# Math and the Obvious

My #1 goal, when I teach a math class, is to convey a certain way of thinking about math. It’s quite different from what my students have done before, and many of them find it difficult and frustrating. But in the end, many of them start to see math the way I do–and I count that as a major success even if they eventually forget all the details, facts, and techniques we learned along the way.

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÷(1/4) 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÷(1/4) 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,” they would all be correct–by definition they would be correct, *simply because they agreed on it*–and the world would go on just as before. But if all the mathematicians in the world decided that 3^{7} was the same thing as 7^{3}, 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.

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 Common Core 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.”

Want to read more? Here are a couple of links from excellent educators. At the level of details, their messages are quite different from mine (and from each other’s). But in terms of the big pictures, I think we all have the same core message: it’s more important to get students actively engaged in formulating the math, and applying their own thinking, than working through given processes to find a right answer.

- Lockhart’s Lament (https://www.maa.org/external_archive/devlin/LockhartsLament.pdf) is much longer than my essay, and a lot more fun and interesting. 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.
- Dan Meyer’s TEDx talk (http://www.youtube.com/watch?v=BlvKWEvKSi8) is something I have not only watched, but I have also shown it to my students.

Kenny Felder

When we talk about building a "number sense" in students, I think this is exactly what we are referring to, the common sense of numbers–making certain things obvious by teaching our children to put numbers in a context that makes sense to them–just like Siri does when you ask her what zero divided by zero is. Nice article.

Fantastic article, Kenny Felder! It is so upsetting that mathematics isn't taught intuitively but rather it is taught through rote learning. I love math because I like puzzles. Whenever I'm given a new math concept or problem, I try to look at it as a puzzle. Where do the numbers and/or symbols go and why do they go there? That's one way math could be taught.

We (Air Force Academy Math Department) would always hear from the engineering departments when upper level students were no longer adept at computing derivatives and applying log properties.There is a truth to what you are saying, but it needs to be expanded more broadly to quantitative problem solving. My goal was to impart quantitative problem solving confidence and prowess. If they don't have those (or lose those later), they can't look it up like simple mechanical steps in the process.

Excellent and enjoyable read. Thank you.

Thanks Kenny! Very nice!

Very well stated! I was blessed with the insight to "figure" this out at a very young age, and I know many adults for whom this is still not "obvious". I believe one's "faith" and ability to use math as the powerful tool that it is has very broad implications throughout the entire human intellect.

"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."That's a really good point. I used to tutor math, and I'd see people almost get embarrassed when they'd finally see through a problem to a simple solution. It was as if they felt that since the solution was simple, they were silly for struggling for so long, they should have seen it right away. Of course, in reality its as you say: The problem is complex until its really solved/understood, at which point it always becomes simple in some way.

Good article, but the jab at elementary teachers was very uncalled for…I'd say it (lack of interest in learning the why behind things) has more to do with lack of parent involvement in a child's early schooling than it does the teachers. Your attitude towards elementary teachers is rude and insulting.

I think every person has it very own way to understand the world that sourrounds him/her. I studied law some years ago, not my desire but by default, due to some circumstances. Along with highschool I took English, then allong with college I took French, German and I could not finish Iitalian due to family and personal circumstances. My native tongue is Spanish. A few years ago, having some spare time (or Sabbatical), I decided to learn Math as a language, I defined that the numbers is the substance and the other symbols as plus or minus signs are the procedure ordered or requested to be performed, I also defined that de the actual problem is telling me what it wants or what is need it to be done, by means of observation, analysis and practice I could understand how to solve from simple to complex equations. so after all this learning, I got my certificate on Calculus Derivatives and soon I hope given time I should be taking Integration. I love being able of seeing math simbols and understand them in the way I do, it is a great skill or knowledge. And above all it is fun because it is challenging.

Is it any wonder that they come out thinking math is pointless and boring?Well math is boring if one is not interested in math. I do wonder though if that is the teacher's fault for not making it exciting for all students, or if it is rather an innate feature of a student him/herself. Or does it become boring because the student is not excelling and receiving positive reinforcement by encouragement and marks. Large classrooms do not allow as much time spent per child as necessary for detection of problem areas that a student can have, and they fall behind and this becomes only noticeable at quiz time or examination, when it is surely sometimes too late. Every student does not learn at the same rate, nor ready to learn the same material at the same age. Yet, the standard is to group all students by age and give to them a curriculum to infuse into their brains. One can definitely see the deviation from the "norm" in physical appearance of young people, so why does it not seem also obvious that the mental maturity and intellect also has a variance amongst the young people. The simplest explanation I know is this picture:A part I especially like. Knowing that, a student can feel "one up" on the subject. Other tiny tidbits placed in strategic locations surely should, I think, feel that the student is being let in on little secrets ( and what student doesn't like secrets, which they will remember better than the main textual explanations ) of math.