Below I will give what I think are controversial opinions on mathematics education. It isn’t my intention to offend anyone, only to motivate discussion. I also want to say in advance that when I talk about higher math skills I am not talking about problems with basic numeracy.
I subscribe to a heresy. The heresy is this.
The idea that many people need to know a lot of higher mathematics for utilitarian purposes is patently not correct. The mathematics that the vast majority of people need for work is summarized in tables, or done by algorithms prepared by others. Certainly, it is beneficial for a small number of self-selecting and creative people to know mathematics. But there is generally no materialistic benefit to society for teaching most college students one mathematical fact rather than another – to factor quadratics rather than compute toy probabilities, for example.
Of course, materialistic results are arguably not the principal value to society of mathematics education, though we often present them as such.
Humanity has long believed that something important happens when a person learns to perform abstract reasoning. Plato’s inscription above the Academy stated that only those who had studied mathematics could enter. The Rhind Papyrus, written 4000 years ago, asserts that accurate reckoning is the entrance into the knowledge of all existing things, and all obscure secrets. There are many other authorities who say something equally grand and mystical.
Somerset Maugham observed that intensive training in any subject changes one’s view of the world. To justify mathematical education, after abandoning claims for its utility, the change that attends mathematics must not only really exist, but also be unique and desirable. This presents the question: what is the effect of a mathematical education?
The effect of a mathematical education is the ability to process abstraction–to solve problems which are not specified all the way down to the sense-particulars. This is the ability to handle things which have, rather than sensory qualities, only some kind of blank existence: objects which are ontological (and perhaps literal) ciphers.
The idea of a placeholder for a real-world entity is very powerful and mysterious. Some people seem able to handle this with amazing facility and others not at all. Those who can do it are generally those who are able to solve problems.
What is a problem? A problem is an emergence, an exception, something for which there is no known protocol, at least not known to us. It is not always clear how the entities in a problem should be understood. It is fortunate then that there are those who are able to understand things without understanding them– that there are those who are able to move inchoate entities with numb hands and blind eyes into felicitous configurations that are intuited by a seemingly miraculous 6th sense. And then, when correctly arranged, the problematical objects tend to leap into real existence, to develop sounds and colors and textures. They have been turned and displayed artfully, and sanded so that our intuitions can gain traction. At this stage, a problem has been solved.
There is a notable dissimilarity between what I describe above and what takes place in a typical mathematics class, particularly at the low level. I am not sure that Plato would have much use for a graduate of a college algebra course.
We substitute algorithms for thought. A glance at a typical pre-calculus syllabus shows a numbing litany of processes and procedures that lay so thick on the ground that a view of anything interesting is completely impossible. And this characterizes all the mathematical education that most people ever see, with the possible exception of high-school geometry (which is perennially under threat).
This is a difficult question to answer. My conviction is that teaching algorithms is easier. It is easier for the educational agencies because it creates an objective measure of success, and enforces a minimum amount of achievement. It is easier for the teacher, who may be unsuited for adventuring. It is certainly easier for the students, though they trade exhilaration and difficulty for agonizing monotony and boredom.
The ability to solve problems varies wildly; requiring genuine problem solving in a class can instantly give rise to a satisfied elite of competent students, and a miserable complement of failures. This situation is such an anathema that we prefer the frigid status quo, which at least puts a balm on the unpleasant and difficult realities which result from compulsory mathematics classes. The effect is that even though we receive more training in mathematics than any society in history, it is sapped of its real worth.
To the extent that students are allowed to think, however, they certainly intuit the difference between the sunshine of open-ended investigation and the pabulum of algorithmic methods. At certain times we can, despite everything, find our way through the endless jungle of canonical tautologies in introductory courses, and introduce a little light.