9 x 7 = 63

I was teaching Billy how to add and multiply today. We use tokens as a manipulative, so the kids can show me what they are thinking when they do arithmetic.

I asked Billy to share with me how he multiplies. He then explained that he does a lot of flash cards at home, and that he knows how to use his fingers to multiply.

When talking to kids, my premise is talk less-listen more, so I asked him to explain. He told me about how he remembers what 9 times 7 is. I later Googled this trick and here's what he basically said:

This may be an easier way to do 9's with finger math. Choose the number you will multiply by nine. Count to that number beginning with the pinky finger of the left hand with palms facing down. Once you get to that number, fold that finger down. The numbers to the left of the folded finger are tens. The numbers to the right of the folded finger are ones. Example: 9 x 7 = 63 Count to seven starting with the left pinkie finger. That should put you to the pointer finger of the right hand. Fold that pointer finger under. To the left you have six digits or 60. To the right you have 3 fingers or 3. 63!

When I asked Billy what 9 times 7 was he responded with 63! I then asked him how he knew that. He felt content with just holding his fingers up and nodding at them - as if to say, "I just showed you, silly". So I asked him to prove the answer was 63. He thought about it and said:

Well, 9 times 5 is 45, and that's two groups of 9 short of 7, and I know that 9 plus 9 is 18, so 18 plus 45 is 63.

Someone might hear this and say: "Look, the trick worked. He understood what he was doing." My response: it is far more likely that Billy has developed this number sense not because of the trick but in spite of it. Also, if teachers explicitly say or just implicitly hint that the most important ability in math is quickly knowing the right answer, then kids will sacrifice thinking for precision at the cost of understanding. In other words, Billy was able to reason why 9 times 7 was 63, but I had to invite him to need to do it.

After experiencing all this, I noticed that an anonymous parent left the following comment on my post about Alberta's new math curriculum:

As a parent of a child in Grade 4, I have serious concerns about this "new" math. They are not learning basic foundational skills, such as multiplication. Without consistent practise of these skills, how can a child go on to apply their knowledge to various forms, like this math insists? I am heading into parent teacher interviews to discuss why my previously math-loving child now hates math, and we find that we are doing the activities she LOVES (like math facts) at home because she isn't learning them at school...

If a teacher is provided the appropriate professional development and they understand the theory behind Jean Piaget's constructivism, then this "new" math actually reflects the very essence of how people learned arithmetic before we had all these tricks and alogrithms - essentially making this "new" math a very "old" math.

Constance Kamii explains why this kind of learning is actually the best way to learn the basic foundational skills such as multiplication:

It took centuries for mathematicians to invent, or construct, “carrying” and “borrowing.” When we teach these algorithms to children without letting them go through a left-to-right process, we are requiring them to skip a step in their development.

Honeslty, if you want to immerse yourself in understanding this "new" constructivist math, there are three books by Constance Kamii that you really need to read:


Young Children Reinvent Arithmetic

Young Children Continue to Reinvent Arithmetic (2nd Grade)

Young Children Continue to Reinvent Arithmetic (3rd Grade)