She makes a good point about algebra students not understanding prerequisites. I worked for two years at a high school where students were having trouble with algebra. Almost invariably this was due to (a) an abysmal grasp of arithmetic; many students couldn't divide or even subtract reliably, leading to insurmountable frustation, and (b) the hand-wavy means by which algebra, and especially its archaic conventions (such as order of operations and elision of the times symbol) were taught.
I once solved these problems for a small group of these children to whom I taught algebra from first principles: all grammar was explicit (no elision of parentheses or operators) and we used no numerals. The reduction rules then were able to fit down one side of a whiteboard, and after five one-hour sessions, self-professed math phobes were solving nontrivial equations.
Interesting! This same problem occurs in calculus as well. Leibniz's notation is pretty offensive, for example: in basic calculus, d/dx is an operator. In differential equations, you suddenly begin treating dx as a separable algebraic term — and way too many teachers suck at explaining why this is suddenly permissible.
I recommend Elementary Calculus: An Infinitesimal Approach to those interested in calculus. It treats dx as a term from the start, and matches my (and I suspect most people's) intuition of the subject much better than epsilon-delta. And it is actually completely rigorous, not hand-waving infinitesimals.
My high school has a relatively progressive math program, and we never learned d/dx as a function. We began calculus with limits, then started solving Lim[dx->0](dy/dx), then we started using d/dx as a shorthand so we did not need to pull a nested expression out to take its derivative. I never thought to think of d/dx as a function until I started doing functional programming.
+1 on explicit grammar. In my recollection, that was the biggest source of frustration in studying math subjects: decoding the actual meaning of a math text fragment and separating random elisions from errors.
It's funny how in grade school the multiplication symbol is omitted, only to be (more confusingly replaced) by a period later on when the expressions get complicated.
I suspect it's the lack of arithmetic skill more than the conventions that are the heart of the problem. I think kids should be required to learn to do mental arithmetic before they're allowed to use calculators, but this may be impractical.
In the United States denying children access to things seems to be quite impractical :-)
I also worry that smart kids will work out that calculators are ubiquitous before they force themselves to become proficient at mental arithmetic (e.g. Google and Spotlight are both far more powerful calculators than anything I had access to as a student).
I once solved these problems for a small group of these children to whom I taught algebra from first principles: all grammar was explicit (no elision of parentheses or operators) and we used no numerals. The reduction rules then were able to fit down one side of a whiteboard, and after five one-hour sessions, self-professed math phobes were solving nontrivial equations.