Monday, May 11, 2009

How to teach math

> On Sat, 09 May 2009 15:30:52 -0400, Kenneth Tilton wrote:
Well, i was not really trolling, I was forking the thread to make fun of  the New Math that tried to get the numeral/number distinction across to five year olds.

Someone responded:

You find it better to start with medieval concepts working gradually on to the mathematics of XIX century, while explaining each next year what was wrong with the things they learnt a year ago?

I said all that? Ma's gonna be right proud. But... 

Funny you should ask. Yes, I suspect the path society took to get to what it knows now about math is the path an individual neuronal mass should follow. ie, kids should encounter zero and roman numerals and place value and algebraic variables in the same order society developed those ideas.  The history of math is your math curriculum guide.

The New Math erred by selecting the logical organization of mathematical concepts as its curricular pole star. Next came Constructivism, which wanted kids to reinvent math. From scratch. Cool idea, but too slow.

Instead, let the history of mathematics dictate the order in which things are directly taught. Maybe go further and teach math as history with less emphasis on computation. Math often advanced when needed to solve real problems. Maybe we can shut up the little devils asking why they need to learn this stuff.

As for explaining all along the way what was wrong with the ideas taught the day before, hey, ever read a book on programming? They typically develop a chunk of code iteratively, presenting ever more improved variations on a primitive original. Come to think of it, ever develop some software? Same thing.

Here's the deal: most folks do not even know zero had to be invented. One understands zero better if one has done without it and then the teacher invents it for you. Something like that.


Dmitry A. Kazakov said...

Poor kids!

Well, in my view this one of the major problems of modern education system: pupils are learning history of mathematics/physics/chemistry instead of the subject. I think this is all wrong.

I myself, as a kid, played a guinea-pig in an education program which started mathematics with some basics of the set theory and used logical formalism in all proofs. If I correctly remember the program was developed on the initiative of Kolmogorov himself. That time I had no particular interest in mathematics, I was a typical humanitarian interested only in arts and ancient history. Nevertheless it was so easy, and what a contrast to the way physics was later taught. Unfortunately the program was quickly scrapped.

There is no reason to learn others errors. Modern mathematics is so much simpler than the old one! Littlewood in his "Miscellany" gives a horrific example of the definition of function from textbooks of XIX century:

Why should anybody learn that mess?

spongefrob said...

All teaching is the teaching of history. I didn't much understand Maxwells equations ata all(er.. still don't 'fully' understand... :-) until I understood what Faraday was thinking.

Dmitry A. Kazakov said...

I remember a textbook on physics for "advanced schoolboys". It contained multiple pages of extremely complicated calculations of the waves a moving electron should radiate. The horror of this can be imagined from the fact that authors carefully avoided differential equations (since even "advanced schoolboys" should not have known that). When I finally had struggled through this jungle, having understand far less than a half of it, I read an amazing notice by the authors, that all this was actually rubbish, since there is quantum mechanics which "you (little idiot) will probably have to learn later".

Slobodan Blazeski said...

hey, ever read a book on programming? They typically develop a chunk of code iteratively, presenting ever more improved variations on a primitive original.

There is a great tale in How buildings learn where Christopher Alexander was suggesting that incremental approach in architecture is the right thing. So maybe the the old saying poem is never finished A poem is never finished, only abandoned spans to many domains. Maybe even math learning.

Kenny Tilton said...

Bobi, if you can find the thread on c.l.lisp you will find I use the same example (iteratively better/more complex treatments of the same programming task). ie, Great minds think alike.

TruePath said...

Speaking as a mathematics postdoc it's my view that what really went wrong is the educators drank their own cool-aid.

Everyone in math education performs at least lip service to the idea that understanding is the ultimate goal. After all the thought that we are just teaching mindless algorithms better done on a computer to encourage good work habits is just too depressing.

But realistically real mathematical understanding has almost nothing to do with what is practically and politically necessery to teach. Practically people need to be able to count their change, multiply daily costs to obtain monthly costs and so forth and maybe do some interest rate calculations. Even mathmaticians just do these kind of calculuations on thoughtless autopilot so understanding is hardly necessery. This is the extent of the math that virtually all lawyers, doctors and even MBAs can perform. Reasonably, therefore, a certain practical segment of society insists we prioritize teaching these useful skills rather than ones most students will immediatly forget.

Really understanding mathematics, however, requires approaching the entire subject from a different perspective. Indeed, the biggest impediment to real understanding for most students is their expectation that everything has a mindless algorithmic answer. Most students, however, don't have the desire or interest (and hence lack the capability) to understand math in this fashion.

It's like the difference between understanding how to program and knowing the CS theory that lets you design your own decent languages. Only the later students truly understand what's going on in a certain sense but insisting on giving everyone a course in CS theory before letting them code would be a failing strategy for most students.

Kenny Tilton said...

(1) TruePath mentions "mindless algorithms" repeatedly, but only the worst teachers fail to provide the motivation behind rules such as (to pick one) ax + bx = (a + b)x.

eg, The hints and annotations in my emerging Algebra trainer eventually get around to motivating the rules they recommend (even though the trainer is not meant to replace a teacher). As a student myself when I forgot rules I would work backwards from trivial concrete examples I could solve to recover the forgotten rule. "Lessee, 2x+3x=5x, so the rule is add coefficients." I was always careful not to pick two 2's because 2+2 equals 2*2!

(2) Here's a nice piece by Underwood Dudley from the May AMS Notices on why we learn Algebra. Fairly relevant to "How to teach..."