This is a post about owning your mathematical experience, problem-solving, and flexibility. It also involves our goats—yes, we have goats—and a story of what happened when we took them hiking this summer.

What you need to know about our goats is that they hate to get wet (notice their bleats when they encounter the river in the video above). Still, if we, their herd, are on the other side of a river, they’ll figure out how to get to us.

Sid, the white goat, is a problem solver, and he takes full control of the problem at hand. When he crosses a river, he’ll stay still for a long time, considering his options, sometimes walking up and down the bank to scout for other ways across. Myshkin—the black goat—on the other hand, complains (via bleats) about the fact that he has to cross a river at all, then panics when he realizes he’s on one side and everyone else is on the other.

Here’s the thing, though: Myshkin is far more athletic than Sid; he can jump farther, higher, run faster, pull harder, etc. And yet, despite his athletic advantage, this river crossing goes much, much better for Sid.

Allow me to make an analogy here. We’ve got two approaches to problem solving here, one of which involves an owned, thoughtful, flexible approach, and one that involves avoidance and panic. This is one reason owning your mathematical experience is crucial: it makes math, and hence life, easier. Sometimes, the difference between easy and impossible is as simple as whether you feel you’re allowed to be a little flexible with a problem, and as a rule, we’re much more likely to mess with things if we feel like they’re ours.

Here’s a contrived example, which I normally wouldn’t mention, except kids see stuff like this all the time: without a calculator, determine the product 499 x 501.

Kid 1, who doesn’t own their mathematical experience, multiplies out this product using one of the long multiplication algorithms, and likely makes a mistake. When Conrad Wolfram talks about how we’re training kids to be third rate calculators, this is what he’s talking about (and you’d never trust the kid’s answer to this problem over the calculator’s).

Kid 2, who feels like they’re in charge, says, “Too bad they didn’t ask me 500 x 500, because that would be much easier.” Just in noticing that 499 and 501 are close to 500, this kid is already opening the door to something momentous: connecting hard new problems to easier ones you’ve already solved is at the heart of mathematics. This kid, who knows that 5 x 5 = 25 and therefore 500 x 500 = 250,000, now figures that the answer to the other problem must be close to 250,000. Maybe it’s even the same!

If Kid 2 is really on it, maybe they’ll check this out by doing some simple examples to see what happens when you make the numbers in a product one larger and smaller. Like:

1 x 3 vs. 2 x 2

2 x 4 vs. 3 x 3

3 x 5 vs. 4 x 4

and so on. And what this kid notices is that, amazingly, all the numbers on the left column are 1 less than the numbers on the right column.

1 x 3 = 3, which is one less than 2 x 2 = 4

2 x 4 = 8, which is one less than 3 x 3 = 9

3 x 5 = 15, which is one less than 4 x 4 = 16

Now Kid 2 is really excited. They’re raising their hand wildly, and they can’t wait to tell you that 499 x 501 must be one less than 500 x 500 = 250,000, that is, 499 x 501 = 249,999. Not only that, they want another one. They can tell you, they claim, what 499,999 x 500,001 is. They can tell you what 29 x 31 is. They’re all over it.

(They haven’t actually proven anything, just gone on the basis of a pattern, but let’s acknowledge that they’ve discovered something pretty cool here. Not only that, there are plenty of places to go from here.)

Here’s the thing about these two kids, though: Kid 1 has to contend with a much harder problem. Kid 2, you’ll notice, never does the tough long multiplication. There’s nothing in this whole though process that requires more than one-digit multiplication, plus adding on some zeroes at the end. What this means is that kids who don’t make connections to easier problems, who don’t feel enough in control of what they’re doing to take a step back and look at other ways it could have been or it’s connected to, those kids are actually doing math that is technically harder, as well as less illuminating and rewarding. This arithmetic isn’t just more interesting to Kid 2; the arithmetic is actually far easier.

In our analogy, Myshkin (the black goat) is Kid 1, better at straight arithmetic, but not thinking about how to set up the problem to his advantage; Sid is Kid 2, weaker when it comes to raw power, but in control and thoughtful about the situation. Kid 2, like Sid, finds the stepping stones that turns an impossible crossing into an exhilarating one. And Sid is the one who makes it dry across the river.

(PS You thought this blog post would be about this problem, didn’t you?)

EXCELLENT – I have kid number 1 and we have goats, I am excited to show her this and hope that she grasps the concept. Thank you for sharing! I love that you hike with your goats, I think ours would hike with us if we took them. They follow my kid around everywhere.