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A Mathematician's Lament
April 10, 2008 7:34 AM   Subscribe

“…if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.”
“A complete prescription for permanently disabling young minds — a proven cure for curiosity. What have they done to mathematics! There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could all be having so much more fun.”
It's a beautiful 25-page PDF, just 392k. (via)
posted by blasdelf (79 comments total) 77 users marked this as a favorite

Leland MacInnes has some constructive curriculum proposals...
posted by anthill at 7:40 AM on April 10, 2008

I think the author underrates their own creativity -- I'm pretty sure they could design a superior curiosity-crushing machine.
posted by aramaic at 7:41 AM on April 10, 2008 [1 favorite]

This reminds me of Feynman's story about reviewing math textbooks.
posted by horsemuth at 7:44 AM on April 10, 2008 [1 favorite]

Word, not LaTeX? What kind of mathematician is he?
posted by demiurge at 7:44 AM on April 10, 2008 [2 favorites]

The schools of math taught during high school are strange as well; why is calculus typically favored over discrete math, or probability and statistics? Maybe schools are worried that a population educated in probability won't buy any more education-funding lottery tickets...
posted by jenkinsEar at 7:44 AM on April 10, 2008 [9 favorites]

The (via) in the post was found via this CS blog +followup
posted by blasdelf at 7:48 AM on April 10, 2008

demiurge: Unless he had used pdflatex, the resulting PDF out of vanilla LaTeX would have been ugly, inelegant, and HUEG. He doesn't need the equation formatting, and Word doesn't have to suck if you use it correctly (unlike using Excel for statistics).
posted by blasdelf at 7:53 AM on April 10, 2008

It's a beautiful 25-page PDF, just 392k.

A detail which should have been mentioned on the front page.

(will go read now)
posted by [NOT HERMITOSIS-IST] at 7:53 AM on April 10, 2008

The problem isn't that math education is designed to crush the enjoyment of math (although it certainly does) it's that math education is designed to prepare children for a lifetime of doing arithmetic without a computer.

And a lot of it, it's really ridiculous, is just based on parents thinking that their children ought to be taught math the same way we were. I remember when I was a kid in 7th grade we had an 'experimental' section that taught the powers of ten, logarithms, etc in a fun way. Half the kids complained because they thought it was too easy, and they sort of felt like they were being condensed too or something.

And there was actually a lot of complaints from parents as well, thinking that their kids were somehow missing the 'hard' work of doing arithmetic on paper, they called it "McMath" etc. It was all absurd. My mom said she was one of the few people who actually advocated for it at a school board meeting, most of the parents hated it.

And it was good stuff too! Very informative, I thought at the time.

I also remember a time in highschool, when I was taking a calc class our teacher was venting about a school board meeting where one of the board members actually said "I add and subtract factions all the time, but I can't remember the last time I used integers!" Now, aside from the fact that you need integers to do fractions, the comment is absurd because in today's world you have computers to do all that drudgery. You're never going to need it, so teaching kids this way really does build up a hatred of math in a lot of them.

We have computers, they're not going away. But there is a real resistance out there to changing the way we teach math, and that resistance comes from people who barely know it!
posted by delmoi at 7:55 AM on April 10, 2008 [1 favorite]

The schools of math taught during high school are strange as well; why is calculus typically favored over discrete math, or probability and statistics?

Well probability and statistics would be pretty useless without calculus. You could teach people the formulas, but they wouldn't be able to understand how they are derived.
posted by delmoi at 7:57 AM on April 10, 2008

This is not a very new observation. In "Computer Lib", Ted Nelson tosses off an aphorism that went something like this: The goal of education is to teach students to hate subjects. The one the student learns to hate last becomes the student's career.
posted by Class Goat at 7:59 AM on April 10, 2008 [25 favorites]

See also this AskMe thread and Clifford Stoll's TED talk[video, autoplays]. And of course Feynman's observation that "there is no science being taught in Brazil."

This isn't an enormous pet peeve of mine, no siree.
posted by Skorgu at 8:01 AM on April 10, 2008 [2 favorites]

Delmoi - you use a computer to figure out your money in a grocery store?

There's a reason - a damn good one - to teach arithmetic Fractions even. If you don't have some idea what the answer is going to be -and you won't if you always rely on a calculator -you have no idea if the answer is correct. I get kids come into the math lab at the community college, that have no idea that something times ten is gonna have a zero at the end. They have no idea that a half, and another half, is *one*. They can type it into a calculator, sure as shit, but 15 + 8, without a calculator, takes them a couple minutes, as they add it out on their fingers. They can't figure out 6% tax on a hundred dollars without plugging it in.

I used to joke that word problems had an advantage over regular algebra - you can at least tell if your answer makes sense. YOu know, if you get the volume of your swimming pool to be five gallons, you know you screwed up somewhere! Lately, though, I've had students honestly ask me, what's wrong with that answer?
posted by notsnot at 8:08 AM on April 10, 2008 [6 favorites]

There is a distinction between calculating and thinking. All education emphasizes is calculation: memorization of arbitrary rules for symbolic manipulation to perform the calculation properly. This is because education was dumbed down in the 50s to more efficiently engender a tax farm for the powers that be in which the world's populations are the obedient cash flow generating herd. It's sort of like the Matrix except with cash instead of btus. Those who excel in school are those who are most amenable to stimulus-response conditioning.
posted by norabarnacl3 at 8:16 AM on April 10, 2008 [4 favorites]

notsnot: Be careful with that. I teach at a state university, and the students usually just don't care if their answer "makes sense". 90% of the time, they're not even thinking about it.

For instance, a student was doing a trig problem, one of these "how tall is the tree, if its shadow is this long, and the sun is at this angle" and so forth. He got that the tree was something like 5,000 miles tall. He didn't bat an eyelash. He simply complained for "losing points" because he thought he "had all the steps right".

When I pointed out to him that such a tree would be absurdly large, he gave me a blank stare.

I don't blame the student for this, I blame the school system. It seems to me that kids are taught to divorce themselves from their intuition when doing any kind of math problem. Years later, when I come along and tell a student that his answer is absurd and he "should have known" it was wrong, he gets extremely frustrated.
posted by King Bee at 8:18 AM on April 10, 2008 [2 favorites]

> You could teach people the formulas, but they wouldn't be able to understand how they are derived.

I'm not sure that teaching the derivation of the formulas is really important in a high-school mathematics class; I'd rather have instructors spend the time explaining the concepts, and I don't think you need a lot of formalism for that.

Obviously it's been a while, but when I went through school, the conceptual basis of 'calculus' (the idea of limits, and the relationship between "slope" and a derivative and "area under the curve" and an integral, and some real-world examples of why that's useful) were all covered in pre-calculus. Calculus itself was actually performing the calculations, and I think it was of questionable utility for most of the class.

There's obviously a point where you have to be careful that your teaching isn't so high-level that it hides too much complexity from the students and makes them think they understand something that they really don't, but there's also a lot of needless formalism and derivation in introductory classes. I'd say this is a problem both in mathematics and physics, the latter being more my field. At least in an introductory, high-school level class, aimed at students who have never thought about the ideas before, there are more than enough conceptual topics to fill a semester; there's no reason to get bogged down in formalism, notation, or the derivation of anything but the most trivial formulas. All that stuff comes pretty naturally once you understand the underlying concepts, but it's absolutely impenetrable until then.

I also have an ongoing hatred of physics teachers and professors who imply through their instruction that equations and models answer "why" questions; I suppose this is less of a problem in mathematics, where the models really are the topic being studied, at least in many cases.
posted by Kadin2048 at 8:19 AM on April 10, 2008

By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject. It is like saying that Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be inspired by that?
Damn right! In a parallel, a lot of people nowaday have tried at least one to prepare and bake bread at home, following a number of recipes and certainly with very mixed results. The process itself is rather easy, but figuring out the right combination isn't ; for instance, a rather famous NYT article on quick'n'dirt breadmaking made "waves" in breadbaking enthusiast for its sheer simplicty.

If one was to start breadmaking from that simple NYT solution , most of the enjoyment and appreciation for the ingenuity would be lost, as it immediately would become just another easily accomplished thing, nice but boring. In math, a parallel of a recipe would be a plugin function such as f(X)=2x+2 , for all X belonging to N. If you aren't shown why it's 2x and not 3x , why +2 and not +2.0001 and why the .0001 difference may become huge with an interated sum, you are less likely to be trilled by an f(x).

Whereas, people that have experiment with pulverized and fresh yeast, variable fermentation times and temperature, kneading or lack thereof , had an entirely different experience. At times more frustrating, but also more rewarding when you pull out of the oven a bread with a crust so damn delicious it redefines sliced bread. In maths, a parallel could be obtaining an understanding of how rigorous the definition of a limit can be.

Similarly, demonstration of some theorem can be utterly engrossing, but more often then not it is just spitted out by many (not all) college professors/assitants. It's formally excellent, but it's rather flat and bleak like unsalted unleavened bread, which is palatable only because it's necessarily somehow sweet in mouth.
posted by elpapacito at 8:20 AM on April 10, 2008

you use a computer to figure out your money in a grocery store? you use a flint knife to skin the cow? Why not?
posted by aramaic at 8:22 AM on April 10, 2008

The cashier in a grocery store certainly uses a computer to calculate what you owe.
posted by garlic at 8:32 AM on April 10, 2008 [2 favorites]

Also got mentioned here.

I don't know what to make of this. On one hand, I sortof generally agree with Lockhart. I went through a Math Education program up to the point of doing student teaching, and then walked away in no small part because the curriculum seemed empty to me (*I* was bored more than half the time. How could I expect more from the students?), and I was far from sure I could escape that problem just by trying to do better.

But I'm skeptical of the idea that if we didn't systematically destroy people's interest in the subject via the school system we'd see numbers of people spontaneously becoming mathematicians in the same way we see people become musicians and painters. I don't think most people play with imaginary triangles the same way he's talking about in the article. There's an identification of patterns and rations and such that I think comes naturally, but the inclination to formalize things and get into the mathematicians aesthetic is rarer.

And in particular, I have a huge problem with this statement:

"In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide — a relief from daily life, an anodyne to the practical workaday world. "

Bologna. I can take the point that as with many arts, the reason you learn them isn't necessarily to do something practical -- the "music can lead an army into battle, but that's not why we do it" argument works fine for me. But (a) part of the beauty of math is its unreasonable usefulness and (b) lots of students are in fact fairly interested in how it can be used, and I don't think there's a damn thing wrong with teaching them about it. Yeah, compound interest isn't going to dazzle everybody, but the idea of leaving it out just because it's practical is wrong.

All that said, I think the spirit of his lament is correct. Even if not everyone is a natural mathematician waiting to happen if only not lobotomized by the schools, even if he overstates his case about how practicability is less important than art, I think he's quite right in a number of areas.

In particular:


A) pharmaceutical companies : doctors
B) record companies : disk jockeys
C) corporations : congressmen
D) all of the above"

(I've long theorized one way to improve the quality of instruction would be to chuck textbooks as a requirement, perhaps ban them outright.)

"The trouble is that math, like painting or poetry, is hard creative work. That makes it very difficult to teach. Mathematics is a slow, contemplative process. It takes time to produce a work of art, and it takes a skilled teacher to recognize one. Of course it’s easier to post a set of rule than to guide aspiring young artists, and it’s easier to write a VCR manual than to write an actual book with a point of view.

Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognize it when they see it. It is not necessary that you learn music from a professional composer, but would you want yourself or your child to be taught by someone who doesn’t even play an instrument, and has never listened to a piece of music in their lives?"

There is a larger problem here, I think. The educational system as it is now produces a class of professional educators who may (but much more often, may not) have practiced in their field. This is another huge reason why I walked away from education. Because I wanted to spend at least part of my life being a practitioner if I could, and if I was ever going to teach, I wanted to teach as one.

This is not only a problem in the field of math. How much writing instruction consists of "VCR manual" rules and procedures? I had some good instruction in public school, but a lot of it fell in the VCR manual category.
posted by weston at 8:33 AM on April 10, 2008 [1 favorite]

The really exciting math that we want schools to teach can't be done if you're innumerate. If you can't do basic math without a calculator, you're innumerate. Luckily, that won't prevent you from living a full and happy life. But the idea that we can just skip the "boring" stuff in math makes no sense to me. You can't do it in any other part of life. I don't have time to read the full essay right now, but I did look at the intro, which relies on an analogy to music. Here's an analogy to music that I think is more apt - to be able to play the really complex and amazing compositions on any instrument requires practice, often years of practice, and practice is really boring. But it would be absurd to get rid of it because it may be killing the joy of music for children, even though that is in fact often the case. No boring practice, no mastery.

I'm sorry, I just don't get it! Understanding basic mathematical operations and algebra -- that stuff is a prerequisite for everything else.

Also, from my skim, it seems that this guy is saying we should get rid of math as a curricular requirement because we've failed to teach it well. It would be consistent, in that case, for him to call for the dismantling of the public school system, but it's far from clear that this would result in a better education for all (or most, or some).

I'm sorry to be brief. I will collect some good essays on this topic and post them later. I promised myself I would do work today.
posted by prefpara at 8:38 AM on April 10, 2008 [9 favorites]

A little off topic, but what the heck:

I teach introductory painting at a large public university, known primarily for it's engineering programs. This university tends to attract hard-working but incurious students who view higher education as vocational training and are easily turned off by fruity intellectual stuff.

And boy, do they suck at painting.

As a result, the studio art department is pushing a curriculum based on rote, decontextualized color theory, just like the nightmare scenario described in this article.

Life imitates art!
posted by ducky l'orange at 8:42 AM on April 10, 2008 [4 favorites]

I don't skin the cow with an obsidian stone, no, but at least I know it's a fucking mammal!
posted by notsnot at 8:45 AM on April 10, 2008

Anecdata here, but after years of undergraduate stats in social and life science courses with some some rough calculus thrown in, I find myself in a graduate stats class aimed at master's of education students (due to a fluke in the curricula, blah blah blah). There are two weeks left in class, and we haven't done a single actual statistical problem. The professor claims to be afraid of "losing us", so he's dumbed the down the course so that it's all basically "real world" examples of the concepts he's talking about. For a few classes I assumed this was leading somewhere and took careful notes, but at this point I study for other courses and play Tetris on my cell phone. The teachers I'm in the course with do the same.

This is apparently the norm for math classes aimed at teachers.

This school is fairly well ranked (70th percentile, I think) when the quality of our education graduates is assessed.

I have no hope for math education improving anytime soon.
posted by Benjy at 8:59 AM on April 10, 2008

You know, part of the issue here is that you can't really make someone think independently. At best you can recognize it when it happens, and give the guy doing it a thumbs-up. And in fact, recognizing independent thought is pretty tricky — by definition it's going to be something you don't expect, so it often goes right over your head.

All of which is to say that, while it would be nice if schools could somehow churn out creative, independent thinkers, it's not likely to happen — they just aren't the sort of thing one can churn out.
posted by nebulawindphone at 9:03 AM on April 10, 2008 [2 favorites]

I absolutely could not stand music education when I was very young it was either being forced to sing songs I hated or learning classical theory and being made to write a certain melody by a set of rules that didn't make sense to me because the kind of music I liked never followed that set of rules.

It seems that what delays so many peoples' curiosity in math is the grueling repetition of poorly conceptualized algorithms at a young age. For me, it wasn't until Calculus that I really understood the power and beauty of mathematics in its own right, and not as a tool for solving boring and trivial problems. Young idealistic children seem to be receptive to some of the beautiful abstract aspects of science and math as long as they aren't too cryptic.

Additionally, in our climate of declining student performance, educators and administrators at the top levels are approaching the problem by enforcing greater accountability and assessment rather than creativity so that they can prove that it is actually the students and parents fault instead of lousy curricula.
posted by hellslinger at 9:06 AM on April 10, 2008

delmoi writes "Now, aside from the fact that you need integers to do fractions, the comment is absurd because in today's world you have computers to do all that drudgery. You're never going to need it, so teaching kids this way really does build up a hatred of math in a lot of them."

Part of my problems with math are described by what you just suggested, learned hate of trite calculation. Some of my nightmareish calculus and financial math exams appeared to me to be so because of an anxiety I developed over producing the correct results, the goddamn "right number". While the subjects remain fascinating and informative, I spent probably too much time over training myself to calculate _accurately_ and in a limited amount of time.

Yet if the objective of the exam is to measure how quick and accurae you are, then I could certainly have used a graphical calculator with a number of preformed function and deliver the result in a fraction of a minute. But no sir, I had the usual solar powered chinese calc. Hell, I would have killed for an entire math course developed on Maple/mathematica or similia, so that I could have also learned how to exploit these immensely powerful programs and eventually PLAY with maths, which is fun.

Indeed, as King Bee points out about one student

King Bee writes "He got that the tree was something like 5,000 miles tall. He didn't bat an eyelash. He simply complained for 'losing points' because he thought he 'had all the steps right'"

which may suggest that all he really cared about was spit the goddamn number and be done with it , as he didn't bother checking if the results were sound, and that he was under the impression that following a sequence of steps is what math is all about.

In a parallel, as I am teaching accounting right now, I see the same anxiety in pupils who seems to more just concerned with writing the right numbers in the right box, left or right box who cares what's the difference? They entirely miss the utility of accounting because they only see the tax-evasion and balance sheet aspects, which are infinitely boring unless you have your own company.
posted by elpapacito at 9:07 AM on April 10, 2008

I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics science literature history art music etc.... education.
posted by ZenMasterThis at 9:10 AM on April 10, 2008

Some comic relief with a very fitting MP's Village Idiot in Rural Society video.
posted by elpapacito at 9:21 AM on April 10, 2008

Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognize it when they see it.

The problem with this, and the same is true of many other technical fields, is that a love of math and the ability to practice are valuable skills in many fields. Would Feynman have taught elementary school? Is the teaching of children valuable enough to move scientists from the job of figuring stuff out to teaching children, and are we willing to monetize enough for them to make the choice? $50000, maybe more, if we are to lure bright eyed college grads with physics degrees into small, culturally barren rural towns. While the old saw of "those who can, do. those who can't, teach" is overly dismissive of the abilities of a good teacher, when a love of math and science can get you an engineering or science degree, and a well paying job in the city, how do we lure people to teach?

The cynic in me also thinks that the author severely overestimates the creative abilities of people. While the story of the large authoritarian, corporate behemoth squeezing the creativity out of small children makes for a wonderful picture, I think it gives a disservice to the tenacity of creativity, and overestimates the number of people who have that inquisitive spark. Many people I talk to don't know how lights work. As in a light bulb, and the mechanics therein. This does not distress them in the slightest.

Yes, a certain amount of mathematical
shorthand has evolved over the centuries, but it is in no way essential.
Most mathematics is done with a friend over a cup of coffee, with a
diagram scribbled on a napkin.

A certain amount? He makes moll hills out of mountains.
posted by zabuni at 9:23 AM on April 10, 2008 [1 favorite]

prefpara, thank you for that comment. There was another vague objection I felt hovering in the background of my thinking about Lockhart's essay, and I think you captured it perfectly.

I also think your answer is a little dangerous -- it's easy to imagine an educator bureaucrat hiding behind that answer as an excuse to change little. And it's a good justification because it's true. I expect Lockhart's answer might be to agree that it's true, but that we've let the cart get before the horse in this case, and the drills and practice aren't serving the end anymore.
posted by weston at 9:24 AM on April 10, 2008

While the old saw of "those who can, do. those who can't, teach" is overly dismissive of the abilities of a good teacher, when a love of math and science can get you an engineering or science degree, and a well paying job in the city, how do we lure people to teach?

I think part of the answer has to be to have a career cycle available where people can practice as well as teach.
posted by weston at 9:28 AM on April 10, 2008 [3 favorites]

Curricula are designed so that they can be taught by bad teachers. Because, to be honest, most teachers are pretty bad.

There are always better ways to teach a subject, but those methods require good teachers, and they're pretty scarce. So all the teachers get a standardized curriculum that can be taught by just about anyone.

Next to the education our kids are getting in foreign languages, our mathematics education looks downright wonderful. We've got students who get straight A's in a language throughout high school and college, and couldn't hold a conversation or write a letter if their lives depended on it. It quite simply doesn't work, it's widely-acknowledged not to work, and still we have an entire sub-industry devoted to teaching it the way we've always taught it. Why? Because inadequate teachers can grasp that curriculum.

There's a solution, but it requires a large-scale application of sense. So it's not going to happen for a long, long time.
posted by MrVisible at 9:31 AM on April 10, 2008 [2 favorites]

Curricula are designed so that they can be taught by bad teachers. Because, to be honest, most teachers are pretty bad.

Yes. Yes, thank fucking god yes someone else sees this. Curricula have evolved from "you should probably have seen most of this by the time you graduate" to the McDonald's fry cook training manual.

If you look at the best school systems in the world (Canada, Finland, Singapore) you find they all do basically the same thing. They make sure they have excellent teachers. Finland tracks future teachers from high school and keeps a very small pool of excellent teachers available, gives them some money and lets them teach. With no exceptions I can think of every subject has fantastic stories of drama, confusion, discovery and conflict underlying the now-obvious "facts." Take something as dry and boring as Maxwell's equations:
So you do the arithmetic. You take the values you measured for εo and μo, multiply them together, take the square root, and then take the reciprocal. The answer is a speed, so it has units of speed, in this case metres per second. And the answer is very close to 300,000,000 metres per second. Converted into miles, that's a tad over 186,000 miles per second. Being James Clerk Maxwell, and a brilliant physicist, you immediately recognise what this number is.

The speed of light.
Nobody knows what light actually is.

You stop.

Nobody in the world - except you - knows what light actually is.

When you were writing down your equations, you were thinking about electricity and magnetism. Light was the farthest thing from your mind. You had not the slightest clue (and nor did anyone else) that light was related to elecricity or magnetism. But there it is, falling out of your equations.

You realise that you are the first person in all of history to know what light is made of. Can you imagine that feeling? — David Morgan-Mar
Science, math and their ilk are nothing less than the fundamental nature of the universe, the discovery of truth in the world and the distillation of knowledge from the vapor of nuance. That our educational system has managed to make the mechanisms of the world we live in boring is a feat unequaled in sadness. How dare we cripple our children by reducing the real magic that surrounds us to bubbles on a fucking scantron.

“Education is not filling a bucket, but lighting a fire.” —Yeats
posted by Skorgu at 9:52 AM on April 10, 2008 [33 favorites]

delmoi writes "And there was actually a lot of complaints from parents as well, thinking that their kids were somehow missing the 'hard' work of doing arithmetic on paper, they called it "McMath" etc. It was all absurd. My mom said she was one of the few people who actually advocated for it at a school board meeting, most of the parents hated it."

I agree that most of the rote work of math is no longer necessary, but at the same time, the mechanics should be understood. There is no shame in being able to do long division on paper, for example. My systems administrator (and boss) at work has a slide rule hanging on his wall, which he doesn't use a lot but can. I'd also mention that being able to use an abacus is a very handy skill.
posted by krinklyfig at 9:52 AM on April 10, 2008

I'm more and more coming around to the opinion that kids go to school to keep you from going crazy and to learn socialisation. Anything else they have to learn for themselves or you have to teach them.
posted by Artw at 9:58 AM on April 10, 2008 [1 favorite]

Teaching them, to learn for themselves of course being the ideal double-win.
posted by Artw at 9:58 AM on April 10, 2008

I'm more and more coming around to the opinion that kids go to school to keep [the student's parents] from going crazy and to learn socialisation.

My schooling failed in both of these things.
posted by Faint of Butt at 10:03 AM on April 10, 2008 [1 favorite]

I'm sure everyone heres got room to be less social and with crazier parents, to a greater or lesser degree.
posted by Artw at 10:14 AM on April 10, 2008

Ugh. The cynicism in this thread is crushing my will to go get my 6th grade students from art class.

I have a degree in math from the University of Chicago, so I've done some math, real as it comes. I also happen to teach math to 6th graders here in North Lawndale.

Many, many math teachers are not adequately comfortable with the creative side of math, that's one problem. Another is the curriculum, which does emphasize the how over the why. Because I care about this stuff, I do a lot of work to try to supplement it, but it isn't easy and I won't claim my lessons are always exciting.

But I don't for a second believe this problem is insurmountable. It might require very radical changes in the way we educate children, but even from where I stand I can begin to see what those changes should be. I don't necessarily see how we can get to them, but I think it's possible. If I didn't I would quit now.

Gotta go get the kids now, but I would like to write more of my thoughts later if anyone is interested.
posted by mai at 10:17 AM on April 10, 2008 [2 favorites]

The problem with math education, I believe, is less the material that is taught than the attitude with which it is taught. To illustrate that point, I offer some of my own experiences:

I am a freshman mathematics major. I decided I wanted to be a research mathematician when I was in tenth grade. My subsequent high school math classes (pre-calculus, calculus BC, and IBH math) were so boring that I nearly changed my mind.

I found every math class I ever took before this year (my first year at college) to be absurdly boring. It is not that they were easy; on the contrary, BC calc was one of the hardest courses my school offered. The problem was that they were (and probably still are) difficult in an extremely uninteresting way. Sure, I was taught plenty of stuff in those classes, some of it useful, even, but most of the "learning" was based on rote memorization. There was no discussion of why any of it worked, or where it was going, or what you could use it for. It was all absolutely isolated from both the real world and the rest of mathematics.

For that matter, there was never, and I mean never, any hint that there is a "rest of mathematics". I suspect a lot of otherwise intelligent and well-educated people go through high school (and probably their whole lives) without ever hearing the words "set theory" or "abstract algebra". Apart from being a darned shame in itself, this leads to quite a lot of confusion as to what, exactly, mathematics is. I once got into a small argument with my IBH math teacher about that. As part of an exercise packet, she asked us to prove that for rectangles of fixed perimeter, a square has the greatest area. She expected us to regurgitate a calculus-based proof she showed us in class. I didn't want to do this, so I said (or something to this effect):

x + y = P

x is greater than or equal to y

xy = A

To maximize A, make y = x.

She gave me no points for it, because (this is a direct quote) "it's not math." Sure, it's not perfect; I'd be fine if she took off points for hand-waving and general lack of rigor. But "it's not math"? I asked her was math was, and she said it was "numbers and equations and stuff like that." Coming from the teacher of what is supposedly a college-level course, that is not an adequate answer.

I dropped that class a couple weeks later. When a future math major is dropping math classes out of frustration, that is a problem. Granted, part of that problem may be on my end. I can be awfully stubborn about these things: when I was in third grade, I flat refused to memorize the multiplication table (and still haven't) on the theory that that is what calculators are for. However, stubborn or not, I should have been encouraged by my math classes, not dissuaded. The fact that I received such a shallow and frustrating math education, at what is supposedly one of the best high schools in the country, is very, very troubling to me.

No, probably high schools shouldn't go replacing AP calculus with "intro to group theory," but to not even mention that things like group theory exist is to contribute to mass ignorance. It is this mass ignorance that creates situations like mine: students who would otherwise love math being driven away from it by their teachers' attitudes (e.g. "math is nothing more than numbers", "shut up and memorize your tables," etc.). Yes, zabuni is right, the creative spirit is both rare and tenacious, but right now creative students have to fight their teachers in order to use it in their math classes, and that is not OK.

Sorry to be so long winded, but I'm afraid I feel rather strongly about this.
posted by Commander Rachek at 10:36 AM on April 10, 2008 [7 favorites]

I don't blame the student for this, I blame the school system. It seems to me that kids are taught to divorce themselves from their intuition when doing any kind of math problem. Years later, when I come along and tell a student that his answer is absurd and he "should have known" it was wrong, he gets extremely frustrated.

I had a somewhat opposite experience in school. I could figure geometry problems logically, calculating the volume of prisms we hadn't seen before. No one understand how I could, while for me it was dead obvious. Well, it's a cylinder, so take the circle's area and multiply it by the height and that's all. "But how does that work?" It was the same as the formula, but no other student could apparently see how that formula actually had been built.

The really, really sad and depressing thing was that my teachers were the same. I would show them my work, explain how I figured things out, they would hum and haw and then tell me to just stick to the formula and not try to reason things out.

I grew to really, really hate lazy, dumb math teachers.
posted by splice at 10:50 AM on April 10, 2008 [2 favorites]

I asked her was math was, and she said it was "numbers and equations and stuff like that." Coming from the teacher of what is supposedly a college-level course, that is not an adequate answer.

Yeah, exactly my experience. If it's not the actual exact formula as described in the books, the teacher is completely clueless.

No sir, don't try to understand. Just read the page, grade according to the answer sheet, and just let everyone think math doesn't make sense and doesn't matter.
posted by splice at 10:52 AM on April 10, 2008 [1 favorite]

In my opinion, the analogy between learning math and music is spot-on; you can't perform complex music well (or at all) without spending years practicing scales, developing muscle memory, teaching your fingers what to do so you don't have to consciously think about it. Similarly, you can't perform complex mathematics well (or at all) without spending years practicing basic arithmetic, algebra, developing a sense for what concepts relate to each other and how to use ideas from one area of mathematics in a different area, and teaching your mind to think rigorously and symbolically, while also remembering the meaning behind the symbols.

It is unfortunate for the many who will never have the inclination or ability to perform advanced mathematics that they have to undergo 'math boot camp' for several years in childhood. If we could develop some unfailing magic test that would determine who's going to be a scientist or mathematician, and who's not, we could develop two different types of math education and everyone would be happier. Without such a test though, it seems only proper that we should prepare every student for every eventuality.

I'm not trying to justify the sorry state of mathematics education in America; I teach math at a local community college and am well acquainted with what our high school graduates can and cannot do mathematically. But I do have to say that I think the 'best' kind of math education will necessarily involve great deals of tedium and rote calculation. Obviously we should combine that with insight and justification and great examples of applications of what they're learning, but there is no substitute for getting one's hands dirty and actually performing lengthy computations -- it is (partially) through the process of working through laborious (and sometimes boring!) computations that one develops the facility to do complex mathematical operations.
posted by evinrude at 11:32 AM on April 10, 2008 [4 favorites]

Leibniz had a vision of amazing scope and grandeur. The notation he had developed for the differential and integral calculus, the notation still used today, made it easy to do complicated calculations with little thought. It was as though the notation did the work. In Leibniz's vision, something similar could be done for the whole scope of human knowledge. He dreamt of an encyclopedic compilation, of a universal artificial mathematical language in which each facet of knowledge could be expressed, of calculational rules which would reveal all the logical interrelationships among these propositions. Finally, he dreamed of machines capable of carrying out calculations, freeing the mind for creative thought. Even with his optimism, Leibniz knew that the task of transforming this dream to reality was not something he could accomplish alone. But he did believe that a small number of capable people working together in a scientific academy could accomplish much of it in a few years. It was to fund such an academy that Leibniz had embarked on his Harz mountain project.
posted by elpapacito at 11:54 AM on April 10, 2008 [3 favorites]

blasdelf writes "demiurge: Unless he had used pdflatex, the resulting PDF out of vanilla LaTeX would have been ugly, inelegant, and HUEG. He doesn't need the equation formatting, and Word doesn't have to suck if you use it correctly (unlike using Excel for statistics)."

One way to generate compact pdf files from latex. Not HUEG at all :-)
dvips -t letter -Ppdf -G0 $1 -o - | \
        ps2pdf -dMaxSubsetPct=100 \
        -dSubsetFonts=true \
        -dEmbedAllFonts=true \
        -dPDFsettings=/prepress - $(basename $1 .dvi).pdf

posted by Araucaria at 12:12 PM on April 10, 2008

In an earlier thread, a book by Liping Ma was recommended. It compares the math understanding of American and Chinese teachers. The American teachers ended up relying a lot more on standardized rules and most couldn't explain the rational behind division by fractions, for instance.

One point the book glossed over was that most of the Chinese math teachers taught only math while the American teachers taught every subject (this was elementary school). The Chinese teachers had a chance to focus on the curriculum and have it build year to year. The American teachers weren't necessarily good at math to begin with and only had to deal with it a few hours a week.

This changes by high school where a lot more American math teachers teach only math. But by that point a lot of students don't understand the basic principles.
posted by Gary at 12:17 PM on April 10, 2008 [2 favorites]

I was the only kid in my 4th grade class to have been skipped a grade. I was also the only kid to have passed the multiplication tables test on the first try. I had recently moved to the area from a different school district where it was not taught, and the teacher had to stop me when I started going: "13 × 1 = 13, 13 × 2 = 26..." I mean, shoot, it's just math.

The same stuff that titillates me about math nauseates my girlfriend. I'm still pondering showing her the ".999... = 1" thing but I fear having to clean up after headexplosionness.

I still remember how proud my Math Team coach was of me when I got a 1 on the AIME. "Remember, the median is a zero," he kept saying to me. I was like, shoot, I only got one out of 15! But I've always been slow on standardized tests, and I came up with a really cool but convoluted proof to arrive at my solution, and I had fun doing it. I had similar fun taking a stab at the Erdos-Strauss Conjecture last year.

Math has always just been fun to me. Completely interesting, fascinating stuff. I found my father's calculator when I was 3 or 4; I remember asking him what the "x thing" was. He told me about multiplication, and how it was shorthand for adding numbers of things a certain amount of times. It was so neat to me that instead of just counting things linearly, you could arrange them in a square or rectangle and count them that way.

"The cashier in a grocery store certainly uses a computer to calculate what you owe."

Not me, man. For fun, I'd try to make change before the register. I was pretty good at it, too, at least for denominations of ten/twenty. Was a heck of a way to entertain the customers.

I always hear I should teach math. I don't know. Not with the present curriculum. My high school made us read "Build-a-Book Geometry" the summer before our freshmen year. A lot of that book has stuck with me. I'm reminded of it by a lot of comments in this thread. Actually making your own math, your own internally-consistent formal system is worthwhile. I think I'll do alright one on one, with my own kids. I've got more of a mentoring temperament than a teaching one. You kinda just have to go out into the world and smell the math, you know? Point it out when it's happening. Math is in the moment.
posted by Eideteker at 12:26 PM on April 10, 2008 [3 favorites]

This reminded me of the experience I had some years back of dating a mathematician. Being a musician, the rather grand realisation that we were doing very similar sorts of things mentally was a bit of a bombshell. I even grew to enjoy reading descriptions of "higher" math written for the layperson. Before this, I had always wondered why proper mathematicians were kinda weirdo nutcases, just like the rest of the artists. Methinks that there is a reason why standard curricula tend to de-emphasize the creative process in math; creative sub-cultures are scary.
posted by nonreflectiveobject at 12:38 PM on April 10, 2008 [2 favorites]

If word gets out that math is a creative art, all math departments everywhere will start getting their funding cut instantly.
posted by MrVisible at 12:42 PM on April 10, 2008 [3 favorites]

I get into discussions with my girlfriend (and many of my friends in the humanities) about this all the time. Although I'm in Econ, and not math, I'm much math-ier than most of my friends.

I've always been pretty good at math, particularly when I could understand its application. I never wanted to be a mathematician, but I got to college and was told "if you want to go anywhere in Economics, you need math," so I enrolled in Calculus and realized "holy cra, math has application to EVERYTHING." So I've been taking math since, although I'll probably stop before I get to fundamentals of abstract algebra because that class will both ruin my GPA

I think the trap kids fall into - at least the people I know who are not "math people" - is twofold. First, they get it into their heads that the just aren't good at math and they will never be good at math. This is probably a result of shitty teaching. Second, somewhere along the line, we start teaching people that being ilnumerate or illiterate in mathemathics or whatever you want to call it is completely acceptable, and that being good at math trade off with social skills and the ability to be interesting. Of course, when someone who has this ground into their brains has to take a math class, they get anxious and freak out and perform poorly because they're too busy freaking out to test well, or too afraid of "looking dumb" to ask questions about things they don't understand.

To be honest, I'm not sure where I was going with this, except that I think it's a shame that more people don't take calculus because it's fucking mind blowing. Can you imagine Leibniz and Newton when they figured out the fundamental theorem of calculus? God damn.
posted by dismas at 12:52 PM on April 10, 2008 [1 favorite]

It seems to me that kids are taught to divorce themselves from their intuition when doing any kind of math problem.

The reason for this, as the recent discussion of the Monty Hall problem shows, is that our intuition is invariably bad.

There are a few mutants who can "see" a lot of the solutions very easily, but for the rest of us, it's an intuition that has to be developed over time, with lots of practice.

I never had a lot of natural aptitude at math. What I can do I can do as a result of years and years of practice. Doing pages and pages of grunge work ended up working out pretty well for me. I still can't give immediate answers to problems, but I can look at a problem and say to myself, "I know how to solve this."
posted by deanc at 1:14 PM on April 10, 2008

Following up on what I just said, I want to say that it's a reason that it is a tragedy that math classes can be such a bad experience for people. The truth is that few of us have the intuition to pick up difficult math easily. However, if students were able to stay engaged with the material long enough, they'd develop the intuition. As it is, the system seems to single out the people who "just get it" and nurture them, while if you can't pick it up that quickly, you're considered "not a math person," and the system discontinues its interest in you when it comes to math classes.
posted by deanc at 1:17 PM on April 10, 2008 [1 favorite]

Delmoi - you use a computer to figure out your money in a grocery store?

Sure, when I use the automated checkout machine, but most of the time the time it's a cashier using the computer.

f you don't have some idea what the answer is going to be -and you won't if you always rely on a calculator -you have no idea if the answer is correct.

And of course, I estimate stuff in my head all the time. But here's the thing all of the estimation skills I have I learned on my own they were never taught in school. I never sit there and perform a pencil and paper computation algorithm to figure out the exact amount, I round up or down to get numbers that are more compatible and come up with an estimate. Now that actually would be a useful skill to teach kids, but I don't think it's done very often.

It's much more useful to be able to come up with a ballpark answer and find an exact one using a calculator then memorize some algorithm for finding exact numbers with a pencil and paper.

I get kids come into the math lab at the community college, that have no idea that something times ten is gonna have a zero at the end.

And it's exactly that kind of thing that the parents in my story were opposed to teaching. They wanted kids to learn how to do calculations on pencil and paper, not how to actually use math.

I'm all for teaching kids to do math in their heads. I'm opposed drilling them on paper systems like long division. You can teach them how it works, but you don't need to drill it into them so they'll never forget because it's not something they'll ever need to use.

x + y = P

x is greater than or equal to y

xy = A

To maximize A, make y = x.

Well, in this example Limx→∞x*y = ∞ when y ≠ 0. So rather then a square, you get an infinitely long box of arbitrary non-zero height. Not what you want.
posted by delmoi at 2:23 PM on April 10, 2008 [1 favorite]

As it is, the system seems to single out the people who "just get it" and nurture them, while if you can't pick it up that quickly, you're considered "not a math person," and the system discontinues its interest in you when it comes to math classes.

Actually, by my experience (see above), people who are intuitively good at math aren't given a much easier time by the system unless they are content with not doing much in the way of developing that talent (which, to be fair, seems to include a fairly large proportion of mathematically talented people). I was not content with that, and frequently butted heads with my teachers because of it. I have been fortunate enough to have a pair of mathematicians as grandparents to show me what math really is. Without this extra source of knowledge, I would have just gotten frustrated with math, and turned away from it. As I tried to make clear, this almost happened anyway. That, to me, is the greatest sign that something is wrong--they can't even hang on to the kids who like math.

However, you're completely spot-on about how kids who don't show an immediate aptitude for certain concepts tend to be given up for dead. This is indeed extremely unfair, but I think it comes from the same root as my trials and tribulations as an AP/IB student: the subject is taught as though the trivial tools are all that is there. I'm willing to bet at least part of your "just not getting it" came from not being able to see what it all meant. Lockhart is wrong on one thing: math is a language, among other things, and if no teacher ever explains what the heck it means, so that you really, intuitively understand what the heck is going on on the chalkboard, that's not your fault. In fact, I suspect a lot of people in AP calc classes don't actually understand it on an intuitive level (I'm pretty sure my teacher didn't); they're just really good at remembering abstract formulae.
posted by Commander Rachek at 2:27 PM on April 10, 2008

Araucaria: It may be more compact, but it'll still be super inelegant, and ugly in most PDF viewers (especially if you're using the METAFONT bitmaps, the default).

You're converting bitmap DVI -> Postscript -> PDF. Each letterform basically ends up as it's own layer in the PDF, as a bitmap that is usually smoothed very poorly. Adobe and Apple's PDF libraries do a decent (if slow) job of smoothing things out, but every other PDF rendering engine I've used does a terrible job. The weirdness is very evident if you select text in a DVI-generated PDF.

I've used both XeTeX (mac only) and tetex+pdflatex to generate beautiful PDFs directly, using truetype/opentype fonts.

DVI is a relic designed around generating bitmap output ready for phototypesetting, it predates Postscript by several years. There's a reason DVI and METAFONT never got any use outside of the TeX community :)
posted by blasdelf at 2:31 PM on April 10, 2008

As for cashiers giving you your change, I'm one of those people like dilbert that some cashiers hate. They ring up 7.12 and I have them a $10 bill, two loonies (2 one dollar coins), a quarter and 2 cents to make things easy. About half of the time cashiers will try and either give back the pennies or the loonies, not knowing that I want a $5 bill and to rid myself of pennies. About 40% of the time the cashier will roll their eyes at me in the way that says that they know I know what I'm doing, what why the fuck don't I just use my debit card. But there's that 10% of the time, where the cashier stops for a half second, and one can hear them mumble $5.15 under their breath before the punch in the amount on the cash register. I really like those 10% times.

As well, there are times where the cashier mis-types the amount that you gave them. Being able to do the math and know the change you're getting allows you to save a lot of time from them calling someone with "keys" over to void the transaction and reenter an amount.

Otherwise, regarding math and it being "hard," I think it's mainly a confidence thing, and it being common knowledge that math is "hard." My 10 year old says that he's bad at math. But when I check over his math work, he'll only have simple arithmetic errors. I'll tell him which ones are wrong, and then he'll check them and double check the new answer he got before showing it to me. If he'd double check his work the first time, I doubt he'd have issue; he's just racing to get things done because "math is hard." But admittedly the math concepts that he's exposed to really aren't that far beyond simple integer arithmetic, but he so far has not had a problem grokking anything. I keep pointing this out to him. And he still says that he's not good at math. Argh.

His class just finished a probability unit; I'm going to make time to introduce him to the monty hall problem this weekend.
posted by nobeagle at 2:40 PM on April 10, 2008

Well, in this example Lim x→∞ x*y = ∞ when y ≠ 0. So rather then a square, you get an infinitely long box of arbitrary non-zero height. Not what you want.

Well, I could argue that by saying x + y = P (which really should be 2(x + y) = P; whoops) I am establishing that x and y are real, bounded (is that the word I want here?) values. Of course, I never explicitly defined P (or x or y), which is just one of several problems with the rigor of the proof. But that's not why my teacher didn't like it. She didn't think it was incorrect mathematics, or sloppy mathematics, she thought it was not mathematics at all. She managed to turn what could have been a really nice teaching moment into one of my bitterest memories of high school. Why? Because she didn't understand the subject she was trying to teach.
posted by Commander Rachek at 2:42 PM on April 10, 2008

Silly people. The most important thing about Math in the USA isn't how the problem is solved! It is what solution the kid comes up with when he takes the NCLB standardized tests.

Understanding why the answer is correct is not relevant. The only thing that matters is that every child figures out how to get exactly the right answer so that your schools' funding isn't cut.

The relevant word problems would be something like:

Assume you have a class of 30 students. What is the minimum average score that those students would have to achieve on the NCLB tests in order to preserve your job and secure further funding for your school?

Mr. Johnson has 5 sections of Math with 30 students in each section. Assuming that 14% of the total number of students in Mr. Johnson's classes will fail the NCLB exams, how many students will his school system need to fabricate expulsions excuses against in order to make it appear as if their school system has a 100% pass rate?
posted by Joey Michaels at 2:48 PM on April 10, 2008 [6 favorites]

delmoi writes " x + y = P

"x is greater than or equal to y

"xy = A

"To maximize A, make y = x.

"Well, in this example Limx→∞x*y = ∞ when y ≠ 0. So rather then a square, you get an infinitely long box of arbitrary non-zero height. Not what you want."

And that's exactly why I don't really need to calculate f(z)= x^2 > f(k)=x^3 for all -∞<> x^3 and the equation is verified , for instance, by diving both for x^2, so that I obtain 1>x^1 , which is verified only when x<1>f(k) for x<1>

posted by elpapacito at 2:50 PM on April 10, 2008

Dammit, I fail at code tag
posted by elpapacito at 2:51 PM on April 10, 2008

delmoi: estimation - Here in Canada, in Feb 5th graders had a specific unit on estimation, and since then during probability, and factions/decimals, there've also been a few questions involving estimating the answers. I don't seem to remember that in the states in WI growing up.

re: limits - However, y is defined as X, so your limit is setup too poorly to approach correctness. Heck, "estimate" it - setup P as 10, and try a huge X of 4.999 and y of 0.001 - the area is .004999 instead of 6.25 that one gets with X = Y = 2.5 .

Besides, first of all P=2x+2y so y=0.5P-x so the limit is of X*(0.5P-X) which is 0.5XP-X^2 ... if X can't be zero, that's equivalent to 0.5P-X , so clearly one doesn't want a value approaching infinity . At anyrate, one would be taking the limit as X approaches 0.5P, not as X approaches infinity. They're not asking for the largest geometrical shape, they're asking for how to maximize the area for a given perimeter.

But the OP of that comment was also a bit lazy; there's no "why" in the answer, and while there's experience of previous knowledge, he could have easily have guessed "x=y+1" scribbled it out and put in "x=y" ... the x=y+1 would have been just as wrong, and without showing *why* it is so, to the teacher it looks just like a guess.

There's people earlier upthread who were annoyed about needing "the right answer" being more stressed over the hows and whys, and this comment annoyed my like my kid's filling out a reading comprehension test, that asks "What will happen, and why do you think so" and ignoring the "and what do you think so."
posted by nobeagle at 3:03 PM on April 10, 2008 [1 favorite]

math is good drugs

after hours and hours of shrieking high school teachers pointing at poles of real functions and spooking kids about complex numbers I still got through okay.

I had one session of impressionistic group theory. I was so sleep depped afterwards, and I dreamt of the most amazing symmetric objects...

math, to me, is not about numbers. it's about playing with structure and symmetry. it's a dance. I show nonmathematicians what I do and their eyes glaze over and it's suddenly all the stuff I'm doing is a "math problem".
posted by oonh at 3:09 PM on April 10, 2008

P= x+y

P = 2x + 2y

Uh ?

I'd have expected 2P=2x+2y ?
posted by elpapacito at 3:18 PM on April 10, 2008

elpapacito: I was saying that P=X+y isn't correct. I was (naturally?) assuming that P = perimeter. I guess that since one is using rectangular shapes, that one could say P = 1/2 the perimeter and thus P=x+y ... of note as well, using the 0.5-x only works for taking the limit to infinity (which as I mentioned one can't do with this problem); it's obviously not going to give the area, one still has to use either A=0.5XP-X^2 or drop the 0.5 if one is defining P as 1/2 the perimeter.

I wish my job had more math involved.
posted by nobeagle at 3:26 PM on April 10, 2008

But the OP of that comment was also a bit lazy; there's no "why" in the answer

But don't you see? That's precisely my point. It was a lazy, hand-wavy argument (though I think the concept is sound, I simply did not then, and do not now, have the language to explain it properly), but my teacher didn't understand that was what was wrong with it. She didn't know of any other way to prove the hypothesis besides what came out of the book, so as soon as she saw something else, she assumed it wasn't "real math." If she'd said "I gave you no points because it's a lazy argument, here's how you can make your proofs better in the future," that would have been different.

By the way, that should be "she [not he] could easily have guessed..."
posted by Commander Rachek at 3:28 PM on April 10, 2008

Also, I did correct myself earlier about x + y = P vs. 2(x + y) = P, but it really doesn't matter. The point is that x + y (and thus x and y) are real and bounded. I ought to have stated this explicitly in the proof, but sadly did not.
posted by Commander Rachek at 3:38 PM on April 10, 2008

Ah, I had missed the point about why exactly you were upset. For some reason I got that you were upset that the lazy "this is right" wasn't accepted.

I agree with your point, that a high school math teacher should know multiple approaches to most problems, and be able to recognize which one a student is going down, so they can coach them along that approach. However, sometimes one is concentrating on working with a certain approach, rather than on getting the answer. But the "not real math" line seems a bit off for her.
posted by nobeagle at 4:06 PM on April 10, 2008

Commander Rachek writes "The point is that x + y (and thus x and y) are real and bounded."

Ah, domain definition. My best teacher busted our balls on accurate domain definition at all times.
posted by elpapacito at 4:14 PM on April 10, 2008

As someone who does math for a living, I can tell you: Arithmetic is not mathematics.

(Oh, sure, eventually you need to know how to do some arithmetic, and even Lockhart himself alluded to that, but if what you are trying to teach kids about is mathematics, then drilling times tables---or even learning algorithms to estimate quantities---won't accomplish that.)
posted by leahwrenn at 4:54 PM on April 10, 2008

A couple of years ago I embarked on a fairly large project to build a tornado resistant but inexpensive house. This required teaching myself Strength of Materials, which I would've taken the semester after I dropped out of college back in 1982.

Real engineers don't use much math, certainly not calculus; everything is worked out to tables and simple formulas. But when you are being experimental, a lot of those tables don't apply. And as I considered the large number of arbitrary-looking constants I was using, I got nervous. Yes, I know how those equations are derived and I know the people who test materials are professionals, but I started to feel like I was standing atop a 37 story building looking at the ground through a telescope whose construction I wasn't thoroughly familiar with.

So I cut myself a 14-inch length of Southern Yellow Pine 2x4 and took it to work, where we have a machine capable of testing up to 160,000 pounds in tension and compression. I blocked up the 2x4 with supports 12 inches apart and arranged to press down on it with a bit of one inch diameter steel rod at the center. And I screwed the machine down in small increments, measuring the deflection and recording the force until the first popping sounds told me the wood was failing.

The fact that all the measurements came out to within 5 percent of what the calculations predicted greatly enhanced my confidence in the math I was using -- and gave me a little burst of pleasure I can scarcely put into words.
posted by localroger at 5:06 PM on April 10, 2008 [2 favorites]

My worst, laziest math teacher spent fully half of our class hours reading people's calculator manuals to them. Really. When he wasn't doing that, he was dinging points for solving problems by numeric methods that he happened to know had exact, analytic answers, never mind that the numeric method was accurate to six decimal places or that a comparable problem from the real world would only ever have a numeric solution, and he was doing other bull like that. One day I had a moment like splice describes.

On our assignment, with no real justification, was a problem along the lines of 'give an equation for an ellipse that passes through these two points'—only peripherally related to the main point of the lesson, although you could certainly brain it out if you understood the definition of an ellipse and if you cared to. Most of my classmates punted because they didn't understand the definition. Their form of punting was to write nothing, or just make a mess of mathematical symbols. I punted because I was sick to death of putting forth my best efforts for this twit and getting a big heap of nothing in return. (My calculator was not among those for which he had manuals.*) My form of punting was to give the easiest solution: the equation for a circle that passed through the points.

Sure enough, when I get my paper back, it's marked minus a point for a wrong answer. Not minus a point for smart aleck, not minus a point for lazy, just minus a point for wrong. And being an irritable little punk who'd had self-important nitwits wasting his time all day long for years, I went and called him on it. First I established that he knew there were multiple solutions to the problem. Then I asked him what was wrong with my equation. He said it wasn't an ellipse. I pulled the canonical generic-ellipse equation from the book and showed that mine could be put in that form. This had been his standard argument for those sorts of things: Put it in the right form. But this time he said it wasn't an ellipse. I asked him how. He said it was a circle. I protested that every circle was an ellipse. I said I'd prove it. He said go ahead. I showed him that two parameters in my ellipse equation were equal, and that if you set them equal in the ellipse equation, the circle equation fell out. That was why mine was a circle: Circles are a special case of ellipses. The problem hadn't said a non-circular ellipse. He kind of sat there silently for a moment, and then he objected that my thing wasn't an ellipse, it was a circle. At that moment I happened to remember that our textbook had said, explicitly, that circles are ellipses, a few pages before. I flipped back through the pages, scanning, and there it was. I pointed it out and he studied it for a few long seconds.

All right, he said, and marked the point. And a new chapter was added to the mythology of my half-dozen geek friends: There is actually someone so stupid he can hear the growl of authority but not the song of algebra. It was as though we'd shown him his face reflected in a pool, and he'd looked on quizzically, and we'd scratched a frowny face in the mud and he'd proclaimed it a fine likeness. It was worse: It was as though Thomas had touched the marks of nails and thrust his hand into the wounded side, and then wandered off muttering that he guessed he'd better ask a Pharisee about this resurrection business.

I'll say it: I really, really hate lazy, dumb math teachers.

*Actually it was an HP with reverse Polish notation, which I hold responsible for training me to know the quantities and operations behind the notation.
posted by eritain at 7:28 PM on April 10, 2008 [8 favorites]

Speaking as an elementary-school math teacher, I can say the following:

-- Very, very few elementary school teachers like math. Some of my cohort in teaching school boasted of how much they hated math, that they would just hold their noses and teach it. How do you think most of these teachers teach math?
-- I used to paint, some. I took a very powerful class in college that was a bit like what Lockhart describes as the painter's nightmare: Painting color swatches in 10-step gradients; painting very methodically, the same subject under different-color lights, etc. It made me capable of painting -- opened me way up from where I was before.
-- Kids (most of them) like the chewy, complicated challenges better than the easy stuff. It is, for instance, much easier to get a 10-year-old interesting in multiplying 34x7 in his head (7x30=210 + 7x4=28 = 238) than to rattle off 7x3. The former induces the student to bother with the latter, as a bridge to showoffy stuff. The showoffy stuff -- sectioning and multiplying numbers very rapidly in your head -- makes you suddenly able to see patterns you wouldn't see otherwise.
-- I once walked a student through tic-tac-toe, showing him it was a "closed" (unlosable, if you go first) game. We had to consider symmetry (no point considering all the corners when they're really the same). The result was essentially a proof. When we had exhausted all possibilities, I remember him just geeked out. He said: "This has been one of the greatest afternoons of my life."
-- I remember teaching kids functions. If you say that "x" is "whatever number you say," you can put a card saying "3x+4" in a box. Now the box is a machine: You say "3" into it and it says "13" back. Etc. Then they wore functions on their foreheads and became machines. Much socializing. Then I showed them that if you use a cartesian grid, you can map those answers on it. 3 (on the x line) gets you 13 (on the y line). Do a lot of them, and you get a slope. Then you can ask the slope. Say "7" and the slope will show you that the answer is 25. Any number you come up with, the line gives you the right answer. I will never forget one girl saying: "Why does this work?" It still gives me chills to think of it. I couldn't answer that. We were both contemplating something uncanny, almost divine. Those kinds of math moments come along once in a blue moon, but damn they're sweet.
-- I have a kid who can't get his math work done because he wants to come up with answers that are "funny." For instance, if the question is "a number between 0 and 1," he believes that .737373.... is a funny answer. This is a mathematician of a sort unrecognizable to standardized tests.
-- Kids don't give the tiniest damn if math is practical. They do care if it's mysterious, divine or in any way beautiful (and funny is beautiful).
posted by argybarg at 9:45 PM on April 10, 2008 [12 favorites]

This is the only thing I read on metafilter today & I admit to not being done with the thread yet. It's been worth it. I have a 9 year old who is Freaking Out about the state standardized tests that start this year for her, particularly about the math portions... and yet, she thinks tesseracts are really cool (yeah, we were reading M.L'Engle's A Wrinkle in Time) & we spent 45 or so minutes one night talking about math & neato geometry stuff & her eyes got all lit up with excitement... it was just so amazing to see. I concluded with, "So see? Math is really cool!" And she concluded with, "SOME math is really cool -- THIS math is really cool." Silent assertion that what she was doing for homework currently was distinctly not cool. So I've been thinking about math a lot lately & this fit into all that thinking very nicely. Great post. Thanks. :)

Totally got a courtesy D in geometry in 10th grade because I took every test between 2 & 7 times until I didn't fail each test... I think I probably actually failed, but the teacher was annoyed that I was taking up his time at lunches & after school week after week after week all year long taking & retaking & retaking each test in a desperate attempt to at least pass it that he was afraid I'd be back the following year if he didn't "pass" me... Hardest fucking D I ever earned.
posted by susanbeeswax at 12:05 AM on April 11, 2008

x + y = P

x is greater than or equal to y

xy = A

To maximize A, make y = x.

this merely restates the question. There is no proof here whatsoever. no wonder you got zero.
posted by mary8nne at 5:13 AM on April 11, 2008

Thanks for this post; it really made me think.
posted by lunit at 7:52 AM on April 11, 2008

I'm about to head out to go get my masters in teaching, focused in math in the fall. This conversation gives me both hope and fills me with dread. I teach at a private school right now and realized that almost all of the real math that I've shown students has been outside of the classroom. But at least I'm showing it to them.

AP Calculus has so much to get through that there's very little time for seeing anything else. Algebra I on the other hand- I remember how wide their eyes became when we worked out the probability of winning the lottery. I'd like to think that I saved them some money somewhere.
posted by Hactar at 10:01 AM on April 11, 2008

The study found that college students who learned a mathematical concept with concrete examples couldn't apply that knowledge to new situations, while students who learned the same concept through abstract examples were much more likely to be able to transfer that knowledge to different situations.
posted by Skorgu at 12:36 PM on April 24, 2008 [1 favorite]

Most of my first and second graders say that math is their favorite subject, and they almost all do it really well... but the math program we use is just horrible.

It's a 'spiral' program, meaning they spend a day on plane shapes, then a day on 3-d shapes, then a day on perimeter, then area, then they won't get anything beyond a few review problems until next year. It covers every concept very shallowly, every year.

The problem for teachers is that the tests that we have to give them to prove that they're learning assumes that they've learned these concepts in the order they appear in the book, so we can't rearrange things to cover concepts in more depth. If I throw the book away and cover math concepts in depth as I'd like to, my kids will fail the test.

One interesting (to me) observation...

Our math series never teaches the kids what the symbol '=' actually means. The people who wrote the series assumed that 6 year olds know what it is, because who doesn't? Then the test they take presents the problem 3 + 2 = __ + 3, and most of my kids my first year got it wrong because they thought the '=' symbol meant 'answer goes after this.'

I started teaching them that the two sides of the '=' have to work out to be the same, and when we do daily math equations I get all kinds of interesting things... I have kids write 7=4+3 and 2x2=2+2 and all kinds of cool stuff. It leads to learning about order of operations and ≠ and <>... I could spend weeks on it all but they just don't give me the time, because I have to move on and cover standard and decimal measurement of length, weight, volume, and temperature in one week, no joke.
posted by Huck500 at 4:53 PM on April 26, 2008

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