JJ’s knock-out question
JJ’s question was a bit of a knock-out blow. Not just because it left me stumped for an answer — nor because I didn’t have the self confidence to let a class discussion find the answer — but because… well, to be frank, I hadn’t thought up till then that JJ could ask this level of question.
My mistake. My assumptions. My ignorance. Sorry JJ.
We’d been looking at writing an explicit equation for an arithmetic sequence. We’d done the recursive routine, now we were finding the equation based on the n, the position of the nth term of the sequence. Arithmetic sequences involve repeated addition — or subtraction — of a constant, the common difference.
Over several terms of the sequence I’d tried to demonstrate, in true traditionalist didactic style, that this repeated addition was the same as multiplication.
Out of the blue JJ asked, “does that mean repeated subtraction is division?”
Wow! We’d bashed away at inverses. The inverse of addition is subtraction, the inverse of multiplication is division… I could see where JJ was coming from. But I was thrown, I didn’t even feel confident enough to say that the answer was yes.
But if JJ asked that level of question, he didn’t want a yes, he wanted the explanation. He wanted to see it built up on the board. And he deserved his explanation.
This is what is called a teachable moment. Teachable moments are supposed to be gifts for teachers. They are also tests of teachers.
Well, I failed the test.
But I was determined to try to seize the moment, even if the moment was to go through a time shift… and a couple of weeks later JJ did get his explanation (see the end of this post).
It wasn’t so much about the question or JJ’s mathematical curiosity. It was about JJ’s belief that he could build some meaning for himself in a subject he would happily tell you isn’t his favorite.
I’d first taught JJ in Algebra 1. He wanted to get it. And he’d try, he really would, but getting three or four consecutive steps correct and then getting the right answer… discouragement was always near at hand.
It was the first year I’d started to think about and experiment with throwing grades — damned confidence and learning killers — out of the window. JJ so desperately didn’t want to get the F, but he just didn’t know how not to get the F he expected. It came up again and again… his mom. It wasn’t the fear of parental anger, but he just didn’t want to disappoint his mom. JJ felt unable to deliver.
That year, I announced: if you turn up, try, don’t smash up the classroom… there’d be no Fs. I can’t remember JJ’s grade that year, but it wasn’t an F.
First lesson this year, JJ was quick to re-check the rules hadn’t changed. Smiles, thumbs up… telling the class it was going to be cool.
Since then JJ has never had the facial expression of early in his first year with me… that look of overwhelming worry, the furrowed brow, has gone. Sometimes, too often, JJ the football star talks too much about football or basketball or something to do with balls. But some work comes in, and increasingly frequently the attempts to make contributions in class discussion.
And then came the question. JJ was thinking mathematically. And he felt sufficiently confident to form the thought and say it in front of a class.
On his end-of-quarter progress report, I can remember JJ got a B. But more important were the sentences on his report card describing that he’d demonstrated mathematical thinking skills. He came up and shook my hand and thanked me for the words… his mom had been delighted he said with a huge grin.
❏ Start at 0… repeatedly add 5
0 + 5 + 5 + 5 + 5 → 20
0 → +5 → (x 4) → 20
0 ← −5 − 5 − 5 − 5 ← 20
0 ← − 5 ← 5 ← (÷ 4) ← 20
Of course, repeated subtractions are multiplications. But division is also multiplication… multiplication by the reciprocal.
My colleague, Roscoe, said that multiplication counts the number of additions from zero… and division counts the subtractions back to zero.
Any other explanations out there?