@paulgarrett They are the worst! ;)

@JiK So let's work over the Dual Numbers instead of the Reals (add z non-zero with z^2=0). We don't have ab=0 <=> to a=0 or b=0 now. Do we have x^2=9 imply x=-3 or +3? I'm thinking yes; so what we actually need for this trick to work isn't something I'm exactly sure of. But it isn't "free", you can have very real-number like things that don't have the property you want.

The fact you can uniquely factorize polynomials within the real numbers isn't "free"... and uniquely factoring to linear terms isn't even true!

I mean, nothing in any of the arguments presented show that -3 and +3 are the only solutions. The actual argument about uniqueness requires proving (or knowing about) properties about the real numbers that nobody is mentioning.

I mean, Calculus without calculation starts looking like Analysis.

@user95017 I mean, I use formulas, but I am not providing a proof. Rather, I am breaking down why it is actually true and thinking about why it might not be true. If we are going to teach intuition we should teach intuition that doesn't mislead! The underlying reason the original problem works relies on those 3 hidden quirks of the question: that there is no real bigger than zero but smaller than every other real bigger than zero, that |a-b|=0 only when a=b, and that |•| is positive. Get those 3 facts intuitively, and the overall fact becomes obvious.

"Also when I give him some homework and he does some parts wrong, then when I just point them out without saying anything more, he knows what is wrong and how to correct it." - um, to succeed at a test, you have to self-correct.

Infinitesimal calculus is both consistent and proves/generates "the same" results as epsilon-delta based calculus at the high school level. You can learn and do calculus without using epsilon deltas, even doing it with rigor, via the path of nonstandard analysis/calculus. en.wikipedia.org/wiki/Nonstandard_calculus

Was your undergraduate degree in relatively applied mathematics? Or was it in pure "proof based" mathematics? More concretely, are you trying to memorize the proofs?