Posts Tagged ‘math education’

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.


\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.

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 2

October 19, 2008

Davisson-Germer Experiment Chad Orzel at Uncertain Principles
The first observation of the wave properties of electrons came by accident. Just like you.

A Beautiful New Theory of Everything Garrett Lisi on
In case you were wondering how everything works…

Didn’t quite catch that? Don’t worry. You can always read the paper.

Infinity is NOT a Number Mark Chu-Carroll at Good Math, Bad Math
More comprehensible than the previous post, if less profound. The fundamental problem with making infinity a number seems to be that it lets you prove all manner of foolishness, such as 1=2.

The Universal Declaration of Human Rights

It’s awfully pretentious to claim your document to be “universal”. Who has the authority? Further, what does it mean for everyone to be equal in rights? Clearly, we are not equal in many senses. Separating out “these things are rights” and “these things are what you have to deal with because of the circumstances of your life” is a tough task. For example, according to this declaration, everyone has a right to marry. But marriage is simply not a universal concept among humans. It’s perfectly conceivable to have viable, righteous societies with absolutely no concept of marriage. The concepts of privacy and property ownership could be sacrificed in righteous societies, under the right circumstances. Creating a list of rights that’s simultaneously universal and specific seems nearly impossible. But the visualization is nice.

Dead Waters Romain Vasseur et. al
Boats that get stuck in plain water. I don’t understand why this works, but the video is really cool.

Chimpanzees Make Spears to Hunt Bushbabies Not Exactly Rocket Science
Like it says, chimps make weapons and kill shit with them. In case you were wondering where we get it from.

Late Bloomers Malcolm Gladwell in The New Yorker
Just because you’re old doesn’t mean you’re useless. Therefore, you might as well slack off for another year or two before beginning that “great life’s work” stuff.

Where’s the Algebra? Michael Alison Chandler on X = Why?
Some chick with a seriously ugly smile asks whether algebra is important. But her “education” from her brother sadly misses the point. She asks, “what good are equations?”, and he replies “We have to learn equations to install lights.” But the entire article is written with the attitude that these equations are magical things that pop out of nowhere to describe lighting systems, their goal being to confuse blue-collar workers to the greatest extent possible. I don’t think there’s any understanding here the equations actually come from somewhere. Someone used a more basic set of principles to derive the equations, or else conducted experiments and then found equations to describe the results. Applying equations to describe real situations is not supposed to be a matter of plugging numbers into formulas.

The Cartoon-Off Farley Katz at The Cartoon Lounge
Normally, I wouldn’t bother linking to something that’s already been Slashdotted, but I bookmarked this page for “Let’s Read the Internet on Wednesday, and then the Slashdot post comes up just hours before I compile my links for the week. I guess the fact that the entire geek culture already knows about doesn’t really impact how funny it is.

The Sun
The web page that makes you go blind if you stare directly at it.

Fabry, Perot, and Their Wonderful Interferometer Skulls In The Stars
The author consistently produces wonderful posts explaining concepts in optics from a historical point of view. I actually used a Fabry-Perot interferometer in physics lab once. What I learned there is that they make surprisingly bad hammers.