# 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.

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. Chris (@absvalteaching) says:

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.

3. Christopher says:

Re: “I’m a he”…Ha! Duly noted. I shall correct the record.

4. Wendi says:

Great advice, except my son Is a perfectionist and demands the right answer ha!ha!

5. 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.

• That was over at Michael Pershan’s Math Mistakes. [http://mathmistakes.org/?p=1526]

• Thanks for that catch. Brain not so good today

6. 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…