Calculus Pedagogy

What is taught in calc classes,
as on the AP syllabus,
is:
- formal, algebraic manipulations

Doing calculus is easy if you:
- can do the operations
- can see which to do

To solve a problem, you:
- recognize what tools to use
- use 'em

Notably, you must:
- break up a problem into small steps
- and do all the steps

The main reason that calculus is difficult for some is that it is not
presented as algorithmically as it's expected to be -done-.
Further, the individual steps often aren't presented as such.
This is especially the case for things like:
- u-substitutions
- integration by parts
- change of variables
...where there is some -choice-.
This is why differentiation is easier than integration:
you can differentiate by just applying deterministic rules,
while when integrating, you have to make -choices-,
and maybe follow wrong paths.

For instance, in change of variables in multiple dimensional
integration, one part is:
- rewriting functions in terms of new variables
Another is:
- rewriting domains in terms of new variables

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Is this (mechanics) valuable?
-Somewhat-:
+ just being able to do the steps is often -enough-
  (for applications, like Economics or much Physics)
+ and it's certainly -useful- (likely -necessary-)
  for understanding or deeper work

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-Should- you make teaching this mechanical?
-If- what you want is for students to simply
 -be able to do computations-, then -yes-,
 this is likely the best way to teach -just that-.

If you want -understanding of the math-,
  then (-some-) technical skills are necessary, but by no means
  -sufficient-: you should do much more interesting stuff
  (like parametrize tori and look at potential functions).

If you want "self-starting students",
  ones who can "fill in the gaps" for themselves,
  who can -learn- from usual books,
  then just giving them shoddy books and telling 'em to sink or swim
  is ridiculous:
  you should -actually teach- -learning skills-.
  IE, at -first- you should cut everything up into bite-sized chunks,
  and -gradually- have them do the bite-sizing for themselves.

If you teach effectively, you can be more efficient about things,
and there are synergies, but there's only so much you can teach:
if you work on "learning skills" or "mathematical understanding",
they won't (-on the whole-) learn computations as well.