What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
"Mathematical knowledge is contained and communicated primarily by documents."
I'm not sure if this is a heuristic, but in terms of beliefs that inhibit learning, this is definitely the one that hurt my mathematical development the most.
I would say the correct statement is "Mathematical knowledge is contained primarily in the minds of mathematicians and communicated primarily by informal oral communication."
This problematic belief grew out of the way that I (and pretty much everyone else) was taught mathematics at the undergraduate and beginning graduate level. In this setting texts are a central authority and a complete, well-written resource for the knowledge needed to solve any mathematical problem encountered.
In the world of mathematical research, this is no longer the case. I finally figured this out by reading Thurston's essay "On proof and progress in mathematics", which I would strongly recommend for any beginning mathematician.
Maybe it is possible to do research mathematics using papers as a primary resource, but I believe this is highly inefficient. I spent several years trying to learn the noncommutative standard model by reading the available papers on the subject and made no real progress. Looking back, I don't think I ever had a chance of succeeding with this approach.
I would guess that to be successful in mathematics, it is absolutely vital to become regularly involved in conversations with working mathematicians, as awkward and intimidating as that might be.
Talking about "functions" when we are actually talking about equivalent classes of functions almost everywhere equal
An element of a $L^p(X)$ space is usually called a "function", and is usually denoted by letters that are used typically for functions ($f$, $g$, $h$, etc.).
It seems to be a harmful heuristic to act "as if" $L^p(X)$ is made of functions, as a function is really something that should give you a value for each point $x$ in $X$. I am aware that it is now common practice, but I am sure it would help to actually introduce an actual name besides "function" to call "equivalent classes of functions modulo equality almost-everywhere". This concept is fundamental, and should be given a proper name. I don't have a proposition for such name, but I kind of wish someone in the past did.
The following refers to the school system in Germany, it may be different in other countries:
In my opinion, one really bad heuristic happens in elementary school, when children learn arithmetics with natural numbers. They learn that addition and subtraction are two entirely different things, because they are taught $a+b=b+a$ but $a-b\neq b-a$. Thus addition is commutative, and subtraction is not. At that level, numbers are solely understood as enumerations of objects.
Then they learn about numbers with units, such as lengths or prices or weights. Also they learn that numbers might have geometric meaning, e.g. as lengths of line segments. But still no concept of negative numbers.
Years later, when they finally get to know negative numbers as well, they have so much incorporated that subtraction is something different from addition that they have difficulties to grasp that $a-b=a+(-b)=(-b)+a$, i.e. that subtraction is nothing else than addition of a negative number.
I think that postponing negative numbers so long is a mistake, and that children in elementary school would very well be capable to understand them.
"Basic (and useful) mathematics is about calculations and higher (pure) mathematics is about proof."
One reason I think this is harmful is that there is no sharp line between calculations and proofs. Very often a certain calculation is essentially the proof except for a few logical connectives. Conversely, in formal logic, one can create a "calculus" that makes proofs appear to be calculations.
Another reason is that it leads students (and more importantly teachers!) to think that a drastic change of mindset is required to learn higher mathematics.
It is indeed true that analysis is quite different from calculus even though there is strong linkage. However, the former also leads to better techniques to calculate things. Putting too much emphasis on the "proof" aspect of analysis tends to put off a lot students who enjoyed playing with polynomials, trigonometry and calculus. Conversely, many students who like to work with proofs are encouraged to believe that what they are doing is somehow "superior" (higher) to "mere" calculus; they then do not do enough computational drills which ill-serves them if they actually take up mathematics!
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.