(Note: this is an excerpt from a blog post that
was originally available at
http://blog.oreillyschool.com/2010/05/the-story-of-the-oreilly-school-of-technology---part-2.html
Some parts of it have been made bold to emphasize
them.)
Because of my own "ah ha" learning experience in mathematics, I was convinced that everyone could learn math well if they had a passionate teacher with the communication skills to get them there. I certainly had the passion, so I set out to transform students into mathematicians.
By all the standard measurements and expectations, I became a great teacher. At least my students thought I was great; nothing feeds the ego like teaching. My night time math review sessions had become so popular that I had to hold them in the largest auditoriums on campus. One of my classes even presented me with a large engraved trophy to thank me for my teaching. They did well on exams, and they were very happy. All was well.
But despite all of this positive reinforcement, my point of view (confidence?) and self-esteem began to suffer a bit after a visit in 1993 to Moscow. I visited the mathematics department at Moscow State University, and stayed with a couple who were both professors there. They had two wonderful daughters, the younger of whom was still in high school. At one point during my stay, she asked me to help her with homework in mathematical mechanics. Of course, I was proud to help, since Mechanics was my research (specialty?). Her homework seemed to me to be rather difficult (advanced) for high school; I had a bit of trouble with it myself. Suspecting that this high level of work might be a strange exception in the Russian educational system, I asked her parents if she was in a school for gifted students or some other advanced learning program. They told me that, no, she was actually an average student in an average school. This confused me. Mathematical mechanics is historically a major focus in Russian Academics, but that didn't explain how their students could be so much more advanced than the students in the United States at this level. After a lot of probing questions, I eventually found out that all of the math exams in Russia are oral exams. Students in Russia have to be able to explain mathematical problems and their solutions out loud.
Intrigued and inspired, I came back to the states, and started giving my students oral exams after their written exams. This was a long and arduous process, because I had to schedule an hour for testing for each student. However, I discovered something that depressed the hell out of me. None of my students knew what they were talking about. Even students who got perfect scores on my written exams didn't really understand what it was that they were doing.
It became clear that students were simply emulating calculation techniques, without understanding where those techniques came from, or how to create them themselves. Then it became clear to me why my review sessions were so popular. In those sessions, students would ask me to solve every type of problem they could find in the text book. Even though I'd have them try the problems before showing them the solution, they were really preparing a decision matrix for a matching game. If the problem was like this, then they would do this; if it was like that then they'd do that, and so on. I also realized that the problems I was asking them to do, were designed with this system in mind. It seemed that most of the calculation techniques were designed to help students pass tests, but did not illuminate the true nature of the mathematical structures.
In the American system of teaching mathematics, we are actually teaching algorithms for getting answers from synthetically designed problems, but not teaching students the art and science of mathematics. Sure, some students get through school, and become great mathematicians despite this system, but we are losing most students through attrition. Ask your friends and neighbors about their experiences with math education, and most will say they hated math in school. Others might say they were good at it, but hated word problems, which I always thought was a curious thing to say (what those people are really saying is that they were better at the matching game when the thing to match was given in a simple form).