A quote from Ralph P. Boas:
…by a phenomenon that everybody who teaches mathematics has observed: the students always have to be taught what they should have learned in the preceding course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.) The average student does not really learn to add fractions in an arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in a calculus class either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. And so on throughout the hierarchy of courses; the most advanced course, naturally, is learned only by teaching it. This is not just because each previous teacher did such a rotten job. It is because there is not time for enough practice on each new topic; and even it there were, it would be insufferably dull.
It is an important lesson — and one that I have been finding hard to absorb — that it’s usually fruitful to move on when one has enough of an understanding of something to be able to read further; that there is nothing to be gained by reading the same thing over and over, just because one feels that there might be something that one has missed. I am not making an argument for shoddy work; I’m only remarking that there is often nothing to be gained from being paranoid/obsessed/fixated with an idea. I wonder if I just need to learn to be more confident.
The quote seems to be open to interpretation: this post (which is where I first saw it) uses it to observe that teachers should be judged on how their students perform at the next level of education.
The other lesson, explicit in
he learns it in calculus, when he is forced to use it
is that one learns by doing. This may be a cliché, but that is exactly why is too easy to forget its meaning. One learns when one is forced to do…
I have observed this in my picking up of programming languages etc., where I’m usually trying less hard to “learn it perfectly”, and therefore (surprsingly?) have better results. It’s time to apply this insight to academic learning, I guess.