Multiplication Mailbag

I got a lot of wonderful responses to my last post which have been percolating in my head over the past few days. Two images of multiplication seemed to emerge:

1. Multiplication is a change of unit.
2. Multiplication is a change of scale.
   

These are equivalent, but each provides a slightly different picture which I think would make a difference to my students.

The idea of multiplication as a change of unit is essentialy that when you say "4 times 6" is "four sixes", you are making "six" the new "one". It's the new unit. The difference between this and just saying multiplication is counting groups is subtle, but important. Substantially, introducing multiplication as counting groups (or repeated addition) reduces it to a mechanical trick one might use to simplify an otherwise lengthy sum or counting exercise. There is no need for the student to adjust his concept of unit. In fact, since the purpose of counting groups is generally just to get an answer in terms of the original unit, it renders such an adjustment inherently useless. What multiplication really allows though, is for us to leave the old unit behind and talk purely in terms of a more convenient one. Burt from Hawaii sent this ingenious lesson plan for drawing out this point.

In the Russian curriculum of Davydov, a follower of Lev Vygotsky, multiplication is taught as taking an indirect count using an intermediate unit size. In one lesson, students are asked to find how many tiny cups of water can be served from a large container. They start pouring the water out and counting how many tiny cups there are, transferring the water into another large container. But it is tedious and difficult to pour into such a tiny cup. The teacher asks them to think of a better solution than pouring into the tiny cup. The next day the teacher suggests using a large cup. They pour 8 tiny cups into the large cup, so they know that each large cupful is 8 tiny cups at once. They record 7 large cups, and he teaches them to write it as 8 x 7, and reads it as “8 taken 7 times”. They don’t compute the product right away. They do more exercises measuring different things with a larger unit, measuring how many smaller units are in the larger unit, and writing the result as 4 x 5, for example. The teacher wants them to understand the concept of using a larger unit size and counting in that unit size. Then later they compute the product using repeated addition. Repeated addition is a useful computational strategy, but the concept is changing the unit size and counting in a larger unit size.

If I taught elementary students, I would absolutely do this activity. Maybe I should even recommend it to the elementary teachers at my school. Unfortunately, my ninth-graders are are too cool to measure water cups. Instead, I think I may show them something like this image that David put together. 'A' is the new one. That's what multiplication does.

The other image I now have in my mind for multiplication was suggested to me by a good friend who called me just to talk about it. Thinking of multiplication as a change of scale lead him to picture the real number line getting stretched like a rubber band. Zero is the center of course; it doesn't move, and the distance you move is directly proportional to the distance you were from zero in the first place. This is similar to picturing multiplication as zooming in or out. I like this visualization because it's just as easy to see what happens to non-integers as integers and shows that multiplication is truly the same operation across the real numbers, whereas when I think of multiplication as changing the unit I find myself still stuck thinking in terms of discrete groups.

The actual approach I plan to take in my classroom is to let my students propose their own definitions at first. I am fairly confident they will sway towards repeated addition. I will absolutely go along with this at first, because I want the understanding of how inadequate this definition is to arise naturally, when we start talking about rational numbers. At that point, I will prod them to fix their definition, and share both of these suggestions.

Making the move to exponentiation still proves a challenge. It's a change of scale, but not a consistent change of scale. I can just barely form a mental picture of the number line as transformed by x^2. Numbers far from zero fly off to extremes, while numbers close to zero huddle closer. I can't imagine getting my students to share the same visualization before they are all ready comfortable with exponentiation; it's simply not very intuitive. I presume I will try the same tact: let the students define it inadequately, then put them in places where they need a better definition.

Further reading on the topic from Keith Devlin: It's Still Not Repeated Addition! Keith actually argues against the tactic of introducing inferior explanations first and revising them later, but in my defense, I am hardly forming my students first impressions of multiplication, and in order to properly shock them out of their complacency with repeated addition, I believe it would be most effective to have them tell me what's wrong with it, rather than simply lecture them about how they've been lied to all these years.




While thinking about this, I've restricted myself to the goal of leading my class to a better understanding. However, the problem hiding in the margins of my mind the entire time has been that this isn't really my issue. This is an elementary school math issue. After nine years of math education, my students should already have an excellent grasp of multiplication. However, in general, the teachers who have the greatest opportunity to explore the concept of multiplication are not part of this conversation.