One of those principles is:
Create an intellectual need for new mathematical skills.
Nowhere is that principle more necessary, in our view, than in the instruction of algebraic expressions. Three of my least favorite words in the English language are “write an expression” because they so often mean we’re asking the student to do the difficult work of variable manipulation without experiencing any of the fruit of that work.
In both of the questions below, students are likely to experience the work of writing an expression as punishment, not power.
Given the width of the lawnmower (W) and the length of the rope (L), write an expression for the pole radius (R) that will make the lawnmower cut the lawn in a perfect spiral.
Given the width of the pool in tiles (n), write an expression for the number of tiles that will fit around the pool border.
We recognize that one reason variables give us power is that they let us complete lots of versions of the same task quickly and reliably. So in our version of both of the above problems, we asked students first to work numerically, both to acclimate them to the task, but especially to establish the feeling that, “Okay, doing a lot of these could get tedious.”
And then we use their expression to power ten pool borders.
And ten lawnmowers.
Those activities are Pool Border Problem and Lawnmower Math. In Picture Perfect, for another example of this principle, we give students the option of either a) filling in a table with 24 rows, or b) writing an algebraic expression once.
In each case, students are more likely to see algebra as power than punishment.