math education

[cellio]

Being of a certain age, I learned arithmetic the conventional way and neatly dodged New Math. I knew things had changed since then -- at least in the ability of high-school graduates to do arithmetic unassisted -- but I didn't realize just how strange things had gotten. amergina posted a link to this (longish) news story broadcast: math education: an inconvenient truth. Sigh.

To summarize, some (apparently-big-name) published curricula are now skipping conventional methods to teach new ways of doing arithmetic. Some are different ways of breaking down the problems; others are primarily notational differences. All of them seem, on average, slower and more error-prone.

Now granted, I sometimes do arithmetic by the "reason through it" process the reporter dislikes (what did they call that, clusters?), but it's kind of specialized. For example, a 15% tip reduces to a 10% tip and half again; that's fast and easy. If I'm multiplying by a number ending in 9 or 1, it's often easier to reduce to another problem and then deal with the leftovers. If I need the square root of 4862 (I just pulled that number out of thin air), I can't tell you exactly what it is but I know it's a bit less than 70. Sometimes I think in patterns like that. I think this is a fine thing to teach people after they have mastered conventional write-it-down-and-work-it-out methods. Not before, and certainly not instead of. (And I think it's better if you can give them an educational environment in which they figure out these "tricks" for themselves, like I did.)

I assume these new teaching methods (which include "use calculators") are largely responsible for many people being unable to get order of magnitude right. Those of the previous generation undoubtedly said that about the move away from slide-rules, but I never used a slide-rule (except as a novelty) and I can approximate... I once had a calculator-armed teenage clerk at a produce stand insist that my bag of vegetables came to over $200. Even if he had no instincts about what vegetables cost, he should have been able to tell that the price codes he'd read off the list didn't add up to that and maybe he'd mistyped something.

(When shopping I tend to keep rough a mental tally, so when I get to the check-out I know approximately what the total should be. I gather that this is unusual. It's just the way I learned to shop, probably from a time when you had to make sure you didn't exceed cash on hand. Now I use plastic for everything, but the habit remained.)

Well, I guess I can take comfort in one thing: if what they say about mental exercise is correct, I should be pretty close to immune to Alzheimer's. :-)

Comments

[dvarin.livejournal.com]

This seems kind of related to a talk we had at the Amazon developer conference day thingy a while ago. It was on how to utilize intuitive thinking, but it started off with just the seems-obvious statement that novices rely more on rules, and experts rely more on intuition, and as you go up the scale of training and experience from novice to expert you stop relying on rules and start relying on intuition. This is why not all experts are good teachers, because they've been using intution so long they don't think about rules for how to get a result, so they can't explain it to a novice.

Looking at these methods for doing problems, it seems that they're trying to teach intution, sometimes without teaching rules first. (So.. I'm pretty sure that basic addition and subtraction have to be memorized. It seems hard to break 4+7 down into some simpler problem.) The question here that isn't getting mentioned is, how is the teacher teaching it? If they don't understand how it works and can't express that to the students, then it's pretty much equivalent to just teaching them rules, only it's a more complex than necessary set of rules. If it's rules that can generate intution with less requisite experience then they;'re good, but the anecdotal non-evidence suggests that it's hit-or-miss.

Cluster problems... ugh. This sort of backward hole-filling reasoning is really dependent on intuition to go anywhere, and if a particular student can work it, it's good, and if they can't, it's dead. It reminds me of teaching imperative programming proofs when I was a TA--I had no problems coming up with loop invariants, but some of the students just couldn't. The way the professors taught it was the way I thought about it, that the loop invariant was some expression which satisfied a few conditions and enabled proving the conclusion, and as for how you determine what it is...they said nothing. I had no problems with it--stare at a loop, figure what it was supposed to do, think about the shape of the logical hole that needed to be filled... flash! Loop invariant! But several students didn't get it so well, and I'm wishing now that I could have come up with rules for it to tell them until they could get their own intuition. Hole-filling (by which I mean, given the start state and the end state come up with the middle) in some cases may just inherently be a heuristic search problem rather than one susceptible to deterministic rules, which means it's totally intution, which means you just need a lot of experience with seeing all the pieces together. Bleh. It's late, I'm babbling. :)

[cellio]

It was on how to utilize intuitive thinking, but it started off with just the seems-obvious statement that novices rely more on rules, and experts rely more on intuition, and as you go up the scale of training and experience from novice to expert you stop relying on rules and start relying on intuition. This is why not all experts are good teachers, because they've been using intution so long they don't think about rules for how to get a result, so they can't explain it to a novice.

Very true, and a likely explanation for some of my college experiences. (You could always tell which professors didn't grok undergrads.)