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October 12, 2008


Possibly you are correct about multiple-choice testing having a (negative) impact on learning math, however I'm interested in hearing more about what kinds of theorem proving you think is necessary to excel in math (to the level that one might be considered the peer of an Olympiad contestant).

For example, when I was in HS, I had to do a lot of math proofs, not just from the standard curriculum, but in topics introduced in the experimental math program I was in, such as affine geometry and abstract algebra. This, presumably, was to bolster certain math skills that would be used later, ie. in college. However, it didn't help me as much as I thought (or hoped) it would. There were still classes I struggled in, whereas other people who didn't have comparable math backgrounds (in terms of proving theorems) seemed to do better with ease. After many years of thinking about this, I've come to the conclusion that these students had better problem-solving strategies than I did. Early out in the problem-solving process of a given problem, they were able to determine which initial steps were likely to result in an answer, and were less likely to get stuck or tripped up. Since exams were mostly one or two hours, this quickness was crucial to doing well. I've noted that some coaching books for math competitions that are now on the market emphasize problem-solving strategies over other things, like actual proofs, even.

Do you think it's more valuable to spend a lot of time on proofs, or problem-solving strategies? For example, a question I like to ask people sometimes, just to see if they can figure it out, is which set is larger, the integers or rationals? There are proofs that one can do to answer this question, but as far as I know, the proof technique is not generalizable to any set (e.g. reals). So a problem-solving strategy is also important to recognize which proof technique is likely to more quickly result in an answer. As this relates to engineering majors, for example, some EEs take a class at MIT called Analysis I (18.100), which is known at other universities as "advanced calculus". Depending upon which version is taken, one might have to do that type of proof. But because MIT is a "firehose", a lot of topics are covered in the class (not all equally well), so a student with a weaker problem-solving technique (but experience doing more rigorous proofs) might have problems that a student with the opposite background (stronger problem-solving, but less rigor).

Some students become frustrated and feel like giving up on math (and some even do so).

On a somewhat related subject, there is a school of thought that suggests that the way that math (and some related subjects like computer science) are taught is not so much based on fundamental principles, but an "agenda" - some intended result the instructors want to accomplish. As this relates to girls learning math, there is a feeling that the common agenda put forth does not value the sorts of skills girls are likely to bring to bear in solving certain types of engineering problems. You can find out more about this sort of thing in papers such as M.S. Mahoney's "BOY'S TOYS AND WOMEN'S WORK: FEMINISM ENGAGES SOFTWARE" ( http://www.princeton.edu/%7Emike/boystoys.htm ).

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