appropriate point during the lesson.

3.1. Participants and Setting

All four instructors are algebraists and teach at public universities. David and Christina were two and three years, respectively, into a tenure track position. Edward and Alice, both tenured professors, had over 25 years of experience each. In a typical semester, both David and Christina teach four courses, Edward teaches three courses, and Alice teaches two courses. As seen in Table 2, the universities range from a small public institution to a large R1 institution. Three of the institutions are designated Hispanic-serving institutions (greater than 25% of the student body is of Hispanic origin). To protect the anonymity of the participants, we do not further break down their ethnic and racial background by institution. However, students in the courses observed were representative of the overall student body and, in some cases, students from backgrounds historically excluded from mathematics comprised most of the students interviewed.

Alice and David used A First Course in Abstract Algebra (7th edition) [Citation8] in their respective abstract algebra courses. Both instructors previously received secondary school teaching certification, and Alice had also taught at that level. Christina taught her university's elementary mathematics teacher preparation course and, in her abstract algebra course, used Abstract Algebra: An Introduction (3rd edition) [Citation14] as the course textbook. Edward previously received some professional development designed for mathematics teacher educators but described himself as unfamiliar with high school curricula. He taught from Contemporary Abstract Algebra (9th edition) [Citation10].

3.2. Procedures

We conducted post-lesson interviews with each instructor within a week after they had implemented a lesson. These interviews focused on how they used the materials, their general perceptions of the lesson, and the connections to teaching high school mathematics they and their students made during the lesson. We also conducted a final interview at the end of the semester to discuss their views on the purpose of the project as a whole, mathematical ideas in the lessons, whether or not the undergraduates learned the mathematical content and how it connects to school mathematics, their use of group work and active learning during the lessons, and other general questions about mathematics teacher preparation.

Instructors implemented the pre-activity, class activity, homework questions and assessment questions for each lesson as part of the regular coursework required for all undergraduates. Informed consent was also obtained from all four instructors. For undergraduates who consented to be part of the study, we analyzed their work from the lessons and invited a subset of consenting undergraduates from each class to participate in a semi-structured interview. Some effort was made to invite undergraduates for interviews based upon their answers on exit tickets after each lesson. However, due to limited class sizes, interviewed undergraduates necessarily represent a convenience sample. We collected twenty undergraduate interviews; seven participants expressed interest in teaching mathematics at the secondary level as a career path, and five reported that they could see themselves in a teaching position in graduate school or as an instructor at the university level. During the interviews, undergraduates re-examined their work on the assessment items and reflected on the mathematics learned and its connection to teaching. They also discussed the class environment (e.g., use of active learning strategies) during the lessons in comparison to the normal routine of the course.

All interviews were transcribed. Although initial transcriptions were verbatim, the select quotes presented below have been lightly edited for clarity by removing speech disfluencies, except when doing so could affect the meaning of the sentence. Pseudonyms were assigned to both instructors and their students. We analyzed the qualitative data for themes that arose related to connections to teaching to determine how our guiding features supported instructors and undergraduates in making connections to teaching (e.g., [Citation5,Citation16]).

4.1. Lesson Content

Pre-activity assignments are designed to achieve several goals: first, they remind undergraduates of foundational prerequisite ideas; second, they include important tasks that might require more time to complete than is available in one class period. Thus, instructors are encouraged to use pre-activities as homework assignments some time before the day in which they implement the class activity. Because students are not constrained by the limitations of a classroom activity, pre-activities can vary in length and complexity.

Class activities, designed for implementation in one 80-minute class period, build on the groundwork laid by the pre-activities. They feature tasks that have been designed with collaborative groups in mind and that can be enhanced with facilitation by an experienced instructor. We note that, by virtue of addressing important mathematical ideas, the tasks in both types of activity may not be entirely unique to this project. For example, solving for the roots of a polynomial in ${\mathbb{Z}}_n$ (see Figure 2) is a common exercise in texts, even those used by our instructors (c.f. [Citation10, p. 246];[Citation8, p. 182]). However, we use this task prior to the introduction of any prescribed explanations (c.f. [Citation10, p. 242]; [Citation8, pp. 177–178]). That is, instructors do not first provide an example at the board that illustrates the failure of the traditional factoring process; instead, undergraduates are the ones that work to discover a first-hand need for important abstract algebra terms. What ultimately makes our class activities unique is not the novelty of the tasks themselves, but that undergraduates explore how an advanced mathematical perspective can be used to extend and explain secondary school algebra techniques.

4.2. Using the Instructional Materials

Like the class activities, pre-activities include important connections to teaching. To make them explicit, we strongly encouraged instructors to spend some amount of class time to draw their students' attention to these connections.

Instructors found different opportunities to make these connections. They also used various implementations of the pre-activity-class activity sequence based on the extent to which they anticipated their students would have to rely on the pre-activity for prerequisite knowledge. For example, David assigned each pre-activity the period before the corresponding class activity. Then, before handing out the class activity, he reviewed his students' responses to the pre-activity in a class discussion to address any misconceptions. When Edward and Christina implemented Solving Equations in ${\mathbb{Z}}_n$, they both chose to assign the pre-activity several days in advance and collect them the period before the class activity. Upon reviewing students' work, Christina elected to spend an extra in-class period before the class activity working as a group on pre-requisite skills that she determined needed extra attention. Like Christina, Alice also decided to use a portion of the preceding class period to talk with her students about the pre-activity when she implemented Groups of Transformations. However, this decision was made before reviewing undergraduate work on the pre-activity; Alice decided to extend the total length of time allotted for the lesson to ensure there would be sufficient opportunity for classroom discussion about connections to teaching.

Instructors chose how best to place our lessons within their usual course structure. Those who used Groups of Transformations positioned that lesson after their usual introduction to $S_3$ and $D_4$ as the groups of symmetries of the triangle and square, respectively. They felt that this sequence provided a sense of continuity in that both lessons dealt with reflections, rotations, and group structure. Neither David nor Edward fit the entire class activity into the suggested 80-minute time frame, but both felt that the material they were able to cover provided their classes with a sufficient overview of the topic. Alice completed the entire lesson within the expected 80 minutes.

Both David and Christina made accommodations for Solving Equations in ${\mathbb{Z}}_n$ in their curricula. David, whose textbook handles groups before rings, taught the lesson at the end of his semester. He adjusted the lesson to be less discovery-oriented by defining integral domains and zero divisors for his students after the pre-activity but before the class activity. Christina, whose textbook handles rings before groups, chose to emphasize the discovery aspects of the lesson. To do so, she skipped a chapter of her textbook that she would normally teach. That chapter explains the properties of ${\mathbb{Z}}_n$ that depend on the modulus, and Christina wanted her class to explore those ideas independently during the lesson. While neither David nor Edward fit the entire class activity into the suggested 80-minute time frame, Christina completed the class activity with her students over two 50-minute class periods.

Every instructor, during the implementation of either lesson, brought with them a copy of the annotated solution document (see Figure 2). The extent to which they used this document varied. For example, Alice added additional, personalized annotations specific to her class. She also color-coded existing annotations for easier reference. On the other hand, David utilized the annotated solution documents primarily for the solutions rather than the annotations: he valued the ability to quickly check students' work with the sample student responses provided in the document.

5.1. Choice of Content

Alice explained the objective of the lessons as introducing “all these abstract terms applied in specific situations where, in-well, in the hope that students be able to see those concepts come alive much better.” That is, the connections to teaching benefited prospective teachers by providing them with applications. Importantly, though, the lessons also leveraged these applications to present all undergraduates with familiar “settings and scenarios as a means to relate the theory to concepts students might've seen, say, in high school or middle school.” David and Christina both appreciated that Solving Equations in ${\mathbb{Z}}_n$ introduced familiar definitions with an emphasis on deeper understanding. David noted that “this is a section that I have always used in the past as a building block to lead up to these later sections,” such as unique factorization of polynomials; similarly, Christina characterized her usual treatment of solving equations as “more of a footnote,” and said that the lesson enhanced her abstract algebra course in that topic. David also noted that he encountered a different set of undergraduate misconceptions when facilitating discussion during Solving Equations in ${\mathbb{Z}}_n$: during the class activity, he was surprised to find that his students struggled with determining allowable algebraic manipulations when solving equations outside of a field. By asking undergraduates to connect abstract algebra concepts to secondary school applications, David came to question if, under his previous presentation of the material, “they [his students] don’t get a really easy intuitive top layer understanding but then they get the real, deep understanding that I'm assuming they're getting.”

Edward mentioned that he always wanted to teach ring elements as solutions to equations but never had time to craft a meaningful lesson with that conceptual slant. He appreciated that Solving Equations in ${\mathbb{Z}}_n$ filled that gap, recalling that his students said that they “were glad to have had the chance to think about the concepts of solutions rather than the process of solutions.” He added that “solving equations is, I think, a very good approach to get them to thinking about inverses and why they're necessary.” Undergraduates echoed this sentiment, framing their understanding via connections to secondary mathematics. One observed that, in high school, “I knew that multiplying or dividing by inverses gave you the identity and stuff. But I didn't call it an identity. I was just dividing by four.” Another undergraduate also said they “didn't realize that I was learning this kind of stuff in high school and it helped-it made it really cool that I had done all this stuff and not realized that I was doing it. You know, it is very humbling realizing how much math went into solving a linear equation.”

5.2. Connections to School Mathematics

One way in which our lessons make explicit connections to teaching is through student thinking tasks. Not typically found in traditional abstract algebra texts, these tasks (c.f. Figures 1, 3) highlight the mathematical reasoning of a hypothetical student. Undergraduates are then tasked with analyzing this student's work and the mathematical understanding that it exhibits. In some student thinking tasks, undergraduates must then choose an appropriate questioning strategy to build on the student's existing understanding (Figure 5). Student thinking tasks often embody all five types of connections (for reference, see Table 1).

Whereas Christina previously used student thinking tasks in courses explicitly designed for teacher education, she had not integrated these applications to teaching into her general mathematics courses. At best, she might ask an abstract algebra class to anticipate an error, but not to analyze or respond to one. Christina found that student thinking tasks did not seem out of place on her assessments; she previously “hadn't thought about how it could fit in an upper-level course, so that was a good thing for me to see too coming out of this.” She realized, “you can adapt your question style and still ask something meaningful.” Undergraduates valued student thinking tasks both as an application to teaching and as a tool for improving their own understanding. One undergraduate, a prospective teacher, described the advantage of student thinking questions as follows:

Many other undergraduates also argued that teaching an idea promoted their own understanding of that concept. For others, successfully guiding a hypothetical student's thinking was affirmation of their mastery of a topic; one undergraduate argued that “if you don't know how to ask the question or …what you're going to ask to try to help them, then you don't understand the material yourself.” Still others appreciated that student thinking tasks often exhibited common misconceptions from abstract algebra that they would now know to avoid.

Broadly, Christina described how the lessons prompted undergraduates to link advanced abstract algebra topics to secondary school. She said, “I think it gave motivation behind things and made them think through topics, to take that abstract stuff and hopefully, maybe, bring it down to a level where they could understand …before applying it back to the abstract setting.” She gave a specific example from Solving Equations in ${\mathbb{Z}}_n$ by highlighting a common misconception from secondary mathematics. Secondary mathematics students might

David described one such circumstance in which content from Groups of Transformations would benefit teachers, explaining that “the connections between the idea of, for example, inverses and identities with regards to rotations and reflections and dilations was really important. Knowing when you could, for example, pick something up and put it down and still call it a rotation.” One of his undergraduates alluded to the same connection:

She claimed that her awareness of vertical connections between advanced mathematics and secondary geometry would benefit her as a teacher by improving her ability to explain mathematical phenomena. It would also help her guide students who are interested in extending their learning. Other undergraduates shared these observations. For example, one noted that prospective teachers with an advanced mathematical perspective “would be able to take that information and say, if they have a student who's really not getting it, maybe provide some sort of explanation that isn't …the same ones that they've seen in their standard textbooks.”

5.3. Fostering Student Engagement Through Active Learning

David commended the active group work embedded in the lessons. He observed that in “upper division [courses], it's harder for me to integrate structured group work” and that most of his attempts to do so had been informal and disorganized. For him, the format of the lessons “opened my eyes a little bit in terms of how you could implement discovery-oriented group work lessons in these more advanced lessons.” After some time spent acclimating to the more collaborative learning style, Christina noted that undergraduates in her class “were appreciative of the opportunity to work together on something in class. I think it was a needed break from the lectures we have been doing.” Edward observed that “simply having the discussion with them [his students] helps to have a better understanding of where they're coming from or what they're lacking.” David attributed the unexpected length of his implementation to extensive, but valuable, classroom discussion, but also noted that this kind of interactive classroom helps motivate undergraduates to both learn and participate in classes that are typically perceived as rigidly constructed. He valued the group work “not only for student knowledge but kind of getting them more excited about the course.”

Group work, or any other student-centered learning technique, is not dependent on connections to teaching. Still, instructors found that student thinking tasks in particular promoted an environment that fostered discussion and participation. Christina noticed that her undergraduates perceived a “low risk” in contributing to mathematical conversations about a hypothetical student's mistakes. One of her undergraduates later described a student thinking task as a microcosm of productive mathematical collaboration:

Several other undergraduates emphasized that the combination of peer working groups and student thinking tasks allowed them to see a greater variety of approaches. Furthermore, some acknowledged that knowing multiple ways to tackle a problem is particularly helpful for teachers: it “helps to understand how different people think, because obviously the way that you tackle a question isn't going to be the way somebody else tackles the question. So being able to help somebody means you have to be able to take a different avenue of thought.”

5.4. Lesson Annotations

Broadly, David observed that the inclusion of lesson annotations provided sufficient detail and advice. He commented that they were akin to a “lesson in a box.” Instructors also offered feedback on each of the types of annotations individually. For example, Christina and David both felt that the sample student responses facilitated their ability to prepare for classroom discussions. These annotations also allowed them to quickly prepare for a lesson that they did not write themselves. Christina appreciated how the side notes in the margins offered additional questions to ask undergraduates which could be used to pace student groups when one group works through the problem too quickly. She said, “[t]hose tips and insights, especially for somebody who isn't used to doing active learning, are really useful.” Edward also appreciated the suggestions, which he viewed as “reminders,” regarding when and how to facilitate whole-class discussions and guide the undergraduates. Instructors found suggested discussion prompts helpful but also effective; Alice “definitely felt that when we did ask some of those prompts that they did generate good thought in students.”

In line with our other guiding features, Alice also pointed out that the connections to teaching annotations were “very important, and I think it would be an empty handout if you don't include that. I think those need to be said.” Christina agreed, noting that not only did she appreciate the connections to teaching as a pedagogical tool, but that her students appreciated it too: “I think they often get in these major math courses and they lose sight of why they're there and why they need to know this stuff.” Edward thought that the connections to teaching annotations often encapsulated ideas that he had seen in professional development for teacher educators. He added that, while these annotations were not necessarily new to him, they contain important information about applications to teaching for instructors “who are not as familiar with what's important.”