> My wife went to school for Architecture, where she learned "basic" structural mechanics, and some Calculus, but still cannot explain to me in simple words what an integral or a derivative is. Not her fault at all: her Calculus professor had them calculate polynomial derivatives for 3 months, without ever making them understand the concept of "rate or change"
I've taught Calculus close to a dozen times now. I talked about integrals. I talked about derivatives. I tried to get into the intuitive nature of rates of change and the limiting process of Riemann sums. I used analogies and real-world examples and illustrations and interactive computer demonstrations.
Students don't listen to that part of the lecture. They just don't. They listen to the part where I start doing problems on the board, because to them math is about doing things, not concepts. That's only reinforced when they're asked on tests to just do things rather than demonstrate understanding of concepts. If, in a college class, you try to change that up by asking concept questions, you get bad end-of-semester evaluations because you've challenged them in a way they didn't expect and their parents aren't paying good money for them to be challenged.
(And realistically it's hard to ask concept questions at that level of mathematical maturity anyway. As answers you get mostly word salad with a smattering of misused jargon. I've tried.)
If you try to mix it up and use techniques from inquiry-based learning (Moore-style, flipped classroom, etc.) then you get a stern talking to from administration.
> , or what "infinitesimal" means.
Probably because even defining infinitesimals with anything resembling rigor is tricky, let alone using them productively. I wouldn't even try it in less than a junior or senior class. There is no such real number that is smaller (in absolute value) than every other real number yet bigger than zero. Infinitesimals, at the calculus level, are a convenient hand-wavy fiction to mask limits and a notational artifact from simpler times.
> For me that's a big failure of our current "science" education system: too much focus on stupid application of equations and formulas, and too little focus on actually comprehending the abstract concepts behind them.
I tried. Every single time I taught the class I tried to teach concepts. It's really easy to fall to the temptation of just working problems on the board every day, because trying anything else will just give you grief. Grief from the students, grief from your colleagues, grief from the administration. Grief from people like you who constantly chastise us for not being able to explain to teenagers that they should pay attention for the whole class. (I love my adult students though. Non-traditional students are the best.)
It's not worth rocking the educational boat if you don't already have tenure. If you do rock the boat you get a reputation for being hard or weird and students avoid your class anyway.
You hit a nerve. I've left academia for this and other reasons, but as you can tell I'm bitter about it.