Wednesday, August 20, 2008

Too much mathematics homework

Recently I commented over at En Tequila Es Verdad, saying in part that I thought too much mathematics homework was a bad thing, education wise.

The response to headlines about US falling behind in education (say, like this one) is usually to increase homework.

Well, a paper in Econometrics Journal apparently concludes that for average students (about half of them, speaking roughly), lots of mathematics homework is not productive.

[Of course, this is looking at relatively short term effects. What will be the effects of too much homework five years down the line? My guess is that long term it will probably be unproductive for an even larger percentage.]

The linked news article says: According to Henderson, the learning process needs to remain a rich, broad experience.

Which is one of the main points I was getting at in my lengthy comment over at Dana's blog. Nice to see I'm not talking complete bullshit.

Think of it this way:
Imagine art class consisted of having to practice drawing a duck, over and over, until you could produce a good outline of a few very particular kinds of duck, drawn just so. You would do half an hour of ducks every night for homework. Then back to school the next day for more ducks. Then you'd move on to chickens. The generalization to all birds would be sort of handwaved, because the curriculum is kind of packed. It's time to move on to drawing fish! If you didn't learn to draw ducks, you would get even more work on drawing ducks. Some aspects of what you learned in drawing ducks could be used in drawing fish, but the relationships aren't very intuitive, and anyway, there's just so many bits to remember and it's all so confusing and WTF, now I have to go home and do fish for an HOUR?

And then suddenly you're drawing battleships, and while drawing kind of made sense before, suddenly it makes no sense. You never quite got the hang of ducks and now you're trying to catch up that, fish and now battleships? How on earth are you ever going to remember all the parts of a battleship? And god forbid you should draw the parts in the wrong order!

If art was like that, most people would hate it.

Imagine Rembrandt at a party, who desperately wants to convey something of the beauty and importance of chiaroscuro. What would he hear, over and over, as he brought up the topic of art?

"Art? I was never any good at that. I always hated art! My worst subject. All those ducks! You must be very strange."

Most people - if you forced them - would be able to draw a fairly reasonable-looking duck, but there'd be precious little art in their lives. They'd certainly have no sense that it could be moving and beautiful - or indeed that it was about anything other than ducks and fish, and maybe something painful about battleships.


Anonymous said...

Hear, hear! (Or is it "here, here"?)

Not all homework is rote algorithm-following, though, especially when you get into higher levels of math. A homework assignment that consists of a couple of seemingly different problems which both turn out to have similar solutions, followed by a series of questions about why these solutions work, what different methods could be used, and what similiarities there are in the two problems, could be very beneficial indeed!

Also, there are benefits to doing the same kinds of problems over and over again, especially in lower levels of math. At some point you've got to memorize your multiplication tables, otherwise later on you'll get so bogged down in working out simple multiplication that you won't have time to solve the big problem.

Anonymous said...

Whether or not a student understands a math concept is apparent with two or three problems; you don't need to give them an entire page to figure this out. Also, there are other (better) ways, besides doing worksheets, to memorize multiplication tables.

Blake Stacey said...

My gut feeling is that kids need to be doing more work, but of a different kind; adding more problems of an unproductive sort is not going to be an illuminating experiment.

Of course, we all have more nerve endings in our guts than in our brains, [citation needed] so you know I'm right.

Efrique said...

Susan and Angela: Yes, you're right, my hastily-formed analogy breaks down. I could write a more sophisticated version that would hold water better, though, and still convey something of why I think over-doing the routine homework can be counterproductive.

(oh, and it's "Hear, hear!" - you're asking others to listen to the words of another person. Wikipedia agrees with me)

Efrique said...

To be more specific - my flawed analogy is not a particularly accurate representation of reality, so much as a reasonably accurate representation of how a lot of people seem to recall their mathematics education. If the only parts that stick in their mind are the dull and routine parts, perhaps there's some room for doing this in other ways.

I don't put the blame for this on teachers, but on parents who actively push for more rote learning, particularly in mathematics.

With my own children, I've tried very hard to reduce the "rote" elements of learning in their homework (not just the mathematics, but all of it), any time there was any hint that it was getting dull. Sometimes its hard work finding ways to achieve that, but it's usually worth the investment of some time to do so.

Anonymous said...

You know, my department head said something interesting yesterday. "Homework is to the math teacher what alcohol is to the alcoholic."

It is very true, and why I am making my assignments 2 part: Required (between 3 and 8 questions depending on the chapter, some chapters are really diverse) and Suggested (to help move a learner along).

Only the Required will be collected, so once a learner "Gets It" they don't have to do a bunch of problems.

Anonymous said...

Your painting analogy is eerily similar to this article, which Sam Shah just pointed out to me in response to a related post on my blog. (I don't totally agree with the author's overall conclusion, but he makes some very good points.)

The real problem with actually reducing homework is that different students need massively different amounts of homework on each topic. You need enough similar problems that students begin to think of them as routine and simple and are ready to move on to something in which those problems are just one step. For some students, this has already happened before they even sit down for homework, while for others it'll take hours. It's a good attempt to say that half the problems are optional, but I'm doubtful that most students who need the optional section actually do it. Short of some sort of computerized thing that makes you do problems just until you can answer routinely without errors, I see no real possible solution. It's similarly difficult to give abstract, non-routine problems for homework, since it's unfair to penalize the lack of a correct solution to such problems, but impossible to otherwise confirm that students really tried them.

Dana Hunter said...

That's the world's most perfect analogy for what math's like to those of us who suffered the endless rote b.s. in school.

From now on, all those who ask me why I don't like math are getting sent here.

Efrique said...

Thanks for that link to the pdf, thoughtcounts_a - and yes, that's a surprisingly similar analogy to mine.

I have sat for many hours (well into the hundreds, at least) watching an artist work, and I believe I see a great deal in common between art and mathematics - not just in the beauty and impact of the outcome, but in the process of its creation.

(Of course there are differences as well!)