Richard Feynman was an American physicist who contributed greatly to the field of quantum mechanics, eventually winning a Nobel Prize for his work. He also worked on the atomic bomb project and wrote a number of books about his colorful life and his views on science and technology, and in his time he was nearly as well-known a scientist as Carl Sagan. Toward the end of his life he notably served on the panel that investigated the explosion of the space shuttle Challenger, most notably bringing focus to a hypothesis that a failed O-ring seal led to the shuttle’s demise–a hypothesis which went on to become the generally accepted idea of the disaster’s cause.
Feynman’s memoir Surely You’re Joking, Mr. Feynman! (Adventures of a Curious Character) is a collection of anecdotes from Feynman’s life, drawn from recorded conversations he had with a friend, Ralph Leighton. The book is a fun, entertaining, and lively look at various episodes from Feynman’s life, though there are sober moments throughout, such as Feynman’s experiences with the educational system in Brazil, which is the passage I quote below.
But we’ll return to Feynman in a moment. First, though, the thing that made me think of this passage at all.
I recently saw a video on social media, shared by someone who has an axe to grind with the way arithmetic is taught to children these days under the guidelines of the “Common Core” curriculum. The video was a split-screen of a multiplication problem. I don’t recall the exact problem, but it was something along the lines of 36×52.
On the left side of the video, there was a teacher and she was teaching her students (who we never see, so I have no idea what age group or learning level she was teaching) to solve this particular problem. Having never undergone the Common Core pedagogy, I can only summarize it in unfair terms: she was saying things like “OK, we start by figuring out how many groups of tens we have, and then we draw a big box so we can start to group it all together.” But on the right side of the video, someone (we don’t see who, we only see their hand holding a pencil) solves the math problem using the method familiar to probably most people: “2 times 6 is 12, put a 2 down here, carry the one, 2 times 3 plus the one we carried, put a 7 down here….” And so on. The person on the right whips through the calculation and has the answer before the teacher on the left has even started processing through the problem.
The point of the video, obviously, is to show how much more obviously superior the “old” way of teaching arithmetic is to the “new” way. Of course, anyone looking at this knows, or should know, that the video is utterly dishonest. Showing a person who has developed a proficiency on one hand is not remotely an indictment of showing someone teaching the thing on the other, for one thing. For another, I can remember by grade school math classes, and I can tell you with a high degree of confidence that the proficiency demonstrated by someone just whipping off the correct answer using the “Multiply the ones digit, carry the tens digit, multiply the tens and add what was carried, lather rinse repeat” method did not come as a result of a few minutes’ instruction by a math teacher followed by an hour of doodle time. We spent days on that shit, and we got to take home page upon page of homework problems to practice the concept: “All right, we’ve spent an hour doing multiplications, so tonight do problems 1-50 in your workbook.”
(An honest admission: more often than not I did enough problems to assure myself that I knew what I was doing, and then I blew off the rest. This strategy was not always successful, in several ways. First, I got caught by teachers not having done all the problems; second, my self-assessment as to my grasp of the material was not always as accurate as I hoped.)
A fair version of this video would show a teacher doing the Common Core method on one side and a teacher actually teaching the old method on the other, but I strongly suspect that wouldn’t be as compelling, would it? Or maybe an adult doing the problem who learned via the Common Core method versus an adult who learned via the “old” method…but I expect that wouldn’t be all that interesting, either. As always it would boil down to who was faster with math. I knew plenty of kids who weren’t very fast doing sums in their heads, versus others who could glance at a problem and tell you the answer without even opening their pencil box. Again, this would not demonstrate anything at all.
(As I’ve been working on this post, this Sunday’s Calvin and Hobbes repeat from many years ago managed to be relevant here:
Yes, that’s what it felt like.
But here’s the thing with the Common Core method as I understand it, versus the “old” method (and I promise we’re getting to a book excerpt eventually): the “old” method was a teaching of a rote algorithmic process to arrive at an answer. You learned multiplication first by being drilled until you know all your single-digit products: 7 times 9, 4 times 5, et cetera. Then you used those to work through a process that would give an answer. That’s what an algorithm is, after all: a sequence of steps that, executed correctly, will yield the same result, each and every time. Kids in my day were being taught algorithms to complete multiplication assignments.
Today, though, it’s clear to me that the idea is to teach a more conceptual approach, a way to think about what numbers are, what they mean, and what is actually happening mathematically when one does multiplication. (Or division, or whatever.) We were taught a process whereby we could calculate 36 times 52, but it appears to me that kids today are being taught what it means to multiply 36 by 52. Those are not the same thing. In my day, the “conceptual” stuff would come into play with annoying word problems (“Suzie has to give one toothpick to 52 kids every day for 36 days, how many toothpicks does Suzie need?”), but even that was more an applicative thing than a conceptual thing.
My father, who taught mathematics in college, got frustrated a lot of the time when students would arrive in his collegiate classes with good algorithmic skills but little ability to conceptualize just what it all meant in terms of mathematical concepts. In calculus classes he’d ask “What’s the derivative of x,” and students would say, “Oh, the derivative of x is [insert whatever the derivative is].” Then he’s ask, “Good! Now what does that mean?”
And that is where we get to Dr. Feynman. One of the chapters in Surely You’re Joking, Mr. Feynman relates Feynman’s experiences in traveling to Brazil to see science education in action in that country, which at the time of his visit was just starting to really develop. Here is Dr. Feynman:
In regard to education in Brazil, I had a very interesting experience. I was teaching a group of students who would ultimately become teachers, since at that time there were not many opportunities in Brazil for a highly trained person in science. These students had already had many course, and this was to be their most advanced course in electricity and magnetism–Maxwell’s equations, and so on.
The university was located in various office buildings throughout the city, and the course I taught met in a building which overlooked the bay.
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 os 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 idex, 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 polarois.”
“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”!
[Feynman has several more experiences like this, after which he is to give a talk to students, professors, administrators, and government officials about his year in Brazil’s science education system.]
The lecture hall was full. I started out by defining science as understanding the behavior of nature. Then I asked, “What is a good reason for reaching science? Of course, no country can consider itself civilized unless…yak, yak, yak.” They were all sitting there nodding, because I know that’s the way they think.
Then I say, “That, of course, is absurd, because why should we feel we have to keep up with another country? We have to do it for a good reason, a sensible reason; not just because other countries do.” Then I talked about the utility of science, and its contribution to the improvement of the human condition, and all that–I really teased them a little bit.
Then I say, “The main purpose of my talk is to demonstrate that no science is being taught in Brazil!”
I can see them stir, thinking, “What? No science? This is absolutely crazy! We have all these classes.”
So I tell them that one of the first things to strike me when I came to Brazil was to see elementary school kids in bookstores, buying physics books. There are so many kids learning physics in Brazil, beginning much earlier than kids so in the United States, that it’s amazing you don’t find many physicists in Brazil–why is that? So many kids are working so hard, and nothing comes of it.
Then I gave the analogy of a Greek scholar who loves the Greek language, who knows that in his own country there aren’t many children studying Greek. But he comes to another country, where he is delighted to find everybody studying Greek–even the smaller kids in the elementary schools. He goes to the examination of a student who is coming to get his degree in Greek, and asks him, “What are Socrates’s ideas on the relationship between Truth and Beauty?”–and the student can’t answer. Then he asks the student, “What did Socrates say to Plato in the Third Symposium?” and the student lights up and goes, “Brrrrrrr-up”–he tells you everything, word for word, that Socrates said, in beautiful Greek.
But what Socrates was talking about in the Third Symposium was the relationship between Truth and Beauty!
What this Greek scholar discovers is, the students in another country learn Greek by first learning to pronounce the letters, then the words, and then sentences and paragraphs. They can recite, word for word, what Socrates said, without realizing that those Gr eek words actually mean somthing. To the student they are all artificial sounds. Nobody has every translated them into words the students can understand.
I said, “That’s how it looks to me, when I see you teaching the kids ‘science’ here in Brazil.”
This, it seems to me, is largely the difference between teaching the “old” way of arithmetic, with its rote memorization of times-tables and all that “Multiply the tens digit, carry the other number into the hundreds column” stuff, and the likely approach to however the Common Core method works. It’s teaching what is actually happening in the mathematics, as opposed to teaching a method that can be robotized.
Here’s some bonus Richard Feynman content, by the way: Ever wonder how a train stays on the tracks? Feynman answers, and it’s not the answer everyone thinks: