Manim: animation engine for explanatory math videos https://github.com/3b1b/manim
The videos also make me angry because it frustrates me that such explanations were not available to me earlier in my life. What is it about the state of math education that this kind of explanation is not there in every class?
One of the flaws of common core is that it seems that the proponents did a poor job of marketing it to the broader adult community. Doing so would have:
1) Given parents the answer to "why is this change happening at all? I learned math just fine as a kid!"
2) Engaged some people in trying to think about the best ways to explain that material and accelerated the formation of a community that gains status with each other by coming up with better and better explanations of the common core curriculum.
Computational skill is used as a proxy for understanding because educators, just like the rest of us, are lazy-ass human beings. The authors of standardized tests don't care if you know what an eigenvalue is, only that you can calculate one.
Or, put more charitably: teaching math is hard as hell.
Like so many other aspects of our lives, modern math education in the US grew out of a knee-jerk response to a perceived crisis -- the launch of Sputnik, in this case -- that wasn't very well thought out. Schools were required to measure kids' progress in math and science quantitatively, precisely, and repeatably. And just as in other fields, once a measurement becomes a target, it ceases to be a good measurement.
If you value this kind of material please consider supporting via their Patreon page.
You can follow this along here: https://www.youtube.com/watch?v=XkY2DOUCWMU&list=PLZHQObOWTQ...
He also has a Statistics series, as well as some pre-calculus / basic algebra stuff.
I do wish the formulas for the dot product and determinant were derived from the geometric explanation, rather than justified with it afterwards. I always appreciated that in classes.
There are some more advanced topics in lin alg that I would have loved to see get the full visual intuitive treatment when I was learning these things.
- SVD, because it's more general and less pathological than eigenvalue decomposition, and often more useful.
- A linear transformation as consisting of (I think, it's been a little while) a choice of eigenvectors and eigenvalues, "divided" by the extra degrees of freedom from duplicate eigenvalues.
- The "taxonomy" of normal matrices and the polar decomposition (obviously comes after complex matrices)
And there's a nice visualization of the mechanical algorithm of matrix multiplication that looks way more "plausible" than the normal one: draw your two input matrices and your output matrix on grids on 3 sides of a rectangular prism around a corner. Then each value in the output matrix is the dot product of the vectors that intersect at that coordinate, and the whole thing only looks right if all the dimensions match up correctly.
I'm still not sure what the essence of the cross product is: how it is related to divisions algebra (quaternions and octonions) and how bivectors fit in the context of general vector spaces. This was not in the scope of the course however.
Dave Ackley's series "Hyperspace Academy", "Robust First Computing", and "Artificial Life" are well animated: https://www.youtube.com/user/DaveAckley/
Vsause's video on "The Banach–Tarski Paradox": https://www.youtube.com/watch?v=s86-Z-CbaHA (mindblown)
XylyXylyX channel's series on "What is a Tensor?" and "What is a Manifold?": https://www.youtube.com/user/XylyXylyX/playlists
Socratica's series on "Abstract Algebra": https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...
Mathologer's channel on all-things math: https://www.youtube.com/channel/UC1_uAIS3r8Vu6JjXWvastJg
Ben Garside's series "Vector Spaces": https://www.youtube.com/channel/UCu5cg_Jd9XSJL_CHUskgkGw/pla...
PatrickJMT's channel on calculus, game theory...(too many to list): https://www.youtube.com/user/patrickJMT/playlists
Numenta's (http://numenta.com) series "HTM School" on the Hierarchical temporal memory (https://en.wikipedia.org/wiki/Hierarchical_temporal_memory) model for ANNs: https://www.youtube.com/playlist?list=PL3yXMgtrZmDqhsFQzwUC9...
Dan Shiffman's series "The Nature of Code: Simulating Natural Systems with Processing": https://www.youtube.com/user/shiffman/playlists?shelf_id=6&s...
YouTube Link: https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...
Specifically their "Learning to See" series.