Friday, March 7 (70 minutes, sprinting)

Yesterday one of my students asked me, "How do I know if I'll ever be good at math?" At least she was asking a question, rather than simply stating she believed herself to be a lost cause.
My first reaction was that I don't consider myself to be "good at math", and so I'm hardly the person to ask. But my view is skewed by being at Caltech and by constantly being bombarded with math I don't understand. By any normal person's measure, yes I'm good at math.
So I had to think for a minute before answering. I wanted to tell her she was already getting pretty good at math, and that if she continued studying she would also continue improving. That innate talent is only a small part of the picture, and that effortful practice is what truly needed.

But I wasn't sure it was true. I want to be encouraging, but I also want to be honest. She's pretty dismal at math. She's slow to catch on to new concepts, and quickly forgets most of what she learns, even the more fundamental things like "this is why we use inverse functions" or "this is what an equation means". Occasionally, she tries to take her knowledge and extend it to solve a new type of problem she hasn't seen before, but I've never seen her do that and actually get the right answer. She simply doesn't have the ability to visualize things the way many people can. She is unable to keep herself mentally organized enough to keep track of more than two or three little parts that have to fit together. Without those skills, her chances of becoming "good at math" are dim.

But being "good at math" is not the bottom line. In her adult life, it will not matter much how good she is at trigonometry. So teaching her trigonometry is not actually my goal. Instead, it's to see if I can get her personally involved and excited. Eager to meet with me and further, eager to work things out on her own.

So instead of answering the question she asked, I told her I wanted to show her something. I explained that you can always tell whether or not a number is divisible by three, without doing the division. You add up the digits, and if the sum is divisible by three, then so is the number. We worked through a couple of examples until she got it.

You may be surprised she hadn't learned this school-kid trick before, but she is really so oblivious that I constantly need to remind myself that I'm working with a sixteen-year-old human being when she claims never to have heard of: solar panels, Mozart, the word "status", the big bang, mirages, and elastic, among other things.
Anyway, she hadn't, and my thought was that most people, when they learn this fact, will give one of three responses:

1. Who cares?
2. That's interesting.
3. Why does that work?

People who will be good at math will give response three. That was my hope, because sometimes she does ask conceptual questions that reveal she's innately curious and questioning the logic of what's she's been told. My favorite was, "Do atoms have shadows?"

In this case, I got response two. But even though my little test didn't give the result I was hoping for, as soon as I asked it I realized I had unwittingly given it to myself as well. Because I didn't know the answer off-hand. And I did, in fact, wonder why it was true. So here we go (it's short)

Take, for example, the number 76,223. This can be written as a sum:
7*10^4 +
6*10^3 +
2*10^2 +
2*10^1 +
3*10^0

Now divide the number by three, which is the same as dividing each part of the sum by three:
7*(10⁴/3) +
6*(10³/3) +
2*(10²/3) +
2*(10¹/3) +
3*(10⁰/3)

Which equals

7*(3333 + 1/3) +
6*(333 + 1/3) +
2*(33 + 1/3) +
2*(3 + 1/3) +
3*(1/3)

We aren't interested in the answer, which just want to know whether 76,223 is a multiple of three. In other words, whether the division problem above gives a whole number.
The parts like 6*333, which are two whole numbers multiplied, don't matter then. All we need in order for that sum to come out a whole number is that the multiples of 1/3 collectively leave us with a whole number. Explicitly

7/3 + 6/3 + 2/3 + 2/3 + 3/3 = whole number
(7+6+2+2+3)/3 = whole number

Which is what I set out to show. Ta-Da!

(This goes deeper. More at a future date.)

As for running, I came to the track today intending to work out, but my right knee still felt a little sore or achy, so I scrapped it in favor of 70 minutes on the infield.

Halfway through, a group of sprinters was about to start a 300, so I jumped in (which I do so frequently when I'm running on the infield and I happen to jog by as they're lining up, that it's almost instinctual). Julie built it up, admonishing them not to let the old man with gray hair beat them (I don't have gray hear, do I?), and so of course I took off pretty fast and ran a 41 (beating them, although they weren't at the fast part of the workout yet.)

Fifteen minutes later they were finally ready to go again (I think I may have missed one in between), so I lined up again. This time people seemed to be watching, since Yezdan was getting near the end of his workout and Julie was telling him to quit sandbagging as he had on the first half and run some quick repeats. So the spectators were making bets on who would win, shortest shorts or tightest shorts?
I made it interesting, but Yez was too fast in the last 100, and we both ran a 40. I was planning on calling it quits there, but Julie wanted me to go again to pull Yez through his last one. About a mile and a half of infield running later we were finally lined up for the last one (Julie told Yezdan he had to beat me, or would have to run more repeats). I took off hard and started really kicking around the turn. We were even coming into the last hundred, but he pulled away again. I think I squeaked just under 40 seconds.

This was probably stupid. Considering I had already decided not to work out that day because something was hurting a bit, top-end sprinting was not what I needed to work on. But it was fun. And the only ill effect I notice now is that I tore some skin off the bottoms of my toes (I was barefoot of course).

Also, 39 seconds for 300m is not too slow...