Revised Edition New Secondary Schools Mathematics Algebric Sentences Pdf

1. INTRODUCTION

After implementing a lesson about integral domains in your undergraduate abstract algebra course, your students can likely reproduce the relevant definitions, cite several examples and non-examples of integral domains, and identify if a ring element acts as a zero divisor. Can your undergraduates who plan to be secondary school mathematics teachers further connect and apply this technical knowledge of abstract algebra to the mathematics they will one day teach? How effective are questions, such as the one in Figure 1, at providing prospective secondary mathematics teachers with opportunities to apply their knowledge to teaching situations? Abstract algebra is often one of the advanced mathematics courses that prospective secondary mathematics teachers are expected to take. Integrating applications for mathematics teaching into such courses supplements the existing collection of applications to other fields such as engineering or physics.

Including School Mathematics Teaching Applications in an Undergraduate Abstract Algebra Course

Published online:

04 May 2021

Figure 1. Applying knowledge of integral domains and zero divisors to student thinking.

The Mathematical Education of Teachers II (MET II) Report by the Conference Board of the Mathematical Sciences (CBMS) [6] recommends that colleges and universities improve the education of prospective secondary mathematics teachers by explicitly integrating connections between the mathematics in undergraduate courses to the secondary school content that prospective teachers will eventually teach. This recommendation arises from the observation that "the mathematics courses they [prospective secondary mathematics teachers] take emphasize preparation for graduate study or careers in business rather than advanced perspectives on the mathematics that is taught in high school" [6, p. 5]. Other researchers note that "prospective high school mathematics teachers, who earn a mathematics major or its equivalent, do not have sufficiently deep understanding of the mathematics of the high school curriculum" [17, p. 107]. These observations highlight the fact that mathematics teacher preparation does not adequately address vital practices of mathematics teaching, such as "making and explaining connections among mathematical ideas" or "maintaining essential features of a mathematical idea while simplifying other aspects to help students understand the idea" [7, p. 34]. This causes prospective secondary teachers to perceive their undergraduate mathematical preparation as unconnected to their teaching [11]. Furthermore, even practicing secondary teachers with first-hand classroom experience have difficulty pinpointing circumstances in which their advanced mathematical knowledge influences their pedagogical decisions [20].

Not enough resources exist that attempt to rectify these shortcomings (for some that do, see [12,18,19]). This is due, in part, to the unique challenges encountered when translating the MET II recommendations into practical undergraduate classroom materials. The goal of the META Math (Mathematical Education of Teachers as an Application of Undergraduate Mathematics) project is to directly address the MET II recommendations by creating lessons that link advanced mathematics content to school mathematics content. The lessons feature teaching applications of the advanced mathematics, integration of active learning practices, and detailed notes to guide their implementation. META Math focused its initial phase of lesson development on calculus, discrete mathematics, abstract algebra, and statistics, typical mainstream undergraduate mathematics courses that include prospective secondary mathematics teachers.

The purpose of this article is to describe the guiding features and implementation of two undergraduate abstract algebra lessons that focus on applications to teaching. Our goal is to attend to the call in the MET II recommendations by providing abstract algebra instructors with resources for helping prospective secondary mathematics teachers make connections between the mathematics they are studying and the mathematics they will teach. What we present is neither a case study nor an assessment study, but instead, an examination of four mathematicians' use of the lessons. We describe their perspectives on the benefits of the lessons for their students, lesson planning, and their own understanding of the material and connections to high school mathematics. We also provide excerpts of their undergraduates' perspectives that reflect aspects of the guiding features in the lessons as well as their understandings of the mathematics and its connection to teaching. Finally, we combine these accounts to discuss what we learned from the instructors and their students. The full lessons are available via https://tinyurl.com/METAMathLessons.

2. GUIDING FEATURES FOR META MATH LESSONS

The MET II Report suggests that mathematics courses for prospective mathematics teachers "should be tailored to the work of teaching, examining connections … between high school and college" [6, p. 54]. We formulated the most relevant types of connections and featured them in practical classroom materials. The design of these lessons encompassed four guiding features: (1) choosing content already embedded in an undergraduate mathematics course that is relevant for prospective teachers to understand at a deep level; (2) making explicit connections between the advanced undergraduate mathematics and school mathematics; (3) fostering undergraduates' engagement through active learning; and (4) writing useful notes in the materials to facilitate mathematics instructors' implementations of applications to high school teaching in a manner that makes explicit connections.

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 $R^{2}$ . 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 $R^{2}$ via matrices. This lesson addresses the Common Core State Standards for Mathematics [15] (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 $Z_{n}$ , undergraduates attempt to find solutions to linear equations and roots of quadratic equations in $Z_{n}$ . Solving equations is a significant component of high school algebra. This lesson highlights the properties of $R$ 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 [15] 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 [4]. 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. [1], we further describe how we designed activities that embed our five connections to teaching.

Including School Mathematics Teaching Applications in an Undergraduate Abstract Algebra Course

Published online:

04 May 2021

Table 1. Five types of connections to teaching [3].

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 [9]. 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 [13]. 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.

Including School Mathematics Teaching Applications in an Undergraduate Abstract Algebra Course

Published online:

04 May 2021

Figure 2. Excerpt from the Solving Equations in $Z_{n}$ annotated solution document.

3. DATA SOURCES

Two abstract algebra instructors at different universities each implemented both lessons in Spring 2019. Two more abstract algebra instructors each implemented one of the two lessons in Fall 2019. The Fall instructors each also implemented a third lesson developed to accommodate a wider variety of curricula. This lesson received less testing, feedback, and revision; as a result, we do not describe it or include details on its implementation.

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.

Including School Mathematics Teaching Applications in an Undergraduate Abstract Algebra Course

Published online:

04 May 2021

Table 2. Details on participants and settings.

Alice and David used A First Course in Abstract Algebra (7th edition) [8] 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) [14] 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) [10].

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., [5,16]).

4. IMPLEMENTATION OF THE LESSONS

Before field testing these lessons, each instructor participated in an orientation to the META Math project via an online, one-hour training session. They became acquainted with the guiding features of a lesson (especially the importance of emphasizing connections to teaching), each type of annotation, and recommended strategies for successfully implementing a lesson. The orientation also underscored that the lessons should not be treated as scripts for them to follow to the letter. Instead, we encouraged instructors to adapt their implementation of the lessons to accommodate their curriculum and classroom practice so long as their choices aligned with our guiding features.

With this in mind, instructors reported preparing for their implementation of the lessons in different ways that best suited their pedagogical style. For example, one instructor chose to only make a cursory reading of the provided annotated lesson plans so that he would be less likely to appear to his class as an absolute epistemological authority, thereby inadvertently discouraging student-led discussion. Other instructors read the provided documents, but also worked through enough of each activity to get a better sense for appropriate pacing and where they would most be needed to facilitate appropriate discourse.

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 $Z_{n}$ (see Figure 2) is a common exercise in texts, even those used by our instructors (c.f. [10, p. 246];[8, p. 182]). However, we use this task prior to the introduction of any prescribed explanations (c.f. [10, p. 242]; [8, 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.

4.1.1. Groups of Transformations

The pre-activity assignment for Groups of Transformations begins with an examination of what a secondary mathematics teacher might mean if she tells her class that the order of an operation "doesn't matter." Undergraduates discuss how the phrase "order doesn't matter" might encompass the formal mathematical ideas of closure, associativity, and commutativity. They then apply their understanding to explain whether the order in which one applies rotations to a vector in $R^{2}$ "matters." Finally, undergraduates explore the geometric effect that certain matrices have on vectors in $R^{2}$ to identify the general form of rotation and reflection matrices.

In the Groups of Transformations class activity, undergraduates assess the rotation and reflection matrices that they identified in the pre-activity for (abelian) group structure. Whether or not the sequence in which an operation is applied "matters" is then reframed in this context. Finally, undergraduates apply their advanced understanding of these transformations to explore a secondary school geometry student's reasoning and help him understand congruency (Figure 3). In this task, an undergraduate response to 5(a) might hypothesize that Todd thinks reversing the order in which transformations are applied would also "reverse" the effect they have on a vector. In 5(b), undergraduates might use matrices to represent the transformations in question; then, they could use critical observations about group inverses and commutativity from previous tasks to identify one (or more) ways to provide a correct alternative to Todd's work. This task includes the Content Knowledge, Explaining Mathematical Content, Looking Back/Looking Forward, and School Student Thinking types of connections.

Including School Mathematics Teaching Applications in an Undergraduate Abstract Algebra Course

Published online:

04 May 2021

Figure 3. A task from the class activity of Groups of Transformations.

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 $Z_{n}$ , 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 $Z_{n}$ 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 $Z_{n}$ 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. FINDINGS

We organize our findings based upon our four guiding features for the lessons: (1) choosing content connected to teaching; (2) making explicit connections to school mathematics; (3) fostering undergraduate student engagement; and (4) including annotated lesson plans (see Section 2).

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 $Z_{n}$ 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 $Z_{n}$ : 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 $Z_{n}$ 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).

Including School Mathematics Teaching Applications in an Undergraduate Abstract Algebra Course

Published online:

04 May 2021

Figure 5. A student thinking task used as an assessment question in Groups of Transformations, including sample responses from the solution document.

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:

Let's say you can memorize the answer. You just know the simplest path to the right answer. You'd be like, OK, I can get it. You can just find a simple answer or something that you remember your teacher told you, right? If you're presented with a wrong answer, then in fact you actually have to justify why this is wrong. You'd actually have to go back to the main thing. Find a definition or find some reasoning as to why that's wrong.

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 $Z_{n}$ by highlighting a common misconception from secondary mathematics. Secondary mathematics students might

go to solve a quadratic and get to the factoring-well, they think they have factored it. They factor one side and it's equal to, say, four, and then they set each of the factors equal to four and then solve. And if we don't emphasize that we're using this zero product property, with or without saying those words to the high school students, they could have-they could potentially fall into this trap, right?

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:

When I was in high school, I didn't ever think about reflections like that [as matrices]. But I guess if I were to teach in a high school setting …I would definitely be able to have the working background knowledge to be able to give more thorough explanations and not just be like, well, it happens because it happens. I could say, if you look into it, matrices say this, this, and this.

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:

I feel like, after you have that time for independent thought, it's important to bounce those ideas off of other people. Not just – especially not just the teacher, because then you're able to see many different people's points of view from it. Just like we were talking about analyzing the different students' points of view, because I might be Omar [the name of a hypothetical student in a task] and I might be wrong, or someone else might be Omar and they might be wrong, and being able to explain that, or have that be explained to you, and having those explanations and arguments bounce off each other is very important in the learning process.

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."

6. DISCUSSION

Several ideas emerge regarding promising practices for instructors of upper-division mathematics courses who want to include applications to teaching in their courses.

First, a certain amount of lesson framing and probing of student knowledge needs to precede the main activities that include applications to teaching. In our materials, this was addressed in the pre-activity. Critically, instructors used undergraduates' work on these materials to determine where content knowledge support might be needed and to scaffold the main class activity. Instructors were more comfortable implementing the class activity if they had given their students several days to first complete and/or discuss the pre-activity.

Second, instructors used the explicit connections to teaching, included as instructor notes in the annotated lesson plans, to support their implementation of the lesson. These annotations also served as memorable landmarks that instructors could use to orient themselves to a different approach to teaching content. When asked what advice they would give to someone who will be implementing the materials for the first time, most instructors recommended carefully reading the entire lesson, including annotations, to familiarize oneself with the timing and structure of important connections.

Next, employing the active engagement techniques suggested in the lesson plans provided a model for implementing these techniques in upper division courses and promoted an inquiry-oriented learning environment. Undergraduates responded positively to opportunities to collaborate on tasks.

Finally, using tasks that involve student work and understanding student thinking provides meaningful ways to probe undergraduates' mathematical reasoning and understandings. This approach, scaffolded by the annotated lesson plans and solution documents in our materials, can be a refreshing way to engage undergraduates with common misconceptions in mathematics. As some instructors noted, student thinking tasks are relatively novel in general mathematics courses. Undergraduates sometimes feel uncomfortable analyzing student thinking when they themselves are not confident in their own mathematical thinking [2]. As such, instructors should take special care in their facilitation of student thinking tasks when the tasks are first introduced to their undergraduates.

7. CONCLUSION

Accounts of the lessons given by instructors and their undergraduates indicate that our guiding features for the materials supported the integration of applications to teaching in abstract algebra. Instructors with various levels of teaching experience and training in teacher education found the lessons appropriate for their existing courses. They applied our connections to teaching to facilitate active learning strategies and enrich content. Providing these resources to instructors of abstract algebra addresses the MET II call for more applications to mathematics teaching in mainstream courses and offers another way to assess whether undergraduates can use their technical knowledge in practical ways. As more instructors use the META Math materials, we hope to build a body of knowledge of best practices and key features for designing more lessons like these. In doing so, we can enhance the ways in which we address the needs of prospective secondary mathematics teachers in mainstream undergraduate mathematics courses.

Additional information

Funding

This research is based upon work partially supported by the National Science Foundation (NSF) under grant number DUE-1726624. Any opinions, findings, conclusions or recommendations are those of the authors and do not necessarily reflect the views of the NSF.

Notes on contributors

James A. M. Álvarez

James A. M. Álvarez is a Mathematics professor and a Distinguished Teaching Professor at the University of Texas at Arlington where he also serves as the graduate director of the Master of Arts in Mathematics program for practicing secondary mathematics teachers. His primary research interests are in mathematical problem solving and the development of mathematical knowledge for teaching for both prospective and practicing mathematics teachers. He is also active in providing professional development for mathematics faculty associated with implementing Emerging Scholars Programs and using active learning strategies in the classroom.

Andrew Kercher

Andrew Kercher is a Mathematics Ph.D. candidate at the University of Texas at Arlington. His primary research interest is in the development of mathematical problem solving strategies in post-secondary mathematics students, which is the topic of his dissertation. After graduation he plans to pursue a career in academia.

Kyle Turner

Kyle Turner is a graduate student in the Department of Mathematics at the University of Texas at Arlington. His research interests are in undergraduate mathematics education focusing on the transition from secondary to tertiary mathematics. After graduation he plans to pursue a career in academia.

Elizabeth G. Arnold

Elizabeth G. Arnold is an assistant professor in the Department of Mathematics at Colorado State University. Her research interests primarily focus on the use of technology in the classroom and the mathematical and statistical preparation and development of K–12 teachers.

Elizabeth A. Burroughs

Elizabeth A. Burroughs is professor and Department Head in the Department of Mathematical Sciences at Montana State University. Her primary research interests are in the teaching of mathematical modeling and in pre-service mathematics teacher preparation.

Elizabeth W. Fulton

Elizabeth W. Fulton is an assistant research professor in the Department of Mathematical Sciences at Montana State University. Her research interests include mathematical modeling at elementary and secondary grades and development of mathematical knowledge for teaching mathematics for pre-service teachers.

References

Álvarez, J. A. M. , E. G. Arnold, E. A. Burroughs, E. W. Fulton, and A. C. Kercher. 2020. The design of tasks that address applications to teaching secondary mathematics for use in undergraduate mathematics courses. Journal of Mathematical Behavior. 60: 100814.[Crossref], [Web of Science ®] , [Google Scholar]
Álvarez, J. A. M. , A. C. Kercher, and K. R. Turner. 2020. Undergraduates' perceptions of the benefits of working tasks focused on analyzing student thinking as an application for teaching in abstract algebra. In S. Karunakaran, Z. Reed, and A. Higgins (Eds), 23rd Annual Conference on Research in Undergraduate Mathematics Education, pp. 10–17. Boston, MA: Mathematical Association of America. [Google Scholar]
Arnold, E. G. , E. A. Burroughs, E. W. Fulton, and J. A. M. Álvarez. 2020. Applications of teaching secondary mathematics in undergraduate mathematics courses. In TSG33 of the 14th International Congress on Mathematical Education. Shanghai, China: International Mathematical Union. [Paper delayed until July 2021 due to COVID-19 pandemic]. arXiv:2102.04537. [Google Scholar]
Ball, D. L. , M. H. Thames, and G. Phelps. 2008. Content knowledge for teaching: What makes it special? Journal of Teacher Education. 59(5): 389–407.[Crossref], [Web of Science ®] , [Google Scholar]
Braun, V. and V. Clarke. 2006. Using thematic analysis in psychology. Qualitative Research in Psychology. 3: 77–101.[Taylor & Francis Online] , [Google Scholar]
Conference Board of the Mathematical Sciences. 2012. Mathematical Education of Teachers II. Providence RI and Washington, DC: American Mathematical Society and Mathematical Association of America.[Crossref] , [Google Scholar]
Ferrini-Mundy, J. and B. Findell. 2001. The mathematical education of prospective teachers of secondary school mathematics: Old assumptions, new challenges. In T. Rishel (Ed.), Mathematics and the Mathematical Sciences in 2010: What Should Students Know?, pp. 31–41. Washington, DC: Mathematical Association of America. [Google Scholar]
Fraleigh, J. B. 2003. A First Course in Abstract Algebra, Seventh Edition. Reading, MA: Addison-Wesley. [Google Scholar]
Freeman, S. , S. L. Eddy, M. McDonough, M. K. Smith, N. Okoroafor, H. Jordt, and M. P. Wenderoth. 2014. Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences. 111(23): 8410–8415.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]
Gallian, J. A. 2017. Contemporary Abstract Algebra, Ninth Edition . Boston, MA: Cengage. [Google Scholar]
Goulding, M. , G. Hatch, and M. Rodd. 2003. Undergraduate mathematics experience: Its significance in secondary mathematics teacher preparation. Journal of Mathematics Teacher Education. 6(4): 361–393.[Crossref] , [Google Scholar]
Heid, M. K. , P. S. Wilson, and G. W. Blume, Eds. 2015. Mathematical Understanding for Secondary Teaching: A Framework and Classroom-Based Situations. Charlotte, NC: Information Age Publishing. [Google Scholar]
Hiebert, J. and A. K. Morris. 2012. Teaching, rather than teachers, as a path toward improving classroom instruction. Journal of Teacher Education. 63(2): 92–102.[Crossref], [Web of Science ®] , [Google Scholar]
Hungerford, T. W. 2017. Abstract Algebra: An Introduction, Third Edition. Boston, MA: Cengage. [Google Scholar]
National Governors Association Center for Best Practices & Council of Chief State School Officers. 2010. Common Core State Standards for Mathematics. Washington, DC: Authors. [Google Scholar]
Nowell, L. S. , J. M. Norris, D. E. White, and N. J. Moules. 2017. Thematic analysis: Striving to meet the trustworthiness criteria. International Journal of Qualitative Methods. 16: 1–13.[Crossref], [Web of Science ®] , [Google Scholar]
Speer, N. M. , K. D. King, and H. Howell. 2014. Definitions of mathematical knowledge for teaching: Using these constructs in research on secondary and college mathematics teachers. Journal of Mathematics Teacher Education. 18(2): 105–122.[Crossref], [Web of Science ®] , [Google Scholar]
Strayer, J. F. , A. E. Lischka, C. M. Quinn, and L. A. Watson. 2018. Developing preservice secondary mathematics teachers' (PSMTs') mathematical knowledge for teaching (MKT) in content courses. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, and S. Brown (Eds), Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics Education, pp. 826–833. San Diego, CA: Mathematical Association of America. [Google Scholar]
Wasserman, N. , K. Weber, T. Fukawa-Connelly, and W. McGuffey. 2019. Designing advanced mathematics courses to influence secondary teaching: Fostering mathematics teachers' 'attention to scope'. Journal of Mathematics Teacher Education. 22(4): 379–406.[Crossref], [Web of Science ®] , [Google Scholar]
Zazkis, R. and R. Leikin. 2010. Advanced mathematical knowledge in teaching practice: Perceptions of secondary mathematics teachers. Mathematical Thinking and Learning. 12(4): 263–281.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]