Question: Every good teacher knows that giving an *informal explanation* (that is, an explanation not based on strict definitions and maybe making use of metaphors, intuition, or everyday language) of a mathematical concept can be really helpful to students. And every good teacher has at least one of these informal explanations for each of the main mathematical concepts they teaches. However, I believe that some of these informal explanations happen to actually cause misunderstanding of the mathematical concept, to the point that it would be much better avoiding them (or replacing them with better informal explanations).

I think it would be interesting to list some examples of these *bad* informal explanations. Here my example:

**Bad:** *"Continuous functions are those functions that can be drawn without lifting the pencil from the paper."*

I heard this one many times from many teachers, and I find it causes misconceptions for multiple reasons: 1) It is actually about the arc connectedness of the graph of the function, not the continuity 2) It convinces the students that many continuous functions are not continuous because they can't be drawn without lifting the pencil, like 1/x on its domain of definition 3) It has no connection with the delta-epsilon definition of continuity that the student need to learn.

To make my point more clear, I also give an example of a *good* informal explanation:

**Good:** *"The concept of group encodes the properties of transformations of an object. The identity element corresponds to 'doing nothing', the inverse element to 'reverting the transformation', associative laws means... etc."*

What are your examples of informal explanations that cause misconceptions?

Answer: 
For a much lower-level topic, consider explaining to beginning algebra students why "like terms" can be combined. On a few occasions, I have resorted to reasoning with students that **adding algebraic expressions is like adding quantities with units**. [Our curriculum begins with units and geometry before algebra, so this is usually safe ground in my class.]

If they already believe that $5cm$ + $7cm$ makes physical sense, and $5cm^2$ + $7cm$ does not, then we can make an argument that $5x + 7x$ can reasonably be called $12x$, but $5x^2 + 7x$ cannot be combined nicely. However, I always regret this line of conversation, because I feel I have to backpedal for the sake of honesty. Since students recognize that $5cm^2 + 7cm$ isn't even computable (forgoing the possibility that $5$ and $7$ have convenient units), they will sometimes say that $5x^2 + 7x$ is also a "bad" expression that cannot represent anything real. This is obviously a problem. [Hence, I avoid this informal explanation if possible.]
**Modular arythmetic works just like a clock**

I've seen several students expect "half-hours" making an appearance in $\mathbb{Z}\_{12}$. This leads them to be confused about statements like $5^{-1}=5$ rather than $5^{-1}=0:12$