Multiplication.

Question

Is there any other way to visualise multiplication apart from repetitive addition. I've been thinking about it for quite a while now that we're introduced to multiplication as solely repetitive addition in the lower grades. Is that approach correct? Is there any other way to visualise it?

Answer 1

I have recently been teaching this concept to my own child (about first grade math). My explanations have revolved around a couple of concrete representations.

1. Length of rods

   We use Cuisenaire rods to make arithmetic operations more concrete. So, for multiplication if you have five rods of length 2, say the child can see and feel that that is the "same" as one rod of length 10. I'm amazed at how quickly my child has learned the corresponding lengths of these rods and can quickly do arithmetic by saying something to the effect of "a brown, and two reds makes ..."

2. Area of geometric figures

   This is very similar to 1. This may use the rods again, but the pattern would be arranged in a rectangle. The question I ask in this case is something to the effect, "if we have five rows of green rods, how many white rods (unit length) is that equivalent to?"

3. Counting groups of fingers

   This has actually worked the best for my child. For $3 \times 2$ for example, I just hold up two hands with three fingers each. We can then count them. Of course, you don't need to use fingers, apples work just as well.

I recognize that many of these have the potential to fall back on the idea of repeated addition but I don't think that is necessarily a bad thing. Mathematics is all about building abstractions. As one concept becomes more concrete (addition in this case), it becomes a block to build upon with further abstraction.

Answer 2

a geometric way could use Thales' theorem and look at the multiplication as a "stretch" of the original value. This would be useful when you start multiplying for numbers between 0 and 1, since it shows that the result could be less than the number you start with.