Teaching mathematics as though consciousness matters

[nancylebov]

http://www.maa.org/devlin/LockhartsLament.pdf

The main problem with school mathematics is that there are no problems. Oh, I know what
passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here
is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad
way to learn mathematics: to be a trained chimpanzee.
But a problem, a genuine honest-to-goodness natural human question— that’s another thing.
How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a
number? How many ways can I symmetrically tile a surface? The history of mathematics is the
history of mankind’s engagement with questions like these, not the mindless regurgitation of
formulas and algorithms (together with contrived exercises designed to make use of them).
A good problem is something you don’t know how to solve. That’s what makes it a good
puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as
a springboard to other interesting questions. A triangle takes up half its box. What about a
pyramid inside its three-dimensional box? Can we handle this problem in a similar way?
I can understand the idea of training students to master certain techniques— I do that too.
But not as an end in itself. Technique in mathematics, as in any art, should be learned in context.
The great problems, their history, the creative process— that is the proper setting. Give your
students a good problem, let them struggle and get frustrated. See what they come up with.
Wait until they are dying for an idea, then give them some technique. But not too much.

Link thanks to andrewducker.

Comments

[whswhs.livejournal.com]

As somebody who calculates things for pleasure, and who uses mathematical relationships in various projects (for example, I just worked out the volume and weight of a limestone pyramid of a specific size for GURPS Low-Tech, on which I'm the lead author), I've always hated those "now let's have you struggle with this problem for a while, trying to reinvent the wheel, and then if you can't figure anything out we'll give you some hints" approaches to learning mathematics. If I have a problem I want to solve, I'm quite capable of appreciating a technique of solving it without needing that kind of buildup, and I may go looking for such a technique, or ask online if anyone can point me at one.

And looking at the matter from the other side, I spent a year tutoring a high school student who was enrolled in a "let's have you discover natural laws for yourself" physics course. His mind was completely unengaged, because he wasn't interested in the first place. One of his recurrent problems was that in a later week he would be trying to figure something out that built on what he had done in an earlier week . . . but he didn't remember what he had done earlier, and hadn't kept careful records of it. He had no sense of accumulating knowledge; he was just dealing with arbitrary externally imposed problems.

As to the argument that people don't need to learn how to balance a checkbook because they have calculators, I've long since stopped believing that one. I remember the young woman I was tutoring in accounting many years ago. As an illustrative problem, I proposed a situation that involved figuring 1% of a million dollars. She said, "A hundred thousand dollars." I questioned that, and she said, "A thousand dollars?" My experience is that if someone doesn't know how to figure out the right answer for themselves—indeed, if the approximate right answer doesn't jump into their mind—then they are quite capable of pushing the wrong buttons on a calculator, getting an absurd answer, and not understanding that it's absurd. And this certainly applies to balancing a checkbook, where the real issue isn't adding and subtracting correctly, but grasping the underlying algebraic relationships between debits and credits that let you set up the calculation you actually need to do.

I was able to figure out multiplication, division, fractions, decimals, and percentages in third grade, from one reading of a short review of basic math that I happened to encounter. But a lot of students can't do that. They need to learn through repetition. And the kind of instruction that's useful to someone mathematically gifted, or even someone who just has a feel for basic numerical relationships such as I have, isn't going to help the other kind of students.

[nancylebov.livejournal.com]

Thanks. The essay in my link probably underestimates just how good teachers need to be for a question-oriented approach to teaching math and/or neglects students' different learning styles. For that matter, sometimes I learn best (or at least do learn) through repetition.

[dcseain.livejournal.com]

...After class I spoke with the teacher. “So your students don’t actually do any painting?” I
asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main
Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and
apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that. ...

This matches my mathematcs expcerience. My 6th grade math teacher taught us algebra. In 7th grade, i hade to take pre-algebra, which covered less than the 6th grade math class had, then went on to algebra, which covered what my 6th grade math class did. End result, by end of middle school, i considered math useless, in part because one is never taught useful application.

Got to HS, where application was combined with the lessons, and it got more interesting. Still, in the end, only statistics is enjoyable for me.

[richardthinks.livejournal.com]

I have exactly the same reaction to the way history is "taught," even at a professional/academic level. History is an exercise in narrative construction and problem-identifying (not even problem solving, much of the time, but trying to figure out questions that might be answerable and that might illuminate something about "the bare facts," to give them some meaning). It is almost never taught as this, though. Instead you get the results of other historians' work to look at - either suspiciously neat narratives or slender theories backed up by volumes of hand-selected data, and it's left to you to infer what the motivation might have been for constructing the historical narrative in just this way.

As far as maths goes, I learned to apply formulae quite well but never made the leap to knowing why or how I should choose a certain formula, much less how to derive new formulae as needed. I remain convinced that it's a field not everyone is suited for, and that many students feel that it might just be outside their grasp. The dispiriting realisation that you have to work hard and still may not get it is, I think, what lies behind the often-repeated complaint that maths "isn't useful." Hardly anything you learn in school is "useful," but most of it is achievable in some solid way, so you can finish the task and get your prize.