While it is still common (and frankly, necessary) to rail at the limitations of learning mathematics via watching videos, my personal umbrage has more to do with presentation than with educational philosophy.
The mathematical video genre is still in its infancy. I am reminded of early films that were, essentially, canned plays.
(From L’Assassinat du Duc de Guise in 1908.)
Oftentimes in videos teaching mathematics with notation they simply duplicate what could be done on a blackboard, without fully utilizing the medium.
However, there are techniques particular to the video format which can strengthen presentation of even mundane notation. For instance, in my Q*Bert Teaches the Binomial Theorem video I made crude use of a split-screen parallel action to reinforce working an abstract level of mathematics simultaneously with a concrete level.
For now, I want to focus on applying animation to the notation itself for clarity.
The video is chock-full of interesting animated moments, but I want to take apart a small section at 5:43. In particular the video shows some algebra peformed on .
Step 1: Multiply the left side by . The variable “falls from the sky” and is enlonged to convey the gravity of motion.
Step 2: Once the variable has fallen, the equation “tilts” to show how it is imbalanced. A second falls onto the right side of the equation.
Step 3: The equation comes back into balance, and the two variables on the left side of the equal side divide.
Step 4: The variables on the right hand side start to multiply, conveyed by a “merge” effect …
Step 5: … forming .
Here’s a much more recent example from TED-Ed:
When adding matrices, the positions are not only emphasized by color but by bouncing balls.
When mentioning the term “2×2 matrix” meaning “2 rows by 2 columns” the vocabulary use is emphasized by motion across the rows and columns.
The second matrix is “translated up a bit” by doing a full animation of the matrix sliding to the position.
When the video discusses “the first row” and the “the first column” not only are the relevant numbers highlighted, but they shrink and enlarge as a strong visual signal.
When discussing the problem of why matrix multiplication sometimes doesn’t work, the “shrink-and-enlarge” signal moves along the row-matched-with-column progression in such a way it becomes visually clear why the narrator becomes stuck at “3 x ….”
These are work-heavy to make, yes, but what if there was some application customized to create animation with mathematics notation? At the very least, there’s a whole vocabulary of cinematic technique that has gone unexplored in the presentation of mathematics.