# Making the Connection: Research and Teaching in Undergraduate Mathematics Education

Marilyn P. Carlson
Chris Rasmussen
Series: MAA Notes
Volume: 73
Edition: 1
Pages: 333
https://www.jstor.org/stable/10.4169/j.ctt5hh8kb

1. Front Matter
(pp. i-vi)
2. Preface
(pp. vii-viii)
(pp. ix-xii)

• ### 1a. Foundations for Beginning Calculus

• 1 On Developing a Rich Conception of Variable
(pp. 3-14)
Maria Trigueros and Sally Jacobs

Have you ever considered that what mathematicians call ‘variable’ is not a mathematically well-defined concept? And that variable can have different meanings in different settings? Unlike the concept of function, for example, variable has no precise mathematical definition. It has come to be a “catch all” term to cover a variety of uses of letters in expressions and equations. As a result, students are often unclear about the different ways letters are used in mathematics.

Later in this chapter, we provide practical suggestions to help students develop a rich conception of variable as called for by the National Council of...

• 2 Rethinking Change
(pp. 15-26)
Bob Speiser and Chuck Walter

A cat springs from a walk into a run. A spiral shell takes form through growth. With suitable mathematics, what might we learn about the motion or the form? Calculus is said to help with questions about movement, growth and change. How, exactly, does it help us? What might we need to understand? How could we gain such understanding? In this paper we explore two tasks that we designed for students, one about a moving cat and one about a spiral shell. We have written extensively about these tasks (Speiser & Walter, 1994, 1996, 2004; Speiser, Walter & Maher, 2003), as well...

• 3 Foundational Reasoning Abilities that Promote Coherence in Students’ Function Understanding
(pp. 27-42)
Michael Oehrtman, Marilyn Carlson and Patrick W. Thompson

The concept of function is central to undergraduate mathematics, foundational to modern mathematics, and essential in related areas of the sciences. A strong understanding of the function concept is also essential for any student hoping to understand calculus — a critical course for the development of future scientists, engineers, and mathematicians.

Since 1888, there have been repeated calls for school curricula to place greater emphasis on functions (College Entrance Examination Board, 1959; Hamley, 1934; Hedrick, 1922; Klein, 1883; National Council of Teachers of Mathematics, 1934, 1989, 2000). Despite these and other calls, students continue to emerge from high school and...

• 4 The Concept of Accumulation in Calculus
(pp. 43-52)
Patrick W. Thompson and Jason Silverman

The concept of accumulation is central to the idea of integration, and therefore is at the core of understanding many ideas and applications in calculus. On one hand, the idea of accumulation is trivial. You accumulate a quantity by getting more of it. We accumulate injuries as we exercise. We accumulate junk as we grow older. We accumulate wealth by gaining more of it. There are some details to consider, such as whether it makes sense to think of accumulating a negative amount of a quantity, but the main idea is straightforward.

On the other hand, the idea of accumulation...

• ### 1b. Infinity, Limit and Divisibility

• 5 Developing Notions of Infinity
(pp. 55-64)
Michael A. McDonald and Anne Brown

The various notions of infinity are among the most intriguing and challenging concepts in mathematics. Despite their important role in the undergraduate mathematics curriculum, concepts related to infinity typically receive little direct instructional attention. While a number of mathematics education researchers have examined students’ understanding of topics related to infinity, this is still an area rich with unanswered questions. In this chapter, we do not attempt to summarize all of the research in this area and its possible implications for instruction. Rather we look at three specific examples — comparing infinite sets, determining limits of sequences, and constructing infinite iterative...

• 6 Layers of Abstraction: Theory and Design for the Instruction of Limit Concepts
(pp. 65-80)
Michael Oehrtman

Imagine asking a first-semester calculus student to explain the definition of the derivative using the epsilon-delta definition of a limit. Given the difficulty of each of these concepts for students in such a course, you might not be surprised at the array of confused responses generated by a question requiring understanding of both. Since the central ideas in calculus are defined in terms of limits, research on students’ understanding of limits and the ways in which they can develop more powerful ways of reasoning about them has significant implications for instructional design. Throughout this paper we will focus on calculus...

• 7 Divisibility and Transparency of Number Representations
(pp. 81-92)
Rina Zazkis

It is easy for colleagues to agree that students’ understanding is one of the main goals of instruction. It is considerably more difficult to agree on what good understanding of a specific concept entails and how it is possible to achieve it or to assess it. I believe that understanding of any mathematical concept includes the ability to deal with various representations of this concept. As suggested by the title of this article, I focus here on the concept of divisibility and how it may be understood by considering various representations of natural numbers.

Divisibility is one of the main...

• ### 1c. Proving Theorems

• 8 Overcoming Students’ Difficulties in Learning to Understand and Construct Proofs
(pp. 95-110)
Annie Selden and John Selden

When a topologist colleague was asked to teach remedial geometry, he usedSchaum’s Outline of Geometryand also wrote proofs on the blackboard. One day a student, who was familiar with two-column proofs having statements such as ΔABD≅ ΔBCDand reasons such asSAS, blurted out in utter surprise, “You mean proofs can have words!”

This geometry student’s previous experience had led him to an unfortunate view of proof. Other students experience epiphanies about themselves and proof. Asked what she (personally) got out of a transition-to-proof course, one of our students answered, “I learned that I could wake up...

• 9 Mathematical Induction: Cognitive and Instructional Considerations
(pp. 111-124)
Guershon Harel and Stacy Brown

The principle of mathematical induction (MI) is a prominent proof technique used to justify theorems involving properties of the set of natural numbers. The principle can be stated in different, yet equivalent, versions. The following are two versions common in textbooks:¹

Version 1: LetSbe a subset of$\mathbb{N}$(the set of natural numbers). If the following two properties hold, then${S = \mathbb{N}}$.

(i) 1 ∈S.

(ii)kS, thenk+ 1 ∈S.

Version 2: Suppose we have a sequence of mathematical statementsP(1),P(2),… (one for each natural number). If the following two properties hold,...

• 10 Proving Starting from Informal Notions of Symmetry and Transformations
(pp. 125-138)
Michelle Zandieh, Sean Larsen and Denise Nunley

In this chapter we consider the challenge of promoting students’ ability to develop their own proofs of geometry theorems. We have found that students can make use of transformations and symmetries of geometric figures to gain insight into why a particular theorem is true. These insights often have the potential to form the basis for rigorous proofs. In the following classroom vignette, we see the excitement that comes from discovering an idea that seems to explain exactly why a theorem is true, followed by the realization that there is significant work to be done in order to develop a rigorous...

• 11 Teaching and Learning Group Theory
(pp. 139-152)
Keith Weber and Sean Larsen

Abstract algebra is an important course in the undergraduate mathematics curriculum. For some undergraduates, abstract algebra is the first mathematics course in which they must move beyond learning templates and procedures for solving common classes of problems (Dubinsky, Dautermann, Leron, and Zazkis, 1994). For most undergraduates, this course is one of their earliest experiences in coping with the difficult notions of mathematical abstraction and formal proof. Empirical research studies attest to students’ difficulties in abstract algebra; these studies have shown that many students do not understand fundamental concepts in group theory (e.g., Leron, Hazzan, and Zazkis, 1995; Asiala, Dubinsky, Mathews,...

• 12 Teaching for Understanding: A Case of Students’ Learning to Use the Uniqueness Theorem as a Tool in Differential Equations
(pp. 153-164)
Chris Rasmussen and Wei Ruan

Students in many undergraduate mathematics courses tend not to readily and appropriately use theorems as tools for making arguments and solving problems (Schoenfeld, 1989; Hazzan & Leron, 1996). Students’ reluctance to use theorems as tools is a problem that is not only cognitive in nature (that is, the difficulty is in how students conceptualize particular mathematical ideas), but also social in nature (that is, the nature of class discussion, the interpretation of tasks and ideas, etc.). In this chapter we highlight results from a classroom-based research program in differential equations that has resulted in some positive progress on the problem of...

• ### 2a. Interacting with Students

• 13 Meeting New Teaching Challenges: Teaching Strategies that Mediate between All Lecture and All Student Discovery
(pp. 167-178)
Karen Marrongelle and Chris Rasmussen

A growing number of postsecondary mathematics educators are exploring teaching strategies other than lecture (Holton, 2001). The motivations for such change include personal dissatisfaction with student learning, students’ poor retention of knowledge, student dissatisfaction with their undergraduate experiences in science, mathematics, and engineering (National Science Foundation, 1996; Seymour & Hewitt, 1997), as well as efforts to rethink core courses such as calculus, linear algebra, and differential equations. As postsecondary educators make changes to their practice they often struggle with many of the same issues that K–12 mathematics teachers encounter as they attempt to change their practice. In this chapter we...

• 14 Examining Interaction Patterns in College-Level Mathematics Classes: A Case Study
(pp. 179-190)
Susan Nickerson and Janet Bowers

While discussing the pedagogical challenges of teaching an undergraduate discrete math course, one of our colleagues recently lamented that

Students are ill-prepared for this course…but this ill-preparation is a curious issue. I think it has more to do with the way they learned mathematics than with the content of the previous courses.

In this chapter, we propose a response to his comment. In particular, the goal of our discussion is to illustrate that the ways in which teachers and students interact can profoundly affect the attitudes students formas well asthe content they learn.

This view of the importance...

• 15 Mathematics as a Constructive Activity: Exploiting Dimensions of Possible Variation
(pp. 191-204)
John Mason and Anne Watson

Mathematics is often seen by learners as a collection of concepts and techniques for solving problems assigned as homework. Learners, especially in cognate disciplines such as engineering, computer science, geography, management, economics, and the social sciences, see mathematics as a toolbox on which they are forced to draw at times in order to pursue their own discipline. They want familiarity and fluency with necessary techniques as tools to get the answers they seek. For them, learning mathematics is seen as a matter of training behaviour sufficiently to be able to perform fluently and competently on tests, and to use mathematics...

• 16 Supporting High Achievement in Introductory Mathematics Courses: What We Have Learned from 30 Years of the Emerging Scholars Program
(pp. 205-220)
Eric Hsu, Teri J. Murphy and Uri Treisman

This article is aimed toward faculty in mathematics departments who are working to increase the number of high-achieving mathematics students from racial and ethnic minorities and for researchers investigating these endeavors. The Emerging Scholars Program (ESP) is one of the most widespread models for supporting such increases. It is also one of the oldest, so there is a considerable body of research, both quantitative and qualitative, related to its impact. Whether or not one chooses to implement an ESP, this discussion of the history, philosophy, structure, impact, and future of the program will highlight important and emerging themes that any...

• ### 2b. Using Definitions, Examples and Technology

• 17 The Role of Mathematical Definitions in Mathematics and in Undergraduate Mathematics Courses
(pp. 223-232)
Barbara Edwards and Michael B. Ward

One of the earliest subjects of undergraduate mathematics education research was students’ difficulties in writing formal mathematical proofs. Some research focused on the heuristics involved in proof-writing, but early attempts to show that the teaching of heuristics and strategies benefited students’ proof-writing skills (Bittinger, 1968; Goldberg, 1973) failed to produce statistically significant results. Other difficulties have been identified, including students’ weak understanding of logic and/or mathematical concepts and their definitions (cf. Hart, 1986; Moore, 1994). Several recent studies have looked further at students’ proof-writing skills (cf. Dreyfus, 1999, Harel & Sowder, 1998; Selden & Selden, 2003); this topic is also addressed in...

• 18 Computer-Based Technologies and Plausible Reasoning
(pp. 233-244)
Nathalie Sinclair

The purpose of this chapter is to describe how computer-based tools can help students in the doing and learning of mathematics, and to provide specific examples that illustrate the way in which well-designed technologies can support mathematical discovery and understanding. I begin with an example.

Task. Take the three vertices of a triangleABCand reflect them each across the opposite side of the triangle to obtain a new “reflex” triangleDEF(convince yourself you always do indeed get a new triangle!). Repeat the process. Most people who have seen this problem conjecture that the reflex triangle, after several iterations,...

• 19 Worked Examples and Concept Example Usage in Understanding Mathematical Concepts and Proofs
(pp. 245-252)
Keith Weber, Mary Porter and David Housman

Elsewhere in this volume, Watson and Mason discuss example generation from the students’ perspective by highlighting some of the ways that example generation can be used to increase students’ understanding of mathematics and improve their attitudes toward mathematics. This chapter complements this work by describing ways that teachers and textbooks might use examples to help undergraduates better understand mathematics. We distinguish between using worked examples to solve exercises and problems and using examples to help promote students’ understanding of mathematical concepts and proofs. We begin with worked examples provided by the teacher or textbook. We then discuss the role of...

• ### 2c. Knowledge, Assumptions, and Problem Solving Behaviors for Teaching

• 20 From Concept Images to Pedagogic Structure for a Mathematical Topic
(pp. 255-274)
John Mason

The principal aim of this chapter is to provide a structure for mathematical topics as an aid to ‘psychologizing the subject matter’, as Dewey (1933) put it. The secondary aim is to reveal just how complex a matter preparing to teach a topic effectively can be, beyond trying to make the definitions and theorems as clear as possible.

The chapter develops the notion of aconcept image(Tall & Vinner 1981) into a description of a framework based on a threefold structure of the psyche. Two mathematical topics, quotient groups and L’Hôpital’s rule, are used to illustrate how the framework can...

• 21 Promoting Effective Mathematical Practices in Students: Insights from Problem Solving Research
(pp. 275-288)
Marilyn Carlson, Irene Bloom and Peggy Glick

Mathematicians and mathematics educators have been curious about the processes and attributes of problem solving for over 50 years. As mathematics teachers at any level of education, we want to know what teaching practices we can employ to help our students develop effective problem solving abilities. This curiosity has led to numerous investigations of the attributes and processes of problem solving. In this chapter, we describe insights from a study we conducted of the mathematical practices of 12 research mathematicians. We believe these insights are useful to teachers striving to promote mathematical practices in students at all levels—from first-grade...

• 22 When Students Don’t Apply the Knowledge You Think They Have, Rethink Your Assumptions about Transfer
(pp. 289-304)
Joanne Lobato

Teaching so that knowledge generalizes beyond initial learning experiences is a central goal of education. Yet teachers frequently bemoan the inability of students to use their mathematical knowledge to solve real world applications or to successfully tackle novel extension problems. Furthermore, researchers have been more successful in showing how people fail totransferlearning (i.e., apply knowledge learned in one setting to a new situation) than they have been in producing it (McKeough, Lupart, & Marini, 1995). Because we are most frequently prompted to reflect upon transfer when it doesn’t occur, this chapter begins with an undergraduate teaching vignette in which...

• 23 How do Mathematicians Learn to Teach? Implications from Research on Teachers and Teaching for Graduate Student Professional Development
(pp. 305-318)
Natasha Speer and Ole Hald

Scenario 1¹ At a pre-semester orientation session for mathematics graduate students, the following question and student work were presented:\begin{align*} & \text{If }y={{(3-{{x}^{2}})}^{3}},\ \text{what is }\frac{dy}{dx}? \\ & \text{Student work: }\frac{dy}{dx}=3{{(3-{{x}^{2}})}^{2}}\cdot (-2x)\cdot (-2). \\ \end{align*}

The graduate students were asked to describe what a student might have been thinking when producing such an answer. After a few moments, the question was repeated, but none of the graduate students offered a potential explanation. Then, a professor who was sitting in the room said, “Well, it’s not a bad answer.” He then explained how the student’s answer showed a pretty solid understanding of the chain rule, but that the student had applied the rule repeatedly instead of...

6. About the Editors
(pp. 319-320)