It's important to appreciate how useful it might be to make math "tangible". Sure, someone who can define a manifold by saying "oh, put charts on it, locally diffeo blah blah" probably has a good set of mental models that help them find analogies and even "tangibilize" (word?) new ideas. Once you learn the way abstraction works, broadly speaking, you can take the training wheels off: but lots of people never get past that stage. On one hand, I see a lot of people on HN talk about how the complicated notation of academic math/CS keep people out (and there is an understandable amount of resentment at people keeping "outsiders" out with this), and on the other hand, I sort of reflexively bristle (it's gotten lesser now) at people integrating the notion of an inner product into a vector space, because it is important to not stumble later when you find out your basic intuition for something is broken. (Of course, intuition can be incrementally "patched": Terence Tao's essay talks about this, from the perspective of someone who is a brilliant educator in addition to also being one of the most versatile mathematicians around.)
Maybe presentations of basic mathematics that are
- free of half-truths
can be made accessible by using such visualizations and interactive techniques to decrease the perceived unfamiliarity of the ideas? I don't think there are many treatments of mathematical topics that satisfy these criteria and yet manage to be approachable: one either skimps on a clean presentation (Khan Academy), or assumes a lot of mathematical maturity (shoutout to Aluffi!) from the reader. "Manipulable resources" might help fill this gap. It's an exciting time!
In the section where they give examples of matrix inverses, to give people a sense of how important multiplication order is, they give an example of RHR'H' (using a prime for inverse, R for a rotation matrix, and H for a shear matrix). One of the most beautiful illustrations in the book follows, with the four corners of a square moving independently in circles, and then the book states that
It is quite close, but it is not at all useful.
: http://immersivemath.com/ila/ch06_matrices/ch06.html ("Example 6.12: Matrix Product Inverse au Faux")
: Visual Complex Analysis is brilliant, though.