Math and the Obvious

I begin on Day 1 by writing this on the board:

3 + 7 = 7 + 3

This is called the "Commutative Property of Addition," but I have another word. I call it: obvious. If you have three apples and I have seven...or, if you have seven apples and I have three...the total we have together is the same either way, isn't it?

Now, I put something else on the board:

3 × 7 = 7 × 3

You can guess that this is the "Commutative Property of Multiplication." But I call it: not obvious.

Let's think about what that's saying. 3×7 means you have three groups, each of which has seven items. Let's count that out: 7, 14, 21.

7×3 means seven groups of three each. Let's count that out: 3, 6, 9, 12, 15, 18...21. Hey, it came out the same. It worked!

But is it just a coincidence? Will it also work for 6×9, and for 12×137, and for every other possible multiplication we could do? What I'm asking is, can you think of a reason that will make it obvious that these two things—three groups of seven, and seven groups of three—had to come out the same?

This is not a rhetorical question. I give the class some time to think about it, and I get a variety of answers. The simplest explanation I know is this picture:

Can you see it? You can look at this as three groups (rows) with seven squares each: 7+7+7. Or, you can look at it as seven groups (columns) with three squares each: 3+3+3+3+3+3+3. Either way, you get the same number of squares. That picture convinces me that it will also work for 12×137: I don't have to count it out, and I don't have to take anyone's word for it either. It's...well, obvious.

So I'm not using the word "obvious" the way that most people do. When I say something is "obvious" I do not mean "anyone would figure it out quickly." Sometimes I spend hours and hours trying to make something obvious to myself. But if I succeed, I eventually get to the point where I can "Well, of course it's that way. It couldn't possibly be any other way." That, to me, is what math is all about.

Math is often compared to a foreign language. I've even heard math teachers say "math is just another language." I think this is a very misleading analogy.

Math has a language. Certain words ("root") are used very differently from how they are conventionally used, and other words ("polynomial") are never used outside of math. But these words express the math: they are not the math itself. If we started saying "fizbot" instead of root, the math ("you can't take the square fizbot of a negative number") would be the same.

What math has in common with a foreign language is that you have to memorize rules of how things fit together. "Je vais, tu vas, il va..." "Negative b plus or minus the square root of b-squared minus..." Apply these rules scrupulously or you will go in the wrong direction.

But here's the key difference. If you ask your French teacher "Why does it go Je vais, tu vas, il va?" the only answer you might expect is historical ("Here's how it evolved from the Latin roots"). You're not looking for "why it makes sense" because it isn't really supposed to. It's just how they talk in France.

In math, on the other hand, there is a reason why you can't take the square root of a negative number. There is a reason why 12÷4 is smaller than 12, but 12÷¼ is bigger than 12. And that reason has to go deeper than saying "When we divide by a fraction, we flip-and-multiply." It has to go to the heart of what division means, until you say "Well of course 12÷¼ has to be 48, and nothing but 48 would make sense. It's obvious!" (Hint: what does this problem mean in terms of pizza?)

To put it another way: if all the English speakers in the world decided tomorrow that we would use the word "thurple" to mean "lunch," the world would go on just as before. But if all the mathematicians in the world decided that 3⁷ was the same thing as 7³, they would all, unanimously, be wrong. If they convinced the engineers, bridges would fall down. No one decided that there is no "Commutative Property of Exponents": someone figured it out. You can figure it out too.

So why don't people get that? Why do my students look terrified and cry out "Just tell us the answer!" when I suggest they figure something out?

For the most part, I blame their elementary school teachers. Very few people go into elementary school teaching because they love math. They go because they love children, they love playing games and telling stories, or (in some cases) it was the only professional job they could possibly get and keep (but that's a subject for another essay).

So the teacher learns the rules of math from the book, and teaches them from the book. And if a student asks "Why do we need a common denominator when we add fractions, but not when we multiply them," the teacher has no clue. Rather than risk embarrassment in front of the class, she glowers and says "Because that's how we do it."

It doesn't take long for the students to get the idea: don't try to think, just learn the rules. Or, to put it another way, all of math is glorified long division. We divide, multiply, subtract, bring down, divide, multiply, subtract, bring down, and some day when we're in high school we'll do pages and pages of dividing, multiplying, subtracting, bring downing.

Is it any wonder that they come out thinking math is pointless and boring?

In five years, my current crop of Algebra II students will not remember the laws of logarithms, and my Calculus students will not remember the quotient rule. I don't mind that a bit. If they ever need those things, they can look them up.

What I do hope they remember—and many of them have told me that they do—is that math is not at all what they used to think it was. Math is not a foreign language, or a set of rules that you learn and apply, or glorified long division. Math is the most perfect elaboration of common sense. It has no rules except the rules that we were all born with, built into our brains. And any time anyone tells you "This is the way it is" in math, you have the right—even the obligation—to ask why, and to keep asking why. You're not done when you can say "I can do it now." You're done when you say "Of course. Now that I see it this way...it's obvious."

Note added later: About a year after I posted this essay, Meg Perry sent me a link to a different essay on a similar theme. Lockhart's Lament is much longer than my essay, and I don't agree with 100% of what he says, but I think he's pushing in the exact right direction, and doing so in a much more fun and interesting way than my essay. If you don't want to read his whole essay, at least read the first few pages about the musician's nightmare: that really says it all.

One final thought: I've noticed both your description of "obvious" and my comment rely quite a bit on language implying visions (i.e.: you "look" at the rectangle, "visualize" the rectangle, "observe" 3+7, etc.). The implication is that we need to "see" math in order for it to be obvious. This is another difference between math and a foreign (or domestic) language—in the latter you would ask whether a particular sentence "sounds" right.

I think you're raising a great point. The counter-argument would be, as anyone who has ever taken multivariate Calculus can assure you, that sometimes visualizing things properly is one of the most difficult and confusing parts of math. But, that being said, a huge amount of our brain is dedicated to visual processing. So as a general rule, I think that when we can make things more visual, we give our intuition a huge boost. I sometimes point out to my students that this is the only reason we bother graphing things: it never gives you information you didn't already have, but it allows you to lean on your brain's natural strength (visual pattern recognition) instead of its weakness (raw linear computation).

This might not just be an habitual thing (as it often is, senselessly, in languages): The difference between sound and vision is in that the former deals with intake of sequential information whereas the latter is about simultaneous regard, and understanding is essentially about recognizing commonness of features, which can only be done if the compared elements are regarded simultaneously.

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