The Dangers of Abstraction

Here's an excerpt from Surely You're Joking, Mr. Feynman. Feynman is talking with a group of physics students from Brazil.

I discovered a very strange phenomenon: I could ask a question, which the students would answer immediately. But the next time I would ask the question-the same subject, and the same question, as far as I could tell-they couldn't answer it at all! For instance, one time I was talking about polarized light, and I gave them all some strips of polaroid. Polaroid passes only light whose electric vector is in a certain direction, so I explained how you could tell which way the light is polarized from whether the polaroid is dark or light.

We first took two strips of polaroid and rotated them until they let the most light through. From doing that we could tell that the two strips were now admitting light polarized in the same direction-what passed through one piece of polaroid could also pass through the other. But then I asked them how one could tell the absolute direction of polarization, for a single piece of polaroid. They hadn't any idea. I knew this took a certain amount of ingenuity, so I gave them a hint: "Look at the light reflected from the bay outside." Nobody said anything. Then I said, "Have you ever heard of Brewster's Angle?"

"Yes, sir! Brewster's Angle is the angle at which light reflected from a medium with an index of refraction is completely polarized."

"And which way is the light polarized when it's reflected?"

"The light is polarized perpendicular to the plane of reflection, sir."

Even now, I have to think about it; they knew it cold! They even knew the tangent of the angle equals the index! I said, "Well?" Still nothing. They had just told me that light reflected from a medium with an index, such as the bay outside, was polarized; they had even told me which way it was polarized. I said,

"Look at the bay outside, through the polaroid. Now turn the polaroid." "Ooh, it's polarized!" they said.

After a lot of investigation, I finally figured out that the students had memorized everything, but they didn't know what anything meant. When they heard "light that is reflected from a medium with an index," they didn't know that it meant a material such as water. They didn't know that the "direction of the light" is the direction in which you see something when you're looking at it, and so on. Everything was entirely memorized, yet nothing had been translated into meaningful words. So if I asked, "What is Brewster's Angle?" I'm going into the computer with the right keywords. But if I say, "Look at the water," nothing happens-they don't have anything under "Look at the water"!

This sort of thing happens to lots of people who use Anki. It used to happen a lot to me. You're reading a textbook which neatly states, "Brewster's Angle is the angle at which light reflected from a medium with an index of refraction is completely polarized." So you make an Anki card:

What is Brewster's Angle? :: The angle at which light reflected from a medium with an index of refraction is completely polarized.

and perhaps another:

Which way is light polarized, after ...? ::  Perpendicular to the plane of reflection.

Five years after adding these cards, you'll be exactly like the Brazilian physics students: you'll have the words, but the meaning will have long since fallen away. This will be true even if once you deeply understood Brewster's Angle, since you're not doing anything to refresh that deep understanding, only the words that surround it.

The way to avoid this is to make your cards more concrete. A much better group of cards asks things like "Here's a situation with a polaroid and some light. Which way is the light polarized after reflecting?" and "What angle should we turn the polaroid to to achieve such and such effect?" and "Joe's polarized sunglasses cause this effect; how much does he have to turn them to avoid it" and so on and so forth. If you possess the true understanding of Brewster's Angle required to answer these cards, practicing them will let you retain that understanding far into the future.

"But I want to know the real definition!", you might say. Here's an unexpected result. By practicing the concrete examples, you don't sacrifice the abstract principle. In fact, if someone asks you for the definition of Brewster's Angle after studying a bunch of cards like that, you'll think for a second and easily reconstruct it from your examples.

A usable understanding can easily be translated into a definition, but a definition is almost impossible to translate into a usable understanding. So practice cards which require usable understanding, not definitions.