This Week’s Skill
Rules like these are too quickly abstracted, memorized, confused, and forgotten.
We can attach to them meaning and purpose by asking ourselves, why did we come up with these shortcuts? If these shortcuts are aspirin, then how do we create the headache?
What a Theory of Need Recommends
Again, with Harel’s “need for computation,” students need to experience the “longcut” before they learn the shortcut. Otherwise it’s just another trick in the endless series of tricks students call “math class.”
Several people suggested the same in last week’s thread, of course. Ask students to calculate expressions like these the long way before discussing shortcuts the students may have noticed (or that you may have noticed as a member of the class also).
Chris Hulitt was one of my workshop participants in Norristown, PA, and his group suggested an important addition to this idea. Ask students to calculate this expression instead
Looks the same as the last, right? But whereas the last expression resolves to 16, this expression resolves to 1. That headache is a little bit sharper. How did this gangly mess of numbers result in such a simple answer? Could I have realized that in advance?
Again, this isn’t real world, or relevant, per our usual definitions of the term. And yet this approach may still endow exponent rules with a purpose they often lack.
Next Week’s Skill
Determining if a relationship is a function or not.
This is another skill that can become quickly instrumental (run a vertical line over the graph, etc.) and obscure why it is aspirin for a particular kind of headache.
Let us know your ideas for motivating the definition of a function in the comments.
What You Recommended
I think of my 6th grade students writing down the prime factorizations of whole numbers (Why learn that? Oh yeah, to improve number sense and seeing structure behind numbers, among other reasons). When you work with something simple like 24, writing out 2*2*2*3 is not so bad. When you up the stakes to something like 256, writing out 2*2*2*2*2*2*2*2 becomes annoying. It’s not so much a headache as a tedious process that any normal person would like to make quicker and easier. At this point, I discuss exponents as a means of communicating all of that multiplication without having to write it all.