excerpted from "What do we know about mathematics curricula?" Journal of Mathematical Behavior, Volume 13, #1 (1994), pp. 55-80
What do we want students to learn in mathematics courses, and how do we characterize what they have learned? I begin with a story that casts the issue in sharp relief. The story is true, with proper names replaced by variables because the specifics of people and place are irrelevant - and because the conversation described might well have taken place at dozens of universities around the country.
When I joined the mathematics department at University X some years ago I had a chat with the department chair, Professor Y, about my problem solving course. Y was truly supportive. He endorsed the idea of offering the course and helped me decide the level at which it should be offered. (Previous incarnations of the course had been given for student groups that ranged from first year liberal arts majors to advanced mathematics majors.) We decided the course should be offered as a lower division elective, so that it would be accessible both to potential mathematics majors and to non-majors who might want a dose of mathematical thinking above and beyond formal course requirements. After we had assigned it a course number, and right after Professor Y mentioned that the course might become an attractive option for mathematics majors, the following dialogue took place.
Y: Of course we can't give credit toward the major for taking the course.
AHS: Why not?
Y: Because you're not teaching content.
I said that there was indeed content to the course: In solving the problems I assigned in the course discussions the students did a reasonable amount of elementary number theory, geometry, and so on. But Y didn't buy the argument, because the content wasn't identified and packaged in the standard ways; I didn't cover specific theorems, bodies of knowledge, etc. (And, it is true that the students and I work through material in the problem solving course at a much slower pace than we would in a lecture class.) So I tried a different tack.
AHS: I'll tell you what. The goal of the course is simple: I want the students to be able to solve problems I haven't explicitly taught them how to solve. The real test is my final exam, which is a collection of difficult problems that don't look very much like the ones I discuss in the course. Here's a bet. Suppose I have 20 kids in the class, just the ones who've enrolled for it. You can hand pick 20 junior and senior math majors and give them the exam. I'll bet you my semester's salary that my kids will outperform yours on the final.
Y: I'm sure you're right, but we still can't give credit toward the major. You're not teaching them content - you're just teaching them to think.
What can one say? Only that three years later Professor Z, chair of the computer science department, asked me if I would be willing to have his department require my course of all their majors. It was the only elementary mathematics course in which they learned anything, he said, and the department was interested in the possibility of removing the calculus requirement and requiring my course instead. (Since I was about to leave for another university, the swap didn't happen.)
The moral of this story and the reason that I tell it is that it demonstrates clearly that what counts as mathematical "content" depends on one's point of view. From Professor Y's perspective, the mathematical content of a course (or a lesson, or a whole curriculum) is the sum total of the topics covered. This, of course, is the traditional view. It lies behind the use of standard "scope and sequence" charts to characterize school curricula, the traditional "Math 9 is Algebra I, Math 10 is Geometry, and Math 11 is Algebra II/Trig" curriculum labels, and, for example, the delineation of a proposed new course outline for Math 1A (first-semester majors' calculus) at Berkeley: "Limits and rate of change, 4 lectures; derivatives, 7 lectures; exp., log, and inverse trigonometric functions, 8 lectures. . . ." It is functional in curricular terms, because it fits with the standard, hierarchical notion of curriculum structure: Topics in course A are prerequisites for course B, and so on. This point of view is familiar. I grew up with it, as did the vast majority of my colleagues. It is comfortable, and it is dangerous.
The danger in this kind of "content inventory" point of view comes from what it leaves out: the critically important point that mathematical thinking consists of a lot more than knowing facts, theorems, techniques, etc. This understanding was at the heart of Polya's work, and it has been significantly elaborated over the past two decades. In a nutshell, the emerging view of mathematics learning differs significantly in perspective, scope, and detail from the traditional view. In terms of overall perspective, the difference can be captured in a simple phrase: I would characterize the mathematics a person understands by describing what that person can do mathematically, rather than by an inventory of what the person "knows." That is, when confronted with both familiar and novel situations that call for producing or using mathematical ideas, can someone do so effectively? Note that this performance standard is the one that Professor Y lives by in his professional life, and the one that he uses to judge his colleagues. University mathematicians get promotion and tenure by producing new mathematics - i.e., by developing new or deeper understandings, seeing new connections, solving unsolved problems, etc. In or out of the university, mathematicians earn their keep by posing problems and solving them, either by developing new approaches or by recognizing that a new problem, looked at in the right way, is closely related to a familiar (and solvable) one. The key point here is that mathematicians have to use what they know; just knowing isn't enough. It is ironic that this is precisely the standard I proposed for the evaluation of my problem solving class - could my students outperform others, who had far more training? - and that he rejected, because it didn't fit with his notion of content. He was happy to agree that I taught students to think mathematically, and impressed by the students' work: My students have produced results that led to a paper published in the College Mathematics Journal, also some minor results in number theory that are unfamiliar to most mathematicians. But, trapped by a view that - at least in classroom settings - understanding should be measured in terms of "the quantity of recognizable mathematics known," he had no way to assign value to the things my students had learned to do.
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