Sam Shah's been writing a lot of thoughtful material about calculus instruction lately, including this piece on related rates.
He includes a worksheet with that post and two items struck me. One, this is a pretty charming illustration of a rocketship climbing into space.
Two, it asks students to climb down, not up, the ladder of abstraction. Check it out. It asks students to calculate a table of values for the rocket …
… then it asks for a prediction about the graph.
It asks students to calculate the instantaneous rate of change …
… and then make a prediction about the instantaneous rate of change.
Calculation is something you can do once you've ascended the ladder and turned a concrete situation (a rocketship lifting off) into an equation (h = 50t2). Prediction is something students can do while they mill around at the bottom of the ladder and it'll make their eventual ascent up the ladder easier.
So I'm here, again, wondering what would happen if the worksheet had asked the prediction questions first and then moved on to calculation. Would the students be more successful? Would they have enjoyed the work more?