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.

Duke of Guise

(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.

First, the Project MATHEMATICS! video The Theorem of Pythagoras from 1988.

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.


Netflix Prize (The Lesson)

So a while back I posted a stumper: How Can You Use This (Ultrahard Edition).

The pictures were part of a larger set of 30. Here are two more examples:


Beijing Olympics Opening Ceremony


Lego Zombie Apocalypse

The only criterion for the images was that they be interesting in some way.

I showed the series of 30 images to my class and had them rate the images from 1 to 5. They could use any system they wanted.

I collected those ratings, then put them in a spreadsheet, selectively removing some of the ratings.


I gave the spreadsheet back to the students, then challenged them to work out the missing ratings.

Students could use any math trick they wanted (taking the mean of the known ratings is a good starting point; median and standard deviation are also recommended), but also psychology, media studies, or any other discipline, as long as their answers were justified.

This is all based on the Netflix Prize, a contest run by Netflix with a $1 million dollar prize attached. Winning the contest requires improving their existing “recommendation algorithm” by 10%. This article from the New York Times has a good summary of current progress, and why it is terribly hard to predict if someone will like Napoleon Dynamite.

Here’s one excerpt I find fascinating:

Interestingly, the Netflix Prize competitors do not know anything about the demographics of the customers whose taste they’re trying to predict. The teams sometimes argue on the discussion board about whether their predictions would be better if they knew that customer No. 465 is, for example, a 23-year-old woman in Arizona. Yet most of the leading teams say that personal information is not very useful, because it’s too crude. As one team pointed out to me, the fact that I’m a 40-year-old West Village resident is not very predictive. There’s little reason to think the other 40-year-old men on my block enjoy the same movies as I do. In contrast, the Netflix data are much more rich in meaning. When I tell Netflix that I think Woody Allen’s black comedy “Match Point” deserves three stars but the Joss Whedon sci-fi film “Serenity” is a five-star masterpiece, this reveals quite a lot about my taste. Indeed, Reed Hastings told me that even though Netflix has a good deal of demographic information about its users the company does not currently use it much to generate movie recommendations; merely knowing who people are, paradoxically, isn’t very predictive of their movie tastes.