Many fans and critics consider The Simpsons to be one of the smartest and most literate shows on the air, with many references to scholars and academic subjects, including mathematics and its own mathematician, Professor Frink. One reason for this is that many of the show's writers are Harvard graduates. In addition, some of the writers have significant mathematical backgrounds.

Sarah and I will talk about how we have used The Simpsons to introduce and illustrate substantial mathematics in college classes. In addition, we have documented all of the one hundred or so instances of mathematics on The Simpsons, from arithmetic to calculus to Riemannian geometry. This catalog is posted on my webpage. The address is on the handout we'll provide at the end.

First, here is a scene from the 1939 film "The Wizard of Oz". The Scarecrow has been looking for a brain, and the Wizard is about to help him out.
Scarecrow receiving his brain and reciting the Pythagorean Theorem incorrectly

Now, here is a scene from The Simpsons
Similar scene from The Simpsons
$pringfield (Or, How I Learned to Stop Worrying and Love Legalized Gambling) 1F08 12/16/93:

[After putting on Dr. Kissinger's glasses, found in a men's room toilet]
    Homer:  The sum of the square roots of any two sides of an 
            isosceles triangle is equal to the square root of the remaining side.
    Man in stall:  That's a right triangle, you idiot!
    Homer:  D'oh!

I played this scene on the first day of a pre-calculus class. I wrote Homer's statement on the board and had my students identify and correct the 3 errors. This was a fun, non-stressful and productive way to break the ice and review some prerequisite geometry, and there were other benefits as well. It required students to translate into written English an important result often memorized as simply a^2 +b^2=c^2, which they now see is clearly insufficient. They also see an example of how a mathematical statement can appear or sound true, or authoritative, without being valid, and so they must learn to check their claims very carefully.

Sarah will describe the next activity.
Sarah
People ask us how we began using The Simpsons in class. As graduate students at the University of Pennsylvania, we were huge fans of the show. As we were watching, we noticed the math references. At the same time, as a TA, I began to think about how I could motivate and engage my students, especially those who were math phobic. After I obtained my PhD from Penn, I began teaching at Appalachian State University and was given a liberal arts math course to teach. Since I am a Riemannian geometer, I created a segment on the geometry of the earth and universe.

In a special Halloween episode, there is an entire segment in which Homer Simpson enters the third dimension. Without even being prompted, students immediately question whether The Simpsons live in two or three dimensions. Hence this is a great way to begin a discussion of what dimension means and lead up to the shape of space.

I'll talk about how I've used it in the classroom setting.
Part of Homer³ segment in Treehouse of Horror VI ⁴ 3F04 (10/30/90) Text transcript by James A. Cherry

[End of Act Two.  Time: 14:03]
Third segment title: "Homer^3"
Outside, thunder crashes as Marge calls out to Homer.
Marge: Homer!  Get ready!...
Homer3 text transcript was cut here.
Selma: Have we got a family activity for you:
Patty: A pillowcase full of seashells from our trip to Sulfur Bay.
Selma: You can help us clean and organize them.
Patty: And pry out all the dead hermit crabs.  Get a screwdriver.
Homer will have none of it.
Homer: I'll take my chances in the mystery wall!
        [goes into it]
        [steps out into 3D land, looks around]
        [thinking] Oh, glory of glories.  Oh heavenly testament to the
       eternal majesty of God's creation.
        [out loud] Holy macaroni!
Homer3 text transcript was cut here.
Homer: What's going on here?  I'm so bulgy.
        [taps his belly; it ripples for a while]
       My stomach sticks way out in front and my -- aah!
        [checks out his bulgy behind]
Homer3 text transcript was cut here.
Frink draws a strange diagram on the wall.
Lisa: Well, where's my Dad?
Frink: Well, it should be obvious to even the most dim-witted
          individual who holds an advanced degree in hyperbolic
          topology, n'gee, that Homer Simpson has stumbled into...[the
          lights go off] the third dimension.
Lisa: [turning the lights back on] Sorry.
Frink: [drawing on a blackboard] Here is an ordinary square --
Wiggum: Whoa, whoa -- slow down, egghead!
Frink: -- but suppose we exte-end the square beyond the two
          dimensions of our universe (along the hypothetical Z axis,
          there).
Everyone: [gasps]
Frink: This forms a three-dimensional object known as a "cube", or a
          "Frinkahedron" in honor of its discoverer, n'hey, n'hey.
Homer: [disembodied] Help me!  Are you helping me, or are you going
          on and on?
Frink: Oh, right.  And, of course, within, we find the doomed
          individual.
Homer3 text transcript continues.

There are equations flying around in the new dimension and Bart trying to save his father by using an x, y, z axis lamppost sign.
To begin with, the students must pretend that the Simpsons really were 2D and that Homer did transform into 3D. They explore what it is like to be a 2D creature with help from Davide Cervone's movies of a cube passing through Flatland. For example, they are asked what a 2D Marge would see when a cube passes through her plane of existence at various angles, how a 2D Marge could pass a 2D Lisa, and what a 2D Marge would have to look like in order to eat or see. After answering these questions, students are asked to write a letter from 3D Homer to 2D Marge explaining what happened to him, including changes in appearance and mathematical explanations in their own words, such as Prof Frinks explanation of the 3rd dimension. I direct them to websites containing a text transcript of the complete Homer³ segment and a description of the animation techniques used for Homer³.

The students find this challenging but fun, and come up with creative explanations and great ideas. But, one of the dangers of this assignment is that there are some students who don't take it seriously and come up with explanations that are nonsensical and have no mathematical or logical basis. Since this writing assignment is part of satisfying a writing designator on the course, I ask the students to revise their work, which takes care of the problem. For example, one student had 3D Homer explaining to 2D Marge that he was flying. For revisions, I told the student that I didn't understand what he meant and asked him to explain.

For the next lab, usually 1-2 weeks later, I give the students directions to pretend that the Simpsons had been 3D all along and that Homer had transformed to become a 4th spatial dimensional creature. They answer questions about the 4th physical dimension and then write a letter from 4D Homer to 3D Marge. This motivates a discussion on how we can understand the fourth physical dimension and we examine the shape of space with help from Jeff Week's research and activities.

Even the small details of Homer's new dimension are mathematically interesting. For example, since the equation 1782¹² +1841¹²=1922¹² flies by Homer, students are led through reasons why it is incorrect. Notice that the left hand side of this equation is odd since it is a sum of an even number and an odd number, while the right hand side is even. Also, it is an apparent counterexample to Fermat's Last Theorem, which we had discussed earlier in the semester. On the final exam, I ask students to calculate the left hand side and the right hand side of the equation on their calculator. Most brands of calculators will show that the sides are equal. Students are then asked to resolve the apparent contradiction.

Now back to Andrew...
Andrew
The title of our talk comes from a 1st season episode in which 4th grader Bart Simpson cheats on a math test and consequently is enrolled in a school for gifted children.

r dr r joke from Bart the Genius ⁵ 7G02 (1/14/90) by Raymond Chen

% Back at school...
Ms. M: So y = r^3/3.  And if you determine the rate of change in this
        curve correctly, I think you'll be pleasantly surprised.
Class: [the entire class chuckles except Bart who looks pained]
Ms. M: Don't you get it, Bart?  Derivative dy = 3 r^2 dr / 3, or r^2 dr, or r dr r.  
        Har-de-har-har, get it?
Bart: [not amused]  Oh, yeah.  [forced laugh]

Now, back to Sarah!
Sarah
Imagine two girls at a gifted school playing patty-cake while chanting digits of Pi:

Digits of Pi Lisa's Sax (3G02, 10/19/97)

Two girls at a gifted school play patty-cake while chanting the digits of pi:
Cross my heart and hope to die
Here's the digits that make pi
3.1415926535897932384...

We have mentioned only a few of the many examples of mathematics on the Simpsons. The diversity and depth of the mathematics are quite remarkable, in particular for an animated show. Because the students laugh, but then start to ask questions and engage the mathematics on their own, we can help them to enjoy learning significant mathematics with creative classroom activities related to the The Simpsons.

While the video format is the most effective and fun for the students in conveying all the information, sometimes audio and/or text combined with some still pictures, or just a verbal description of the relevant portion will do just as nicely and is less distracting.

The is also a website which has text transcripts of many episodes, including the Homer³ episode. This has been quite useful to me - that is how I found out about the would-be counterexample to Fermat's Last Theorem. A Simpsons fan had stopped the video to write down all the equations flying around. I find the transcripts quite helpful in order to reinforce learning even when I play video or audio.

I wanted to mention the Cyberworld IMAX movie. The movie is still ⁶   in IMAX theatres and is actually a 3D movie. A dvd of season 1 of The Simpsons is due out in September ⁷. with other season releases to follow. Andrew has compiled a guide of mathematical segments of The Simpsons - it is on the web and the address is on the handout.

To conclude, because it contains so many instances of mathematics, The Simpsons is an ideal source of fun ways to introduce important concepts to students, to motivate them and to encourage deep understanding. Over a period of years, only one of my hundreds of students commented on an evaluation that using cartoons was "stupid." Most students find it interesting, entertaining and creative. Recently a student in evaluations said that "the cartoons were great and fun ways to get them excited about the mathematics". And of course that is the whole point of using them! Thank you.

Handout

Footnotes