PROOFS - October 13, 2011

The struggle over the teaching of school, particularly HS, continues to confront us, and everyone who has actually been to one imagines that he is an expert on what needs to be learned there and how to present that. Even in the ranks of professors of Mathematics in U, there remains a disagreement on how that should be accomplished. Thus, like most of those in that line of work, I find that entry level undergraduates have no idea of what constitutes a proof in elementary Euclidean geometry or any other part of Mathematics, or in any other discipline. This reduces technical argument to something just barely amounting to plausibility, or even less. In noting this failing, teachers of Calculus are unable to find the time or opportunity to teach this aspect of proof within the Calculus agenda. As a result, the most important feature separating proof from plausibility does not reach many students. There is simply no space, time or opportunity to take up the two disciplines in the same format. The discipline enforced by the traditional 2-column proof separating assertions and justifications is denounced as anachronistic and stultifying by those possibly tired of exacting that kind of effort from unwilling adolescents. Yet it is the keystone upon which the entire edifice of Mathematics (which is much more than the mere art of calculation) stands. Computers have been built to replace human thought in calculation, at least on the undergraduate level, but the matter of logical proof, once passed by, is highly unlikely to reappear in the later levels of learning. And in USA, we are even suffering in the HS a shortage of teachers who have learned that and understand it. The penetration of a faulty narrative proof entails a search of the mind and intent of the putative prover, and if one faces the attempted work of novices, is exhausting. At the same time, attempts by teacher unions to obtain space and time for that in school has always fallen before school board intransigence.