A Consequence of Time Lag in Learning Mathematics

Time lag in learning mathematics is a well-known phenomenon:

I think the answer is supplied 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 has a corollary: Nobody understands cutting-edge mathematics, not even the people discovering it. That might explain why people took the Copenhagen Interpretation of quantum mechanics seriously. Quantum mechanics could not be understood until it was used to discover other things.