David M. Bressoud August, 2008
MAA has just published an important Notes volume that I hope will get the attention it deserves: Making the Connection: Research and Teaching in Undergraduate Mathematics Education, edited by Marilyn Carlson and Chris Rasmussen. It is a compendium of 23 research articles in undergraduate mathematics education, and it is important enough that I will be devoting two columns to its discussion.
None of the articles describes controlled studies that can be used to convince your colleagues that one particular curriculum or pedagogical approach is more effective—however that might be measured—than another. Somewhat apologetically, the editors explain why such studies are very difficult to pull off. In fact, what these articles do reveal is much richer and more interesting . They provide windows into how students think about mathematics and suggest how to overcome obstacles to student understanding. Many of the chapters offer insights about the reasoning abilities that support students in understanding and using central ideas of precalculus and calculus. The results are sometimes obvious, at least in retrospect, but that can be the mark of a profound insight. And just because it is obvious does not necessarily imply that we who teach pay enough attention to it.
The lessons from these articles were reinforced by a fortuitous coincidence. I am just back from the 11th Annual Legacy of R. L. Moore Conference in Austin, an invigorating and inspiring experience. The Moore method, strictly interpreted, is purely self-constructed discovery-based learning. Students are given definitions and axioms, and are then individually challenged to prove a sequence of theorems that leads them into and through the material of the course. The instructor can be coach and critic, but is never allowed to explain a proof or provide a solution. Recourse to texts or other reference sources is strictly forbidden. Lecturing is out of the question. The conference, in fact, was much broader than this, encompassing a range of approaches to active learning. The common thread was that all speakers recognized and emphasized the need for students to use exploration and discovery to construct a personal understanding of key mathematical concepts.
In Bob Eslinger's introductory remarks to the conference, he noted how research into student learning supports active learning. On the second day, when Keith Weber talked about his research into how students learn to prove mathematical theorems, he turned this around, remarking how well inquiry-based learned accords with the insights coming out of educational research. The importance of active, exploratory learning is the consistent message that comes through this Notes volume.
The first half of the book collects articles that examine how students think about basic mathematical concepts: divisibility, variables, functions, limits, definite integrals, infinity, proofs, symmetry arguments, proof by induction, groups and cosets, and solutions of differential equations. These articles examine the errors students make, the flawed or inadequate reasoning that leads to these errors, and some promising methods for overcoming these student difficulties.
There is a common and consistent observation that students approach new mathematical ideas by falling back on what has worked for them in the past. Once a means of solving a certain type of problem is mastered, there is a great reluctance to break out and to think about how a similar but distinctly different problem might better be approached. For example, in the article on divisibility by Rina Zazkis , she found that pre-service elementary teachers when asked whether 3^3 * 5^2 * 7 is divisible by 15 would pull out their calculator, multiply out the number, divide by 15, and check that the quotient is an integer.
In Michael Oehrtman's article on limits , he discusses a Tall and Vinner  problem that that they showed creates great difficulty, even for senior mathematics majors in the British system who had had at least two years of experience with epsilon-delta proofs:
True or false: Suppose as x —> a then f(x) —> b, and as y —> b then g(y) —> c. Then it follows that as x —> a then g(f(x)) —> c.
Twenty-one of twenty-two students insisted this is true, even when pressed if they were really certain. The problem here is the intuitive understanding of limit as a process of approaching a value and the failure to recognize that this notion of approaching has different meanings in the hypothesis and conclusion. In the hypothesis, equality is excluded, but not in the conclusion. Thus, the conclusion of the first premise is not equivalent to the hypothesis of the second, but students had great trouble seeing this because they were wedded to an intuitive understanding of limit that had served them well in most situations. The lesson is that mere exposure to precise definitions and theorems, even if repeated over an extended period of time, is not sufficient to make them part of a student's working knowledge.
The most effective strategy for overcoming these self-imposed blinders is to force students to explore precisely the sorts of situations that cannot be handled by the methods with which they are most comfortable, that force them to expand their working knowledge. In an enlightening passage, Zazkis describes taking the same students who had pulled out the calculator to determine divisibility by 15 and asking them whether 3^30 * 5^20 * 7 is divisible by 15. The calculator procedure is useless here. Once students realized this, they looked for an alternate approach, and now recognized that because both 3 and 5 are factors of this number, so must 15 be a factor. Having broken through on their own to see this, they now approached similar problems that could be done with the calculator, but now recognized that the calculator was not needed. I strongly suspect that telling students that there are counter-examples to the Tall and Vinner statement, requiring them to find one, and then asking them to reflect on why their initial analysis was incorrect, would succeed in expanding their understanding of limits.
Keith Weber and Sean Larsen in their article on "Teaching and Learning Group Theory"  explain Gravemeijer's  structured pedagogical approach using "developmental research projects" to accomplish this expansion of understanding. The process begins in recognizing the relevant informal knowledge and strategies that students are likely to bring to a new mathematical topic. It continues as they are presented with problems or projects that draw on this knowledge and these strategies. It culminates in forced reflection that leads to the formal definitions and theorems. Many of the articles in this volume take precisely this approach and show how it can be implemented. Unlike the strict Moore method, there is a role for lecture and explanation by the instructor, but it is focused on this reflective piece that formalizes the insights students have already achieved. It reminds me of my first experience with Project CALC materials in my last year at Penn State. There was a consistent pattern of introducing a new idea, spending time exploring it on computers, and then coming back to class to formalize the insights that had been gained in the laboratory. At the end of the semester, one student summarized this by saying that the initial lecture was always very confusing, but she was never worried because she knew she would get a chance in the lab to figure out what was really going on.
Guershon Harel and Stacy Brown  explain how this approach can work to develop student understanding of proof by induction. Traditionally, induction is taught by first giving the formal statement of the principle of mathematical induction. Students are given problems in which the inductive relationship is explicit, problems such as proving that the sum of the first n positive odd integers is equal to n^2. The inductive relationship is "explicit" because it is readily apparent in reading the problem that the sum of the first n odd integers is equal to the sum of the first n – 1 odd integers plus 2n – 1. Eventually, students are challenged to tackle problems in which the inductive relationship is implicit, problems such as determining the minimum number of moves needed to transfer n discs in a Tower of Hanoi problem. But students have been so conditioned to tying their working knowledge of the principle of induction to an explicit recursion that they are at a loss when they try to apply this principle in solving a problem with an implicit relationship.
After discussing some of the errors of understanding that students bring to applications of the principle of induction, including trying to apply it in cases where it is not relevant, Harel and Brown argue for a complete inversion of the traditional progression. Students come to the study of induction with an intuitive appreciation for building one case upon a previous case. This is where the teaching should begin. There are two problems that Harel and Brown have found to be effective at getting students started working toward the principle of induction:
- Find an upper bound for the sequence sqrt(2), sqrt(2 + sqrt(2)), sqrt( 2 + sqrt(2 + sqrt(2))), ...
- Given 3^n coins, all identical except for one that is heavier, and given a balance, show that you can find the heavy coin in n weighings.
Both of these problems involve an implicit recursive relationship. The real challenge is to find this relationship. Once the relationship is found, students sense that they are done. This is the point at which they need to reflect on why that finishes the problem, and this is the point at which the instructor can introduce the principle of induction as a means of formalizing their intuitive understanding.
I particularly resonated with this article because it confirms what I have observed in my own discrete math class. In a previous column, Pólya's Art of Guessing, I talked about the project based on Pólya's video "Let Us Teach Guessing" that I give to my students, finding a formula in terms of binomial coefficients for the number of regions in space formed by n random planes. After checking the first few cases and making their conjectures, my students quickly realize that the key to proving their conjecture is to find the inductive relationship. They struggle with this, but they have no trouble recognizing that once they know how to go from the formula for n – 1 planes to the formula for n planes, they are done. What I've learned from the Harel and Brown article is that I need to build in a reflective piece for the end of this project to ensure that my students see how they are using the formal principle of induction.
In next month's column, I'll explore the second half of this book.
 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.
 Rina Zazkis, Divisibility and transparency of number representations, pages 81–91 in .
 Michael Oehrtman, Layers of abstraction: Theory and design for the instruction of limit concepts, pages 65–80 in .
 D. Tall and S. Vinner, Concept image and concept definition in mathematics with particular reference to limits and continuity, Educational Studies in Mathematics, vol 12, 1981, pages 151–169.
 Keith Weber and Sean Larsen, Teaching and learning group theory, pages 139–151 in .
 K. Gravemeijer, Developmental research as a research method, pages 277–296 in A. Sierpinska and J. Kilpatrick, editors, Mathematics Education as a Research Domain: A Search for Identity, Kluwer, Dordrecht, The Netherlands, 1998.
 Guershon Harel and Stacy Brown, Mathematical induction: Cognitive and instructional considerations, pages 111–123 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 email@example.com. This column does not reflect an official position of the MAA.