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

[baron-steffan.livejournal.com]

This is a huge "button" for me. I should mention that I'm a pharmacist, and that pharmaceutical calculation (of which subject my professor wrote the bible) is highly dependent on plugging figures into set formulas. Ratios are our life, doncha know.

Well, I'm here to tell you that set formulas are, or can be, death. No, really, I mean that literally. Because you have to understand the logic of what you're calculating. We got zero credit for "decimal point errors" in pharmacy school, thank goodness! "Sorry: your patient is dead."

An Illustrative Anecdote
Every semester, in one class or another, we got a version of the same problem. You get a prescription for one pint of potassium permanganate solution, such that, when the patient takes a tablespoonful and
dilutes it to a gallon, he gets a 1:5000 concentration. How much potassium permanganate do you need? It's the sort of problem most students figure you "need to know for the test" becasue you'll never see it in real life.

Guess again.

When I worked at the hospital, my director was an incompetent Captain Queeg from "the Caine Mutiny". And one day, we got this problem. Everyone recognized it, and Queeg shut down the department while he set everyone working on it. I mean, really, c'mon, it's just a basic ratio problem but NOOOoooo, it was that ratio problem! Work on it and show your work.

Well, what I figured was this. I needed to know how much KMnO4 had to be in that tablespoon. And I needed to know how many tablespoons I had to make. So I told him the answer, and when he asked for my work I told him what I just said. Wouldn't buy it. Where's your work, where's your formula? I just told you. Nope, unacceptable. And how come you didn't use the metric system, you know we use the metric system here.

That's where dependence on formulas can lead you. What you need to realize is that disciplines reduce to simpler disciplines. Medicine is biology. Biology is chemistry. Chemistry is physics. Physics is math. And math is logic. What kids need to be taught is how to solve math problems with logic. And from what I see, that isn't being done any more.

[cellio]

Medicine, engineering, and probably other disciplines shouldn't give partial credit. Dead is dead, after all.

I think everyone needs to be taught logic. For some, alas, it won't stick, and I'd rather everyone have some established techniques to fall back on.

[benzado.livejournal.com]

Shouldn't give partial credit?

Mathematics is about problem solving. When you have solved a problem, and you just give me the answer in a vacuum, why should I believe that the answer is correct? Hopefully you can convince me by presenting an argument and walking me through your reasoning. This is sort of what school calls "show your work" but most classes this means "write out the steps in the formula we gave you". Classes where students actually have to reason the problem out and present their reasoning look like "kids are struggling and teacher won't show them how to do the problem" and students/parents/sometimes teachers get frustrated and don't see the point.

Anyway, if "the right answer" is the goal of every math problem, and reasoning/understanding is just a means to an end that the class is not interested in, then there is no need for a concept like partial credit, because your answer is either right or wrong. Hell, it would be easier to grade.

On the other hand, if having some sense of what is going on is important, and maybe the student doesn't arrive at the right answer because of a mistake in computation, but she did write out what she was doing, maybe she should get more credit than the student who left the answer space blank?