>>12855552Ok I agree with you as well on the value of the limit approach for intiution and I am happy we have reached this point of agreement which is the only way a debate can have a productive output.
>Most calculus teachers are NOT anal about Step 2 and that's my problem. Maybe you are not and that is a good, but most teachers certainly are anal about it. Especially AP teachers and so forth, but also a minority of professors forced to teach lower level undergrad. Some Anon was being bitchy earlier in the thread, but I have literally seen the exact same discussion here go down between faculty members of my department in the break room (and by the way that discussion was also unironically less mature).
I think maybe the final point, of possible contention, is that if a student like OP asks "can I treat this as a fraction [and implicitly asking if they can split the nominator with the denominator]" then the correct answer to give is:
"Only if interpret the differentials as ... [while also making it clear that only certain operations are permitted which they will learn more about later and it is not important in single variable calculus]"
and not
"No you can never split that fraction, it is a notation for differentiation"
Which is a very, very common response and it also the understanding that most students leave calculus with if they go on to study science or whatever. Hence OP's question.
