11–20 of 78 posts
If you're interested in more great animations for a visual understanding of linear algebra, I can't recommend 3Blue1Brown's "Essence of Linear Algebra" series on Youtube highly enough. https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
I'm constantly seeing more people coming out with books for free or just a passive donation link. This makes me immensely happy seeing as how they're leveraging the available free resources (Latex, CC-BY-SA content, free software for graphics) to make more resources available for free.
Open software is one thing, but a book is much more permanent in my opinion. A book like this will never go 'stale' or old like software does. We only need handful of good books for every topic out there at which point we can basically not buy books anymore. For many topics, I hardly have to consider buying a book since I can just use a free book offered by a professor. And watch course lectures.
What I want to say is this: Please write more for free. It doesn't matter if there is not much interest in what you are writing. It will help you too!
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[3]. (Of course, intuition can be incrementally "patched": Terence Tao's essay[3] 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
- simple
- rigorous
- 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[2] 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!
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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[1] 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.
[0]: https://en.wikipedia.org/wiki/Commutator
[1]: http://immersivemath.com/ila/ch06_matrices/ch06.html ("Example 6.12: Matrix Product Inverse au Faux")
[2]: Visual Complex Analysis is brilliant, though.
[3]: https://terrytao.wordpress.com/career-advice/there%E2%80%99s...
I found https://www.lth.se/fileadmin/lth/genombrottet/LUkonf2015/41_... which says "The figures were programmed using JavaScript using a graphics engine that we have developed for the interactive illustrations", but that's all the detail it gives.
This is very nice. As a suggestion for improvement, consider allowing the learner to edit the formulas which represent the figures and have the figures update. Additionally editing the figures could update the formula in real time. This sort of bi-directional instant feedback will aid the understanding and engagement of the learner better than figure manipulation alone.
in short (over simplified, my take): interactive visualizations are dead because nobody interacts with them.
wondering if this holds here as well.
Can't find the reference anymore, but there were also papers in educational sciences that interactive books usually don't increase comprehension of kids (they just play with them instead of depen their understanding).
edit: sorry didn't want to sound too critical. The work is awesome (upvote), was just thinking out loud.
Since then i have seen the Strang videos again and again. Beginning to end. Read the book chapter by chapter and exercises by exercise. And what a delight it had been. And then i jumped upon Joe Blitzstein's probability lectures. What a blast ! Is there a list of teachers like these there, who in the pretext of teaching algebra/probability etc are in reality wiring up our thinking process in ways immaterial to subject they are teaching. Many of us don't want the material to be too casual/layman terms (which hampers self understanding as its no challenging anything within us) and not too rigid (where we cannot break through the challenge).