# Teaching Yourself and Others

In his wonderful new book “Proof and the Art of Mathematics”, Joel David Hamkins asks a question about irrational numbers that stopped me in my tracks:

Prove that the irrational real numbers are exactly those real numbers that are a different distance from every rational number.

I reacted to this problem in three stages:

1. OK, sure.
2. Wait what?
3. Oh cool! Why?

It took me a moment to understand what he was asking, then a moment longer to think about how to approach a solution. In the end, what helped me make sense of it was a problem that I’ve used to teach fractions to my 3rd and 4th Graders. These sort of connections between the math of young students and of sophisticated adults is mathematically exciting. It gives me the same sort of “oh man that’s beautiful” buzz that mathematicians sometimes use to describe their realizations.

The question comes from Marilyn Burns’ fraction books: Put two fractions on the number line, maybe 1/4 and 1/2. What number is exactly halfway between them?

Lots of students would say “1/3.” This isn’t how fractions work, though. (The jumps from 1/4 to 1/3 to 1/2 aren’t constant, an early example of non-linear growth.) This problem often catches my students in stages just as the irrational number question caught me: Oh, it’s obvious! Wait what now?

The question mark is half of the distance from 1/4 to 1/2. That makes it 1/8 away from 1/4, and 1/8 + 1/4 = 3/8.

It’s possible to generalize this result. I have good memories of pre-teen students at math camp, huddled around a chalkboard and trying to express this result algebraically.

The actual solution to this isn’t particularly important at the moment. The point is that you can always find the distance between two fractions, and that distance can always be expressed as a fraction. And half of that distance? Again, a fraction. Pick the lower place on the number line, jump ahead by that half-distance and what do you get? Fraction, fraction, fraction. It’s fractions all down the line. As long as you start with two fractions, you get one in the middle.

Fractions are also known as “rational numbers.”[1] So let’s take another step closer to the original problem, which was about the distance between an irrational number and any rational numbers.

Start with a number on the number line. If it’s rational, then you can write it as a fraction. Then reach out in one direction to another fractional/rational number. What’s waiting for you in the other direction?

Again, you can compute the answer using the distance between 2/5 and 1/2, What’s significant is that this other number will also be … wait for it … a fraction. No way out, they’re everywhere. You’ll be able to express the distance between 2/5 and 1/2 as a fraction, you subtract that distance from 2/5 and … you get the picture. Fractions, across the sky. But what does it mean?

This time, start with an irrational number – something like the square root of 2 – and stick it in the middle slot. Then, stretch out in one direction to reach a rational number. What sort of number is waiting for us an equal distance in the opposite direction?

Suppose for a moment that, as in the 4th Grade question, the other number turns out to be a rational number. Then at the left and the right are rationals. Remember the pre-teen version of the question: if the left and right numbers are expressable as fractions, there is a formula for finding a rational number exactly halfway between them. And that would mean the square root of 2 is rational. Which, no. So! That other number is irrational.

Another, more precise way of saying the above: Let m be an irrational number and suppose it is the same distance to two rational numbers. Then m is exactly halfway between two rational numbers, and based on Marilyn Burns that too is rational…hey no way, man, it’s irrational! So irrational numbers are a different distance away from every rational number, which means if a rational number is on an irrational’s right side, there isn’t a rational number an equal distance to the left. And vice versa.

(Strictly speaking this is only half of what Hamkins asked us to prove. The other half is showing that every number that is a different distance to every rational number is irrational.)

A lot of mathematics seems obvious in retrospect, and this was one of those times for me. That’s also the case for the connections to what I teach my elementary and middle school students. Duh, fractions are fractions and this was a problem about fractions. (Fractions, EVERYWHERE. Let that haunt you.)

What I find wonderful is that when you’re teaching you never know what seeds you’re planting. That’s supposed to be a truism about teaching kids, but it also seems true when applied to yourself. Every new idea you share with someone is an idea that might be useful to you, the teacher, in some new and unexpected context. In a very real sense, teaching is also sometimes teaching yourself.

[1] There is probably a nit to pick about rational numbers being more precise a term than fractions in this context, but I’m going to juuuuuust slip right away from that conversation if you don’t mind.

# People Actually Really Like Mathematics

Sure, people tell you that they don’t like math. They’ll say they’re bad at it, that they hate it. That they’ll do anything to avoid it, that the very thought of it gives them the heebie-jeebies.

Don’t believe them for a second, though, because actually? Everybody likes math. The proof is that even people who profess to hate mathematics do a lot of it by choice, for fun. The issue is what people see as “doing mathematics,” and how disconnected what they enjoy is from what happens at school.

There may be other examples of popular mathematics, but mostly I’m thinking of games and puzzles.

Mathematics is the art of necessity, but not everything is necessarily true. Some things just happen to be true. My name is “Michael” though my parents could have named me “Marvin” or “Melrose.” Other things have to be true; they’re forced to be the way they are. These are the inescapable facts of existence, the ones we can’t wriggle out of. So it’s not some sort of coincidence that ever since I’ve had children, I’ve been a parent. And you may not be shocked to learn that ever since I’ve been a resident of New York City, I’ve lived here. Much as I wish at times, there’s no escaping your home as long as you live there.

One of my favorite games to play with students is Mastermind. It’s a game of code-breaking and logic. It’s a game that puts you in direct contact with necessity and possibility, and I use it in class to help students grasp the difference between those two kinds of reasoning.

It’s not a “math” game. It’s a board game, the kind of thing my friends’ parents would have in their basements when I was a kid. But it’s all about logical necessity. That’s not science. It’s not a sport. If anything deserves to be called math, these sorts of games do. What else would they be?

And once we’re on the lookout for logical necessity, it’s all around us. It’s printed in the newspaper (KenKen, Sodoku), sold at the toy store (Guess Who?), and built into our computers (Minesweeper). The reason it’s everywhere? People love this stuff. They love how it makes them feel and think. It’s mathematics; they like it.

The point of course is that very few people recognize this in their school mathematics, which is dominated by a different experience: that of carefully following steps.

Now is that so bad? Absolutely not, people also love carefully following steps. They love assembling LEGO cars and (even if they won’t admit it) IKEA furniture. There’s joy in successfully executing a tricky procedure…

…but it’s not quite the same as thinking logically, is it? And why can’t people experience both in school?

I’m getting a bit too close to preaching to the choir for my own personal comfort. What I’m trying to emphasize is that there’s no reason to define “mathematics” as an activity that is identical with the math that people study in school. That’s entirely artificial. True, all definitions here are pretty artificial. Still what’s distinctive about mathematics to me is its obsession with what is logically guaranteed to be true. And that’s an obsession shared by billions of game players and puzzle solvers around the world.

Mathematicians and educators can sometimes be heard asking, how do we get more people interested in mathematics? And I think the answer is, mostly they already are. The question is whether we and our institutions are interested in their mathematics. And mostly, we aren’t.

# The Kind of Help My Son Wants

For the last few days my morning has begun at 6:00 on the dot. That’s when my son swings open the door to our bedroom and asks for help with a math question. Fielding questions from a small child who wakes you up and demands help an inch away from your face seems like a good way to sharpen your classroom skills. It’s like some bizarre emergency preparedness drill. This is a million dollar teacher-training idea, but for you I give it away for free.

On the other hand, all that time in the classroom is decent prep for dealing with some of these parental duties. And it gives those of us with young children an N = 1 perspective on a perennial teaching question: what are kids looking for when they ask for help? And I think the answer is that in that moment what they want most of all is to understand. They aren’t primarily interested in having their own original thinking validated. And the implication is that telling kids they’re wrong is mostly trouble when the kids aren’t able to quickly grasp what was incorrect.

This is a bit of a story, but stick with me.

When my son barged into our room this morning, he told us he had two math questions he needed help with, 16 x 9 and 16 x 12. I told him that this was great, but to be quiet so that he didn’t wake up the baby.

Once I picked up the baby (baby woke up) I told my son than I thought 16 x 10 would be a good place to start, since it was so close to 16 x 9. I asked, do you know 16 x 10? He told me it was 160.

Fantastic, now we’re cooking. This is a conversation I’ve had approximately ten billion times in my life as a 3rd and 4th Grade math teacher. I took the next step. Would 16 x 9 be bigger or smaller than that, I asked? He thought and said, smaller. Yes!

I went in for the kill. (Uhh so to speak.) 16 x 9 is smaller than 16 x 10. I asked, how much smaller? This is the hard part, the part that puts a lot of pressure on one’s conceptual understanding of multiplication. If 16 x 10 is ten groups of 16, then we can take away a 16 from 160. If 16 x 10 is sixteen groups of 10, then we have to think about what happens when all those 10s turn to 9s. Without practice thinking this way, kids tend to shrug and guess.

That’s what my son did. He said that we should take ten away from 160. He had zero confidence at all in this. He asked if he was right, and I said it wasn’t.

Now what? I knew that the thing to do would be to draw a picture. Unfortunately, I was horizontal and undressed. I started talking about groups of sixteens … I was shut down by my son who, after all, is six years old and who I recently overheard telling his kindergarten teacher that he is inspired by “vehicles.”

The boy started complaining that he already knew the answer anyway. Cool, I said. So you know the answer is 144. But then he got really annoyed at me, telling me I had spoiled the problem for him. And then I tried to walk this back. I realized that his own fragile competence was on the line. Also he gets loud when he’s worked up and the clock now read 6:01.

Anyway, when he came in he had also said he wanted help with 16 x 12, and I casually changed the subject to that problem. I dragged my sorry butt out of bed and to the table. I poured him a bowl of Cheerios and grabbed a piece of paper and a marker. Then I drew a rectangle in green. I told my son I was going to draw a picture of 16 x 12, but start with 16 x 10. Then I asked him which was bigger, 16 x 12 or 16 x 10, and then how much I would have to add to the green rectangle to make up the difference. This all went much better.

Why did our talk about 16 x 12 go better than 16 x 9? Clearly things started going south when I told him that subtracting 10 wasn’t right. But I also corrected him on things while we were talking about 16 x 12. He told me the wrong dimensions of that green rectangle at first, for instance.

There were two differences when we talked about 16 x 12. First, he was loading up on breakfast cereal and riding a blood sugar high. Second, he understood what I was talking about when I corrected him. That’s thanks to the image. A wonderful thing about people is we turn correction into self-correction. You can’t help but take some credit for recognizing your own mistake, even when someone else points it out.

But when I corrected my son the first time he had no idea why he was wrong, and because of that he freaked out a bit. It’s OK to tell my kid that he’s wrong, but if he doesn’t understand why it’s not going to go well. And it’s the same with students. Don’t focus on whether you’re honoring their original thinking or not — they wouldn’t ask for help if they didn’t want access to your expertise. Focus instead on whether you’re connecting what they don’t yet understand to what they already do.

Is this all there is to it? No. The same boy I’ve written about above just asked for help building something with his toy magnets. He was trying to build a porch, but it kept collapsing. I had an idea and showed it to him. I have to say, my solution was pretty clever. I’m great at toys. The boy took a look at it and decided it was “stupid” and “bad” and that I was “dumb.” Only some of which is true! (It was a great porch.)

It’s not all about understanding, but a lot of it is.