Monday, February 25, 2008

what is maths?

What is maths?

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...

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.

Rafael Espericueta
Professor of Mathematics
Bakersfield College
- what is math?
I think that last sentence is the most interesting insight yet that I've read about what is maths:
"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


bigmaggie said...

I think the guy complaining about the quadratic formula highlights a clear problem, not with math itself, but with math education. I've taught math, but currently teach Physics. It seems that far too many math teachers rely on having kids memorize formulas and processes rather than teaching them to understand the relationships inherent in those formulas and processes. As a result, kids only learn how to solve specific types of problems, and do not master problem solving at all.

Whenever kids ask me "Why do we have to learn this? When will we ever have to use this formula?," I bring up the following analogy. I ask the class if anyone plays a sport. No matter what sport it is, I ask them, "Do you have to do weight training at (fill in the blank sport) practice?" Inevitably they say yes. (You can substitute running drills or stretching or some other exercise.) I then ask them if, in the middle of a game or meet, do they have to suddenly stop and start lifting weights. They will say no, so I then ask them why then it is that they have to learn how to lift weights and spend time doing it if they aren't ever going to use that skill in the actual competition. That's usually the end of the conversation, because they get my point. Math is exercise for your brain. That's why they make you learn it in school.

Bill Kerr said...

hi maggie,

the link from which I obtained the party conversation about quadratics also contains this story, very similar to your own:

" Professional athletes spend hours in gyms working out on equipment of all sorts. Special trainers are hired to advise them on workout schedules. They spend hours running on treadmills. Why do they do that? Are they learning skills necessary for playing their sport, say basketball?

Imagine there're three seconds left in the seventh game of the NBA championship. The score is tied. Time out. The pressure is intense. The coach is huddling with his star players. He says to one, "OK Michael, this is it. You know what to do." And Michael says, "Right coach. Bring in my treadmill!"

Duh! Of course not! But then what was all that treadmill time for? If the treadmill is not seen during the actual game, was it just a waste to use it? Were all those trainers wasting their time? Of course not. It produced (if it was done right!) something of value, namely stamina and aerobic capacity. Those capacities are of enormous value even if they cannot be seen in any immediate sense. So too does mathematics education produce something of value, true mental capacity and the ability to think"

Darius said...

A recent, personal, epiphany came when I realized that the maths (which I experience) are the selective ignoring of detailed qualitative information so that one can concentrate on the quantitative properties of a problem one wishes to solve, or find an answer to a proposed question. I'm constraining my answer to the applying of maths as the purpose for teaching maths, not just as teaching "thinking". Studying rhetoric and philosophy also teaches "thinking".

We measure a coast line for reasons of property ownership or weather effects or such not, regardless that the physical length of where the water meets the land changes numerous times every fraction of a second in numerous places. We compare apples to apples and apples to oranges even though no two apples are precisely the same.

Perhaps the interest in maths might arrive more through the nature and personal relevance of the questions asked rather than the question of the relevance of learning how to solve them. Maybe we should start with probability theory and statistics rather than geometry so the population can better understand when someone asserts something "is" a fact that it really "is" a fact (perhaps drawn from current events). At least, fewer would lose their quantitative money at casinos. ;-)

Geometry was great when the industrial age needed engineers to change our physical world.

Probability is greater in a world where everyone has a global microphone, where everyone competes for global resources, where everyone can make micro level changes in their lives which have an accumulated global impact (in many domains) so that "facts" dissolve into shades of gray in our only, shared, moment-by-moment ever changing world.

Anonymous said...

I gotta say, Bill, this is one of your best posts! I really enjoyed it.

I remember dealing with quadratics in algebra a lot. We'd start with a formula, something like x^2 + 2x - 3, and then we'd decompose it into pairs: (x + 3)(x - 1). I also remember the quadratic formula. We may have even done a proof for it. To this day I don't remember its usefullness, though I just looked it up and it looks like it's used for finding the value of x in a 2nd order equation like x^2 + 2x - 3 = 0.

I've discussed this before with you privately, but for the most part the math I had taught me symbol manipulation. Calculus was pretty much the same way. Of all the math classes I had the one where I felt I learned the most was geometry, because we built knowledge throughout the course. We did proofs all the time. Each theorem was built on knowledge that had been previously acquired via. proofs. It wasn't too abstract for me either.

I don't think it was until years later through the practice of programming that I realized some deeper meanings to what I was taught in algebra. It wasn't because I worked with it more. I just thought about it some more in a larger context. The same goes for Calculus.

For example, I studied the concept of closures for a while on my own not too long ago. People who program in Smalltalk use them all the time. What you learn is the idea of a higher-order function--a function that takes another function as input, and may produce a function as output. This relates to some concepts in Calculus, like the Chain Rule. I remember this is one of those things that used to mystify me. Now it's clearer.

I've seen a couple IT/programmer bloggers say that math helps you be a better programmer, and I think they're right, mostly because of the symbol manipulation aspects. We were informed by the practice of algebra that you can take something that looks complex and simplify its form by decomposing it to its essential elements, without giving up any of the original meaning.