Posts Tagged ‘learning’

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.


‘Simple’ Isn’t ‘Easy’

November 7, 2010

You are probably aware that 3^{1/2} = \sqrt{3}. Sometimes when I’m tutoring I wind up teaching this to young students. Here is the story I use:

You already know that 3^4*3^2 = 3^6 for a very simple reason.

Forget the reason for a moment, and just focus on the rule. When you multiply exponents with the same base, you can add the powers.

That means

3^{1/2}*3^{1/2} = 3^1 = 3

Evidently, 3^{1/2} is a number such that if you multiply it by itself, you get three. But that is exactly the meaning of the square root! Hence 3^{1/2} = \sqrt{3}.

This is a very simple idea, but when I try it on students, it usually fails.

After going through the story, I ask what 16^{1/2} is. I’m hoping to hear “four”, but that’s not what happens. Sometimes they say it’s eight. Sometimes they say they don’t know. But the most common response is to go through the whole thing again. The student writes down

16^{1/2}*16^{1/2} = 16^1 = 16.

They stare it at for a while. Then they look up at me and say, “Is that right?” We discuss it a bit further to clarify. Circuitously, we stumble upon 16^{1/2}=4. After that we do a few more half-powers and they get it right. Then I ask what 8^{1/3} is. The student will write down

8^{1/3}*8^{1/3} = 8^{2/3}.

“It doesn’t work for that one,” they say. “You just get a 2/3 power, and we can’t do that.” So we talk about it some more, until after some time the student can go between roots and exponents.

Then I ask what 4^{3/2} is, but they struggle with this, too. Once that’s down we try for 6^{-1}, but that is also impenetrable (I usually hear that it’s -6). When I suggest trying to figure it out based on the rule of exponent addition, the student feels frustrated and defeated.

It’s curious that I have such difficulty teaching this idea. It is not too complicated or too difficult, even for a young child. It is far simpler than long division and far less abstract than “set the unknown variable equal to x”. The problem is not the sophistication of the idea, but a more fundamental error in communication. When I give my little presentation, the students simply have no idea what I’m doing.

An analogy: I’m teaching someone how to lift weights (this is very hypothetical). I take a dumbbell and I start doing some bicep curls. It’s only a 5-lb dumbbell, and the motion is very simple, so I figure the guy I’m teaching will get it for sure. I hand him the weight and say, “You try.”

When I hand over the weight and the student starts yanking it up and down. He purposely mimics the way I grunt in exertion and copies my facial expressions. He remembers how I looked over my shoulder to talk to him while I demonstrated the exercise, so he looks over his shoulder when trying it out. The weight ultimately does go up and down, but only with a great deal of extraneous commotion. I straighten him out with some effort, but when we move over to the bench press we’ll repeat the whole confused process.

The problem is that before we began, my student didn’t know what weight-lifting is. He didn’t know the point is to make your muscles stronger, or the counter-intuitive idea that to make your muscles stronger, you first have to tire them out by working them hard.

Similarly, my math students watch me do this strange algebraic exercise with exponents not knowing that the goal is to discover new things. They think, instead, that I was simply teaching a new procedure, as in, “This is how you solve problems where the exponent is one half.”

This is not really a big problem. Students can learn new things; that’s what being a student is about. The problem is that students’ ineptitude at this task frustrates me. At times, when watching a student struggle with a problem, I’ve felt ironic wonder at the student’s remarkable creativity – how do they find so many unexpected ways to get everything totally wrong? I wind up concluding that the student is “stupid”, and the student leaves the lesson with only the impression that they have somehow failed at a task they never even understood.

I make these grievous errors in judgment because I assume that since I’ve seen the student handle far more complicated tasks, they should master this one right away. That is not so. ‘Simple’ isn’t ‘easy’. Computing a determinant of a 4×4 matrix isn’t simple, but my students can blaze through it. Showing that the determinant will be zero by noticing that the last row is equal to first row is very simple, but I’ve never had a student use that method.

The things we’re good at are not what’s simplest, but what’s most familiar. The converse also holds: things that are unfamiliar are difficult, even if they’re simple. I personally find it much easier to solve geometry problems using coordinates, algebra, and calculus than using Euclidean geometry, even when the Euclidean approach may be just a few lines of sketching and finding a similar triangle.

When I first noticed that students were having a hard time with problems because they required unfamiliar thinking, and not because they were too hard or because the students were bad, I tried to remedy the situation with speeches. I would talk about how interesting it is to figure out where a formula comes from. I would say over and over that no, I don’t have all the formulas memorized, because as long as I know most of it, I can figure the rest out. I would prove my point by waiting until they embarked on a difficult calculation, and then solving it quickly in my head using some trick or other, supposedly demonstrating how useful it is to be able to approach a problem many different ways. Then I would describe how it’s done. “You’ll like this thing I’m about to show you,” I would say. “It’ll make your life easier.”

This backfired. It mostly led the students to believe that I either gained some ineffable voodoo skills in college or that I am in possession of an extraordinary native intellect that they could never hope to emulate.

I still don’t know quite how to handle the “simple isn’t easy problem”. I have become far more patient when trying to push students’ boundaries, and far less ambitious. I regret the many times I compromised a student’s chance at learning and my own at equanimity by failing to recognize “simple isn’t easy” in practice. I continue to search for simpler and simpler teaching stories, but I don’t spend enough time searching for ways to make the unfamiliar territory easier to navigate. I don’t know how complicated a task that is – to figure out how to build a stepladder to a new level cognition – but I know it isn’t yet easy.

Let’s Read the Internet! Week 11: Good and Bad Explanation

May 14, 2010

Do you think rationally about all the opinions you read, carefully considering why you agree or disagree with any given viewpoint, or is your method for discourse more like the way you sift through a hundred crappy photos of yourself to find the kinda-hot-but-not-too-slutty one that will be your Facebook profile picture? Oh yes, I like this one. All the other can go now.

It’s been a long time since I last read the internet with you, so it’s time to do that again. Hopefully you’ll be entertained, and also question the way you think about facts and reality. Although this is a links dump, incredibly none of it involves cats or pornography.

Via Swans on Tea, Feynman discusses, in a tangential manner, what magnetism is.

When I launch into an explanation, my goal is something is along the lines of, “I’m going to say something to you, and when I’m done, you’ll understand it the way I do.” My guess is that most people implicitly think about explanation the same way. An explainer says some words, possibly along with drawing pictures or doing a demonstration, and the explainee watches, listens, and understands.

We expect some confusion and some back-and-forth questions. Also, the scope of what is explained may be very small, so that the explainer perhaps knows a lot more details, but despite these caveats I think this “I will give you my knowledge” approach is the subtext for most of our explanations.

The strange thing is that if you ask people directly what explanation is, they do not believe this. They believe that explanations are highly context-dependent, and that they’re imperfect, and that their scope is limited. (“I don’t expect the explainee to get everything. The explanation just gives the general idea, and they’ll work out the details in due time…”), but when I watch two people engaged in a explainer/explainee interaction I get the feeling that they will consider the exchange a failure (or at least not wholly successful) if the explainee ultimately does not understand the subject the way the explainer does. Even the drastically different approaches people take when explaining something to an adult or to a child seem based on the principle that in order for the explanation to be effective, it must be worded to suit the audience, but the explainer still hopes to be completely understood. They just need to find the right way to say things.

Feynman points out that this sort of explanation is impossible because knowledge doesn’t consist of tidbits. Feynman cannot take his knowledge of magnetism and “dumb it down” in any sort of accurate way, because that knowledge is couched in the context of everything else he knows about nature. Feynman’s understanding of magnetic forces was much more thorough than the interviewer’s because Feynman understood the fundamental forces involved; he knew all about quantum theory and the interaction of light with matter, and had a feeling for what things were and were not already known and explained by physical models. He also had practical experience with magnets, and had taught students about magnetism and investigated all sorts of magnetic phenomena. But in addition to this knowledge of the theories and models of magnetism, Feynman’s understanding is tempered by his abilities. What separates the scientist from the layperson is not their knowledge of science, but their ability to mathematically manipulate the model, or even create a new one, to derive understanding.

If Feynman were still around and he sat down to tutor me in all aspects of electromagnetism, we could probably make a lot of progress. With enough time, he could teach me everything he knew. But I still wouldn’t understand it the way he did.

With that, let’s look at an explanation I particularly liked:

We Recommend a Singular Value Decomposition
David Austin at the American Mathematical Society.

This is an explanation of the singular value decomposition, a basic tool in linear algebra. I remember learning about it while studying linear algebra, but I didn’t understand it very clearly. I thought about it only formally, and I kept getting the idea of what it was confused with the proof that it exists. As a result, if I were asked to explain singular value decompositions to someone else, I’d have first gone back to my linear algebra book to review, then pretty much repeated what it said there, trying desperately to do things just differently enough that I wasn’t copying.

I got the feeling that Austin did the opposite in writing this article. he did not sit down and say, “Okay, what are all the things I know about SVD and all the good examples of it, and then how can I condense them all and make it appropriate to the audience?”

Instead, it seemed like he said, “I happen to know a couple of good pictures that make this clear in the case of a 2×2 matrix. Based on that, what sort of presentation of the SVD makes sense? What level of detail would muddy the presentation? If I change the order I present the ideas, how will that change the reader’s perception of the SVD’s theoretical and practical importance? What can be left out, and how can I get straight to the heart of the matter and communicate that first?”

Very quickly in the essay, Austin gets to this picture:

Singular value decomposition of [(1,1),(0,1)]

which illustrates the singular value decomposition of

\left[ \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right].

There are only a few short paragraphs before that, but already we’ve walked through a story that motivates it. Austin gives three examples showing how we can understand linear transformations visually, and by the time we finish the third, it was apparent to me that a singular value decomposition is a logical extension of the linear algebra I was already familiar with. He had me hooked for the rest of the article.

After giving his example, Austin builds directly to the equation

M = U \Sigma V^T

which illustrates why it’s a “decomposition”, and what each part of the decomposition means. Only after giving a fairly complete explanation of what a singular value decomposition is did he start to go into how to find it and how to apply it.

Lots of math or physics writing I see doesn’t take this approach. Instead, the first I see a particular equation is at the end of its derivation. That means that all the derivation leading up to it seemed unmotivated to me. Austin doesn’t even include the derivations. There’s enough detail that I could work through the missing parts by myself, ultimately understanding them better than I would if each step were spelled out for me. For example, he writes

In other words, the function |M x| on the unit circle has a maximum at v_1 and a minimum at v_2. This reduces the problem to a rather standard calculus problem in which we wish to optimize a function over the unit circle. It turns out that the critical points of this function occur at the eigenvectors of the matrix M^TM.

That’s actually more effective for me than actually going through the details of the calculus problem. It points me in the right direction to go over it when I’m interested, but in the meantime lets me continue on to the rest of the good stuff.

By reorganizing the material, omitting details, and (literally) illustrating his concepts, Austin finally got me to pay attention to something I ostensibly learned years ago.

Next, I’d like to illustrate my lack of creativity by returning to Feynman, this time his Caltech commencement address from 1974

Cargo Cult Science

Feynman identifies a problem:

In the South Seas there is a Cargo Cult of people. During the war they saw airplanes land with lots of good materials, and they want the same thing to happen now. So they’ve arranged to make things like runways, to put fires along the sides of the runways, to make a wooden hut for a man to sit in, with two wooden pieces on his head like headphones and bars of bamboo sticking out like antennas—he’s the controller—and they wait for the airplanes to land. They’re doing everything right. The form is perfect. It looks exactly the way it looked before. But it doesn’t work. No airplanes land. So I call these things Cargo Cult Science, because they follow all the apparent precepts and forms of scientific investigation, but they’re missing something essential, because the planes don’t land.

and suggests a solution:

Details that could throw doubt on your interpretation must be given, if you know them. You must do the best you can—if you know anything at all wrong, or possibly wrong—to explain it. If you make a theory, for example, and advertise it, or put it out, then you must also put down all the facts that disagree with it, as well as those that agree with it. There is also a more subtle problem. When you have put a lot of ideas together to make an elaborate theory, you want to make sure, when explaining what it fits, that those things it fits are not just the things that gave you the idea for the theory; but that the finished theory makes something else come out right, in addition.

For an example of awful science, take a look at a story that made it to Slashdot a little while ago, Scientists Postulate Extinct Hominid with 150 IQ.

The Slashdot summary says,

Neuroscientists Gary Lynch and Richard Granger have an interesting article in Discover Magazine about the Boskops, an extinct hominid that had big eyes, child-like faces, and forebrains roughly 50% larger than modern man indicating they may have had an average intelligence of around 150, making them geniuses among Homo sapiens. The combination of a large cranium and immature face would look decidedly unusual to modern eyes, but not entirely unfamiliar. Such faces peer out from the covers of countless science fiction books and are often attached to ‘alien abductors’ in movies.

Slashdot is known for being strong on computer news, not for their science coverage, but still it’s surprising to me that such a ridiculous bit of claptrap got so much attention. A few commenters point out how absurd the conclusion that an entire race of people had an average IQ of 150 is, but there is so much white noise in the comments of any large online community that most people usually don’t read them, probably including the people who write the comments in the first place.

And even if Slashdot will publish sensational cargo cult stories like this, what business does it have in Discover Magazine, which I don’t read, but had assumed was fairly reputable? Discover published this quote about the Boskops:

Where your memory of a walk down a Parisian street may include the mental visual image of the street vendor, the bistro, and the charming little church, the Boskop may also have had the music coming from the bistro, the conversations from other strollers, and the peculiar window over the door of the church. Alas, if only the Boskop had had the chance to stroll a Parisian boulevard!

First, that doesn’t sound like high intelligence to me. It sounds like autism. Second, how the fuck would you know that from looking at some skulls? Such conclusions obviously have no place in the science-with-integrity Feynman described.

20 years ago, if I had read that story I would not have gone to the effort to follow up on it. (For one thing I’d have been five years old, and so instead of doing some research I would have drank a juice box, gone outside to play, and pooped myself.) Now we have the internet, and follow-up is very easy. Fortunately, high up on the Google results is John Hawks’ article, The “Amazing” Boskops. Hawks, summarizing his review of literature on the Boskops, writes,

…in fact, what happened is that a small set of large crania were taken from a much larger sample of varied crania, and given the name, “Boskopoid.” This selection was initially done almost without any regard for archaeological or cultural associations — any old, large skull was a “Boskop”. Later, when a more systematic inventory of archaeological associations was entered into evidence, it became clear that the “Boskop race” was entirely a figment of anthropologists’ imaginations. Instead, the MSA-to-LSA population of South Africa had a varied array of features, within the last 20,000 years trending toward those present in historic southern African peoples.

Hawks then followed up with more detail later.

The good news is that the Boskop nonsense will die out because it’s wrong, and our system works well enough that things that are wrong do eventually die out.

In that little vignette, I looked at a big magazine and published book that were nonsense, and debunked by a blog. It’s not always easy to determine the credibility of a source, and its reputation can be misleading. Blogs have a terrible a reputation in general, while some people seem to believe that if it’s in a book, it must be true. (Unfortunately people take this to the extreme with one particularly poorly-documented and self-contradictory bestselling book!)

A more difficult stickier issue is anthropogenic global warming. There is little doubt in my mind that anthropogenic global warming is real, but unlike with evolution, I do not believe that because I have looked at the scientific evidence and thought about the arguments for and against. I haven’t examined the methods of collecting raw data or the factors accounted for in climate models. I don’t even know how accurate those models’ predictions are. I take it all on the word of climate scientists and a cursory review of their reports. I do not see this as a problem or a failure of my rationality. I do withhold judgment on whether global warming is as important an issue as, say, pollution or direct destruction of natural resources, but I do not feel reservation in stating that I think it is very likely that if humans continue on the way they’ve been going, the Earth will warm with severe consequences.

What does this have to do with cargo cult science? Cargo cult science is the reason I believe the climate scientists rather than the climate skeptics. My goal here isn’t to convince you one way or another about climate science, or to link to the best-reasoned discussions about it or to give an accurate cross-section of the blogosphere’s thinking process. These are various opinions on anthropogenic global warming, and my hope is that reading for the underlying decision-making process is an instructive exercise.

Here is Lord Monckton, a prominent global warming critic:

Here he is interviewing a Greenpeace supporter about why she believes in anthropogenic global warming:

Here is the UN group Monckton criticizes, the
Intergovernmental Panel on Climate Change
In particular, their Climate Change 2007 Synthesis Report, a 52-page summary of all things climate science. For more detail, their Publications and Data are available.

Here is a recent letter published in Science. It discusses the process scientists use to create reports on the climate, the uncertainty in scientific results, the fallibility of scientific findings, and the role of integrity in science.
Climate Change and the Integrity of Science

Here is statistician and blogger Andrew Gelman talking about expert opinion and scientific consensus:
How do I form my attitudes and opinions about scientific questions?

Here is famous skeptic James Randi on the pressure for scientific consensus, the fallibility of scientists, the uncertainty in models of complicated phenomena, and his skepticism of anthropogenic global warming:
AGW Revisited

Here is the petition Randi describes, the
Petition Project

Here is a reply to Randi and the Petition Project from PZ Myers, a biologist and well-known angry internet scientist.
Say it ain’t so, Randi!

Here is a graphic by David McCandless. Its goal is to present an example of the arguments one would uncover in an attempt to self-educate about climate science using only the internet.
Global Warming Skeptics vs. The Scientific Consensus

Greg Laden writes about skepticism, rationality, and groupthink in a lengthy post.
Are you a real skeptic? I doubt it.

Here is the Wikipedia Article on anthropogenic global warming, along with tabs to the discussion page for the article and the article history. This is a featured article on Wikipedia.
Global Warming

My focus on the process people are using to come to terms with global warming isn’t meant to deemphasize the importance of this issue and of other aspects of the relationship between humanity and our biome. Our Earth is a fantastically diverse and endlessly beautiful home. Of course I want to understand it better.

Click to watch "Home" by Yann Arthus-Bertrand

Also here is a physics blog story about a mathematical model of cows.

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.