Equations are very confusing if you don't know what "X" represents

I'm starting to believe you can't teach anything well without first asking a lot of questions & mapping out people's mental model

I spent some time with a 13 year old who happened to be struggling with math. He was telling me about how he’s been going to this extra tutoring thing that isn’t very fun.

I cheer him up by telling him about some of my favorite mathematical oddities. “Did you know that I can break maths? Watch this: 1 + 1…sometimes equals 1! How? What do you get when you add two rain drops together? Oh that’s a weird example? I can come up with a million of them. What about if you have two lakes, and you add them together (connect them). Now you have one lake! It doesn’t stop there, sometimes 11 + 2 = 1! (if you’re looking at an analog clock)”

That gets him excited and he opens up. He agrees to show me his math homework.

We look at this equation:

\(x^2 - 1 = 0\)

He says “one half!” I say not quite, he says, “one!!” which is correct, but he’s clearly guessing.

I say, what if it was +1 instead of -1 ?

\(x^2 + 1 = 0\)

He starts guessing again, “two? no, four!” There is something wrong here. Something he’s not getting even though he’s seen a bunch of equations and how they were solved.

I try something else, I start writing, ok, let’s make it simpler. “What if we had…” and I’m still writing this on the board, haven’t finished yet:

\(2x - 1 = \)

And he yells an answer: “negative one!!”

That blew my mind.

The boy is not solving the equation. The boy is pattern matching. He’s trying to follow a bunch of disconnected arcane rules that don’t make sense to him.

What is X?

I ask him this, and he thinks hard again, thinking he just got the answer wrong. He’s really staring at “2x - 1 =” and trying to figure the correct answer for X.

I say, no no, sorry, I meant, what IS x? Like what does it represent? You know it’s like a variable, it can be anything, right?

He says yes of course, he knows that. It can be anything. (but he doesn’t get it. He still thinks that means, it can be anything, because it’s different in every question. He still think in this question, there is one correct answer).

Playing with metaphor

Now that I know what’s happening in his mind (or rather, what’s NOT happening) I tackle that directly. His math problem can never be solved by doing more drills. Everyone tries harder and harder until they give up. The child grows up thinking they are “not good at math” or whatever.

Here is what I tell him:

I’m looking for a number. If I multiply this number by 2, I subtract one, I get one. What might that number be?

He thinks for a moment and immediately gets the right answer. No guessing. It’s “1”. I keep doing that for every equation he was struggling with, he gets it right away.

Crucially, he’s not guessing while the question is clearly incomplete. He can actually tell now, when a question is impossible (there is no one answer for X). In fact, there’s a deeper insight here, that he sees when I try a different metaphor:

There is a bank over there. When you give it money, it doubles it, and charges you 1 dollar.

And I stop there. And I ask him, “how much does the bank give you back?”

And he asks me, “well, how much do YOU give the bank?”

I smiled wide. 😊

He now understands “2x - 1”.

Questions for the reader

What is your takeaway from this story? Is this something you see happen in your life? Now that you’ve seen this, does it change anything about how you see teaching, for yourself, for others?

Do you think this applies to “advice giving” in any way? Is it possible that two people are looking at the same piece of advice, and it’s not working for one of them, because they repeatedly say they get it, and they can even answer questions correctly about it…but they are actually missing something fundamental about it?