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

Re: I work for one of those companies...

[siderea]

One of the really interesting things in all this is the question of what are the differences between how people who are good at math think about math and how people who aren't good at math think about math. What's particularly interesting about it is that as you begin to look into it, all sorts of questions arise about, for instance, causality -- are people who are better at math better because of what they do differently, or do they do things differently because they're better at it? -- and about innate and learned traits -- is there such a thing as having a knack for math, and if so, are the ways of approaching math that such a person uses of any use to someone who doesn't have a knack for it?

These somewhat rarified abstract questions have some amazingly concrete ramifications. I am very good at math, and have always been -- both computational tasks and mathematical reasoning. Is it useful for people who aren't good at math to be instructed in how to think the way I think? Or is there some difference in how my mind works, such that my methods exploit strengths of mine which others lack, and they'd learn math better with a method which draws on their intellectual strengths? Or would teaching to their strengths amount to neglecting their weaknesses? Is the way I (and other highly mathematically adept people) reason about math required for reasoning about math, such that it's necessary? If I sit down and tutor someone in math (as I have done) is it useful for me to use my own reasoning process as a guide for what good mathematical reasoning looks like? What if that way of thinking is really alien to the student?

To be clear, here, I am waaaaaaaaaaaaay off my employers' page here. I think (I don't work for the math department!) that their premise is that taking the way adept math students think about math and trying to teach less adept kids those ways of thinking is precisely the way to go. Indeed, that the premise is to teach little kids, even little kids who are bad at math, how to think about math the way adult mathematicians (and engineers and scientists) think about math.

But I think it's an interesting and open question. MB testing suggests 75% of Americans prefer concrete to abstract reasoning, and are more adept with concrete operations than abstract ones. Let us assume that is true, and that therefore memorizing and rote learning is easier and more familiar to the majority of American kids in elementary school (and their parents!) than is generalizing from cases and finding patterns. For such students, traditional drill-and-kill arithmetic lessons teach to their strength -- or allow them to coast on their strength -- at the complete neglect of their weakness; when they suddenly are expected to reason deductively, to invent or pick approaches to solve problems, or, heaven help them, prepare an equation to solve a word problem, they are completely unprepared. So the question is then how to teach that 75% to reason abstractly all along. The particular challenge is that any situation that they can turn into one of memorization or method, they will, because they prefer it. These are students that, (when you tutor them) when asked to reason out a problem, say crossly, "Just show me how to do it!" So on one hand, you want to exploit their strengths, but on the other hand, you have to fight against the students' attempt to reduce all problems to right-answer-getting.

But then, that leaves the question of what happens to the other 25%! What about the kids who are good at abstraction and weak on concrete operations? I'm one of those, and from your description, I'd guess you are, too. If you are teaching to the other 75%, how does that serve the kids like we were? Does it neglect our weaknesses? Or would it be a mercy not to be required to work exclusively from our weakness?

I don't know the answers to these questions; I find them fascinating. And this doesn't even get into the other learning-styles questions. There's lots more questions where these come from.

Re: I work for one of those companies...

[sui66iy.livejournal.com]

Yeah, and the questions aren't only relevant for young students. Many college professors aren't really teaching per se. They're often better resources than they are teachers, but that leaves success to those students who can figure out how to extract the knowledge.

Here's a (weak and naive) analogy: in elementary school, they teach you how to use the library. Rather than have you read every book, they teach you about card catalogs and where the reference section is and so forth. Supposedly, then you can go and find the knowledge you need when you need it. But teachers are resources like libraries, and so is your own learning style. I wonder if there's a way to teach some sort of meta-learning skill.

I sometimes wonder what college would be like if I went back now. On the one hand, my brain is probably less flexible, but on the other hand, I'm a lot more mature and have a lot more experience teaching myself...

Re: I work for one of those companies...

[siderea]

But teachers are resources like libraries, and so is your own learning style. I wonder if there's a way to teach some sort of meta-learning skill.

Heh. http://siderea.livejournal.com/440902.html

I sometimes wonder what college would be like if I went back now.

Hehheh. I'm running that experiment now. I've returned, at the ripe age of 35, to college to finally get a degree (this time). Things are much better this time, for a whole bunch of reasons, but not least because I've gotten over my past schooling and become an autodidact. Apparently, this is normal in "adult learners", according to the returning student orientation materials the school gave me -- and backed up by my experience in all-"adult-learners" classes. The grown-ups are voracious and keep the profs on their toes. They'll show up with their own learning (and other) agendas and hold the profs accountable for meeting them. They're no little bit cynical and know how the world works, and have an attitude of "I'm paying for this, so you'd better live up to my standards." They butt into lectures to demand clarifications. They raise contradictory examples to challenge professors' examples. They have no problem saying, point blank, that school has to take a back seat to work/kids/etc. when they feel it does. They're intensely involved in the classroom and with the material, just like professors always are saying they want. As one prof put it, "The up-side is you guys bring an incredible wealth of experience and engagement to the classroom. The down-side is, damn, ain't nobody can tell you guys nothing!"

tangent

[cellio]

But then, that leaves the question of what happens to the other 25%! What about the kids who are good at abstraction and weak on concrete operations?

This is, of course, a problem that goes well beyond math, and gets to the whole question of when it's ok to cater to the majority and what you owe the minority. I felt largely cheated by my school system because they had no accommodations for people near the tails of the bell curve; it was very much a "one size fits all" approach. I don't know what the answer is; even if it were practical for every student to have an individual, tailored experience, I don't think that would be ideal either. Somewhere in there you also have to learn about compromise and getting along in community, too.

Re: tangent

[siderea]

Yepyep! And that raises some very fundamental questions about (1) how a society allocates resources in general and (2) just what is universal education trying to achieve.

If we are educating kids to a certain standard.... why do we have (or try to have) all kids stay in school until 12th grade? And why do we then throw kids out at age 18, whether or not they've met the standard?

Or are we providing an educational opportunity from ages X to Y, during which kids are entitled to make the most of themselves, whatever that is?

Are we trying to prepare a mete workforce for the exigencies of the 21st century? Are we trying to unleash human potential? Are we trying to educate an electorate?

What is the purpose of the enterprise? And how will we know if/when we're achieving it?