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

[http://users.livejournal.com/merle_/]

I actually really like the "non-conventional" methods, like approximating tips and the like. That sort of thing can be quite useful. My mom was astounded when I came up for the "knock a digit off, then add half again" method for computing 15% tips, because she could do that, and even thinking about the multiplication would cause her to freeze up. Personally, I look for nearby integers divisble by 6 or 7: the tip on $27 is about $4 since 28/7=4.

However, what you say is certainly correct: you must know how to do the basics before any of these techniques becomes useful. If you do not know your multiplication tables, you do not stand a chance of getting a correct answer. And letting kids use calculators is boneheaded. One may as well allow them dictionaries for spelling tests, or Clif notes for book reports.

Where the methods of estimation are really useful, though, is for people who have some skills but cannot come to appreciate mathematics because it was drilled into them that it was hard, boring, and useless. I may not be able to come up with 20% of $69.34 easily, but 20% of $70 is a snap. And I know enough about transitivity that I could come up with a first-order error correction (say, adjusting my answer by 20% of $.65).

I am told that I learned math in the age of "new math". I don't think so, because, like you, I had to come up with those tricks myself. And that's a useful learning experience in and of itself. But many people are unable to discover those tricks.

[cellio]

I use non-conventional methods; they're often faster. But I wouldn't understand them, and they wouldn't be fast, if I didn't have that solid grounding first. (We agree; I'm just reiterating.)

I have sometimes taught people who only have the basic methods to see a particular class of problems logically. Once I demonstrate I can see the light bulb go off. I think that only happens because they can see why it must be true. Multiplying by 9 seems hard, but multiplying by 10 and then subtracting the number once makes sense once you point it out. Multiplication is just specialized addition, so you're free to break it up into managable chunks.

[http://users.livejournal.com/merle_/]

I think we agree that we agree. ;-)

It is easy for others, though, who "suffered and suffered" through rote memorization to see these tricks and wonder why they weren't taught them to begin with. Imagine all the wasted time! But the problem comes because they simply want to teach the tricks, not the process. There is a big difference between telling someone to divide by six to get a tip and explaining that 1/6 ~= 16%, describing commutativity, and someone really grasping what is going on.

One way to put it is that the joy is in the synthesis of knowledge into these tricks, not just the knowledge of the tricks. And that is what we have to learn how to teach: that learning is enlightening and interesting, not just something to be suffered through so you can get a bigger paycheck. Such a shift in beliefs seems unlikely to me, though, when there is the panacea to be offered by "new new new math".