Along the way, a curious thing happened.  My niece saw a 6×2 rectangle, horizontally oriented, and wrote 6 x 2 next to it, and saw the same size rectangle, vertically oriented, and wrote 6 x 2, then crossed it out to write 2 x 6 (my sister-in-law thought to “correct” her, but I asked to leave it alone).

Last week I went to my niece’s school, where she goes as a 3rd grader, and where I went as a 3rd grader, in a city 70 miles from New York. I visited math classes at several grade levels. And here is what I found in a second grade class:

I also found a woman who was friends with my nursery school teacher’s daughter. And I learned that my teacher is still alive and kicking. I’m going to dig out some old photos and pay her a visit. And thank her.

May 25, 2014 pm31 5:51 pm 5:51 pm

Hmm, what do you think of this? (It seems silly to me to insist that x groups of y be written as a different multiplication problem from y groups of x.)

2. May 26, 2014 am31 8:25 am 8:25 am

I don’t think there is any insisting going on. They are taught G groups of O objects.

I actually watched the class being taught, as the stuff was being added to the chart paper. This was clearly what they’d been taught all along. She was close to bringing them to $7 \times 2 = 2 \times 7$, but they were getting to the point that the quantities were equal, not that the things were the same.

That’s probably why my niece thought $6 \times 2$ and $2 \times 6$ were different things, but knew instantly that they were equal.

It would take insisting to make one of us make that distinction. We think of multiplication as commutative. We might even think of $6 \times 2$ and $2 \times 6$ as not only equal, but “the same” whatever that means. But years after 2nd grade I struggled with non-commutative multiplication (of matrices), and maybe that’s a seed that could have been planted earlier.

May 26, 2014 am31 11:17 am 11:17 am

If a kid already knew about multiplication, and wrote it “backwards”, would they get it wrong?

I see what you are saying – if this is done well, it can get kids seeing something deep. But if it is done as part of the curriculum, without the teachers being very well-trained, then many of them will make it a rule, and kids will learn that math has arbitrary rules. Not the lesson we hope for.

4. May 26, 2014 am31 11:21 am 11:21 am

They were 2nd graders. If they knew, they were playing along. But I don’t think they knew. The teacher was strong.

She circled the rectangular arrays, and said that this was now a whole, and that they would divide it in parts. And the kids did. And she wrote some of the divisions underneath, they wrote others.

But if the teacher had to insist… well, then we are describing a teacher who would have other problems as well, no? I think that is a different discussion.