One of the joys of teaching fractions for the first time is seeing mistakes from my students that, prior to now, I’ve only read about.

(Does that make me a dork?)

(Yes, Michael, that makes you a dork.)

Anyway, the first mistake is an absolute classic. With numbers that show any sort of structural complexity, students regularly treat the components individually. See, decimals, complex numbers, polynomials, and other things I’m sure. (If you have links or suggestions of other sorts of common mistakes that fit this paradigm, drop a comment in the comment hole.)

The second mistake pushes on their part-whole understanding in a direct way. I’m not sure how this mistake plays itself out beyond the geometric context. Thoughts, people?

The third mistake is an interesting one, and one that I suspect is about being used to seeing number lines only of unit length before. Other than that, I don’t have much interesting to say about that. But maybe you do?