Doing and teaching
Watching yourself think takes effort and attention.
Our tutoring consultant, as we’ve mentioned, mostly works with High School students on their classes and standardized college admissions tests. Over the years, he’s developed ways of explaining subjects and guiding students, including showing how to approach some types of problems that appear often.
Sometimes more advanced students appear. A few in their first year of college have come for help, especially those in large classes where access to teachers is limited (and teaching assistants may be less skilled in teaching). Recently, a college student has come in for help on the Graduate Record Examination (GRE), used for admission to graduate programs the same way the SAT is used for undergraduate admission.
The GRE doesn’t actually require a higher level of mathematics than the SAT. It does, however, have problems that require the use of mathematics in a freer and more creative way. The approaches developed for the SAT aren’t useful for these. Now, with a bit of thought our consultant could certainly solve these problems. However, his task was not to solve them, but to explain how to solve them. So he would attack the problem and work through it, then try to show why and how he took the steps he took; or watch his own thinking and comment on it along the way.
Our consultant learned how to do this kind of problem mostly by doing a lot of them over the years. Some approaches worked, some things didn’t. Over time the details of each problem disappeared and the approaches became less the subject of conscious thought and more a sort of near-reflex. Unearthing them so he can explain them, or even watch them at work, is not easy.
Most scientists and mathematicians learn their problem-solving the way our tutor did. So one might ask, why not let all students do the same? Why mess with what works? Well, there are several answers to that. One we favor rests on a quote we remember from a paper on quantum mechanics years ago, by Roland Omnès: “But science does not depend on quotation, however elevated the source. It depends on elucidation, so that the feats of genius may be made ordinary learning for beginners.” In other words, what may cost a scientist a long and difficult effort to discover must be explicable with far less effort. If we all must rediscover things the hard way, we make no progress.
(Quick-witted readers will note the irony of using a quote that advises us not to use quotes. But we think that here the use of a citation is proper: it encapsulates a truth in a well-written expression.)
Of course, scientific problem-solving cannot be reduced to a limited, teachable set of techniques and approaches. If it could be, all would have been finished long ago. Our students will have to learn their own lessons from the problems they’ll find, new ones we never encountered. Then it will be their turn to explain.
