One of my former students is now teaching middle school. Last year she wrote to ask me, “Why are Algebra II and PreCalc the same course, essentially? (If they are not, then I need to fix our curriculum!!.)”
“Because,” I explained, “What should be happening is that all the same manipulations the students learned to do as progressively more involved algebra exercises, get reapplied within the context of understanding how elementary functions work.
“The set up is supposed to aim toward a calculus vision of considering the effect that (small) changes in the variable inputs have on the behavior of the outputs.”
I am fiercely convinced that Precalculus should be the arena in which they get used to moving their concern from advanced arithmetic to relationships. I’d like to be able to focus on number relationships, but calculators have made that much harder to accomplish.
The circular functions provide a splendid forum for introducing number relations which are far removed from the rehash of algebraic manipulation. So I’ve been hammering at the students to hand-draft graphical images which relate a measured distance around the unit circle to the coordinates of the terminal point. Now many arrived having already memorized these “special” pairings. My goal is to cement those pairs to their relation.
A fellow precalculus teacher listened to my ideas, how much time I planned to take on this; what activities I was employing with the students
“Oh,” she replied, “They don’t need all that when they get to calculus!”
On the one hand it is true that their calculus course will revisit the notion of functions — in the same way precalculus revisits algebraic manipulation. On the other hand, answers to their calculus exercises will come more easily to those who can manipulate symbols quickly and accurately, so the temptation is so focus on rehashing skill drills in the precalculus class. Any theoretical background will sink into subconscious, at best.
But I don’t want to merely rehash old stuff and I don’t want merely a room full of bored computers. I want students to take the fundamental ideas and mentally build new things, finishing those computations only as an afterthought.
I want to teach them much more than they’ll ever think they need to know.