(no subject)

[rivka]

I just read an interesting article by a mathematician, lamenting the way his subject is tortured and murdered in schools. (Article is here, in a PDF.) Here's his basic thesis:

All this fussing and primping about which "topics" should be taught in what order, or the use of this notation instead of that notation, or which make and model of calculator to use, for god’s sake— it’s like rearranging the deck chairs on the Titanic! Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion— not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it. Remove this from mathematics and you can have all the conferences you like; it won’t matter. Operate all you want, doctors: your patient is already dead.

The saddest part of all this "reform" are the attempts to “make math interesting” and "relevant to kids’ lives." You don’t need to make math interesting— it’s already more interesting than we can handle! And the glory of it is its complete irrelevance to our lives.

He's passionate, furious, despondent, and very funny, producing gems along the lines of:

All metaphor aside, geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum. Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured. (Apparently I am incapable of putting all metaphor aside.)

The article is long, but I found it a quick read. It's worth reading to the end, if only to get his truth-in-advertising summary of the K-12 math curriculum ("TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds.") I was one of those people who was very good at plugging numbers correctly into formulas but never felt like I had a good conceptual grasp of math. This article makes me feel sad about what I missed.

Comments

[selki.livejournal.com]

I learned more about the interesting aspects of math in the philosophy dept. at college than I ever did in any math class.

[txobserver.livejournal.com]

You would enjoy reading Alan Kay on math education, and you should have a look at Sugar on a Stick (SOAS) at sugarlabs.org

[wcg.livejournal.com]

Thanks for the link. I happen to like geometry and trig, so I disagree somewhat about their utility. But I certainly agree that the standard academic math curriculum does more harm than good.

[laurent-atl.livejournal.com]

i think the point of the article is not to say kids should not take geometry and trigonometry, but that this should not be done following the current curriculum

[naomikritzer]

Molly loves math. (She especially loves number theory, and codes and ciphers.) Because of this, I've discovered there are actually a lot of really kick-ass books about mathematics written for children that emphasize the interesting stuff rather than memorization and arithmetic. (I actually think kids all do need to learn to do arithmetic, even though it turns a lot of kids off to math. It's too generally applicable to skip over. Anything beyond basic algebra, I'm not sure why we require it.)

Some titles, if you're interested for Alex (or yourself):

The Number Devil, which has a little boy who gets kidnapped in his dreams by a demon who gives him a tour of mathematical concepts like prime numbers and the Fibonacci sequence, among other things.

The Man Who Counted, which is more for adults (though Molly read and liked it), which has problems you can work out as you read it.

Math for Smarty Pants and The I Hate Mathematics Book by Marilyn Burns, who is a really interesting math educator. She wrote a book for adults called Math: Facing an American Phobia that talks about how she teaches math and how her methods help kids grasp things like fraction conversions. Alas, if you're at all familiar with the Everyday Mathematics math curriculum, it's kind of what happens if you take her ideas and try to turn them into a curriculum. One of the fascinating things about education is that when you take good ideas and turn them into a curriculum, even if you do so with the best of intentions and lots of advice from smart people, it all goes to hell.

And recently:

How Math Works, which is colorful and not too long.

The Moscow Puzzles, which we have out of the library right now; some are too hard but others are really fun.

Professor Stewart's Cabinet of Mathematical Curiosities

There's a lot out there. This is emphatically not how it's taught, though. I actually think kids do need to learn how to do arithmetic, because it's so generally useful in everyday life. However, I don't know why as a society we've decided that everyone needs to learn trigonometry. I have forgotten nearly all the math I learned beyond basic algebra because I have never ever in my real life used it. And I went all the way through Calculus III (in which I got a D, the lowest grade of my college career).

[abbylee]

*nods* I've always loved math, but found the best way to fuel it was outside of school.

[naomikritzer]

OK, I'm reading a bit further in. I love his ideas about teaching geometry. Some of my thoughts on his commentary are colored by the fact that I've volunteered in an elementary school for three years, and my older kid is not in high school yet (or anywhere near it). He clearly WANT to apply his ideas to grade school, but I don't think he's actually taught in one.

There are kids who come to grade school not knowing how to count. Or who start first grade unable to do addition problems like 3+1. Part of the purpose of teaching early grade school arithmetic is to teach kids things like how to make change, or know if they were given the correct change. This is important stuff.

[naomikritzer]

Well, I stand corrected; I googled, and he teaches K-12, all grades.

That implies he does enrichment, though, rather than being the only math teacher these kids get. So it's someone else who makes sure they can count, add, multiply, etc.

High school math has been in the news here (in MN) a bit, because we had this test that all the high school kids were supposed to pass in order to graduate. However, very few (like less than a third, IIRC) were passing it. They ran some questions from the test in the paper, and there is no way in hell most of the adults I know could pass this test, which raises the question, why are we requiring these students to do so? It's really not clear to me why we have set mathematics (rather than music theory, say, or formal logic, or Classical Greek, or some other not-wide-applicable intellectual exercise) up as one of the major things that all kids should study. Basic arithmetic is useful. Fractions are useful. Basic algebra is useful. Beyond that, you might just as well have everyone do discovery-style math with a teacher like Lockhart because while they might not master trig that way, they will never ever use trig for anything so who really care? and with Lockhart's approach, they'll at least learn a lot about exploring ideas and thinking about problems. There's very little inclination to question why we think students need to learn this stuff -- what the point of it is, whether it's mental calisthenics or if it's just in case they become mechanical engineers or something.

[green-knight.livejournal.com]

Statistics is useful. An understanding of statistics will give you a life tool: can you expect to win the lottery? Do pyramid schemes work? How valuable are newspaper statistics that say '75% of all the people we questioned' when they don't tell you whether they questioned four or four hundred?

One skill they don't tend to teach in school but which is the most useful of all is the kind of mental arithmetic that in German is known as 'pi times thumb' - you round numbers and get an idea of the size of the number you expect to get from any calculation. So when you're asked to pay a supermarket bill that's fifty dollars higher than what you thought was in your trolley, you'll question it immediately, and you'll spot that something went wrong when your bank statement or tax return don't look right. In order to do that, you need multiplication tables up to twelve - and a few mental tricks.

[naomikritzer]

Statistics is very useful and is hardly ever taught in school beyond the most rudimentary level.

They do usually teach some estimation in younger grades.

[almeda.livejournal.com]

The problem I had with estimation as taught in the schools is that they only ever taught us to estimate AFTER we already had the basic math we were estimating down pat ... so I thought it was a huge, useless waste of time and energy, and didn't bother even trying. Much, much later it became clear to me that estimation is very useful (in the 'how to tell if your calculator is lying to you' department -- does this answer look like the sort of answer you ought to get?), but the way it was TAUGHT caused me to instantly write it off.

[panacea1.livejournal.com]

See, I'm one of those weirdoes that thinks a solid general education should include, among other things, music theory, formal logic, and if not classical Greek then at least a riding-the-subway-and-reading-the-newspaper functional proficiency in a language not spoken in one's home. A general education should be a sampling of everything - the useful and the intriguing both - not just a "crank 'em through" process. I'd like to see number theory and intro to prob & stat at the high school level as an alternative to the trig/calculus sequence, though. Most people won't use a whole lot of calculus, it's true. Geometry, otoh, is widely applicable to the visual arts - if you let it be. But I digress and am about to rant in a stranger's journal, so I'd better stop now. ;-)

[almeda.livejournal.com]

My Freshman Comp II english class in college (at a community college, actually) was basically Classical Rhetoric. It was awesome. Stuff like the different kinds of fallacies should totally be taught in 4th or 5th grade English, both as a way of teaching kids to calibrate their b*llsh*t meters, and to help them figure out how to structure a paper.

[almeda.livejournal.com]

I use trig for craft projects (if I want to make something x high at the peak, with a slanty top, and y high at the edge, and it's z wide between those two verticals, (a) what angles are involved for the structural members, and (b) what's the length of the third side I need in materials?

Etc. Anything crafty involving angles can be aided by application of trig, though it's not necessarily REQUIRED.

[nancylebov.livejournal.com]

I liked the essay a lot, too, and linked to it recently. I got this comment (http://whswhs.livejournal.com/) from someone who was already interested in math learns better by being told how to do things.

I still the essay has a lot going for it, but probably underestimated how good the teachers would need to be to get the challenge level right.

[pfctdayelise.livejournal.com]

That's a great read. Almost makes me want to go and be a maths teacher to try and improve things.

I totally agree with his comment about the lack of historical context in maths teaching. Not that students should learn the history of mathematics, pre se, but the history of problems.

thanks for sharing!

[micheinnz.livejournal.com]

My love of mathematics was badly damaged by a bad teacher when I was fifteen (tenth grade, US?). I'd gone from marks in the high 90s to marks in the mid 60s, due to his crap teaching. I told my parents that I was worried I wasn't learning anything (that class was mostly crowd control, and he was crap at that too), so they went to see him. His reply? "I don't see what she's concerned about -- she's top of the class!"

So I switched right off and let my marks drop until by the middle of the next year I was failing maths and put into remedial tuition. Which I basically refused to make any effort in, because I couldn't see the point.

I didn't get my mathematics back until I went to Polytech in 2007 to do my electrical course. We started over again from fractions and negative numbers and went all the way to vector calculus in nine months. It was glorious. And it felt like I'd got part of myself back that I didn't know I'd been missing.

[janetmiles.livejournal.com]

It wasn't until I took pre-Calculus in college that someone pointed out that the distance formula is just the Pythagorean Theorem turned inside out. That turned it from a painfully memorized framework into an AHA!

I hated proofs in high school, because I always seemed to end up going in circles. On the other hand, I loved them when I tool Introduction to Real Math in college. I think I scared a co-worker when I bounced in one morning and asked, "Want to see a really cool proof that the square root of two is an irrational number?"

[guruwench.livejournal.com]

I know so many people who hate math, but I've always liked it (with a couple of exceptions). Geometry I had trouble with, as I had trouble visualizing things. I also hated proofs!

Janet, I would have thought that proof would be cool to see, and I wouldn't have been scared at all. ;)

[janetmiles.livejournal.com]

Here you go -- the link is slow to load, but it gets there eventually.

[johnpalmer.livejournal.com]

Apropos of nothing, one line stuck out at me:

If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself.

It's an interesting thing, because I know that a lot of children (myself included) were deprived of the ability to be wrong safely and happily, or taught that frustration could be a productive thing that lets you ponder a while before you spot the trick.

As for missing things, anyone can learn to enjoy a spot of math. A good beginner's problem is this: How high do you have to go to search for prime factors of a number? For example, to figure out that 30 is 5x3x2, you don't need to check 29, or 28. But how high should you go?

This is a problem that is really simple... but a lot of folks, pondering it (and arriving at the right answer) get their first taste of why math can be fun.

This seems too easy, so I'm probably wrong, but...

[janetmiles.livejournal.com]

For any positive integer X, the largest prime factor of X will be less than or equal to X/N, where N is the smallest prime factor of X.

Re: This seems too easy, so I'm probably wrong, but...

[johnpalmer.livejournal.com]

That is, in fact, almost the perfect solution... you have all of the major important facts down, it's all a matter of figuring out what question you have the answer to. (And that can depend on what you *want* the answer to.)

You're right; the key to the answer is thinking about what happens when you multiply the larger by the smaller... assuming there *is* a larger, and a smaller (e.g. 25 is 5x5), and in thinking about what happens when you divide the number by the factor you've just found.

[miscrants.livejournal.com]

This article makes me feel sad about what I missed.

You might like this video on how to turn a sphere inside out (http://video.google.com/videoplay?docid=-6626464599825291409). Although there are good topological reasons for the those rules, imagining surfaces that can pass through themselves but not crease does have some of that feeling of glorious irrelevance Lockhart talks about.

These visual proofs (http://www.billthelizard.com/2009/07/six-visual-proofs_25.html) are kind of cute too.

(I'm some random person you don't know. I got here via Respectful of Otters and I think a side link on Making Light, and stuck around because you write interesting stuff about parenting. I hope that's okay.)