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!