Including School Mathematics Teaching Applications in an Undergraduate Discrete Mathematics Course

Abstract

This paper introduces lessons designed to incorporate applications to teaching high school mathematics in an undergraduate Discrete Mathematics course. Because many prospective high school teachers do not take courses that are specifically designed for teachers, providing materials aimed at teacher preparation that can be easily integrated into courses that serve a general mathematics major is one strategy for addressing mathematics teacher preparation. We developed lessons using four guiding features: choosing appropriate content; making school mathematics connections; incorporating active learning; and providing robust lesson notes. Our interview-based findings document how the lessons were used by instructors and analyze the mathematical ideas and understandings that arose from the use of the lessons at two different sites.

Keywords:

• Mathematical knowledge for teaching
• discrete mathematics
• prospective high school mathematics teachers

1. INTRODUCTION

You have just finished implementing a lesson on the binomial theorem. Your undergraduates can prove the theorem and apply it to expand binomials. Have they deepened their understanding of the binomial theorem to the extent that they could help a high school student who is learning the theorem? How would your undergraduates answer the following question shown in Figure 1? Because prospective high school teachers are among the population of undergraduates frequently enrolled in a Discrete Mathematics course, there is an opportunity for instructors to attend to the needs of prospective teachers by incorporating applications to teaching into the curriculum. Figure 1 shows an example of such an application.

Figure 1. Henry's Question: An example of a teaching application.

Teacher preparation programs aim to equip prospective high school teachers with the mathematical and pedagogical knowledge needed to teach mathematics to school students. The Mathematical Education of Teachers (MET) II Report [3] sets forth a program of study for the mathematical preparation of high school teachers that includes courses specifically designed for prospective mathematics teachers. Yet, for many teacher preparation programs, the limitations imposed by small programs with low prospective teacher enrollments make it difficult, if not impossible, to deliver a program that includes courses that only enroll prospective teachers. The MET II Report acknowledges that, despite their recommendation of specialized courses, often prospective high school teachers will take standard courses for mathematics majors; they recommend spending time in these undergraduate courses “looking back” at the content of school mathematics and “examining connections” between what prospective teachers are studying in college and what they will be teaching their future students as an important component of a mathematics major that prepares prospective teachers [3, p. 54].

The question of how mathematics instructors should proceed when teaching prospective teachers alongside other students is an important one for teacher preparation programs. It can be challenging, in practice, for instructors to implement the recommendations of the MET II Report in a course that is not exclusively for prospective mathematics teachers. What does it mean to “examine connections” between undergraduate and school mathematics? And, having identified some connections, how should instructors develop a treatment of existing content in order to provide experiences that can lead prospective teachers to an in-depth understanding of high school mathematics?

The difficulty of including teaching applications in mathematics courses can also be attributed, in part, to the culture of mathematics and the value it places on teaching compared to other career paths. Core courses in an undergraduate mathematics major curriculum often have designated examples and exercises attending to specific application areas such as physics, engineering, and business. These problems serve to emphasize the utility of mathematics in these areas, to legitimize the application areas as requiring deep mathematical thinking, and more tacitly to help undergraduates understand the breadth of career opportunities available for those with robust mathematical preparation. Yet, applications specifically tailored to teaching (such as Henry's Question presented in Figure 1) are often minimal or nonexistent in these courses [8].

In response to these needs, the META Math Project is focused on adding secondary mathematics teaching explicitly to the list of application areas of core undergraduate mathematics major courses by creating instructional materials to help instructors integrate the recommendations set forth in the MET II Report for a variety of mathematics major courses. With these materials, we focus on enhancing prospective high school teachers' understanding of the connections between undergraduate mathematics and teaching school mathematics. These connections to teaching address both the mathematical content and interpersonal interactions integral to teaching. Our materials increase awareness of these connections among all undergraduates in the course, even those not intending to pursue teaching as a career, in a way that deepens their own understanding of undergraduate mathematics. Further, our materials prompt undergraduates to consider the mathematics content from a teacher's perspective; we've found that this focus on the learning process leads undergraduates to engage in valuable reflection about their own mathematics learning.

This work complements that of other researchers, who over the last decade have made progress in understanding the benefits of studying advanced mathematics as part of high school teacher preparation. Several researchers have also focused on examining connections to teaching mathematics. Lai and Patterson [8], for instance, have investigated the extent to which tasks that direct attention to the work of teaching are incorporated in textbooks for prospective elementary and secondary mathematics teachers. Patterson [12] analyzed real analysis textbooks and mapped concepts within these textbooks to K–12 content standards. Wasserman [13] has worked on categorizing types of connections between advanced mathematics and school mathematics. He investigated different ways prospective teachers can engage in developing mathematical knowledge for teaching in their undergraduate mathematics major courses and articulated four different points of connections to teaching. These connections extend beyond mathematical content connections and include pedagogical connections that can be made in the mathematics courses taken by prospective teachers. Although researchers have identified connections to teaching, prospective teachers often do not recognize connections to teaching after completing their courses [15], indicating that further attention is needed to support mathematics instructors in teaching mathematics courses as part of teacher preparation programs.

Mathematics teacher educators have focused on creating materials to support instructors in highlighting connections to teaching in their courses. These materials have been developed using different design features and have been created for different purposes. Wasserman et al. [14] focused on designing supplemental tasks, specifically for prospective secondary teachers, in content courses. They studied K–12 mathematics curricula and practices and selected concepts that are related to undergraduate mathematics. Their tasks combine mathematical knowledge and pedagogical knowledge integral to teaching. Heid et al. [6] focused on developing materials for prospective and practicing secondary teachers. They studied school classrooms and identified instances of mathematics content that are related to undergraduate mathematics. These instances are presented as situations that motivate the task and prompt undergraduates to investigate the mathematics contained in these situations. Our goals – making connections to teaching high school mathematics – fit alongside the work of [6,14]. Like Wasserman et al., our tasks combine mathematical knowledge and pedagogical knowledge needed for teaching. What differs are our intended purpose and approach to making connections. We intend for these materials to replace standard lessons in content courses and to be used by all undergraduates, regardless of whether they are prospective teachers. To create connections to teaching, we started by examining undergraduate mathematics curricula and identified topics central to school mathematics curricula and practices.

META Math is a multifaceted project, creating materials for use in Statistics, Calculus, Abstract Algebra, and Discrete Mathematics courses and providing research about their development and use. The full lessons are available via https://tinyurl.com/METAMathLessons. The purpose of this article is to describe the Discrete Mathematics lessons we have created. In order to help other instructors understand how the materials can be used, we discuss how instructors and undergraduates responded to the materials and how they reacted to the teaching applications that are contained in the materials. What we present is not a case study or an assessment study, but rather is a description of the guiding features of the materials along with a demonstration of how two instructors addressed connections between undergraduate and school mathematics and incorporated these connections into their existing course, with the intent to lead prospective teachers to an in-depth understanding of high school mathematics.

2. GUIDING FEATURES FOR META MATH LESSONS

The MET II Report [3] emphasizes that it is important for prospective teachers to “examine connections,” but does not provide specifics of exactly what this examination entails. We have articulated a set of connections between undergraduate mathematics and teaching secondary mathematics (see [2]), and we have developed instructional materials that embed these connections. When designing these materials, we remained focused on four key features: (1) choosing content already embedded in the undergraduate mathematics course that is relevant for prospective teachers to understand at a deep level; (2) making explicit connections between 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 teaching in a manner that makes explicit connections. The first two guiding features enable us to highlight how undergraduate mathematics relates to high school mathematics, while the second two guiding features attend to important aspects of a well-designed and engaging lesson.

2.1. Choosing Content

We have identified content areas that are studied in undergraduate mathematics and that have a clear grounding in the specific topics studied in high school. For example, the undergraduate content might be related to an explicit high school content standard that prospective teachers will need to teach their future students; the binomial theorem lesson described below is an example of this type. Other times, the mathematics that undergraduates learn is not usually a specific high school content standard, but rather, the undergraduate mathematics provides prospective teachers with foundational knowledge that may better help them explain mathematical concepts to their students; the foundations of divisibility lesson described below is an example of this type.

The binomial theorem is a topic that is frequently taught in high school, and it is a theorem frequently taught in an undergraduate Discrete Mathematics course. This lesson (see Figure 2 for activity questions) motivates the binomial theorem by examining how it arises in various contexts such as Pascal's triangle, multiplying polynomials, and binomial coefficient identities. The lesson offers an opportunity for undergraduates to learn combinatorial proof techniques and compare them to more familiar algebraic techniques.

Figure 2. The pre-activity and class activity questions from the binomial theorem lesson.

Notions of divisibility provide an underpinning of much mathematical study, flowing from arithmetic, to the division algorithm, greatest common divisors, and the Euclidean algorithm. Divisibility of the natural numbers is at the beginning of any study of number theory and relies on arithmetic accessible to most fifth-grade students. This understanding of divisibility of integers is extended to divisibility of polynomials in high school algebra. The lesson (see Figure 3 for activity questions) guides undergraduates to provide explanations about divisibility properties of integers using the definition of divisibility, the quotient-remainder theorem, and knowledge about the base-ten representation of numbers. Studying divisibility rules, though not usually a mathematics content standard, provides undergraduates with the opportunity to engage with the mathematical practice of “looking for, and making use of structure” [11].

Figure 3. The pre-activity and class activity questions from the foundations of divisibility lesson.

2.2. School Mathematics Connections

In response to the recommendations set forth in the MET II Report [3], we have defined five tangible components of lessons that connect undergraduate mathematics to school mathematics and can be implemented by mathematicians who teach prospective teachers but have no particular training in teacher preparation [2]. These connections to teaching include the ideas displayed in Table 1. In Álvarez et al. [1], we further describe how we designed tasks that embed these five connections to teaching. These connections are explored in relation to the two lessons throughout the remainder of this paper.

Table 1. Five types of connections to teaching.

Connection	Description
Content Knowledge	Undergraduates use course content in applied contexts or to answer mathematical questions in the course.
Explaining Mathematical Content	Undergraduates justify mathematical procedures or theorems and use of related mathematical concepts.
Looking Back/Looking Forward	Undergraduates explain how mathematics topics are related over a span of K–12 curriculum through undergraduate mathematics.
School Student Thinking	Undergraduates evaluate the mathematics underlying a student's work and explain what that student may understand.
Guiding School Students' Understanding	Undergraduates pose or evaluate guiding questions to help a hypothetical student understand a mathematical concept and explain how the questions may guide the student's learning.

2.3. Active Learning

We incorporate these five types of connections in the lessons by creating materials from an active learning perspective (see [4,5]). Each lesson emphasizes the use of group work, class activities, and allocates time for students to share their solutions, both in small group and large group settings. Because such practices may be new to instructors and undergraduates, we provide robust instructional notes to facilitate the implementation of the lessons. Both lessons consist of inquiry-based pre-activity and class activity assignments, a set of homework problems, and two assessment questions. They offer an opportunity for undergraduates to examine work from a hypothetical high school student and to consider how they can guide high school students' understanding of the mathematical concepts.

2.4. Lesson Notes

We created a guide for instructors, called an “annotated lesson plan” [7, p. 94]. Many instructors create teaching notes containing examples, theorems, and definitions for themselves before teaching a lesson; the annotated lesson plan can be viewed as a generalized way to consolidate comprehensive information about teaching a topic that can be shared among many instructors. An annotated lesson plan is designed to offer more detailed instructional considerations than found in textbooks. It encompasses more than lecture notes; the annotations also include information about “what to do” and “why/how to do it that way” [7, p. 95]. “What to do” prescribes content for instructors to implement in the classroom while “why/how to do it that way” provides sound reasoning for the prescription, together with useful information for achieving its goals [10]. Specific to our project, these annotations also explain how the undergraduate content relates to school mathematics.

Figure 4 highlights some additions brought forth in an annotated lesson plan as seen on a supplemental handout displaying the class activity with annotations and possible solutions. This example contains the questions presented to the students along with four types of annotations: explanations about the intent of the questions; common student responses and common student errors; guidance for how an instructor can implement the activity as intended; and explanations of how the question is an application to teaching high school mathematics that addresses the needs of prospective teachers.

Figure 4. Henry's Question with annotations.

We note that annotated lesson plans are not scripted lessons. Rather, we encourage instructors to follow the spirit of the annotated lesson plan, which focuses on making connections between the undergraduate mathematics and school mathematics.

3. DATA SOURCES

Two instructors at different institutions implemented both lessons in their classes. We conducted two post-lesson interviews with each instructor to discuss their overall reactions to the lessons, how they used the materials, and the connections to teaching high school mathematics they and their students made during the lesson. We also conducted a final interview with each instructor at the end of the term, after they had given a final exam that covered material from the entire course, including the assessment questions we provided. At the interview, instructors discussed their perceptions of the purpose of the lessons, how the lessons connect to school mathematics, and how the lessons support the preparation of prospective high school mathematics teachers. We also asked about how the lessons fit with their usual instruction, including their perceptions of their students' learning and the use of active learning during the lessons.

As part of their regular coursework, all undergraduates in these two instructors' classes completed the pre-activity, class activity, homework questions, and assessment questions for each lesson. Those who consented to participate in our study gave us permission to analyze their work from the lessons, and some undergraduates from each class were invited to participate in a semi-structured interview. During these interviews, we asked undergraduates to discuss their work from the lessons, their perceptions of how the content in each lesson connects to school mathematics, the use of group work, and how the lesson did or did not strengthen their understanding of the mathematical content.

We then transcribed all of the interviews and used a descriptive coding process [9] to qualitatively analyze them to describe how the guiding features supported instructors and undergraduates in making connections to teaching. Both instructors and their students are identified by pseudonyms.

4. IMPLEMENTATION OF THE LESSONS

4.1. Participants and Setting

Two instructors taught both lessons in their Discrete Mathematics classes. Maria is a visiting assistant professor at a small liberal arts college on the west coast. Her college has a small teacher preparation program and there were no identified prospective teachers enrolled in her class. The majority of her students were mathematics or computer science majors, and she typically relies on lecture-based instruction. Nicholas is a professor at a large public university in the Pacific Northwest. His university has a substantial teacher preparation program and he had at least one identified prospective teacher in his class. He indicated that he uses some group work strategies in his class.

4.2. Fitting the Lessons into Their Courses

The binomial theorem lesson covers the binomial theorem and combinatorial proof techniques. Undergraduates are expected to know how to multiply polynomials and that $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ counts the number of ways to select k objects from n objects without repetition. The anticipated length for this lesson was two 50 min class periods. Maria taught the lesson near the end of the combinatorics section in her course and used one 75 min class period to implement the lesson. She had already covered some combinatorial proofs and counting techniques. Nicholas implemented this lesson at the beginning of a counting unit because his course textbook uses the binomial theorem to introduce the combinatorics unit. He taught the lesson during two 50 min class periods. Nicholas thought the lesson worked well prior to learning counting because “there isn't much sophisticated counting [in the lesson].”

The foundations of divisibility lesson offers the opportunity to apply the quotient-remainder theorem, use the definition of divisibility, and write proofs using cases and numbers expressed in base-ten expanded form. Maria taught this lesson after the modular arithmetic and number theory sections of her course, and she found “the timing to be perfect.” She implemented the lesson in one 75 min class period. Nicholas's course did not cover the definition of divisibility or the quotient-remainder theorem, so this lesson was not a replacement lesson like it was for Maria or like the binomial theorem lesson was for both of them. Nicholas used two 50 min periods to implement the lesson and treated it as a review before the final, intending to review ideas of proof and the use of qualifiers and cases.

4.3. How Instructors Used the Instructional Materials

Both Maria and Nicholas reported that they read through the annotated lesson plans several times and took note of the key points and connections to make during each lesson. Both instructors brought a printed copy of the annotated lesson plan with them to class, and while both wanted to have the copy with them, they reported using it as a “safety blanket” and mostly taught from an internalized understanding of the lesson gained from reading the lesson plan multiple times. When asked what advice he has for other instructors using these materials, Nicholas advised others to “sit down and read through the materials. It's worth taking the time and kind of reading through the anticipated [student] responses because they really do happen.”

5. FINDINGS

We present our findings based on our four guiding features.

5.1. Mathematical Content

When discussing the mathematical content embedded in each lesson, both Nicholas and Maria emphasized how it builds on content their undergraduates should have seen prior to college, and they articulated why the undergraduate content was important for teaching school mathematics. Nicholas discussed how both lessons “[occupied] a very nice role between very familiar school mathematics and very (to them) cutting edge upper level math” and provided his students with the opportunity to study a “familiar topic with a slightly more advanced perspective on it.” Nicholas also viewed the binomial theorem as “must-see content” for prospective teachers. His take on the binomial theorem lesson was that his students can see “[binomial] expansions in sort of a new way because of the counting” and they can “now leverage their counting ability to predict how many terms should combine.” He said that it is important for teachers to understand the connection between the binomial theorem and polynomial expansion: “Anyone who is teaching polynomial expansion – I would really want them to know that connection to the binomial theorem.” He even stated, “I got the sense that [some undergraduates] were thinking about why [the binomial theorem] was related to Pascal's triangle.”

Nicholas expressed how there were important mathematical concepts that prospective teachers were exposed to in the foundations of divisibility lesson. This lesson not only exposed undergraduates to the use and visualization of numbers in the base-ten system, it also gave his students an opportunity to use proofs to understand the mechanics behind divisibility rules. He stated,

If you wanted to understand why those familiar divisibility tests work, like summing the digits for divisibility by 9, the answer is right there. If you can leverage that definition of divisibility along with some clever manipulation of the expression for base-ten numbers, then you can kind of backfill some of those gimmicks or number tricks with a little more – well not just like confidence that they're true, but the understanding of the mechanics of why they're true.

He appreciated how this lesson presented proof writing in a way that explains why the divisibility rules really work, rather than just as an activity focusing solely on the purpose of doing proofs “because it's on the syllabus.”

Maria expressed how she found these lessons to be valuable for prospective teachers because of the skills they gained by engaging with the lessons. Her discussion of the importance of the content in these lessons was primarily focused on the foundations underlying the school mathematics to be taught; she also emphasized the opportunity given to undergraduates to see multiple perspectives and approaches to a problem, aiding them in explaining or understanding the mathematics they will be teaching. After the foundation of divisibility lesson, she expressed how prospective teachers will likely not teach their future students the divisibility proofs that they worked on in the activity, but they “might need to use some of the information from what we proved to clarify or justify [mathematics to their students].” Maria also expressed how the content in the binomial theorem lesson was useful because it provides prospective teachers different ways to teach polynomial expansion beyond the “FOIL method.” She discussed how the lesson prompted her students to also “look at the summation or look at Pascal's triangle” and see how they are related. She viewed the opportunity to look at different approaches to a problem as “valuable for a high school algebra teacher to know.” Overall, Maria thought that by viewing mathematics through the perspective of teaching, all undergraduates (non-prospective teachers included) had to think deeply about the mathematical content of the lessons. She described how understanding the mathematics of a school student's work is a skill prospective teachers need and that the applications to teaching embedded throughout the lessons “got [her undergraduates] really thinking.”

Undergraduates in both classes also expressed how these lessons, focused on applications to teaching, were useful. Many, for instance, recalled using the binomial theorem or Pascal's triangle in high school as a “trick” to expand binomials, but stated how they didn't understand the connection. A few undergraduates expressed how these lessons prompted them to get in the mindset of a teacher which helped their mathematical understanding. One, in particular, stated, “Teaching is a universal skill. Everyone should have some teaching skills. You need the skills and they help you understand. There's no better way to understand something than to try to teach someone those things.”

5.2. School Mathematics Connections

Both Maria and Nicholas are mathematicians; they have not been formally trained to prepare prospective teachers. After implementing these lessons, both discussed how the lessons helped them better prepare prospective teachers and make connections to teaching they would not have done otherwise. Furthermore, their students, even those who do not see themselves in a teaching role in the future, also found value in making connections between the mathematics they were studying and the mathematics taught in high school.

Maria specifically discussed her personal challenge connecting undergraduate mathematics to school mathematics and how the annotations helped her in overcoming this challenge. She stated,

[Connecting undergraduate mathematics to school mathematics] was kind of the biggest gap I had to bridge. But I thought [the annotations] were super helpful for that. So, in general, it made some good points [for] me as an educator. I learned a lot [from] reading [the annotations].

She further discussed how she used many of the annotations focusing on connections for her own background knowledge of a high school curriculum. For instance, she was not familiar with how topics such as the binomial theorem are taught in high school prior to implementing this lesson.

Maria said that both lessons emphasized the school student thinking connection and discussed how incorporating this type of connection into her curriculum was a new instructional strategy for her. She enjoyed these types of applications to teaching and from her perspective, they pushed her undergraduates to think more deeply about the mathematical content. Many of Maria's students echoed her comments and stated how the lessons helped them to deepen their understanding of the mathematics because they were pushed to explain a procedure or theorem they knew, or because they were prompted to examine an idea that they had not previously considered, or because they encountered questions (such as Henry's Question) that identified a misconception they held themselves. For example, one student said that the activities made her explain specifically what the student errors were in addition to what was mathematically incorrect. Instead of explaining what the correct answer was, she said she had to describe how to fix the student work to arrive at the correct solution. She said that this required more thinking about the mathematics. Another student said “teaching is a way of learning” and that by viewing the problem though the context of teaching the topic, he was pushed to deepen his own understanding of the material.

Like Maria, Nicholas also described clear connections in the lessons between the college mathematics content and content in high school. He said that the “vertical connections stuck out and seemed most novel and front and center in the activity.” He also discussed how the school student thinking connections were unique to these materials and not something he has previously done in his classes. He thought these lessons encouraged students to engage in deep mathematical thinking by analyzing student work. He particularly referenced an activity problem that focuses on a hypothetical student and a mistake he made when using the binomial theorem (i.e., Henry's Question), acknowledging that it was one of his favorite things about the lesson.

When it came to [Henry] and [his] mistake, that, for me, was the most interesting discussion they had. It was really good to see some students talking through, in third person, about mistakes that I've seen students make themselves. First of all, the groups were engaged on that one. …[The question] was concrete enough and confusing enough mathematically that they really had to parse through and talk about what was happening. …That [conversation] was good, and they were describing solutions and tips for avoiding mistakes that they themselves would typically make.

Nicholas concluded by stating how even though he was aware that topics in Discrete Mathematics relate to high school mathematics, these lessons gave him “more experience at just directly and less suddenly employing [connections to teaching] in the classroom.” He expressed how the strategy of analyzing student work and making sense of it was more beneficial to his undergraduates than “just kind of giving a throwaway line like, by the way this is related to the binomial theorem that you might remember from high school.”

Nicholas's undergraduates also made comments that aligned with his perspective about the benefits of examining student work. The following three students highlighted different ways looking at student work is useful for understanding mathematics. One undergraduate, Zachary, confirmed that analyzing student work encouraged deep thinking beyond calculations and procedures. Referencing a question from the lesson, he stated,

[If I was just asked to compute the expansion,] my calculator can do n choose k, so I don't even have to understand what [the calculator is] doing …I can just plug numbers into the equation. But when you're asked how does [Henry] not understand this? Well now all of a sudden I have to actually understand what is happening here. So it made me do the work to actually understand what the binomial theorem was.

When reflecting on the student thinking questions, Gazelle discussed how it is important to also understand the why behind the mathematics.

These [student thinking questions] are asking more in-depth questions …in case someone asks you more than just how to solve this, but more like why. I feel like …a lot of times we're not given the chance to think about why things are the way they are, why they come out that way. It's nice to take some time to do that and see if you do this, what will happen.

Peter, an undergraduate who is not preparing to be a teacher, stated how analyzing other students' thinking helped prepare him to not make similar mistakes. He expressed how the lessons would help prepare prospective teachers “both for learning [the content] and for practicing teaching.” He elaborated how it was helpful how the activity pushed him to think of questions to help guide hypothetical students' misunderstandings rather than just telling them how to correctly solve the problem. Thinking of these questions made him “think more critically about the problem.”

5.3. Active Learning

Both Maria and Nicholas taught in classrooms with rows of tables, so while students were not seated in groups, they could easily work with the people next to them. Maria did not use group work in her class outside of these two lessons and Nicholas did occasionally use group work in his class. Maria thought that group work mostly went well, despite having some behavioral issues with one group and uncertainty about timing. She thought that the group work aspect probably would have gone better had she included it in her course earlier and had the annotated lesson plans included suggested amounts of time to spend on each part of the activities. Because the use of group work and inquiry-based activities in her class was “newish” to her instruction, she “wasn't sure how [long] to wait for [her students] to get it right and volunteer and do it” versus directly telling her students how to proceed. While group work is not something Maria uses in her class, she liked the activities and stated how being asked to implement group work was “really good for me” and something she would like to use again in her teaching. Nicholas also expressed how the group work component of the lessons would have gone better had he incorporated group work into his course more often prior to implementing these lessons. He discussed how half of his students were eager to work with each other while the other half were reluctant and had to be pushed to talk with each other. Overall, Nicholas felt that the lessons supported student learning and the best way to address the challenges of engaging all students in group work and inquiry-based activities is through repetition that is started early in the semester.

When asked how these two lessons fit in with how Maria and Nicholas teach their classes, most of their undergraduates we interviewed identified how these lessons were different than others but that the lessons “fit very well.” Even in Maria's class, which was typically lecture-based, undergraduates felt that their learning was not generally different from other non-META Math lessons. For example, one undergraduate expressed how these lessons had a different format and it was a different way of learning, but that she learned the material as well as she learned previous concepts from the course. Another undergraduate described how Maria's mode of instruction shifted during these lessons, stating how it “felt like a fun day in class” and how “it very much felt more like an activity that I was doing with someone else.” Another, who generally preferred Maria's style of teaching where he was “just being given the information concisely [and] well” also acknowledged the different role undergraduates had during these lessons and stated how it worked because he felt the content was something they had previously learned: “This is something that we've learned and maybe we're just getting a more in depth of an understandingof it.”

5.4. Lesson Annotations

Both instructors enjoyed implementing the annotated lesson plans and receiving activities that they did not have to create themselves. Nicholas stated how he “learned both content and process.” In terms of content, he learned a new way to motivate and introduce the quotient-remainder theorem. In terms of process, he saw the benefits of incorporating student thinking questions such as Henry's question into the curriculum, referencing the conversations that these types of questions generated among his undergraduates during the lessons. He discussed how these conversations helped him learn about their mathematical knowledge, saying “this material gave me a different kind of window into what they know.”

Maria liked the annotations in the lessons because they first alerted her to questions that undergraduates might have and then helped her navigate conversations with her undergraduates around these questions. For instance, one annotation in the foundations of divisibility lesson emphasized how undergraduates will likely have questions about how the use of cases encompasses both components of an if and only if proof. Maria was thankful for this annotation, stating

I was actually really glad that the [annotations] addressed this with [the cases]. …I appreciate the opportunity to …talk about why the cases were giving you both directions [in the if and only if proof]. …I was glad that the [annotations] spelled that out because that's not something I would have anticipated right out of the gate.

Nicholas also described how helpful the annotations were for him when implementing the lessons. He felt prepared for his undergraduates' questions and how to respond to them. The annotations also helped him know what mathematical ideas to focus on, stating,

And in fact there was a point in the [binomial theorem lesson] where [an annotation] prompted me to make sure that by the time we get to this next part you're going to want them to know the quotient-remainder theorem. And so that was a bacon saver.

Overall, Nicholas is aware that he prepares many prospective teachers, but does not consider himself an expert. He discussed how the annotations easily helped him connect ideas in his course to teaching high school mathematics, stating,

[The lessons] did a good job of getting me in and out with low overhead. I felt comfortable using these specific activities with my particular students who aren't all going to be teachers. It seemed like it was well designed for me to kind of quickly say ‘hey, here's why we're doing it, here's what we're going to do [and] here's how it relates.’ …the lessons definitely did enough for me to feel like I could connect [the course to ideas of teaching].

6. CONCLUSION

This report about how mathematicians use these materials demonstrates some ways that the guiding features we used in our design of these lessons met the needs of the instructors. That is, instructors who have not been specially trained in mathematics teacher preparation find these materials easy to use and find them to enrich their existing Discrete Mathematics courses. Our analysis of their interview transcripts leads us to conclude that the materials provide the focus on applications to teaching that we intend. Furthermore, we find that directly addressing teaching connections has not detracted, in the eyes of undergraduates or of instructors, from the mathematical content that was learned by all undergraduates in the course. In fact, the focus on teaching connections has led to some undergraduates self-reporting that they were pushed to think more deeply about the mathematics. We suggest that META Math lessons can enrich undergraduate Discrete Mathematics courses by providing rich learning opportunities for all students while attending to the specific needs of prospective teachers.

Additional information

Funding

This work was partially supported by the National Science Foundation (NSF) [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

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.

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 mathematical and statistical preparation and development of K–12 teachers and the use of technology in the classroom.

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.

James A. M. Álvarez

James A. Mendoza Álvarez received his Ph.D. in Mathematics from The University of Texas at Austin. He is a professor of mathematics 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 in 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 job in academia.

Kyle Turner

Kyle Turner is a graduate student in the Department of Mathematics at the University of Texas at Arlington. His research interest includes undergraduate mathematics education. After graduation he plans on entering academia.