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Journal of No. 118

March 5th, 2010

Teachers & Books @ 11:31 am

Before the trip, I dashed through Paul Lockhart's A Mathematician's Lament. At first, there was some serious deja vu, since the book is based on a previous essay available online that got a great deal of notoriety when it went viral among the math teachers. I invite you all to read at least the first page of the PDF: the parable of the music teacher.

His basic premise is essentially that math is taught entirely wrong in the schools. Although the music analogy should not be strained too far, he would like math to be taught as an art or an aesthetic, rather than as a drudgery. The ability to read is useful for filling in a DMV form, but that is not why we learn to read. Similarly, the usefulness of mathematics should not be the primary reason offered for the teaching of math. It should be taught as the exploration of ideas... indeed math is the study of pure ideas.
Now, although I'm amenable to a great deal of what he has to say, I think his method or approach would be best suited to an Aristotle teaching Alexander sort of 1-on-1 tutoring. Maybe with a small classroom, it would work. With a normal classroom, even with an excellent teacher, I have grave doubts. Indeed, when I was working as a tutor, I saw the results of similar approaches. One of the schools used a textbook that didn't instruct, it rather offered situation after situation that the students were to explore on their own, constructing their own mathematical ideas. Of course what actually happened was the kids would cry out in anguish, "What do they want me to DO?" They didn't want to explore, they wanted to be instructed. Perhaps that's only because the rigid educational system had already crushed their wee imaginative souls. I tried to play along with its games, and coax the students forward, but I found it maddening to try to get across the concept of, say, factoring a quadratic, from pictures like this (especially since they would introduce the idea at the beginning of the chapter with diagrams like that without any labels on them. It was like an Abbott & Costello routine, with me trying to lead the student by the nose, and the student quite rightly wondering why I wanted to screw up a perfectly clear fact like 5*9 = 45 and breaking it up into little rectangles and what would motivate one to do such a silly thing.
Anyway, I do agree with a lot of what he has to say, and it's clear how much passion he brings to mathematics. If every math teacher were that passionate, they could be teaching math from 19th century McGuffey readers and the kids would turn out fine.

Lockhart's book dovetails nicely with this lengthy article from the NYT Magazine on Building a Better Teacher. It details the obvious fact that education schools don't really teach teachers to teach, but explores how some people are beginning to look into the actual mechanics of teaching.

Once upon a time, teachers were taught at normal schoolsm which focused on "provid[ing] a model school with model classrooms to teach model teaching practices to its student teachers." Somewhere along the way, education school got absorbed by the university, and in some sort of academia-envy, education school turned into spending more focus on Piaget than pupils. Despite student teaching, nascent teachers may not be fully prepared to make a smooth transition to the workforce. And once there, there is even less help (or even knowledge) available to help teachers to improve their teaching. Lots of interesting details in there, like the concept of Mathematical Knowledge for Teaching, or MKT. MKT includes basic mathematical knowledge, but also "math that only teachers need to know, like which visual tools to use to represent fractions (sticks? blocks? a picture of a pizza?) or a sense of the everyday errors students tend to make when they start learning about negative numbers." Tests of math teachers on MKT show that "students whose teacher got an above-average M.K.T. score learned about three more weeks of material over the course of a year than those whose teacher had an average score, a boost equivalent to that of coming from a middle-class family rather than a working-class one." This is signficant in that there are evidently surprisingly few testable measures that differentiate effective teachers from less-effective ones. Anyway, good article.

On the trip to Orlando, I read Margaret Atwood's Alias Grace, a fictional look at the life of a celebrated Canadian murderess. I enjoyed it, particularly the way certain events in the doctor's life strangely mirrored those of Grace. Excellent capture of the historical era, and Atwood does a marvellous job giving individual voices to the characters (bits of the novel is epistolary).

Oh, and as a last tidbit. I was reading the Memoirs of Elizabeth Frankenstein on the plane back from Orlando and the girl across the aisle from me was reading Pride and Prejudice and Zombies, so we had duelling horror literary rewrites. Not yet done with Memoirs, but it's less satisfactory than Alias Grace by far.

Ostensibly a retelling of the events preceding Frankenstein from Elizabeth's point of view, indeed from an aggressively female point of view. But it does not respect the source very well, even in so trivial a detail as the number of brothers that Victor Frankenstein has. And the bulk of the novel (so far) is a rather embarrassing foray into matriarchy & cunning-women & the alchemical Great Work. I think even MZB would blench to write some of this. The only relief is from the occasional editorial hand of Frankenstein's original narrator, Walton. Walton is so antifeminist that his shock at, and deliberate misunderstanding of, Elizabeth's memoirs are perhaps the cleverest bits of writing, and (alas) all too rare.
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Date:March 5th, 2010 07:56 pm (UTC)

You know, I was put up a year in math around 6th grade and I can't fathom what it was that worked or didn't work about how teachers went about getting the concepts across. I do remember liking formal logic in my Geometry class (I sat with a bunch of sophomores who hated everything math-related) precisely because it was a break from the weirdly abstract little symbols but also because it exercised my brain in a new way. I probably would have liked the idea approach. But then, I was a weirdo.
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Date:March 5th, 2010 08:06 pm (UTC)
It's interesting that geometry is the closest in spirit to what math is 'really' like, but it's the area where he sees the greatest failing in the schools:

There is nothing quite so vexing to the author of a scathing indictment as having the primary target of his venom offered up in his support. And never was a wolf in sheep's clothing as insidious, nor a false friend as treacherous, as high school geometry.

But yes, the brain-stretching you experienced was at least part of the good that can still sometimes be found in HS geometry. He derides the formal proof as being unnatural and corrosive of imagination, but all too often logic and the proof are disappearing from geometry to be replaced by memorization of area and volume formulae. As much as he dislikes the formal proof, I think he'd agree with me that that's a step in the wrong direction.
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Date:March 5th, 2010 09:53 pm (UTC)
Funny you should mention this. Geometry was the only 'math' I was ever any good at. For me it was fun and it had a point (no pun intended). As you may imagine, I think I'm part of a very small minority.
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Date:March 5th, 2010 10:30 pm (UTC)
You're not alone. I could see what was going on, unlike in algebra.

Of course, all of my high school math classes were self-paced and and textbook only - unless you needed help on a section, then you could go up to the teacher and he'd go over it. Which was great for the kids who already had a knack for it, much less so for everyone else.
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Date:March 6th, 2010 02:22 am (UTC)
I love that essay. I've been tutoring HS geometry for a few years now, so I've seen good and bad. I'm particularly interested in learning more about how the Japanese teach math, because I tutored one student who'd spent the first few years of his education there, and he was very good. He said they did things like derived pi from scratch, at the elementary school level.

Journal of No. 118