I've been thinking about this for a while as part of my decision to quit focusing exclusively on computing and go back to school and teach maths and science as well
It's also an issue to do with the curriculum wars (content versus process, or whatever), our social mathophobia (bad jokes at staff meetings about mathematicians by humanities teachers who have become adminstrators) and the whites falling behind the asians in international maths competitions (China and India will soon take over the world)
So, what is maths?
There doesn't seem to be a clear cut, definitive answer. I'm happy about that. At this stage I'm writing down some things I have discovered so far. Not definitive, rather preliminary, but a start.
Maths is NOT about formulas and cranking out computations - or rather that's a very small part of maths
Maths is about perceiving and acting in the world in an enhanced way, about perceiving the world in a different way and being able to act more powerfully within it
What is mathematics? Most people would say it has something to do with numbers, but numbers are just one type of mathematical structure. Saying "math is the study of numbers" (or something similar) is like saying that "zoology is the study of giraffes". Math may be better thought of as the study of patterns, but this too falls short...I think that last sentence is the most interesting insight yet that I've read about what is maths:
The more I study math, the more I wonder about what exactly math is. Actually nobody knows. It seems to be a product or our minds, and yet reflects the external universe with uncanny accuracy. A mathematician develops a mathematical theory for its aesthetic unworldly beauty and it's compelling evolution, with no thought of how it might be applied to the world. A century later a physicist finds this theory to be perfect to use as a framework to express his physics (this sort of thing happens frequently). Pretty weird how intimately connected our innermost "mind" and the outermost "universe" really are. This is a profound mystery!
Bruce Bennett, my advisor in grad school, defines mathematics as "unified consciousness theory". As you come to master a branch of mathematics, it's as though you've grown a new abstract organ of perception through which you may then view the world. You've grown a new "mind's eye" that can perceive realities literally inconceivable without this new organ of perception.
Professor of Mathematics
- what is math?
"You've grown a new "mind's eye" that can perceive realities literally inconceivable without this new organ of perception."This is very good but not sufficient. Because it applies equally to other subject domains. It is not mathematical enough.
The other good thing I discovered was what to say to someone at a party when they discover you are a maths teacher and come up to you and say, "they taught me boring quadratics at school and I am a successful businessman and have never used quadratics ... what a waste of time". Here is what you say:
When I was in first grade we read a series of books about Dick and Jane. There were a lot of sentences like "see Dick run" and so forth. Dick and Jane also had a dog called Spot.
What does that have to do with mathematics education? Well, when I occasionally meet people at parties who learn that I am a mathematician and professor, they sometimes show a bit of repressed hostility. One man once said something to me like, "You know, I had to memorize the quadratic formula in school and I've never once done anything with it. I've since forgotten it. What a waste. Have YOU ever had to use it aside from teaching it?"
I was tempted to say, "No, of course not. So what?" Actually though, as a mathematician and computer programmer I do use it, but rarely. Nonetheless the best answer is indeed, "No, of course not. So what?" and that is not a cynical answer.
After all, if I had been the man's first grade teacher, would he have said, "You know, I can't remember anymore what the name of Dick and Jane's dog was. I've never used the fact that their names were Dick and Jane. Therefore you wasted my time when I was six years old."
How absurd! Of course people would never say that. Why? Because they understand intuitively that the details of the story were not the point. The point was to learn to read! Learning to read opens vast new vistas of understanding and leads to all sorts of other competencies. The same thing is true of mathematics. Had the man's mathematics education been a good one he would have seen intuitively what the real point of it all was.
- the most misunderstood subject