In article <firstname.lastname@example.org>, email@example.com (Problem of the Week) writes: |> I have found it to be the case that the best way to test my understanding |> of something is to discuss it, or even better try teaching it to somebody |> else. It is this _discussion_ of mathematics that makes me want to be a |> teacher as opposed to a mathematician who only gets to "do" math. What |> are other ways in which we can test whether or not a student has learned |> something?
I think teaching is by far the best way to learn something. In high school in my Advanced-placement math class, our teacher, Mr. Hoffman, never understood anything, and he was so pathetic that we students took turns teaching him and the rest of the class calculus. At the end of the year, the 14 kids in the class got 11 scores of "5", one of "4", and two of "2" on the advanced-placement exam -- fabulous results.
When I came back to visit my high school the next year, I talked to a student in Mr. Hoffman's calculus class, and was horrified to find that Mr. Hoffman hadn't learned anything from our class, and that years' class was also having to teach him every detail about calculus.
Mr. Hoffman also coached the tennis team, and years later, I met someone who had played on his team. It turns out poor Mr. Hoffman didn't know anything about tennis, either, and the stronger players had had to teach him about the sport from scratch. The amazing thing is that nearly every year, my high school won the state tennis championship!
Mr. Hoffman was clearly the best teacher I ever had in my life, but the sad thing is that even knowing exactly how he did it, I can't imagine myself emulating him. He was so good at feigning ignorance and getting pity that none of us figured him out until years later.
When I taught math in college, I always volunteered to teach classes that I didn't know much about. I probably did twice as much "homework" as the students, terrified because every day I imagine so many questions they could ask that I couldn't answer, and I studied like crazy so I'd know the answers. Looking back on it, those were the classes where my students learned the most, too -- having just thought very hard about the subject, it was very clear to me what parts were difficult and needed lots of explanation, and what parts were "obvious".