In high school our physics professor (and an actual physics PhD) decided that he can't teach us "real physics" without derivatives and integrals (the latter not being in high school curriculum at all), so by the end of the first year, he taught as both. Our maths teacher started tearing her hair out over this, complaining about lack of rigour; eventually she gave up and taught us integrals with full mathematical formality. Having those two takes on the topic helped us understand it really well.
Another science class anecdote - first year electronic engineering, the professor is showing us the derivation of transistor gain formula. He started from pretty much first principles, written out a huge formula, and then started to cross things out - "this in practice is close to 1 so we can ignore it", "in standard operating conditions this is pretty much zero" (and poof, half of the equation gone, multiplied by zero) - and kept going until he arrived at the standard I_c = βI_b. It gave me a new appreciation of just how much engineering is basically handwaving parts of reality away.