Another One of My Periodic Discourses on Mathematics

I was linked today to this article by a Dr. Frank Quinn employed by Virginia Polytechnic University.  Dr. Quinn outlines a philosophical split among mathematicians about a century ago and discusses its outcomes.  He then writes briefly on how the advances made might be introduced into mathematics education with profit.

Dr. Quinn's major contention is that mathematics, understood in a modern sense, is entirely rule-based, with no demand that the rules match any physical results exactly.  When applied educationally - as it commonly is in Geometry - this results in a system which while difficult to learn leaves room for entirely logical conclusions and is therefore in a sense "easy to master" if unlikely to ever be anything remotely close to fully explored.

The primary drawback to such an understanding is of course easy to spot: if it does not actually describe anything, what is the use of mathematics anyway?  It becomes little more than a convoluted and peculiarly abstract art form, at least to the common understanding.  I happen to be attracted to logical puzzles and purity of logic, but in teaching I have been made aware - sometimes more forcefully than others - that many students have no real interest in such things, and more importantly little use for them.

Now Dr. Quinn makes the fair observation that elementary education is still dominated by an earlier view of mathematics, using the teaching of fractions as an example.  That being noted, though, I want to say that there remain at least two possibilities I can come up with:

First, he might be right.  Perhaps at least one part of the reason for, say, falling educational results is that early education goes blithely on with outdated methods while the higher mathematics - starting in, say, high school, where the teachers teaching algebra and calculus have probably at least studied mathematics - is relying on the new framework.  In which case, it would clearly be beneficial to unify the system and use the new and improved math throughout.

On the other hand, the older methods may persist (outside academia) because they are superior in a general sense.  Mathematical proof may frown on analogy, but experience does not.  In a related field, the fact is that elementary science remains mostly Newtonian even though it will yield errors of fact to varying degrees.  Why?  Accounting for relative or quantum effects is simply too difficult.  When we are attempting to train nuclear physicists, then by all means we try get the science correct.  But even an engineer rarely has to worry about such complications, far less a plumber or an accountant.  Similarly, professional (which is at least largely to say, academic) mathematics is certainly very useful as its own thing, but it is hardly a field which everyone needs to be prepared for.  If it turns out that Johnny and his friends can be taught to balance a checkbook more easily by considering half of an apple than by learning about the properties of the ratio of the integers 1 and 2, so much the worse for the integers.

In fact, I suspect - both because I am a cynical person, and from experience - that the old methods have been partly discarded while the new ones have not been fully adopted; or perhaps worse, both methods are attempted simultaneously and the result is confusion.  Let me take a concrete example.

Is a half, after all, a part of a thing, or an arithmetically constructed ratio of two integers whose idea is (almost) entirely man-made?  Even the latter definition is hardly rigorous and results, for instance, in mass confusion when fractional exponents - also known as roots - are introduced.  It now becomes apparent that for these "higher" relations, it would be best if we could have gone back and made sure that the part stressed in the introduction to fractions was their reciprocal relationships - which is not even ratio, per se, but an application of ratio and multiplication.  Meanwhile, three students divide two candy bars equally, and the complete abstraction is suddenly of limited value.  "Three parts of a candy bar" is a simple idea.  "A part of a candy bar which if you could have three candy bars each split into similar parts would make a whole candy bar" is a bit unwieldy, and attempting to ignore the actual candy in order to keep the definition manageable and the numbers pure is not helpful either.

Now, I confess I was not particularly aware there had been any radical changes in the understanding of what mathematics is and how it should be conducted before reading this article.  I would have said that there has been a gradually progressing tightening of definitions, standards, and proofs over the centuries, and certainly "modern" - twentieth century - mathematics is more enamored of these closed system, definitionally-based, completely logical approaches than most, while also yielding some impressive constructions, many of which have even proved useful.

So, maybe for that reason, I fail to see how this "crisis" is to be resolved, or even that it is much of a crisis.  I suspect the problem is not that Johnny and Susan cannot define fractions in a "mathematically correct" fashion, but that they are in ninth or tenth grade and cannot add them because a calculator does their thinking for them, or possibly for worse reasons.

In short I am hesitant about prescribing "modern mathematics" as a solution, for the very reason Dr. Quinn finds it so appealing: its lack of connection to the actual world.  If we are going to set up "science" and "core mathematics" as rivals, I am more inclined to the scientific approach in anything serious.  But even more importantly, Dr. Quinn's worries strike me as those of someone who would complain that each and every driver is not able to design an engine: of course it would be fantastic if it could be managed, but it is also a truly impossible request.