So … dh commented to me that dd told him that she did not know the 6 times table. Worried, he told me that while it is good that she explores freely and deeply etc , *we have to ensure that she doesn’t miss basic things. *

I told him, you know the funny thing is, she was telling me something about 6 x 8, which she found difficult because she knew neither the 6 table nor the 8 table. She said, "Well, 6 x 8 is so hard that I just memorized it."

This was somewhat surprising to me because I thought, *what other way is there? Don’t we memorize all of them? *

But I realized that to her mind, the way she "knew" certain multiplications was different from the way she knew others – for example, if we are doing 6 x 2, we understand that it is 12, we don’t memorize that it is 12. Times two is easy to understand that way. So she has a different approach to multiplication, that varies according to the numbers being multiplied.

Of course dh found all this to be yet another one of my crazy theories.

Later in the evening, he called dd and said, "Stand here, I want to ask you a few questions." Ever ready for a few questions she stood before him. I watched from a distance. His questions are in quotes, her answers are not.

"What is 6 x 8?"

48

"How do you know?"

6 x8 I just memorized because it was so hard.

"What is 6 x 10?"

60

"How do you know?"

When you multiply anything by 10 you just add a zero and move the decimal point one space to the right, if there is a decimal point.

"What is 6 x 7?"

42

"How do you know?"

I know the 7s table.

"What is 6 x 5?"

30

"How do you know?"

To multiply anything by 5 you just cut the number in half and add a zero. Basically you divide by 2 and multiply by 10 because that is the same as multiplying by 5.

"What is 6 x 9?"

54

"How do you know?"

I know the 9s.

"What is 6 x 6?"

36

"How do you know?"

I know all the square numbers.

"What is 6 x 11?"

66

"How do you know?"

I know the 11s!

"What is 6 x 12?"

I don’t know.

(pause)

Well I can just add 66 +6 since 6 x 11 is 66. So it is 72.

The conversation continued with a few more but it was interesting to observe that she had a different answer every time for "How do you know?" Memorization was an exception, not the rule. Even when she said things like "I know the 7s" it was not, by her understanding, a matter of memorizing the answer to a given questions such s 7 x 8, but rather that she had learned to count by 7s – it was part of a skit that she and her friends did some time ago. With 9s she used to use her fingers, but no longer needs to. With 11s it was something else – perhaps visualizing the ones and tens places. Not sure exactly but I realized that she uses multiple processes to multiply.