The Value of Lower-Level Representations

For once, I was able to include some math in my presentation this year. The focus: using "lower level" representations to extend "on level" concepts.

I led participants through a third-grade multiplication lesson in which I used repeated addition (a "lower level" representation of multiplication) to extend the basic concept of multiplication to mental math with 2-digit factors.

First, we covered four of the most common representations of multiplication—array model, equal groups, repeated addition sentence, and multiplication sentence. It looked something like this:


After that, I gave them multiplication sentences such as 5 × 6 = 30, 6 × 6 = 36, 7 × 6 = 42, etc., and we discussed how these sentences highlight the concept of repeated addition (repeatedly adding groups of 6). I continued this pattern, asking participants to use the idea of repeated addition to find each product as I called out the problem: What is 8 × 6? What is 9 × 6?

Once we got to 14 × 6, I had participants think about how difficult that problem would have been for them if I had simply started with, What is 14 × 6? Many of these folks are math-phobic. At the very least, it would have taken them much longer to solve the problem, they would have made stupid mistakes in calculation, and they would not have been confident in their answers.

Next, I had participants prove to me that they could count by 30. I then connected the repeated addition of 6 with the repeated addition of 30, stressing the fact that repeated addition can work with any numbers: What is 1 × 30? What is 2 × 30? What is 3 × 30?

I then told the group that 30 × 30 is 900. And if that's the case . . . What is 31 × 30? What is 32 × 30? What is 33 × 30? The entire group responded immediately and with confidence to each question.

Finally, I had participants place their hands on their heads. I said, "What is 29 × 30? Do the problem in your head. As soon as you know the answer, raise your hand." Once I got everyone's hands up, I went through a few distractors—"If you answered 899, lower your hand," etc. With the exception of one or two people, everyone lowered their hand at the correct answer, 870.

I think participants were pleasantly surprised to learn that they could easily do 2-digit by 2-digit multiplication in their heads, provided they had a benchmark fact to start with. I stressed the fact that I did not lead them to that final answer; they had applied the concept I was teaching and used it in reverse to come up with the correct product.

In conclusion, I said that I had not taught 2-digit by 2-digit multiplication, nor did I teach 2-digit by 1-digit multiplication. Heck, I didn't even teach basic multiplication facts. I taught only the concept of multiplication, focusing on what is really a first- or second-grade representation of multiplication (repeated addition) and extending it into a fourth- or fifth-grade topic area.