What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
This isn't really a heuristic, but I hate "functions are formulas". For most students it takes a really long time to think of a function as anything other than an algebraic expression, even though natural algorithmic examples are everywhere. For example, some students won't think of
\begin{gather}
f(n) = \{\text{1 if $n \bmod 2 = 0$ $\lor$ $-1$ otherwise}\}
\end{gather}
as a function until you write it as $f(n) = (-1)^n$
