What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
Two bad principles that taste worse together: Decimals are the true numbers. Rounding makes no difference.
Since students learn about decimals after they've learned about whole numbers and fractions, they might assume that decimals are always the preferred way to represent real numbers, and so everything should be converted to decimals. Meanwhile, since in generally one cannot be expected to write out an infinite decimal expansion, they might assume that stopping after two decimal places makes no difference.
I'm not saying that approximations are bad. But it's bad to approximate if you have no sense of your error tolerance, or even of the fact that you're introducing an error at all.
Here are two perverse outcomes.
The excluded middle ( A Law or an Heuristic) .
On a more general level given any closed question: Is it A or B ? , the heuristic says it is one or the other disregarding the option : the question is wrong or stupid or irrelevant or incomplete.
The principle of excluded middle disregards intuitionist logic. And has been harmful in not providing direct (constructive) proofs which are often more clear - yet can be harder to find.
Intuitionism is is also rather natural : being against anti-communists does not means you are a communist.
"A set is a collection of elements".
Firstly, this does not distinguish sets and classes. Next in ZFC, sets are characterized more by their relation to each other by $\epsilon$, rather than that they contain anything.