Multiplication and rectangles

I want to suggest a lovely post by somebody else.

It is written by a math teacher who converses with his niece (who is 7 years old) about rectangles and multiplication. As an example, the rectangle below shows that 6×3 is 18. Or is it that 3×6 is 18? That becomes the focus of part of the conversation.

This rectangle shows that 6x3 is 18. Or is it 3x6?

The girls’ parents look on as the discussion unfolds.

At one point, the math teacher stops the mother who is trying to intervene to help the child see that 4×3 is the same as 3×4. And this leads to the lovely sentence in the blog post:

I understand that it is not obvious to non-teachers that not every encounter with mathematics needs to reach “fruition.”

What he means by this is that children can learn from thinking about math, even if they don’t end up with the right answer, and even if they do not experience the full story (here, that multiplication is commutative, which means AxB=BxA for all possible numbers).

Another fabulous math teacher, Fawn Nguyen, told me, “I dare say that it’s not obvious to teachers also.”

Finally, non-math teacher parents may be interested to learn that—consistent with Fawn’s observation—a regular piece of feedback I get from math teachers on my writing here is how impressed they are by my ability to not worry about Tabitha and Griffin getting right answers.

9 thoughts on “Multiplication and rectangles

  1. We learn more from being incorrect than being correct, right? I’ve heard it triggers more synapses. As long as we’re reflecting…

  2. The child wasn’t incorrect (and neither was her mother). The girl was carrying around a mental model of multiplication, probably an array model, but maybe an “m groups of size n” model, and differentiated between m (height) and n (width).

    She has no problem with m x n = n x m, but her mental model makes them two distinct objects, with equal value. At some point she’ll replace that model with something more abstract (and commutative), but I saw no reason to rush that day. I’d rather it grow in her than be shown to her.

    btw, I’m a he.

    Glad you liked the story.

  3. This reminds me of a post I saw recently (wish I remembered where) where a young student saw isosceles triangles on a long base as being triangles, but rejected the image of a right triangle as not being a triangle at all. Perhaps the orientation of the mental rectangle array makes a 3 X 4 rectangle seem to be a different beast than a 4 X 3 rectangle.

  4. My son and I recently built a 3d multiplication model, and he made his own multiplication table. At no point did he use the information that 6 times 3 is the same as 3 times six. When he looked through his multiplication table later, he said that it would have been a lot easier if he had noticed that they were the same…

  5. I would like to further stress this concept of it not needing to come to fruition, especially with math, but with all learning. A stew always tastes better a few days old, same goes for ideas. I first was introduced to this concept when reading a book by John Holt, which one evades me now but he was speaking about learning at your own pace and he gave an example of a boy who spent an entire year laying out blocks every time he multiplied. “Wow”, I thought, “That would drive me nuts” but later it occurred to me that I “know” a lot of math but “get” almost none of it. As my child’s primary educator I take his “getting it” very seriously. This is why I sat on the bus with our Life of Fred book and listened to him “count by 5’s” to one hundred several times. It sounded like this ” 1, 2, 3, 4″ quietly “5” more pronounced and so on. I really wanted to point out the pattern. The second round to 100 I’m pretty sure everyone on the bus wanted to point out the pattern, he’s sort of a loud talker. They were thankfully all to reserved and I was biting my tongue. It took a couple of weeks and then one day he says “Hey mom 5, 10, 15, 20…. it just jumps in a pattern”, he said it as if he had made a huge revelation, and he had. We just went through this again with place value and multiple digit numbers only this time I’m convinced his revelations may be starting to get beyond mine. He had finished the first four texts with me reading the questions and recording his answers. Suddenly though it became work and I got the impression he was looking for the “right” answers instead of going on an adventure life before. So I quit progressing and had him work through the books on his own, writing his own answers. He’s been at it for a season but has been talking about bigger numbers in relation to food, time and his own imaginative play. Sometimes it will take a few minutes, sometimes months, possibly even years. There is magic in truly understanding something that you risk missing when you have someone memorize something. It is difficult but relatively easy to memorize something, to learn to repeat it and even when to apply the things learned by rote, but what sort of foundation is this to true knowledge? When something starts to make true sense it lays a ground work for further learning and comprehension that cannot be forgotten or shaken. Give your children space and time and slow down and let them talk. The best part is that when you do this they draw you into the magic. Who knew math could be fun?

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