Calculus Blurb:
So I'm about to drop my instructional experiment on Calculus. For years I've been really stuck on a few things that haven't made me happy. Considering that I'm starting at a school where apparently the AP Calculus results had been less than impressive, I figure this is my chance to experiment.
Here's the breakdown:
1) Early transcendentals. I know, nothing mind-blowing. I just have felt kind of gross introducing natural log and e so late that it's well beyond when they actually would have cared. It also makes them seem like something special, which they aren't. They're about as special as trig, but I don't want to get off on that tangent. (Sorry)
2) u-notation from the beginning. I've always hated how stupidly difficult substitution has seemed to kids later. I figure if we get it flowing at the start, it would help with SO many things (namely derivatives of composite functions). Again, not mind-blowing, but definitely a step in the right direction.
My thinking is this: Much like Sam Shah, I have always had a weird problem with Chain Rule. It's always been just a kooky rule explained with some analogy that has, in my mind, always been classified as a magic trick to teach the kids that robs them of something. I hate that. In addition, it's even more of a ridiculous moment when u-substitution rears its head.
"Here, kids, learn a thing that's supposed to make integration of a formerly composite function easier."
"This is hard."
"Hmm... Yeeees....It is."
So u-subs to start sets the stage, introduced under the heading of transformations and their effect on derivatives. Chain rule becomes "old hat" after a while, and integrating with u's is a footnote as it's just going backwards. It becomes incredibly simple with natural logs and e's, so there's no need to make a special introduction later.
7th Grade Blurb:
I don't understand Pre-Algebra curriculum layout. It's the same few skills over and over again, but what drives me nuts is the insistence in so many places to teach the skills in isolation. Why do you have to learn how to deal with exponents so long before you actually use exponents? It made way more sense to intro scales of the universe using scientific notation, have that naturally segue to multiplying massive scales by massive scales, and BOOM exponent rules. Not only that? Exponent rules when coupled with a coefficient. Turn that 10 into a variable? Same rules. Yay.
I think my real concern is that I don't feel like there's a sequence to things. It's just algebraic reasoning explorations coupled with occasional geometry. I have to be wrong, but I don't know where. There is so much to be done with more rudimentary skills, like speed in processing order of operations, etc., but I just don't know what my endgame goal is. Everything else I've taught has had a pretty clear endgame goal. Algebra is being able to reason with unknowns and interpret various stimuli algebraically. Algebra 2 is the same but with far more complex relationships. Pre-Calc is about reasoning with angular movement/forces/observation data. Calculus is interpreting instantaneous and accumulated changes from various phases of data/functions.
And 7th grade is.....maaaaaaaaaaaaath? Blargh.
