David M. Bressoud November, 2008
I am finally able to return to a discussion of the MAA Notes volume, Making the Connection: Research and Teaching in Undergraduate Mathematics Education  that was begun in my August column, Student Thinking. The first half of this volume examines what we know about how undergraduates think through the mathematics they are studying and how to identify the obstacles to comprehension. The second half is a more varied collection of reports on research in the teaching of undergraduate mathematics that explores the question of what works in the classroom and what we, as teachers, can do to more effectively facilitate student learning. As in the first half, these articles prove nothing. There are no controlled studies that definitively establish one approach as significantly better than another. But these articles do contain concrete suggestions that are rooted in understanding how students develop their mathematical knowledge. Any thoughtful teacher who wants to improve the learning of her or his students will find big ideas that can help shape a curriculum or program and many small but powerful ideas that can help one decide how to present a new concept and help build student understanding.
One of the big ideas is the Emerging Scholars Program (ESP) that is described in an article by Eric Hsu, Teri J. Murphy, and Uri Triesman . This is a highly successful program for increasing the number of high-achieving mathematics students from racial and ethnic minorities. It stresses building excellence, diversity, and community. Begun at Berkeley in 1974, it has been adopted widely. This article summarizes the lessons that have been learned over many years at many different types of institutions. ESP has not always worked, but the authors now have a pretty good idea of what it takes to make it work and the missteps that can make it ineffective. The article stresses the importance of being honest about how well one's introductory courses actually introduce, how well the preparatory courses prepare. It deals frankly with the sensitive and difficult issue of how narrowly such a program can or should be targeted on minority students. And it talks about how to build a program that will last.
Several articles, including those by Marrongelle and Rasmussen , Nickerson and Bowers , and Mason and Watson  deal with involving students as active participants in the class and helping them to construct their own understanding. Most of us are dissatisfied with straight lecture. We want interaction with our students. These articles provide some insights into when and how to open up the class, how to elicit the exchanges that illuminate ideas for our students, and how to help students build a rich sense of the mathematics.
The articles by Edwards and Ward , Weber, Porter, and Housman , and Mason  all draw on David Tall's ``concept image,'' the working understanding of a concept that mathematicians and students bring to their use of that idea. By comparing and contrasting naive and sophisticated concept images, they help us understand why students fail at certain tasks. Edwards and Ward focus on the use of definitions, Weber, Porter, and Housman on the use of examples, and Mason explores the breadth and importance of concept images. Though not explicit, concept images and the question of how students interpret and remember an example are lurking beneath the surface of Joanne Lobato's discussion  of student difficulties in transfer of mathematical knowledge from one setting to another.
Carlson, Bloom, and Glick  tackle problem-solving, exploring what differentiates those who are successful problem-solvers from those who are not. They discovered certain key habits of successful problem solvers beginning with the importance of spending time on orientation before starting to solve a problem. This includes trying to categorize the problem and then to determine what tools are available that might be helpful. They saw the importance of coping mechanisms when dealing with frustration, mechanisms that enable the persistence that is needed in problem-solving. They also recognized the need for frequent reflection on one's own reasoning. Finally, the authors discuss how to develop these habits in our students.
The article by Sinclair  explores the use of computers to support student exploration and the development of reasoning skills. The last article, by Speer and Hald , looks at the educational training of graduate students and how to help them read student responses in order to understand where students are stumbling as they try to learn mathematics.
Taken as a whole, this book is a useful window into what is and is not known about how undergraduates learn. It is clear that this field is still in its infancy. One is left with far more questions than answers. And yet, there is also a tremendous wealth of good ideas, insights that we do well to keep in mind as each of us thinks about how we can be more effective in our own classrooms.
 Marilyn Carlson and Chris Rasmussen, editors, Making the Connection: Research and Teaching in Undergraduate Mathematics Education, MAA Notes #73, Mathematical Association of America, Washington, DC, 2008.
 Eric Hsu, Teri J. Murphy, and Uri Triesman, Supporting High Achievement in Introductory Mathematics Courses: What We Have Learned from 30 Years of the Emerging Scholars Program, pages 205–220 in .
 Karen Marrongelle and Chris Rasmussen, Meeting New Teaching Challenges: Teaching Strategies that Mediate Between All Lecture and All Student Discovery, pages 167–178 in .
 Susan Nickerson and Janet Bowers, Examining Interaction Patterns in College-Level Mathematics Classes: A Case Study, pages 179–190 in .
 John Mason and Anne Watson, Mathematics as a Constructive Activity: Exploiting Dimensions of Possible Variation, pages 191–204 in .
 Barbara Edwards and Michael Ward, The Role of Mathematical Definition in Mathematics and in Undergraduate Mathematics Courses, pages 223–232 in .
 Keith Weber, Mary Porter, and David Housman, Worked Examples and Concept Example Usage in Understanding Mathematical Concepts and Proofs, pages 245–252 in .
 John Mason, From Concept Images to Pedagogic Structure for a Mathematical Topic, pages 255–274 in .
 Joanne Lobato, When Students Don't Apply the Knowledge You Think They Have, Rethink Your Assumptions about Transfer, pages 289–304 in .
 Marilyn Carlson, Irene Bloom, and Peggy Glick, Promoting Effective Mathematical Practices in Students: Insights from Problem Solving Research, pages 275–288 in .
 Natalie Sinclair, Computer-Based Technologies and Plausible Reasoning, pages 233–244 in .
 Natasha Speer and Ole Hald, How Do Mathematicians Learn To Teach? Implications from Research on Teachers and Teaching for Graduate Student Professional Development, pages 305–317 in .
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David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and president-elect of the MAA. You can reach him at firstname.lastname@example.org. This column does not reflect an official position of the MAA.