|A power of 5|
Is multiplication really repeated addition?
It is a strong and presumptuous claim to say what an idea is. In recent years, I have come to an understanding of why repeated addition is not the strongest foundation on which to build the idea of multiplication. But that is a far cry from making claims about what it is.Similarly, I argue that while there's nothing wrong with thinking of exponents as repeated multiplication, it's not the best way to start thinking about them. Instead, we need a different conception of exponents to build on.
What is that conception? Writing about multiplication, Danielson says "Multiplicative structure is captured better by this idea: A times B means A groups of B."
What's the analogy for exponents? "A to the power of B means, ummm, B nestings of A? B recursions of A? B doublings/triplings of A?"
With multiplication, we can rely on the informal language of groupings to ground multiplication. But every day experience provides less clay to be shaped into the precise language of exponents. Presumably, this is what makes exponents so hard for kids and teachers of kids.
All of this leads me to two claims about teaching exponents meaningfully:
- "A to the power of B means B _______s of A." Effective teaching of exponents will ensure that kids have something to fill in that blank.
- The best noun to fill that blank is simply "powers." As in "A to the power of B means B powers of A."
(I'd say that a "power of A" is an element in the Geometric sequence of A.)
What follows from this is that we have to teach kids what powers are before we can teach them about exponentiation.
There are no special obstacles standing in the way of this task, though. "Power" is essentially a new piece of mathematical language, and that means our job is going to be giving our kids lots and lots of opportunities to talk about powers before showing them exponents.
How do you create situations where kids end up talking about powers? That's the most important teaching problem here, and it's the one that deserves careful and creative thought. I'm sure you have ideas -- I'll share mine soon.
(In case you're keeping score: I'm now recanting this post and its enthusiasm about grounding all of this in Geometry.)