As already noted, when we use ourselves as model learners, we always get it wrong. That is, if we reflect on how we learned mathematics and design our courses accordingly, those courses will be appropriate only for mathematics students who think and work like we did.
However, there is a sense in which we can use ourselves as model learners -- but only if we focus on subjects in which we never became experts. I can comfortably compare the mathematical successes of many of my students with my own experiences in foreign languages, athletics, and music. There is no danger that anyone will consider me a linguist or an athlete, and I am at best an amateur musician. My native talents in these areas range from none to modest, but I have managed some small successes that have been personally satisfying.
Of these three areas, language is the only one in which I have any academic experience. My satisfaction in that area was almost entirely in the form of consistently good grades. I suspect I acquired those grades by twisting the subject into one I understood -- mathematics -- and focusing on structural analysis, patterns, and memorization. Much of what I memorized is now gone, but what I learned of structure and patterns in Latin, French, and German was quite beneficial for understanding English. On the other hand, I failed utterly to achieve one of the primary goals of language instruction: ability to communicate. Furthermore, whenever I entertain the rash idea of uttering a sentence to a native speaker of another language, I panic.
Here is one model for the experience of many mathematics students that we may consider “pretty good”: able to approach the subject in some way on their own terms (often terms not revealed to us), able to get good grades (perhaps by memorization and heroic efforts at test time), perhaps able to use some part of the subject in another area, but essentially unable to achieve our goals (or their own) for “understanding,” and likely terrified at the prospect of having to carry on an intelligent conversation about mathematics.
A second model I draw from my athletic experience. I have little natural athletic ability, but I had good coaching from my father and occasionally from others. I was often the last picked when sides were chosen for a pickup game (any sport), but my father always made sure I got reasonable playing time on teams he coached, and he spent many hours with me on the golf course. There were no special benefits from being related to the coach, other than being in the right place at the right time. In a long and productive coaching career -- with no monetary compensation -- he touched the lives of perhaps two thousand young people, and they were all “his kids.” A few are now professional athletes, but that was not his goal.
Here's the model for student learning: A loving and dedicated coach was able to get me to challenge my deficiencies and achieve beyond my innate abilities -- not by just telling me what to do, but by sticking with me through many frustrations and by convincing me that it was worth the effort to try and to keep trying. In fact, my modest accomplishments with a wide range of individual and team sports have been much more fulfilling than my good grades in language courses. This model has also been much more relevant to my relationships with students in the last decade or so. I'm a slow learner, but eventually I realized that I had a much better career role model in my father than I had in my math professors, most of whom were excellent. With me as student, his job was much harder than theirs.
I get a rather different message from my experience with music. This is an area in which I have a little talent, although certainly not enough to make a living. I also have very little formal training and certainly would have benefited from more. But when I started lessons with a serious musician, neither he nor I could stand each other very long, and I lacked the discipline to practice as seriously as he demanded. The opportunity never came around again. On the other hand, for more than 40 years I have been in instrumental and choral groups directed by true musicians who had a talent for drawing out better performances from amateurs than we could accomplish on our own.
I see in this relationship with music a similarity to what I see with many students who don't think of themselves as mathematicians-in-training, but who are quite capable of individual or group problem solving, as well as conceptual understanding, when properly coached and conducted.
One might argue -- correctly, I think -- that learning mathematics is not like learning in any of the three areas just mentioned. My point is not that we should model mathematics education on language education, but rather that we can use our own experiences ranging from utter failure to satisfaction -- not our areas of professional success -- to get some sense of how most of our students relate to the discipline we are trying to teach.
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Send comments to the author <das@math.duke.edu>Last modified: May 17, 1997