As opposed to a textbook, real maths is highly non-linear. ​ As we will see throughout the post, personalization (and the engagement inherent in it) is essential to the success of the lecture.] ​ [Before the third, I ask the class whether the first two alone are enough. If I get nods, I draw a random collection of dots and lines, with the lines not at all connected to the dots, and they see we need some statement of incidence.] ​ Since we will always draw constellations as a picture, we can just use the picture as our “function.” ​ Compare this to being given the definitions and propositions in the established mathematical language. To an untrained, uninterested student, this is not only confusing, but boring beyond belief! They don’t have the prerequisite intuition for why the definition is needed, and so they are left mindlessly following along at best, and dozing off at worst.

@JadeMasterMath: There are lots of mathematical concepts which don’t have well written resources to learn about them. I think that explaining something in a clear way with a story arc can sometimes be enough. @jeremyjkun: Write about the topics that you learned, where there was a succinct phrase, picture, or idea that suddenly made it clear. Then arrange the whole blog post around getting the reader to that same understanding.

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.

In July 2019, ICMS hosted a workshop on Category Theory.  During the workshop, Eugenia Cheng (School of the Art Institute of Chicago) gave a public lecture entitled Inclusion-Exclusion in mathematics and beyond: who stays in, who falls out, why it happens and what we could do about it. This is a recording of that talkThis talk has captions.  To turn the captions off, press CC on the bottom toolbar.