## Posts Tagged ‘teaching’

### Visualizing Elementary Calculus: Introduction

March 25, 2011

Recently I’ve been trying to be more geometrical when discussing elementary calculus with high school students. I don’t want to write an entire introduction to calculus, but the next few posts will outline some ways I think the geometric view can be helpful.

This series
I – Introduction
II – Trigonometry

You know about $\Delta$, which means “the change in”. For example, if $w$ represents my weight, then $-\Delta w$ represents the weight of the poop I just took.

Let’s say $h$ is your height above sea level. $\Delta h$ is the change in that height, but what change? The change when you climb the stairs? When you jump out of a plane? When you step on a banana peel?

When we think about change, we usually think about two things changing together. You get higher when you climb another stair on the staircase. $h$ is changing, and so is $s$, the number of stairs climbed.

These two changes are related to each other. Say the stairs are 10 cm high. Then you gain 10 cm of height for each stair. We can write that as $\Delta h = 10 {\rm cm} \hspace{.5em} \Delta s$. We can also write it $\Delta h / \Delta s = 10 \hspace{.5em}{\rm cm}$. This says, “the height per stair is ten centimeters.”

This is the goal of calculus – to study the relationships between changing quantities. Let’s do a real example.

### The Area of a Square

Let’s say we have a square whose sides lengths are $x$. Its area is $x^2$. What is the relationship between changes in its area and changes in the length of a side? Draw the square, then expand the sides some. The amount the sides have expanded is $\Delta x$. The new area that’s been added is $\Delta (x^2)$.

We begin with the red square on the left, whose area is x^2. We add an extra amount Delta(x) to the sides, creating all the new green area.

From the picture we see

$\Delta(x^2) = 2x\Delta x + (\Delta x)^2$

This formula relates $\Delta (x^2)$, the change in the area, to $\Delta x$, the change in the length of a side.

### The Derivative of $x^2$

In the picture of the square, there is a little piece in the upper-right corner whose area is $(\Delta x)^2$. It is the smallest bit of area in the whole picture.

Look what happens when we make $\Delta x$ even smaller.

We shrink Delta(x) and observe what happens to the different areas being added on.

In the first picture, $\Delta x$ (no longer marked) is a quarter of $x$. $(\Delta x)^2$ is the dark green area, and it is one quarter as large as $x \Delta x$, the light green area. We see this because the dark patch fits inside the light one four times.

In the second picture, we shrink $\Delta x$ to one eighth of $x$. All the green areas shrink, but the dark patch shrinks on two sides while the light patches shrink on only one. As a result, the dark $(\Delta x)^2$ is now only one eighth the size of the light $x \Delta x$.

If we continued to shrink $\Delta x$, this ratio would continue to decrease. Eventually we could tile the dark patch a million times into the light one. So, as long as $\Delta x$ is very small, we can get a good estimate of the entire green area by ignoring the dark part $(\Delta x)^2$. Thus

$\Delta(x^2) \approx 2x\Delta x$

This approximation becomes better and better as $\Delta x$ shrinks, becoming perfect as $\Delta x$ becomes infinitesimally small.

When we want to indicate these infinitely small changes, we trade in the $\Delta$ for a ${\rm d}$ and write

$\textrm{d}(x^2) = 2x \textrm{d}x$

The terms $\textrm{d}(x^2)$ and $\textrm{d}x$ are called “differentials”. The equation expresses the relationship between two infinitely-small changes, one in $x$ and one in $x^2$.

Frequently, we divide by $\textrm{d}x$ on both sides to get

$\frac{\textrm{d}(x^2)}{\textrm{d}x} = 2x$

This is called “the derivative of $x^2$ with respect to $x$“.

#### Example 1: Estimating Squares

$20^2 = 400$. What is $21^2$?

Here $x$ = 20, and we’re looking at $x^2$. When $x$ goes from 20 to 21, it changes by 1, so $\textrm{d}x = 1$. Our formula tells us

$\textrm{d}(x^2) = 2x \textrm{d}x = 2*20*(1) = 40$

Hence, $x^2$ increases by about 40, from 400 to 440.

The real value is 441. We got the change in $x^2$ wrong by about 2%. That’s because $\textrm{d}x$ wasn’t infinitely small.

Let’s try again, this time estimating the square of 20.00458. Now $\textrm{d}x$ = .00458, so

$\textrm{d}(x^2) = 2 x \textrm{d}x = 2*20*.00458 = .1832$

The estimate is 400.1832. The real value is 400.183221. We did much better, under-estimating the change by only 0.01% this time. Also, it was not much harder to do this problem than the last, but squaring out 20.00458 by hand would be a pain. We saved some work.

#### Example 2: How Far Is the Horizon?

The beach is a good place to think about calculus. If you look out at the ocean, the horizon appears perfectly flat. Nonetheless, we know the Earth is really curved. In fact, we can deduce the curvature of the Earth by standing on the beach and enlisting the help of a friend in a boat.

It works like this: You stand on the beach with your head two meters above the water. Your friend sails away until the boat begins to disappear from sight. The reason the bottom of the boat is disappearing is that it is hidden behind the curvature of Earth.

When the bottom of the boat disappears, measure the distance to some part of the boat you can still see. What’s the relationship between your height, the distance to the boat, and the radius of Earth?

A picture will help. We’ll call your height $h$ and the distance to the horizon $z$.

You are the vertical stick on top, height h. The boat is the brown circle. It's at the horizon, a distance z away. The dotted line shows your line of sight. When the bottom of the boat begins disappearing, a right triangle forms.

Your height, the radius of Earth, and the distance to the horizon are related by the Pythagorean theorem to give

$R^2 + z^2 = (R+h)^2$

this is equivalent to

$z^2 = 2Rh + h^2$

As we have seen, if your height $h$ is small compared to the size of the Earth (and it is), the term $h^2$ drops away and the distance to the horizon is

$z = \sqrt{2Rh}$

You can see about $5 {\rm km}$ at the beach, making the radius of Earth about $6,000 {\rm km}$. (It’s actually $6378.1 {\rm km}$).

Next we want to know how much further you can see if you stand on your tiptoes. That would be a small change $\textrm{d}h$ to your height. It would let you see a small amount $\textrm{d}z$ further. How is $\textrm{d}h$ related to $\textrm{d}z$?

$\textrm{d}(x^2) = 2x\textrm{d}x$

So let $x^2 = h$, or $x = \sqrt{h}$, and we have

$\textrm{d}h = 2\sqrt{h}\hspace{.3em}\textrm{d}(\sqrt{h})$

But we also know

$\sqrt{h} = \frac{z}{\sqrt{2R}}$

so we can substitute that in to $\textrm{d}(\sqrt{h})$ and get

$\textrm{d}h = 2\sqrt{h}\hspace{.3em}\textrm{d}\left(\frac{z}{\sqrt{2R}}\right)$

or

$\frac{\textrm{d}z}{\textrm{d}h} = \sqrt{\frac{R}{2h}}$

This tells us how much further you can see if you get a little higher up. The interesting thing is it depends on $h$. The higher you go, the smaller $\textrm{d}z$. When you’re only two meters up, you get to see almost ten meters further out for every centimeter higher you go. However, if you’re 100m up on top a carousel, you get only 1 meter for each centimeter you rise.

It makes sense that the extra distance you see gets smaller and smaller the higher you go, and eventually shrinks down to zero. No matter how high you go, you can never see more than a quarter way around the globe.

(In reality, light bends due to refraction in the atmosphere, so you can sometimes see a bit further.)

### Circles

Suppose we have a circle with radius $r$. It has a certain area (you undoubtedly know the formula already, but play along). Suppose we increase $r$ by a small amount $\textrm{d}r$. What is the change $\textrm{d}A$ in the area?

The original circle is dark blue with area A and radius R. The radius increases an amount dR, increasing the area by the light blue ring with area dA.

$\textrm{d}A$ is the thin, light-blue ring. Imagine taking that ring and peeling it off the edge of the circle and laying it flat. We’d have a rectangle with width $\textrm{d}R$. Its length comes from the outside edge of the entire circle – the circumference. The circumference is $2 \pi R$, so

$\textrm{d}A = 2\pi R \textrm{d}R$

We saw earlier that $\textrm{d}(x^2) = 2x\textrm{d}x$, so let $x = R$ and we have

$\textrm{d}A = \pi \textrm{d}(R^2)$

Thus the quantities $A$ and $\pi R^2$ change in exactly the same way. Since they also start out the same (both zero when R is zero), we have

$A = \pi R^2$

### Next Post

We’ll look at trigonometry. Geometric arguments about the derivatives of trig functions are very simple ways of visualizing what’s going one, and are usually not introduced in a basic calculus course.

### Exercises

• Draw a cube with sides $x$ and show that $\textrm{d}(x^3) = 3x^2\textrm{d}x$. Thus the derivative of $x^3$ with respect to $x$ is $3x^2$.
• Draw a line with length $x$ and show that $\textrm{d}(x) = \textrm{d}x$, which is of course algebraically obvious. Thus the derivative of $x$ with respect to itself is 1.
• Draw a rectangle with width $w$ and length $c*w$ and show that $\textrm{d}(c*w^2) = 2cw\textrm{d}w = c\textrm{d}(w^2)$. Thus, whenever you have the differential of a variable multiplied by a constant, the constant can pop outside. Where was this property used implicitly in this post?
• Now that you know $\textrm{d}(x^3) = 3x^2\textrm{d}x$, let $x^3 = u$ and find the derivative of $u^{1/3}$ with respect to $u$. (Answer: $\frac{1}{3} u^{-2/3}$)
• What is $\textrm{d}(x^3)/\textrm{d}(x^2)$? Let $u = x^2$ and find the derivative of $u^{3/2}$ with respect to $u$. (Answer: $\frac{3}{2}u^{1/2}$).
• Examine $\textrm{d}(x^4)$ by letting $u = x^2$, so we’re looking at $\textrm{d}(u^2)$. Find the derivative of $x^4$ with respect to $x$. (Answer: $4x^3$)
• Draw an equilateral triangle with sides of length $s$. Increase the sides a small amount $\textrm{d}s$ and relate this to the change in area $\textrm{d}A$. Does this agree with our previous findings?
• Draw an ellipse with a fixed with semi-major axis $a$ and semi-minor axis $b$. Starting with a unit circle, argue by thinking about stretching that the area of the ellipse is $\pi ab$. Increase $a$ by a small amount $\textrm{d}a$ and increase $b$ proportionately. This adds a small area $\textrm{d}A$ to the ellipse. Show that this area is $\pi(a^2+b^2)/b\hspace{.3em}\textrm{d}a$. Does this let us find the circumference of the ellipse by the same thought process as we used for the circle? (Answer: no). Why not?
• Draw a sphere with radius $R$. Use the relationship between $\textrm{d}R$ and $\textrm{d}A$ to find the volume of a sphere, given its surface area is $4\pi R^2$. Check your answer against this post.

### ‘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.

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:

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.

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