## Two Kinds of Understanding

A fixation in some people's minds is that it is important to understand the process of arithmetic. "We have too many students who have been taught to memorize processes rather than understand them," says one. On the surface, it sounds like a statement that is impossible to disagree with. However, you should disagree with it. In the case of basic arithmetic like addition and subtraction, it is a statement that is subtly and profoundly false.

The person is mistaken. Breaking something into steps does not necessarily improve understanding. Sometimes it merely obscures it. To make my point I will bring in two famous scientists to talk about understanding.

I don't know what's the matter with people: they don't learn by understanding, they learn by some other way — by rote or something. Their knowledge is so fragile!

In mathematics you don't understand things. You just get used to them.

There is an apparent contradiction in these two statements. Feynman complains that people don't understand, while von Neumann says that understanding is impossible. What's going on here? The answer is that they are talking about understanding in two different contexts, and I will explain what those two kinds of understanding are.

What Feynman is complaining about is a lack of mastery, or ownership of the concepts. The quote is taken from an anecdote about MIT students who were incapable of applying the calculus they learned to real-life scenarios. Sure, they had memorized the steps necessary to take derivatives, and so passed Calculus, but when confronted with a practical application of the knowledge they were incapable of making the connection. Somehow those steps did not teach them calculus, not in a real sense. What is especially relevant about the Feynman quote is that breaking a rote process into rote steps does not make it any less rote.

John von Neumann is making an observation about something else. What he is saying is that ownership of concepts is not always attended by a comfortable intuitive feeling. Nearly everybody who learns quantum mechanics reports finding it counter-intuitive, yet as far as we know it is a basic truth. You will master many things without being able to "explain" them. Feynman says as much:

What do you mean explain [it]? [...] You just have to take it as an element of the world.

Addition is one such elemental concept. You cannot break it down. To begin with, it must simply be accepted without being understood. In our system we start with arithmetic instead of, say, something "constituent" like digital logic, and with good reason. Arithmetic occupies a special place in the foundation of everything else we do.

## Generations of Crippled Chips

One bad way to subtract is to count down on your fingers and toes. It obviously doesn't work *that* well since you can't start from higher than twenty. What might be less obvious is that the method is more deeply broken than that. A line of people could subtract 1000 from 1000 by putting their fingers together. What would soon be clear though is that the counting down method is very slow.

Replace "a line of people with fingers" by "counting down out loud" here, and you get the same method, only one you can perform yourself. The counting down method might have some pedagogical value, but you wouldn't teach it as the main method. It has too many steps to be useful. Then we ask, does this method deepen our understanding of subtraction? Being aware of its existence may. Performing it repeatedly does not. In fact, doing math this way sounds like a burden. Yet one of the loudest arguments for slower methods is that they improve understanding.

Sometime in the middle of the 20th century, educators in the US mucked with the methods we teach kids. One thing they did, still in effect, was to replace the main subtraction algorithm with an inferior technique. The details of the switch are artfully explained in a song by Tom Lehrer:

I was surprised to learn about this switch. I was more surprised to learn I was on the wrong end of it; that is to say, I was taught the new method. Maybe you will be surprised, as I was, that someone decided to train you in an inferior technique.

The old, pre-1960s method has a simplicity and a clarity that is obvious on first inspection: it is superior. It has fewer steps, and the new method has more. The new method is inferior. Being stuck with the new method harmed me. Speaking from experience, my math was the worse for it. Here is feedback on the experiment. I grew up using this method, and it is bad.

The part in a computer that performs subtraction is known as the ALU, or Arithmetic Logic Unit. What's important to the operation of the chip is that the ALU performs its function quickly, and with a minimum of ceremony. At runtime, the part of computing comprised by arithmetic is not in any way interesting—it is tedium to be completed as efficiently as possible, so that the device can arrive at the more interesting result of the overall computation. If it were possible, you would skip the ALU entirely, but as it generally isn't you must content yourself with an ALU that runs as quickly as it can.

When it comes to problem solving, our brains are structured much the same. The important part is that arithmetic occupies a box that does not get in your way. Teaching kids to use weakened techniques ultimately curbs their problem solving. The unnecessary steps have the same effect as in an ALU: they slow down the computation. We are putting defective chips in kids' heads.

## Why Would You Do That?

I have a theory about how this happened that I will sketch here. Maybe it is true, maybe it is false. I think in the middle of the 20th century Americans became preoccupied with raising average test scores. One problem with this idea is that you may not be able to raise scores at all, especially for basic subjects like arithmetic. Another is, if it is possible to raise the average, you may do it at the expense of what you're actually after, which is good education. What good is a higher average if it came about as a result of faulty material?

If you water down the material, then for sure you make it easier for weak students. All that has to happen then, to raise the average, is that the weak students improve on the test more than the strong students decline. This is a terrible idea. Moreover, a lot of the strong students are probably off the scale when it comes to mastery of arithmetic, and you aren't taking losses beyond proficiency into account. In effect, the weaker techniques are an emergent realization of the mental handicap radio from Harrison Bergeron. You've reverted your overperformers to the mean.

You can't actually want to do this. What you want is for as many kids as possible to learn the old method, and then remedial classes for the kids who can't. For kids who can learn the old way, the old way is better, and every kid who can learn the old way should.