A recent conversation with my younger son, frustrated over his undergraduate math course, reminded me of my long standing objection to how math, and for that matter economics, are often taught. Theorems are proved with a rigor that is more than the students really need—especially in economics, where rigorous proofs can be applied to the real world only by combining them with non-rigorous models. The rigor is not only more than the student needs, it is more than any save the ablest students can understand. It is one thing to follow a proof step by step. It is a different and much more difficult thing to hold the proof in your head and understand why it is right.
My usual example of the problem is the failure to teach students of calculus why the fundamental theorem, that integrating and taking a derivative are inverse operations, is true. It is possible to give a non-rigorous but intuitively persuasive proof of the theorem in about five minutes, one that any student who understands what the two operations are can follow and has a good chance of remembering. If any reader is sufficiently skeptical to warrant the effort, I can do it here.
In my experience, very few of the students who take calculus, even at a good school, are ever shown the proof; I would be surprised if more than one in fifty, a year after taking the course, could reproduce the more rigorous proof that they were, presumably, taught. To check the former impression, I asked my wife for her experience. Her response was that she was taught calculus twice, the first time at a good suburban school (but by an incompetent teacher), the second time at a top liberal arts college. To the best of her memory, she was never shown the proof.
I take as a further piece of evidence the math problems that my son was asking my help with—having assured me that the rules did allow him to discuss homework problems with people. One of them was a problem so trivial that a bright ten year old could probably have solved it without having taken the course, provided only that the problem itself was clearly explained to him. I take the fact that such a problem was assigned as evidence that the instructor believes a substantial fraction of the students do not understand what they are being asked to do, quite aside from knowing how to do it.
It is common, at good schools, to complain against "cookbook mathematics," the sort of course that consists of memorizing the sequence of steps to solve a problem without ever understanding why it works. But it is, I think, an equally serious mistake to present a branch of mathematics in the form in which professional mathematicians structure it after all of the original work in that particular field is done. Not only is it a form in which almost no student not qualified to become a professional mathematician can understand it, it is a form that gives a highly misleading picture of how mathematics, or other forms of theory, are actually done.
I am not a mathematician but I am an economist and know by direct observation how the original parts of my work were done. The process did not start with a step by step proof but with an intuition of how some set of ideas fit together, what characteristics the solution to a problem ought to have. Only after I had groped my way to what was (hopefully) the right answer did I, or someone else, go back and make the argument rigorous.
I am currently working on the third edition of my first book, published about forty years ago. One of the things I am doing is filling in the blanks, working out in more depth and more detail ideas whose essence I understood then and still believe, in most cases, were correct.
Alfred Marshall, arguably the figure most responsible for the creation of neo-classical economics, commented in a letter on the relation between mathematical and verbal arguments in his field. He explained that he worked his arguments out mathematically to make sure they were right. Having done so, he translated them into English. If he could not translate them into English, he burned the mathematics.
There is much to be said for that policy. Mathematics is a more precise language than English for the sort of work Marshall was doing. But it is also a language farther from the intuition of almost all of us. If you have the math and cannot translate it, it is quite likely, although not certain, that the reason is you do not understand it. I sometimes referee journal articles. Occasionally I get one where, if you translate the math into words, it makes no sense—is arguably insane. The author or authors presumably had doctorates in the field. But they were manipulating symbols, not ideas.
My daughter is at the same college as her brother. When she transferred there after two years elsewhere, she was seriously considering majoring in economics. After taking an economics course, she decided on her other alternative major. The reason was not that she does not like economics, or cannot do it—she audited several of my courses while a home schooled student of high school age, and one of my articles contains an idea that I credit in a footnote to her, since it was hers.
The reason was that the course was mostly about the math not the economics. I discussed her experience with a professor at that university of whom I have a high opinion—someone on the short list of people who, when they disagree with me, cause me to seriously consider that I may be making a mistake. He also had a daughter taking economics at the same school. He agreed with my daughter's judgement—that the courses were teaching mathematical rigor instead of economic intuition.
I have let this essay wander from my son's experience to my daughter's by way of mine. But I think the fundamental thread is the same in all. What matters is not remembering but understanding. If you have learned a proof but cannot explain why the result is true, you have been wasting your time.