Taming the Coverage Beast

By Priscilla Bremser, Contributing Editor, Middlebury College

By the end of every workshop and conference session on Inquiry-Based Learning that I’ve attended, someone has raised a hand to ask about coverage. “Don’t you have to sacrifice coverage if you teach this way?” Of course coverage took center stage for many of my professional conversations long before I tested the IBL waters; it’s important. But an equally important question is this: What do we sacrifice when coverage dominates? It may well be conceptual understanding; it’s possible to cover more ground, albeit thinly, if we settle for procedural understanding instead. More than once I’ve settled for even less, delivering a quick lecture just so that my students will have “seen” a particular idea. How do we strike a balance between coverage and other considerations when we are so practiced at reducing a course description to a list of topics? Continue reading

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Inquiry-Oriented Instruction: What It Is and How We Are Trying to Help

By Estrella Johnson, Assistant Professor of Mathematics Education at Virginia Tech University, Karen Keene, Associate Professor of Mathematics Education at North Carolina State University, and Christy Andrews-Larson, Assistant Professor of Mathematics Education at Florida State University

Making fundamental changes to the way you teach is a difficult task. However, with a growing number of students leaving STEM majors, instructors’ dissatisfaction with student learning outcomes, and research indicating positive avenues for improving undergraduate mathematics instruction, some instructors are ready and eager to try something new. In this post, we describe some promising research-based curricular materials, briefly identify specific challenges associated with implementing these materials, and describe a recently funded NSF project aimed at addressing those challenges.

Teaching Inquiry-Oriented Mathematics: Establishing Supports (TIMES) is an NSF-funded project (NFS Awards: #143195, #1431641, #1431393) designed to study how we can support undergraduate instructors as they implement changes in their instruction. A pilot is currently being conducted with a small group of instructors. In the next two years, approximately 35 math instructors will be named TIMES fellows and will participate in the project as they change their teaching of differential equations, linear algebra, or abstract algebra. As project leaders, we will study how to best support these instructors, as well as how their instructional change affects student learning. More details about the project follow later in this blog post.

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The Power of Undergraduate Researchers

By Audrey St. John, Associate Professor of Computer Science at Mount Holyoke College

When I first started teaching, I was mystified (and, frankly, at times panicked) at the thought of having undergraduates work with me on research. I realized this was part of the job, part of my institution’s mission, but I just couldn’t figure out how it would be effective. Sure, these students were bright, eager and motivated to learn, but how much could they contribute with such limited time? A typical research experience might be 8-10 weeks during the summer (full time) or 10 hours a week during a semester; best case, I might find a student who would work with me for a couple years in this way. I had just finished six years in grad school and still felt like I knew nothing. On top of that, my research is at the intersection of computer science and math with applications in the domains of engineering and biology – would I be able to find students with experience in even two of these fields? As it turns out, I would soon discover how powerful research with undergraduates can be, and I’d like to share some of the lessons I’ve learned over the years. Continue reading

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Creating the 2015 CUPM Curriculum Guide

by Martha J. Siegel, Professor, Towson University

I serve as chair of the Mathematical Association of America’s (MAA’s) Committee on the Undergraduate Program in Mathematics (CUPM). Approximately every ten years, CUPM publishes a new curriculum guide, with the primary goal of assisting mathematics departments with their undergraduate offerings. Over five years in the making, the 2015 Curriculum Guide to Majors in the Mathematical Sciences encourages departments to engage in a process of review and renewal, by examining their own beliefs, interests, resources, mission, and particularly their own students in designing or revising a major in mathematics or, more generally, in the mathematical sciences. In the remainder of this blog post, we discuss the history, development, process, and key characteristics and recommendations of the 2015 Guide. Continue reading

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One Reason Fractions (and Many Other Topics) Are Hard: Equivalence Relations Up and Down the Mathematics Curriculum

By Art Duval, Contributing Editor, University of Texas at El Paso 

Why are fractions hard to learn for so many people?  There are many reasons for this, but I like to think about one in particular, a mathematical idea hiding in plain sight, from elementary school to college: equivalence relations.  Consider the fraction sum 2/3 + 1/5, which we of course compute by using 2/3=10/15 and 1/5=3/15, arriving at an answer of 13/15.  This raises a whole host of fundamental questions about equality:  If 2/3 equals 10/15, why can we use one but not the other in evaluating the sum?  Does this mean something is wrong with our idea of “equals”?  Could we have used something else besides 10/15; or, in the other extreme, should we always use 10/15?  This shows that often when we say “equals”, what we really mean is “equivalent”.  Equivalence introduces a number of useful mathematical connections, but we must be careful in how we handle it with our students who just want to know, for instance, how to add two fractions.

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Mathematical education of teachers Part II: What are we doing about Textbook School Mathematics?

By Hung-Hsi Wu

This two-part series is a summary of the longer paper, Textbook School Mathematics and the preparation of mathematics teachers.

TSM (Textbook School Mathematics) has dominated school mathematics curriculum and assessment for the past four decades, yet, in mathematics education, TSM is still the elephant in the room that everybody tries to ignore.

We will look at three examples of this phenomenon. Continue reading

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Mathematical Education of Teachers, Part I: What is Textbook School Mathematics?

By Hung-Hsi Wu

This two-part series is a partial summary of the longer paper, Textbook School Mathematics and the preparation of mathematics teachers.

School mathematics education has been national news for at least two decades. The debate over the adoption of the Common Core State Standards for Mathematics (CCSSM) even became a hot-button issue in the midterm elections of 2014. This surge in the public’s interest in math education stems from one indisputable fact:  school mathematics is in crisis.

From the vantage point of academia, two particular aspects of this crisis are of pressing concern: School textbooks are too often mathematically flawed, and in spite of the heroic efforts of many good teachers, the general level of math teaching in school classrooms is below acceptable.

Mathematicians like to attack problems head-on. To us, the solution is simple: Just write better school textbooks and design better teacher preparation programs. I will concentrate on the latter for now and will not return to the textbook problem until the end of Part 2. Continue reading

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Mathematics Professors and Mathematics Majors’ Expectations of Lectures in Advanced Mathematics

By Keith Weber, associate professor of Mathematics Education at Rutgers’ Graduate School of Education. Dr. Weber is one of the faculty in Rutgers’ Proof Comprehension Research Group.

Introduction
The advanced proof-oriented courses for mathematics majors are typically taught in a lecture format, where much of the lecture is comprised of presenting definitions, theorems, and proofs. There is a general perception amongst mathematicians and mathematics educators that these lectures are not as effective as they could be. However, the issues of why these lectures are not effective and how they might be improved are not discussed often in the mathematics education literature. In my research, I have sought to address this issue. Through task-based interviews with students and discussions about pedagogy with mathematicians, as well as observations of lectures and students’ reactions to them, I have found that mathematics professors and mathematics majors have different expectations of lectures and these different expectations lead to barriers in communication. By expectations, I am referring to (i) what a student is supposed to learn from, or “get out of”, a lecture and (ii) how students should engage in the lecture to understand this content. As a consequence of these different expectations, students do not gain what mathematicians hoped they would from the lectures they attend. Below I describe four such differing expectations and how they might inhibit lecture comprehension, with the hope that discussing these differing expectations might help us improve the teaching of proof at the undergraduate level. Continue reading

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Community, Professional Advice, and Exposure to New Ideas at the Carleton Summer Mathematics Program

By: Sarah Blackwell, mathematics major, Saint Louis University; Rose Kaplan-Kelly, mathematics major, Bryn Mawr College; and Lilly Webster, mathematics major, Grinnell College

Editor’s note: The editorial board believes that in our discussion of teaching and learning, it is important to include the authentic voices of undergraduate students reflecting on their experiences with mathematics. We thank Ms. Blackwell, Ms. Kaplan-Kelly, and Ms. Webster for contributing their essay. The American Mathematical Society maintains a list of summer mathematics programs for undergraduates and has published Proceedings of the Conference on Promoting Undergraduate Research in Mathematics as a resource for mathematicians interested in similar programs.

We were participants in the Summer Math Program for Women Undergraduates (SMP) at Carleton College, a program with the goal of encouraging and supporting women undergraduates in their study of mathematics during their first two years of college. For four weeks we took math classes, listened to math talks, went to problem sessions, and talked about math for fun. We had the opportunity to meet many mathematicians from across the country. The people we met did not fit into the mold of the solitary eccentric that popular culture would have us believe. We met mathematicians who defied negative stereotypes often attributed to people in STEM areas and especially to women who are interested in math. Learning about their projects and interests helped us to see ourselves as capable of becoming mathematicians as well. In talking to them, we started to see what our lives could be like if we pursued math as a career and learned that there was no single “correct” type of person we would need to become. SMP was also an opportunity to meet mathematicians who worked outside of academia, mainly in applied math, which was not an area many of us had been exposed to before. This expanded our view of what being a mathematician might be like and what we could achieve. Continue reading

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Using Primary Source Projects to Teach Mathematics

by Janet Barnett, Colorado State University – Pueblo; Dominic Klyve, Central Washington University; Jerry Lodder, New Mexico State University; Daniel Otero, Xavier University; Nicolas Scoville, Ursinus College; and Diana White, Contributing Editor, University of Colorado Denver

Mathematics faculty and educational researchers are increasingly recognizing the value of the history of mathematics as a support to student learning. The expanding body of literature in this area includes recent special issues of Science & Education and Problems, Resources and Issues in Undergraduate Mathematics Education (PRIMUS), both of which include direct calls for the use of primary historical sources in teaching mathematics. Sessions on the use of primary historical sources in mathematics teaching at venues such as the Joint Mathematics Meetings regularly draw large audiences, and the History of Mathematics Special Interest Group of the Mathematical Association of America (HOMSIGMAA) is one of the largest of the Association’s twelve special interest groups. In this blog post, which is adapted from a recent grant proposal, we explore the rationale for implementing original sources into the teaching and learning of undergraduate mathematics, and then describe in detail one method by which faculty may do so, namely through the use of Primary Source Projects (PSPs).

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