Posts Tagged ‘education’

A Non-mathematician’s Non-apology

March 26, 2011

After finishing this post about the derivative of the sine function, I decided to hunt around online to see how common its approach is.

It’s not common. Most sites take the derivative of sine by considering

\frac{\textrm{d}(\sin\theta)}{\textrm{d}\theta} = \lim_{\Delta\theta \to 0}\frac{\sin(\theta + \Delta \theta) - \sin(\theta)}{\Delta \theta}

and working from there.

Eventually, after wading through three pages of results, I found another write-up of the geometric argument from, of all places, a site called Biblical Christian World View. It is apparently the personal site of a guy who’s good at math and also thinks it makes sense to write things like,

I illustrated Biblical truths with mathematical expressions. For an example, I illustrated the Biblical truth, “With God, nothing is impossible” as “two negatives equate to a ringing positive.” In the arithmetic of negative numbers -(-7) = +7! Two negatives equal a positive.

So. There’s that.

But just a little further along the Google results I found one more presentation of the same idea. This one is from Victor J. Katz, a mathematician who wrote a book about the history of math, and was writing from the historical point of view.

His article is much better than mine. The proof is clearer and surrounded with tons of other insight.

Katz delightfully points out how great a term “arcsine” is – it’s the length of the arc associated with that value of the sine function. Then, at the end, he gives Leibniz’ original argument that y = \sin\theta satisfies \frac{\textrm{d}^2 y}{(\textrm{d}\theta)^2} = -y, and it’s crazy! Differentials are applied willy-nilly and manipulated algebraically in ways nobody does any more. I felt disoriented at first, adapting to this new way of thinking about calculus, and then wondered why I’d never seen it until now.

It’s true that there are a lot of old techniques no one uses, and that’s because now we have better ones. Indeed, modern analysis, with its deltas and epsilons, is much better, mathematically, than manipulating differentials in dubious ways. It’s rigorous and logical.

It’s also hard. I’ve been asked to teach delta-epsilon proofs to quite a few people, and I’ve never been able to get it across. I’m giving up on that for beginners. I am going to teach the geometry stuff, and I’m not going to feel guilty about it.

It is okay to learn a thing the wrong way the first time. That first pass is only there to get you used to the main ideas, and the main idea a calculus is applying derivatives, integrals, and series. It is not the mean value theorem.

Once you learn a rough version, you practice it in the field until you’re comfortable. Do some physics. Learn some differential equations. After all that, it’s nice to come back, study calculus again, and finally understand all that’s really going on.

Actually, I like it better that way. Lots of my college classes made me think, “Oh, wow – so that’s what was behind the curtain!” But if you had shown me all the wheels and gears up front, I’d have been too busy checking how each one fit into the next to see what they accomplished.

A case-in-point is linear algebra. I remember almost nothing from my freshman linear algebra course. It wasn’t a bad course, but it was rigorous, proving theorems from the axioms of vector spaces, and it was beyond the level I was ready for at the time.

A couple years later, I found I really did need to know linear algebra to get through quantum mechanics, so I watched Gilbert Strang’s video lectures, which are far more concrete.

They were wonderful. I understood what was happening. I could do all the calculations and answer all the conceptual questions.

Then, finally, I went back to read Sheldon Axler’s Linear Algebra Done Right, a book that goes back again to the axioms-of-a-vector-space point of view, and thought it was wonderful.

Keith Devlin disagrees. Devlin takes up multiplication, claiming one should not tell young children that multiplication is repeated addition. Multiplication is its own fundamental operation. (The field axioms treat multiplication and addition independently.)

I was taught multiplication as repeated addition as a child, and then retaught multiplication as an fundamental operation in college. Do you know how confused I was by that? None. Zero confusion ever. In fact I never even noticed the discrepancy until Devlin pointed it out. I thought about multiplication as repeated addition when it was convenient, and thought about it as multiplication when that was convenient, and never realized I was switching.

I do the same for the geometric and analytic modes of thinking about calculus now. When I’m solving a physics problem, I don’t even notice whether I’m doing calculus or algebra at a given moment – it’s all just problem solving.

Why, then, do introductory calculus classes spend a month learning limits? Better just to ignore them and press on to the good stuff. There will be time later for learning what the difference between “continuous”, “differentiable”, and “smooth” is – modern medical science is working new miracles all the time.

My Brown Big Spiders

March 21, 2011

Professor: You have to learn to be able to play it blindfolded. The page, for God’s sake! The notes!

David: I’m sorry I was, uh, forgetting them, Professor.

Professor: Would it be asking too much to learn them first?

David: And-And then forget them?

Professor: Precisely.

from the movie Shine



If I want to find the volume and surface area of a sphere, I do it with calculus:

V = \int_{r = 0}^R\int^{2\pi}_{\phi = 0}\int_{\theta = 0}^\pi r^2\sin\theta \textrm{d}\theta \textrm{d}\phi \textrm{d}r  =  \frac{4}{3}\pi R^3

.

S = \int_{\theta = 0}^\pi\int_{\phi = 0}^{2\pi} R^2 \sin\theta\textrm{d}\theta\textrm{d}\phi = 4\pi R^2

This is correct, but I can’t use it with high school geometry students because they don’t know what an integral is, much less a Jacobian.

However, Archimedes came up with a beautiful way of discovering the volume and surface area of a sphere. He did it by relating the sphere to a known shape – a cylinder with a cone cut out of it.

He drew a picture like this:

Archimedes' illustration of the geometry of a sphere

On the left there’s a hemisphere with radius R. On the right, there’s a cylinder with radius and height both also R, so that the hemisphere would fit perfectly inside the cylinder. The cylinder has had a cone cut out from the top down tapering down to the center of the bottom. First, we’ll show that these two shapes have the same volume.

We imagine slicing the hemisphere horizontally at some certain height h. This would reveal a circle as seen in the picture. Call its radius r.

At the same height, we also slice the cylinder, leaving us with a disk. We’ll find the areas of this circle and disk.

The area of the circle is \pi r^2, which by the Pythagorean theorem is also \pi (R^2 - h^2).

Looking at the cylinder, the outer edge of the disk has radius R and the inner edge has radius h, so the area of the disk is also \pi (R^2 - h^2).

Because every horizontal slice of the hemisphere has the same area as the corresponding horizontal slice of the drilled-out cylinder, they must have the same volume. The volume of the cylinder is its original volume minus the volume of the cone, or \pi R^3 - 1/3 \pi R^3 = 2/3 \pi R^3. Hence, the volume of a full sphere is

V = 4/3 \pi R^3

Next, we’ll show that the hemisphere has the same surface area as the outside of the cylinder (the cone is now unimportant).

Take a slice of the outside of the cylinder at height h and of thickness \textrm{d}h. This forms a band around the cylinder whose area is

\textrm{d}S = 2 \pi R \textrm{d}h

Now slice the sphere at the same height with the same \textrm{d}h. This also forms a band. The band is a shorter distance around, but due to the slant of the edge of the circle, it’s also thicker. Let’s call the thickness of this band \textrm{d}x.

Slices of equal thickness dh at equal heights h on a cylinder and sphere.

The area of the band around the hemisphere is the circumference at height h multiplied by the thickness \textrm{d}x.

\textrm{d}S = 2\pi\sqrt{R^2 - h^2}\textrm{d}x

If we draw a tangent line on the sphere, it’s perpendicular to the radius. This gives us similar triangles.

So

\frac{\textrm{d}x}{\textrm{d}h} = \frac{R}{\sqrt{R^2 - h^2}}

Plugging back into the previous expression,

\textrm{d}S = 2\pi\sqrt{R^2 - h^2}*\textrm{d}h * \frac{R}{\sqrt{R^2 - h^2}}  = 2\pi R \textrm{d}h

So the band around the outside of the cylinder and sphere have the same surface area, so the entire shapes have the same surface area. That makes the surface area of a sphere

S = 4 \pi R^2

This is a really lovely argument. The problem is pretty hard, but the solution is simple. (I’m not sure if this is quite how Archimedes did it. To be honest I never even met the guy. I learned the idea from this animation).

I was reviewing solid geometry with a high school junior the other day, so I showed her this argument (but only the volume part). I was proud of myself for offering this little example of how interesting mathematical ideas can be. At least, I was as we began.

“It’s all so complicated!” she moaned a few minutes later when I asked her to identify a certain quantity in our sketch.

Complicated? I had thought the argument was remarkably simple – just draw a sphere and a cylinder next to each other and you’re practically done. What could be simpler? Somehow my student was getting entangled in brambles I couldn’t even see.

I did not draw quite the same picture for her that I drew earlier in this post. I didn’t want to give it all away, so I drew something more like this and asked for r:

Finding r is a simple application of something she knew well – the Pythagorean theorem. She didn’t see it, though, so I showed her this right triangle:

But then she didn’t see how long the new line I just drew was. It’s just R because it’s a radius of the sphere, but although she knew that all radii of a sphere have the same length, she couldn’t easily identify the two lines as radii and call up the relevant information. So I showed her that step, too.

After a bit more prodding, she wrote down r = \sqrt{R^2 + h^2}, a mistake that comes from applying the Pythagorean theorem incorrectly. She knows better, and should have found r^2 = R^2 - h^2, but by this point she was already flustered from her earlier mistakes, confused about what we were trying to do, self-conscious, and generally unable to approach the problem equanimously.

When she realized she had applied the Pythagorean theorem wrong, her frustration mounted, and moments later, at my next question, I was shocked with, “It’s all so complicated!”

Why did this happen? Why did I so horribly misjudge the difficulty of the exercise?

The other day I read this comment on an essay on teaching

I used to teach English as a second language. It was a mind trip.

I remember one of my students saying something like “I saw a brown big spider”. I responded “No, it should be ‘big brown spider'”. He asked why. Not only did I not know the rule involved, I had never even imagined that anyone would ever say it the other way until that moment.

Tutoring has been exposing my own brown big spiders – the little steps and bits of knowledge that I take for granted – for years. I’ve rarely stopped to notice it.

Just to follow each step in the Archimedes argument, you must make an enormous number of mathematical connections behind the scenes in your mind. Here’s a partial list:

  • A “sphere” is a round three-dimensional object like, a ball
  • Every point on the surface of a sphere is the same distance from the center
  • The “surface” of the sphere means its outside edge, or skin
  • A “point” is a little dot with no size at all. It simply marks a place.
  • You can represent three-dimensional figures in two dimensions with certain types of drawing.
  • The point of doing this drawing is to make things easier to visualize.
  • A “hemisphere” is half a sphere – the top half in this case
  • A “cylinder” is basically a tube with constant width.
  • The center of the bottom of the hemisphere is the same point as the center of the sphere it came from.
  • The height of the hemisphere is the same as the distance from the center to the edge horizontally.
  • This means that the cylinder drawn is twice as wide as it is tall.
  • The volume of a cone is one third the area of its base times its height.
  • The volume of a cylinder is its base times its height
  • The area of a circle is \pi times the square of its radius

And so on. I only stopped writing so that I’d eventually finish the rest of this post. Each item I added to that list sparked off several new ones I hadn’t considered.

Try writing your own list and you’ll quickly be overwhelmed by the exponentially-proliferating leaves on your conceptual tree. We didn’t even get close to things like the Cavalieri’s principle.

The items on my brown big spider list are not supposed to be mathematical facts so much as cognitive patterns the reader is required to have. For example, mathematically a point is not, “a little dot with no size at all,” as I called it. It’s a primitive notion and has no definition. The list still calls a point a dot, though, because the mathematically-accurate description isn’t helpful to a student, and isn’t they way most people think of it even when they’ve already learned geometry well.

When I started writing the list, I found myself wanting to say, “A sphere is a set of all points equidistant…”, but that’s no good. It uses the significant brown big spiders of “set” and “equidistant”, as well as the general idea of giving mathematical definitions, something most high schoolers don’t yet understand well. Then I wanted to say, “A sphere is a shape that’s symmetric with respect to rotations about any axis…” but this has all the same problems.

Ultimately, I chose “a sphere is a ball.” It’s imprecise, but it’s the way you think about a sphere before you’ve packaged the concept away so tightly you don’t need to think about it any more. Anyone who tells you a sphere is the two-dimensional manifold S^2 is someone who has forgotten how much they actually know about spheres. They’ve forgotten it in the good way, of course – the way David was supposed to forget the notes to Rachmaninoff. Unfortunately, I experience a crippling side effect when I forget things this way. I forget that other people haven’t yet forgotten them.

This forgetting is the psychological phenomenon of “chunking“. The most famous example involves chess players. Give expert chess players a position from a game between grandmasters and they can easily memorize the positions of thirty pieces. Give them pieces strewn randomly about the board and they’ll remember just a few – no more, in fact, than your average Joe who knows little more about chess than what the real name of the horsey is.

A position from a real game has lots of meaning, if you’re an expert. If you’re an expert you extract order from the position automatically, without consciously processing every detail. The entire task must seem quite simple to a grandmaster. Similarly, the experienced mathematician sees all the important properties of the sphere and the cylinder and the cone without having to list them out one by one, and the process is so automatic they don’t even realize they’re doing it.

In “Simple” Isn’t “Easy”, I learned not to judge the difficulty of new ideas by how simple they are, but by how familiar to the student. Despite this, I have continued to make a similar mistake when dealing with ideas the students have already learned.

“Learned” isn’t “chunked”. My student understood the meaning of “hemisphere” and the formula for the volume of a cone, but she still needed conscious effort to recall and wield those bits of knowledge. Each sat in its own corner in her mind, accessible only by dint of concerted effort, and certainly not ready to flow into a flood of beautiful ideas.

I was trying to dictate a soliloquy for her to transcribe, but I was assuming that because she could see the letters on her keyboard, should could touch-type. It turned out that the effort to hunt-and-peck was so great, all the artistry of the speech was lost.

I want to watch out for my brown big spiders in the future. I want to be more patient when they are discovered and more studious in cataloging, remembering, and working with them. Most of all, I want to look back later, and remember my students forgetting them.

Sync

June 12, 2010

I took a short break from reading Steven Strogatz’s Sync: How Order Emerges From Chaos In the Universe, Nature, and Daily Life earlier today and checked Facebook. Usually, the status updates of my Facebook friends are a seemingly-random menagerie of links to news stories, jokes, anecdotes, and these things: ^_^. Today, though, I found that in just the last twenty minutes, ten or so of my friends had posted nearly identical messages. They had somehow synced.

In this case, it’s not surprising. They were restating the result of the recently-concluded World Cup soccer game, but with more exclamation points than I’d get from Reuters. (Actually, Facebook status updates are the primary way I keep in touch with mainstream sports.) My Facebook synced today because of a strong, external signal influencing all the individual updates. That’s the way we normally think about synchrony. If you want it, you need some sort of a central clock for everyone to follow. A computer chip’s parts sync this way. Coworkers on a project are synced by a manager. Orchestras have conductors. Tug-of-war teams count to three.

By contrast, Strogatz is interested in spontaneous synchrony – synchrony where you won’t expect it and no one’s in charge. A great visual and audio introduction is Strogatz’s own TED talk.

Sync is a broad survey of nonlinear systems from spirals in oscillatory chemical reactions to synchronized menstruation induced by armpit sweat. What’s captivating about it is the story. Like James Gleik’s Chaos or Kip Thorne’s Black Holes and Time Warps, it carries you along from a few researchers diddling around with a curious idea to the creation of a large scientific field. We explore different branches where the original research lead, all the time seeing the different ways scientists and mathematicians approach their problems. From Strogatz, you also get a sense of the way these different approaches contribute to a complete understanding. At different times, Strogatz describes analytical work (solving equations), computer simulations, visualization (including building models from string and clay), laboratory experiments, and field research. Each endeavor feeds back into the others in this story about the science of synchrony.

I was curious, as I read the book, what it would be like if it had been technical as well. What if Strogatz had included didactic discussions of the solvable systems he’d worked on, or outlined the topological proofs he mentioned, or showed the results of the research as he would in a technical scientific talk, all integrated into the same story? A skeptical answer would be that lay readers wouldn’t touch the book and that technical readers would not be interested in the fluff. Strogatz already wrote an introductory textbook on nonlinear dynamics (which I haven’t read, but I’m told it’s good). I’ve seen textbooks that have little biographies inserted here and there, and I’ve seen popular books that use some equations or put technical appendices at the end. I am curious about a book intended to teach an undergraduate course that’s a truly integrated historical story and didactic text. There is an extensive bibliography allowing me to pursue the technical aspect of whatever ideas interest me the most, but that is something quite different from an organized presentation.

I picked up Sync while browsing, and read it because I remembered both the TED talk I linked above and Strogatz’s amusing math columns in the New York Times.

Left as an Exercise to the Society

December 23, 2008

Today I flew across the nation, farting. On the plane, having gorged far beyond satiety on my Dave Eggers anthology, I turned my attention to trying to clarify for myself some points I had been considering in linear algebra.

Later that night, in catching up after a year apart, my older sister asked about my flight. Jul had once considered becoming a math teacher. She took the same AP math classes I did in high school, studied a few technical topics here and there on her way to a linguistics degree. She has some background.

So I told her briefly about my attempts to understand dual spaces. I don’t think it got through, really. The point wasn’t that a vector space is isomorphic to the dual of its dual. The point was that, yes, I had a nice flight, because I sat there with a fresh notebook and step by step watched the algebra of this thing grow out of blank space. The results I had heard about from one source here and another there were materializing right in front of me. It was sloppy. I’m no mathematician. But it was getting increasingly better as I cleaned up a point here and there.

Soon I saw how we could go about associating vectors in the dual spaces, and in one sudden flash of insight, saw that a certain freedom in this choice could lead to Euclidean spaces, or Minkowski spaces, or Hilbert spaces, or, although I can’t claim I actually understand what these are, more complicated Reimann geometries. It all depends on a “metric”, I had been told. But here I was on a bumpy, dry sky-bullet, with stewardesses slamming carts of full of orange juice and assorted Pepsi products against my knee every twenty minutes, serendipitously discovering what the hell a “metric” could be. It was a nice flight.

“Linear algebra”, said Jul, “was a pretty dull class.” Dull? Are we talking about the same linear algebra? And then I realized – no, of course we were not. “So your linear algebra class,” I asked, “was mostly about matrices, and multiplying them and finding determinants and stuff?”

“Yes, that’s right.”

Dammit! Because see, she didn’t take a class on linear algebra. She took a class on formulas. Which is a shame. My sister deserves a lot better than that. She’s smart. Really smart. She was the captain/president/founder of her high school robotics team. Scored like a bajillion points on the SAT (I got a bajillion and one. Sorry, sis.) She taught me how to multiply numbers by 11 when we were this high.

She has a little baby who is grasping after a new syllable or two every day now, and tentatively standing a few momentous seconds at a time on wobbly little legs. Will he sit up straight in his chair at lunch one day and declare through a mouth full of PB&J that it’s obvious a circle is the shortest possible line to enclose a given area, and then laugh and ask to go play Explorers with the kid next door? And if he does, who will notice?

ZapperZ at Physics and Physicists links to a recent paper on physics education. The authors tried to quantify the problem physics teachers are constantly battling – the wide gap in the way they and their students view the nature of the subject.

It’s inevitable that physicists will be more enthralled by their material than physics students on average. If they weren’t enthralled to begin with, the professors would never have gone to grad school. Still, it’s a somewhat saddening that so many students think of physics as a collection of formulas handed down from on high. That’s essentially what the survey shows.

Even at Caltech, I hear the constant complaint, “The problems on the test weren’t the same as the ones we did in class or on the homework.” Or, “the book doesn’t have any worked out examples.” I opened the book. I couldn’t understand, for a while, what they meant. The book definitely did have worked out examples. They were in the paragraphs that began “for example…” and then carried out a calculation. What they meant was, “the book doesn’t do everything for me.”

The other complaint, which I hear more often from younger students, is “I understand the concepts. I just don’t know how to solve the problems.” This has a variant for younger kids, which comes from the parents’ mouth, and is “He understands the math, he just has trouble with the word problems.” Then there is a long, expectant pause, “Can you just help him a bit with the word problems?”

No, not like that. It works the opposite way. I can normally solve the problems well before I understand the concepts. Occasionally I do understand the stuff but not the problem, if there’s some sort of sneaky trick to find. But the mantra of “I understand, but just can’t quite apply,” is some sort of warped refrain that echoes back and forth between students across the nation the way all meaningless idioms of speech do. It’s just something to say about a problem so arcane you aren’t really even sure what it is, or where to look for it.

I want so much to do something. To show them just a bit here or there, to get them started. I don’t know how. I think maybe the best thing to do is to take care of understanding more of this stuff for myself, first.

There are millions of people who really do get it, and can enjoy math on an airplane. Of course I know many of them in person, from school. Over the last few months, as I’ve spent more time on the sorts of places around the internet these people frequent, I see that they’re actually an incredibly strong and interconnected community. Interconnected, but disconnected. Floating in isolation through a nation of anti-intellectualism.

The Asians Are Coming! But I Can’t Count How Many

December 11, 2008

Since I’ve started reading blogs, I’ve seen a lot of instances of people ranting madly about topics they don’t understand very well. These people also don’t understand why they aren’t taken more seriously, or why, in fact, the whole system doesn’t immediately bow to their sagacity. But now that I, too, am a blogger, I’m beginning to understand the severely-debilitating effect the freedom to publish uncensored material has on human judgment. So here I am joining the ranks of men screaming into a hurricane, and unknowingly pointing the wrong direction.

A recent story from the NY Times warns repeatedly that those tricky little Asian people are eating a gazillion tons of fish every day and getting way too good at math. You see, for at least the last ten years both a generic statement and its complement have been considered racist if they involve black people in any way. Further, the whole feeling-generally-uncomfortable-about-anything-Islamic thing has been used as the hook on enough network TV shows that people are starting to get pretty sensitive about that, too. But we haven’t done anything really bad to the Asians since Vietnam, so it’s pretty much okay to treat them as one big group and find reasons to be scared of them.

Apparently, kids in Singapore, Taiwan, and Japan do very well, on average, on standardized math tests. It’s supposed to send off alarm bells and spur us to reform the educational system. But the stat is not what it’s made out to be.

Here are three of the more practical reasons we might want students to be mathematically competent:
1) it helps them balance their checkbook and etc.
2) it’s necessary background for engineers and accountants, etc.
3) it’s necessary for innovation. great technological and scientific breakthroughs are made by people who understand math

But here’s why childrens’ average test scores are irrelevant to these points
1) (math helps with life) It’s increasingly unnecessary for the average person to know math. Computers will do it all for you. Anything that requires a minimal amount of the sort of mathematical, logical, and/or algorithmic thinking employed by a math, science, or computer-type person can now be automated to the point where an intelligent chimpanzee can do it. Want to calculate your BMI? Don’t bother with the formula. Just plug in the numbers to a calculator, which automatically multiplies them to each other for you. Don’t want to figure out your taxes? Plug it into Quicken. Or hire an accountant, who also doesn’t know math very well but can plug things into Quicken more efficiently than you. Don’t know how much longer to boil an ostrich egg than a chicken egg? Don’t bother with dimensional analysis. Just look it up online.

2) (math helps with jobs) Partially, more of the same argument as point 1) applies here. Want to be an airline pilot? Don’t worry yourself too much with the math. Just make sure the numbers from this instrument agree with the numbers from that instrument, and the computers will take care of everything. The percentage of people who really need to be good at math is quite small, so we should be more interested in the scores of the top 5% or top 1% of students than the average score.

3) (math leads to technological and scientific excellence) The average performance of students is simply irrelevant to this one. Big ideas come from people who work hard on problems because they’re intrigued by them and genuinely interested in the work itself. They need a spark of creativity to go with their technical competence, but spark is the really essential thing. It’s far easier to be very good at electrical engineering (for example) than it is to do something important in it. And frankly, hours upon hours drilling practice problems until you’ve memorized all the methods of solution is not going to get you far beyond good test scores. But that, as best I can tell from here, it’s what’s going on with the Asia/West divide in math scores. The Asian kids study longer and work harder. The cuiture is extremely performance-based, so that parents push their kids hard, but they only thing anyone cares about are good grades and good test scores. Since the tests don’t require creativity, why bother encouraging it?

I’ve been teaching American high school kids for a while. Many of them have been first or second generation Americans from Asian families. They grew up bilingual and their households retain most of the traditional values of Korea/Taiwan/Japan, including those relating to education. I’ve also taught kids from America, the UK, India, France, Italy, Turkey, Japan, China, Mexico, Canada, and various places I hadn’t even heard of before i met them. I’ve taught whites, blacks, east asians, south asians, hispanics, polynesians, native americans, and various combinations thereof. And guess what? They’re all the same. Not the kids, I mean, of course they’re quite different from each other. But I do not see any systematic difference in competence, creativity, interest, brilliance, ability to concentrate, or whatever other factors are essential to doing great things with technical material.

It has been my experience that when you look at the top few percent – the ones who are truly gifted at this stuff, and occasionally ask questions that startle me with their insight, or find clearer and more direct explanations of the topic at hand than I had sniffed up myself – are more likely to be male. Not exclusively, of course. The most insightful student I ever had was a girl. But that gender bias is the only systematic tendency that’s stuck out to me.

So the Asian kids kicking American kids’ butts at math is not a clarion call. It may be a benchmark for how effective our educational system is, and how seriously our culture treats education, but not for how many great thinkers we’ll have in this country twenty years from now. If we want to have a home-grown army of thinkers and innovators, we should be more concerned with how much kids like math and want to do it on their own, rather than how many formulas they’ve memorized by age 10. A high schooler’s knowledge of math won’t get you all that far, anyway. It only comes from higher study, and America still has the world’s best system of institutes of higher learning. So it’s not a matter of cramming more into their heads while they’re young. It’s a matter of honestly and fairly presenting to young people what math is and what it can do. As long as grade school doesn’t make kids hate math, it’s doing fine. The ones who have aptitude will naturally gravitate towards it. We need to make sure that when they do, there’s someone there to guide that top 5%, and that we’re not all too busy worrying about the grade of the kid in the middle of the class to notice that the kid at the top just proved a new result in number theory.

My guess is that most of the people who spend their time screaming, “The Asians are coming! They traded their abacuses for TI-89’s and they’re going to swipe the technical carpet from under our fat, complacent feet!” know much more about statistics than about the process of becoming technically competent, one part of which is to learn never to take statistics at face value. If our goal is really to raise the average test score, it has to come as much from a shift in cultural values as a change to the educational system. But if our goal is to be a scientifically and technologically vital society, the masses are not the place to look.

Let’s Read the Internet! week 8

December 8, 2008

Wind-Powered Perpetual Motion
and
Why the Directly-Downwind Faster Than the Wind Car Works”
Mark Chu-Carroll on Good Math, Bad Math

“The only true wisdom is in knowing you know nothing.”

Socrates would have to be a fan of the scientific method. We frequently acclaim the shift towards naturalism in Western thought, as a turning point in our intellectual maturity, but that shift brought with it the less-recognized roots of an even higher goal – the eradication of hubris in the search for understanding. Naturalism, the philosophical position that empirical observation holds the final word in debates on truth, essentially kills the argument of “because I say so.” Truth comes from no one in particular, so there’s at least the faint possibility that people trying to understand the way things work will some day no longer jockey and battle to be “the one who got it right.” That’s a far-out ideal, and maybe if nobody thought they were going to be credited with brilliance, nobody would have the incentive to try to do something brilliant in the first place. But at the very least, when two naturalists have an argument, they can frequently appeal to a common, impartial, higher source – nature – as arbiter.

That’s what’s happened here on Mark Chu-Carroll’s widely-read blog. He initially, and incorrectly, believed a certain device that drives overland into the wind and faster than the wind was a fraud. After long, long debates, he changed his mind, and carefully explained the mistakes in his own reasoning and what he had learned in the process of investigating his own error. Which is pretty much awesome, because such things hardly ever occur in arguments on less savory topics, like abortion. (Oh my God, was that an eating-dead-babies joke?)

I also appreciated the sort of emergent didactic property of the hundred-some post comment thread on Chu-Carroll’s original post. After watching the youtube video of the device (linked from the original post), I wasn’t completely sure whether the treadmill test was fair. It seemed reasonable enough, but I certainly wouldn’t have been prepared to defend it against someone eager to argue the opposite way.

As I read the thread, commenters raised most of the points I was considering. Other people answered those points, and then even more people chimed in with takes that I hadn’t considered at all. The overall effect was for a large amount of white noise and repetition, but also for a strikingly-diverse set of mindsets converging on the same problem. By the time I was done reading what everyone had to say, I felt that I had appreciated more intricacies in the problem than I would ever have discovered thinking about it alone, and I probably understood it better than I would have even if a single skilled author had written a long exposition. The challenge of interpreting each new voice’s arguments, incorporating them with the previous knowledge, and then parsing all of it for myself over and over, trying to find holes in everyone’s logic and patch together a firm understanding piece by piece, was absorbing because it’s so much more interactive than simply reading one single person’s explanation, no matter how clear, detailed, or precise.

It makes me want to argue about physics more often, but only in the good way where your ego doesn’t get too involved.

A Russian Teacher In America
Andre Toom, linked from God Plays Dice

A long essay that’s a borderline sob story about the woes of the American educational system. As a private tutor, I see exactly the sort of problems Toom is discussing on a daily basis – students, even (or perhaps especially) the “good” students, are so maniacally focused on their grade that learning becomes completely lost amidst a sea of test-cramming, and question-memorizing. Students are so wrapped up in the concrete performance markers visible to the world, that they don’t care at all for their true progress, visible chiefly to themselves.

That, at least, is the picture. I only partially buy it. It’s true, to varying degrees, for many students. But it’s not as if this entire nation has no one left interested in math. The sad part is that over two hundred or so students I’ve had, there have been a handful who are truly interested in math and physics, but they seldom have much guidance. Because these kids can gets A’s in math class, no one in public school is very concerned with pushing their limits when there are too many problem kids to worry about first. So I’m more interested in people with plans on how to reach interested young students with extra-curricular math opportunities than I am with people deriding a broken system.

Not everyone is going to love math. In fact, I doubt there’s ever been a society where a majority of people are interested. But the vast majority of our society has to take it in school. So yeah, it’s inevitable that there are lots of people taking math who don’t care about math. But I’ve done the same thing in a literature class before. Ultimately, math is cool enough that some people are going to discover it no matter what the educational system is like, so I’m not all that worried about the alarm bells being rung here.

Blow to Vitamins as Antidote to Ageing
James Randerson at The Guardian

We thought we understood, like, everything. Turns out not. But the next study that comes out will surely reveal the secrets to perfect health once and for all…

Swiss Approve Heroin Scheme but Vote Down Marijuana Law

Sounds like a pretty good plan to me. Administer heroin to addicts in a safe, controlled environment, thereby reducing health risks and driving down the general nastiness associated with black market activity. I can also understand why you wouldn’t want to legalize marijuana in just one small portion of Europe, since everyone would then go there just to smoke. The same argument doesn’t hold as much water for the US with its block-like geography, but I live in California, where marijuana is as good as legal anyway.

Nebulous
Tara Donovan

from Three Quarks Daily

The Not-So-Presidential Debate

The Not So Presidential Debate from aaron sedlak on Vimeo.
also from Three Quarks Daily

Why Punishment Is Worth It In The End
Ed Yong at Not Exactly Rocket Science

Read this article or else! Nah, honestly I would never be able to go through life as someone who tried to understand human interactions by designing toy experiments like this. But It’s nice to get little sixty-second summaries of their months of hard labor.

Over-budget Mars rover mission delayed until 2011
Rachel Courtland at New Scientist

Bad news, since I work at the place where they’re building this thing, and they owe me two months’ back pay already.

15 of the World’s Most Creative Papercraft Artists

You get to feeling a little bit sleazy when you realize all the exposure you’ve had to art in the last two years has come in the form of internet lists with titles like “The Top Ten Totally Badass Avant-Garde Experimental Playdoh Exhibitions of 2008!!” But on the other hand, some of this stuff actually is pretty badass, for being a paper sculpture of a cat.

A Happy/Unhappy New Pair of Studies
Stephen Black at Improbable Research

Among the headlines of news feeds I scanned through this week, there must have been at least ten stories referencing a recent paper purporting to show that happiness is “contagious”, that is, if I were to reach down and magically make your friends happy, you would become happy as well. When I first heard about this, I was intrigued, because I was wondering how you would establish this is a “contagious” effect, and not just correlation. It turns out: you don’t. The researchers, from what I can tell, simply found a correlation and announced that happiness is contagious. News stories are apparently contagious, too, because once word of this paper got out, most of the major science news outlets published something on the story.

But as the link describes, another study found that height was also “contagious”. That is, if your friends are tall, it’s likely you’re tall, too. Just as with happiness.

Sine of an Inscribed Angle
Brent Yorgey on The Math Less Traveled

A cute visualization of the law of sines.