The New York Times has an article in its October 10th edition on the US failing to develop the math skills of its most talented students. While the fact that the US K-12 educational system lags behind many countries is not new, the study stands out in its focus on the best students, especially girls, rather than on the assessment of average trends in the general student population. It achieves that goal by analyzing results from international math competitions, as opposed to SAT scores. The report offers obvious reasons for the situation - "American culture does not highly value talent in math, and so discourages girls" - and reinforces the well-known fact that "many students from the United States in these competitions are immigrants or children of immigrants from countries where education in mathematics is prized." The Times articles profiles former participants to math Olympiads, but frustrated me in its lack of suggestions for improvement. No parent will finish reading the article with the slightest clue on helping foster his or her kid's talent in math. It is also not clear whether the "intensive summer math camps" students have to go through to make the US Olympiad team increase students' preparedness in other scientific disciplines, although anything that forces students to think creatively is always welcome.

Shockingly, the Times article does not make any mention of the type of problems students have to solve in these Olympiads, but you can find such problems from past competitions on this website. It is hard to miss the fact that the Olympiad questions do not have multiple-choice answers; instead, they are about proofs. That's right, proofs. My own opinion about all this is that the US's mediocre performance on math tests is linked to the American obsession with multiple-choice quizzes. Such tests are easy to grade and even machine-readable, but students won't ever be taught to think outside the box if the answer had already been found for them and they just have to identify it correctly. The very fact that students can find the right answer by default if only they can eliminate all the wrong ones is a textbook example of anti-creativity. I would have at least expected the Times journalist to mention that debate.

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 ).

Posted by: gregbo | October 13, 2008 at 05:20 PM