But it is quite difficult to find a level of teaching which is comprehensible and at the same time interesting to an entire class with heterogeneous background. The shape of the mathematics education of a typical student is tall and spindly. It reaches a certain height above which its base can support no more growth, and there it halts or fails. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression.

Learning starts with metacognition, and in order to metacognify (it’s a word) well, your students need you to be transparent in your pedagogical choices. ​ Be honest with yourself about how much time you spend working and how much time you need to work.