When teachers do this, they teach students to think that having a fuzzy understanding about the foundations of the subject is normal rather than something that should raise an internal alarm bell. At the very least, teachers should minimize damage by announcing clearly when they've stopped explaining and started just giving a prescription, and point the students to the place where they can actually learn what's going on.
The same issue comes up on teaching probability, and even worse, elementary stochastic processes. To do it correctly, you need measure theory, but that requires a whole course or courses.
In fact, a lot of the results of freshman calculus require more mathematical analysis than most students really have command of. But we still teach it because the applications are so compelling, and the intuition is important to develop.
Even more generally, as you probably know, there are tricky foundational issues in Mathematics that most professional mathematicians prefer not to pay attention to.
Well I did that. I also gave them a similar warning when I started using the dirac delta, because it is not a proper function and if you want to know more than "it eats integrals", you have to either view it as the limit in a series of function or as a distribution acting on test functions. The thing is you walk a fine line between hiding the truth and not cranking the internal alarm of the students up to a volume where is scares them and prevents further learning.
That last sentence sounds like a euphemism for there simply being a conflict with what students want to learn about and what instructors are required to teach. How would I tell the difference between students being scared versus them healthily resisting the application of mechanical procedures they don't understand?
In the first case you would get outright rejection (typically expressed as "why do we even need that" or "nobody can understand that"), in the latter at least the better students would ask question "why can we do that step", "why does this follow", "earlier you said X, not we do Y, why?".
Yes. And better students tend to want more rigor (or have more brain cycles left over for more rigor in the introduction of new but only tangentially relevant topics) than the weaker students, who are usually more happy to take something at face value as long at it let's them continue with the main topic. The line you have to walk as a teacher is then to find an appropriate level of rigor to not loss (too many) students, while satisfying the curriosity of the more interested students and telling them where to find the extra details they might want to know.
OK but I would describe that as the students having already been poorly partitioned into classes. If you're taking a class where you need to go read independently to understand what's going, it seems much better to either just read fully independently or to take a class in the actual thing you want to learn. Sending a bunch of kids with widely varying levels of skill and interest through the same course is hugely wasteful, however common that may be.
This is not necessarily a wrong thing to teach. As one progresses from old things that are well-understood to new developments,students will have to deal with fuzziness and gaps. They must learn to make progress using a heuristic feel for the subject instead of complete rigour. Too sensitive an alarm for missing foundations will actually stop them from being productive. Think of how much time passed between the invention of calculus and the development of rigorous analysis.
That's probably appropriate for quickly progressing fields, but not for fundamental physics which has utterly stagnated.
People wasting time on important but nearly intractable topics certainly happens (e.g., arguably myself on quantum foundations). But this is absolutely dwarfed by the number of researchers eager to springboard to a cutting edge, which hasn't moved much in decades, but who remain ignorant of the basics.
The terrible incentives for professors to quickly make graduate students useful, rather than invest in fundamental understanding that won't pay off for many years after they graduate, should also lead us to suspect the correct balance isn't being struck.
