Animated mathematical notation and the genre of the mathematical video

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 $\frac{x}{a} = \frac{a}{c}$.

Step 1: Multiply the left side by $a$. The variable “falls from the sky” and is enlonged to convey the gravity of motion.

Step 2: Once the variable $a$ has fallen, the equation “tilts” to show how it is imbalanced. A second $a$ falls onto the right side of the equation.

Step 3: The equation comes back into balance, and the two $a$ variables on the left side of the equal side divide.

Step 4: The $a$ variables on the right hand side start to multiply, conveyed by a “merge” effect …

Step 5: … forming $a^2$.

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.

7 Responses

1. Yes! I just rediscovered that Pythagorean Theorem video myself. So great. I was looking for more online but couldn’t find any. I have a VHS of a trig one from early in my career – I showed it in one of my classes. Pretty sure it was met with a groan.

Incidentally, I’ve noticed that having the lights on in math class is helpful so you can see who is actually awake.

2. Last spring I enrolled in our 2D animation course, thinking I’d pick up some raw skills to put math stuff online. I discovered what you did — that animation principles are essentially untapped. My brain nearly exploded many times… the teacher introduced me to a projectMATHEMATICS! video that was online and it puts most of the self-proclaiming “revolutionary” presentations to abject shame.

3. Jason,
Thanks for this post … and for the others too. The TED-ed is really impressive … makes me going back to programming some examples.

4. As for some language within which to build awesome visuals, you might want to take a look at Steven Wittens’ mathbox at http://acko.net/blog/making-mathbox/

You might also be interested in seeing some of what we’ve done with https://mooculus.osu.edu/

• Anything in particular I should look at in the MOOC? (I tried, at random, “8.06 How large can xy be if x + y = 24? [5:42]” but it seemed fairly conventional; I did appreciate there was actual editing. Also, contrary to what the video says, it is quite possible to draw a picture of the problem.)

5. I am totally not a video-making kind of a dude, but you have completely inspired me to *want* to make a video illustrating matrix multiplication (although I’m not gonna do it), since in my own life the breakthrough in fluent matrix multiplication came when I realized it could be visualized in a dynamic way. I have an image of the rows of the first matrix one at a time turning vertical and raking across the second matrix, dotting themselves with each column, and the answers appearing entry by entry in the product matrix. I’m not sure if this is at all clear. You can also turn the columns in the second matrix horizontal and rake them downward over the first matrix. The only way to communicate this properly would be with a video. I’ve for years wondered if other people look at it the same way.

6. Things like the TED-Ed video really help people grasp concepts in a visual way! I wish there were more! Like you said, there is a medium not being completely grasped with visual mathematics.