This doesn't involve traditional mathematics concepts, but I'm relying heavily on visual explanations in my mathematical explorations of discrete geometry, which you can find at https://quivergeometry.net.
Probably the cleanest examples of where visualization makes things very clear is explaining the groupoid structure of paths [1], and how to generate lattice quivers [2].
But I also enjoy some of the aesthetic qualities of the sections on coverings [3] (intricate), vertex colorings [4] (very colorful!), toroidal lattices [5], and products of quivers [6]. The last in particular involves trying to understand products from both an algebraic and geometric perspective, and 3D visualization is especially helpful there, giving intuition for how a certain product slices off hexagonal pieces of a hypercube.
I forgot to say that if anyone hasn't seen it, RedBlobGames' tutorial on hexagonal grids is an extremely good resource that uses visualization and interactivity to great effect, and really helped me in early stages of my work:
It's quite stunning how far one can get with visualizations. I think it's an indirect discovery of Euler though (who did many visual proofs) that visualizations are really just that and to do proofs textual reasoning is necessary.
> And that the determinant is the volume of the parallelepiped spanned by the vectors:
It always bugs me when people throw that out because it doesn't underscore why that fact is useful. I saw this in my linear algebrea textbook as a practice problem and never got it until 3Blue1Brown spelled it out.
It is quite likely that the determinant was originally defined as the volume of the parallelepiped spanned by the vectors, when the mathematicians were working out linear systems. Then they threw out the intuition and kept the formula for the textbooks. The point is that because the transform is linear, the volume change of the unit square as it is stretched into a parallelepiped is the same transform as will be experienced by any volume in the space, which is quite possibly how the determinant was first identified.
So the perspective where this is an important fact is that the determinant is the volume change of unit volume -> parallelepiped (or higher dimensional equivalent) when a matrix is viewed as a spatial transform, then that turns out to be a remarkably fundamental measure that characterises a lot of useful properties about the entire matrix.
Yeah, this should be emphasized in any linear algebra course as a primary way to think about determinants. Otherwise the determinant seems like just some math formula to memorize that keeps appearing for some reason in other math formulas.
What do you think about a recent Veritassium video that discusses the idea that ‘there is no such thing as a visual learner’ (https://youtu.be/rhgwIhB58PA)
I'm completely open to being wrong or corrected, I will watch the video now. It's likely poor understanding on my part. However, I would reinforce my claim by adding that there is good psychological evidence for verbal vs spacial interest and competency, which is no doubt related.
I think it's basically just arguing semantics. Been a math teacher my whole career, and it's quite plain to me that some kids simply learn better as soon as you draw a picture, and others simply don't. Not sure why it has to be a whole debate.
I would say it's worth debating because it is an inexact and unsound principle on which recent education is built on.
More than that, in the Romanian Master's cycle, if you want to continue your pedagogical studies such that you become a high school teacher, there is an entire course dedicated to learning styles and there is one more dedicated on the other infamous inexact-semantics concept, the multiple intelligences theory.
So I would argue that it has a similarly bad effect on the education domain to any other arbitrary first-principles that lead to sub-optimal research and lack of clarity of what is really going on in some domain, which lead again to sub-optimal methods.
Regarding your observation about how some kids simply learn better after seeing some visual representation of the concepts, there could be many reasons for this (they start paying attention when you draw, they fall somewhere on the Bell curve such that they need that extra representation to get it, as the Veritasium video also presents with the memory experiment), but I would doubt that persistent learning styles of your students would be the differentiator, which is what the video discredits (https://youtu.be/rhgwIhB58PA?t=438).
Probably the cleanest examples of where visualization makes things very clear is explaining the groupoid structure of paths [1], and how to generate lattice quivers [2].
But I also enjoy some of the aesthetic qualities of the sections on coverings [3] (intricate), vertex colorings [4] (very colorful!), toroidal lattices [5], and products of quivers [6]. The last in particular involves trying to understand products from both an algebraic and geometric perspective, and 3D visualization is especially helpful there, giving intuition for how a certain product slices off hexagonal pieces of a hypercube.
[1] https://quivergeometry.net/qg/paths/
[2] https://quivergeometry.net/qg/lattice-quivers/
[3] https://quivergeometry.net/qg/covers/
[4] https://quivergeometry.net/qg/colorings/
[5] https://quivergeometry.net/qg/toroidal-lattices/
[6] https://quivergeometry.net/qg/quiver-products/