Students view themselves as passive consumers of others' mathematics.

One of the most vivid memories of my education comes from an upper division probability class, when my instructor was about to introduce the binomial theorem. She stopped writing the statement of theorem at the point where she needed to write the formula. "I never remember this formula," she said, " but it's so easy to derive that you don't need it anyway." Then she showed us how to derive the formula. What she showed us made sense. To this day I can't remember the formula, but I can derive it, either when I need it (which is rare) or because the thought of it brings back pleasant memories. The idea that was brought home in that class -- that mathematics really makes sense, and that you can figure something out if you need to -- was exhilarating. It is (or should be) part of the pleasure of learning mathematics.

Such moments were rare in my experience as a student, and they were almost completely absent from the classes we observed. The mathematics instruction that we observed consisted almost exclusively of training in skill acquisition. For each of the years K-12 (and beyond; calculus instruction in college is pretty much the same), there was an agreed-upon body of knowledge, consisting of facts and procedures, that comprised the

curriculum. In each course, the task of the teacher was to get students to master the curriculum. That meant that subject matter was presented, explained, and rehearsed; students practiced it until they got it (if they were lucky). There was little sense of exploration, or of the possibility that the students could make sense of the mathematics for themselves. Instead, the students were presented the material in bite-sized pieces so that it would be easy for them to master. As an example, recall the step-by-step procedure for constructions, described above, that was used by the teacher of the target class. Constructions were introduced that way, and students were given practice that way. When, for example, a student had difficulty with a particular problem, the teacher reminded him that the problem called for a construction with which the student was familiar. He then asked: "In your construction, what is step number one?" The student replied correctly. The teacher continued. "Good. In your construction, what is step number two?" And so on. In this way, students got the clear impression that someone else's mathematics was theirs to memorize and spit back. Nor was step-by-step memorization limited to constructions. Recall that the Regents exam had required proofs as well; students were told to commit them to memory. This was standard practice, and was promoted as being both efficient and desirable. For example, an advertisement for a best-selling series of review books for the Regents exams proudly announced: "Students like these books because they offer step-by-step solutions."

The point I wish to stress here is that students develop their understanding of the mathematics from their classroom experience with it. If the "bottom line" is error-free and mechanical performance, students come to believe that that is what mathematics is all about. In the target class, for example, the teacher talked about how important it was for students to think about the mathematics, and to understand it. He pointed out the fact that they should not memorize blindly, because if they did "and forgot a step" they would be in trouble. In truth, however, this rhetoric -- in which the teacher honestly believed -- was contradicted by what took place in his classroom. The classroom structure provided reinforcement for memorization, and the reward structure promoted it. One of the items on our questionnaire, for example, asked students to agree or disagree with the statement "the math that I learn in school is mostly facts and procedures that have to be memorized." With a score of 1 indicating "very true" and a score of 4 indicating "not at all true," this item received an average score of 1.75 -- the third strongest "agree rating" of seventy questions. Yet the statement "When I do a geometry proof I get a better understanding of mathematical thinking" received an average score of 1.99 -- again very strong agreement. These data parallel the secondary school data, where students claimed that mathematics is mostly memorization but that mathematics helps a person to think logically. Our classroom observations supported Carpenter et al.'s (1983, p. 657) conjecture that the "latter attitudes may reflect the beliefs of their teachers or a more general view rather than emerge from their own experience with mathematics." More importantly, the latter attitudes did not influence behavior: when working mathematics problems, the students behaved in accord with the three mathematics beliefs discussed

above.