Tuesday, April 29, 2014

Exponents As Abbreviations, Exponents As Language All Its Own

Most teachers think that exponents are a way to write 4*4*4*4*4*4*4*4*4*4*4*4 in a convenient way. In other words, most teachers think that, fundamentally, exponents are an abbreviation. This is obviously a legitimate mathematical perspective.

If exponents are abbreviations, then your lessons should give kids a chance to feel the need to abbreviate. My favorite version of this way of thinking about teaching exponents comes from Dan.

You want kids to feel the impulse to abbreviate? Give them big numbers and tell them to write 'em down! It's elegant.

There's an entirely different view about what exponents are. In this world, exponents aren't a notational shortcut anymore than multiplication is a notational shortcut of addition. It's a language to describe a sort of pattern, a special sort of number. Instead of just an abbreviation, exponents are the language of successive grouping.

They're the sort of thing that let us say what we see -- what we should be able to see -- when we look at this picture:

The teaching that follows from this view of exponents is going to look a lot different than Dan's (awesome!) lesson. We're going to need to figure out how to create a need for this language in our students. Creating that need is going to need two things: 1) Helping our kids see what's special about these patterns and 2) Running up against the limits of our language for describing these patterns.

But how can we do this? Stay tuned for some thoughts!


  1. I feel like this is definitely a both/and situation. I have been gearing up to (hopefully) teach a calculus class next year and thinking a lot about how differentiation and integration interact and how we often introduce the concepts distinctly and then *discover* that they are related. This moment is magical and powerful for kids.

    I think the same goes for seeing multiplication as grouping, but also as a notational shortcut for adding. Similarly, exponents and multiplication. If we leave that out of our treatment altogether, then we end up missing some massively powerful analogies between add/mult relationships and exp/mult relationships, such as the fact that exponents distribute across multiplication JUST AS multiplication distributes across addition (and for pretty much "the same" reason).

    So I guess, I am mostly just proposing both introductions as distinct, along with some time to discover that these two concepts actually link up. That wonder of different concepts not really being different after all always seems to be a big motivator for my kids.

    1. I definitely agree that if a kid graduated without understanding how multiplication can line up with repeated addition, then we haven't done our job. But, it seems to me, there's a sort-of consensus that repeated addition isn't a helpful way for beginners to think about the operation.

      That's basically how I feel here. There are lots of surprising things about exponents, lots of connections to see, and those connections and surprises will be stronger if we introduce exponents as a way of describing grouping.