Sunday, April 7, 2013

Comics in the Math Classroom

Here are some resources for people that went to my talk on Comics and Graphic Novels in the Math Classroom at this year's NEMATYC conference.

In the presentation, I made reference to the Kuleshov Effect

As well as this really great video about how we learn:

And here is a list of comics that I recommend:


The Cartoon Guide to Calculus
The Cartoon Guide to Statistics
The Manga Guide to Calculus
The Manga Guide to Linear Algebra
The Mystery of the Prime Numbers

Science and Humanities

Wonderful Life with the Elements
The Manga Guide to Physics
The Cartoon Guide to Chemistry
Economix (Highly Recommended)
Action Philosophers


Saturday Morning Breakfast Cereal
Abstruse Goose
Spiked Math
Phd Comics


Logicomix (Highly Recommended)
Suspended in Language

Comics Theory and History

Understanding Comics (Very Highly Recommended)
Making Comics
The Comic Book History of Comics

Friday, March 1, 2013

Ambiguity in Language: Purple People Eater

Well he came down to earth and he lit in a tree,
I said "Mr. Purple People Eater, don't eat me!"
I heard him say in a voice so gruff:
"I wouldn't eat you cuz you're so tough."
This is one of the first posts I wrote when I started this blog, but it never really felt finished it. So it sat un-posted and forgotten until recent events inspired me to finally post it. It still feels unfinished.

Purple People Eater is a song is about a "One eyed, one horned, flying purple people eater" that came to Earth to start a rock 'n roll band, and it goes a little something like this:

Most people imagine some sort of cycloptic, horned, purple monster that flies and eats people. This is a legitimate interpretation, but not the only one. Not even the correct one. Listen again to the verse that starts at 0:50:
I said Mr. Purple People Eater, what's your line?
He said, "Eating purple people, and it sure is fine."
According to these lyrics, the purple people eater eats purple people. This indicates that you or I should be safe (though this guy is pretty much screwed), and that Mr. Purple People Eater will most likely be pretty hungry on a planet without purple people (like ours). This is not how most people interpret it, including the creators of the 1988 Purple People Eater Movie:

The ambiguity stems from the fact that English is not associative and there is no convention for order of operations. Should the expression be evaluated from the right: purple (people eater), or from the left: (purple people) eater. [footnote 1] If we include the entire description, there are actually 5 possibilities:

Once you know to look for it, ambiguity is everywhere: Is a "big, bad dog catcher" a catcher of big, bad dogs, or a dog catcher that's big and bad? Is Dr. Seuss's book about ham and green eggs, or green eggs and green ham? Is Clifford a big, red dog, or a Big Red dog [foot note 2]? What does Candice mean when she asks for an X-Ray of a Kangaroo with three legs?

LEFT: An X-Ray (of a Kangaroo with three legs)
RIGHT: An X-Ray of a Kangaroo (with three legs)

Is this strange ambiguity unique to English, or do other languages get bogged down by purple people eaters as well? I asked around, and here is a short summery of my results:

Purple People Eater
Cantonese Chinese
1. 紫色食人物
2. 食紫色人的物體
Mandarin Chinese
1. 紫色食人者 
2. 食紫色人者 
Spanish [3]
Comedor de gente purpura
(Alternatively: Come gente purpura)
Ο άνθρωπος που τρώει μωβ άνθρωποι
1. Lila Menschenfresser
2. Der Ungeheuer frisst lila menschen
3. Der lila Ungeheuer frisst menschen
Albanian [4]
 Njerzit ngjyrë manushaqe ngrënës



[1] Here the parenthesis are being used to group words, not to indicate a parenthetical statement.


Hooray Puns!

[3] My Spanish speaking friend points out a bonus ambiguity: "Purpura" is neither masculine or feminine, so if "purpura" describes the eater, then it's unclear whether the eater is male or female.

[4] My Albanian friend's handwriting is not fantastic, so there may be some errors

Saturday, February 23, 2013

Let's Get Mobius Up In Here

Here is an applet to explore the function $$\phi(z) = \frac{z - \lambda}{1 - z \bar{\lambda}}$$ for $| \lambda | \leq 1$. In the bottom left is the unit disc with Darth Vader super imposed on it. On the top is also the unit disc. For each point $|z| < 1$, the applet determines the color of $\phi(z)$ and and then colors $z$ that color.

The unit disc on the bottom right allows you to adjust $\lambda$. Just click anywhere in the unit disc and $\lambda$ will change appropriately.

Note that this demonstrates that $\phi$ maps the unit disc to itself. Click here for a proof.

Wednesday, February 20, 2013

Magic Squares

"Alright settle down, Tom, don't you dare throw that paper airplane. (Alright! We have a sub!) You all know the deal, sit down. Mrs. G. left instructions to have you do the Scholastic Math (I hate those!) but she also said that if I wanted, I could do something else (Yeah!).

"So here's the deal, everyone has to draw this grid on piece of paper...

"...and fill in each of the squares with the numbers 1,2,3,4,5,6,7,8, and 9. (Is this like Sudoku?) It's a little like Sudoku. (I hate Sudoku.) Ok, it's nothing like Sudoku. You have to put the numbers so that all the columns add to 15 and all the rows add to 15. If you're really bright (Well that leaves me out.) then try and get it so that the diagonals add to 15 as well.

"(This looks hard.) It is hard. (I suck at math.) That's ok, math is hard, just keep trying."


"(Oh, I think I have it!) Awesome! Now try for the four by four case.

"(What numbers do we use?) You have to use 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, and 16. (What do they add up to). I don't know. (You're not going to tell us what they add up to?) Nope. (Then it's impossible.) No it's not. (How can we make all the rows and columns add to the same number if you don't tell us what they have to add to?) Maybe you should try and figure it out. (How?) I don't know. Anyone have any ideas?

"(Add up all the numbers and then divide four.) That's an awesome idea. (Can we use calculators?) No. (But...) No. (But...) No. (You're mean.) Yep. (So we have to add up all the numbers by hand?) That's not how I would do it. (How would you do it?) I would start by adding all the numbers twice:
 "Why might I do that? (Because you're crazy?) Any other ideas? No? Well, what's 1+16? (17) and what 2+15? (17) and what's 3+14 (17) and what's the pattern here? (They're all 17!) And how many 17s are there? (16)
"(So the answer is 17 times 16?) (That's magic!) Not so quick. Remember that I added all of the numbers twice. (So the answer is 17 times 16 divided by 2?) Yes, that will give you the total. (Can we use calculators now?) Of course not. (So we have to do 17 times 16 by hand?) Not at all. Who can tell me how to simplify this?
"(Cancel the 2) And? (17 times 8... is that the answer?) Well we wanted to divide the answer by 4, remember?
"(So it's 17 times 2) What's that? (34!) Fantastic! Alright, somebody figure out the 4 by 4 case. (And then are you going to make us do the 5 by 5 case?) Yep."


Of the three seventh grade classes I taught (about twenty students a piece), three students managed to solve the 4 by 4 magic square. One student skipped recess to work on it. Many students did not take to this assignment. One girl in particular was straight-up pissed. (Things were thrown.) Eventually though, she figured it out and was elated.

Monday, January 21, 2013

Guess the Function

I substitute teach middle school. Anyone that's ever subbed (especially middle school) knows that one of the cardinal rules of subbing is to not allow any downtime. (I have a story involving downtime in a middle school classroom that ends with a student shouting, "No Brad, you're stapling your face wrong!") Because of this, I'm always prepared with a math game or activity to occupy my students.

My favorite game (and often their's too) is Guess the Function. The rules are to guess the function. Allow me to elaborate:

First I write down a function on a piece of paper. The difficulty of the function varies with the skill level of the class, but usually I start with a linear function.

Next I generate inputs (sometimes I do this by rolling dice, sometimes I ask students to shout numbers out, and sometimes I just pick numbers myself).

Then I write the number and the result of applying the function to that number on the board. Students then have the option of guessing my function, or waiting to see the result of my function on more numbers.

I give one point for right answers and penalize two points for wrong answers.

For example: suppose my function is f(x) = x + 2. I would start by writing 2  4 on the board (because f(2) = 4). Now bold students might start guessing, but that's a bad idea. With the available information, they might guess f(x) = 2x or f(x) = x2, and they'd lose two points. After everyone has either guessed or passed, I would write another pair of numbers on the board (for example 3  5).

Some notes:

  • I use arrow notation (3  5) instead of function notation (f(3) = 5) because I usually play this game with students that haven't been exposed to function notation.
  • I usually have students play this in teams of four or five.
  • Usually, sixth graders are much better at this game than seventh and eighth graders. Typically, seventh and eighth graders tell me that they "haven't learned this yet," whereas sixth graders don't seem to know that they don't know.
  • Mathematicians will point out that for any finite collection of points, there is literally an infinite number of functions that will fit the points. This is true, but I like to think of this game as a game about psychology (what am I thinking?) as much is a game about math.
  • Younger students often give English descriptions of rules (like plus two or times itself), whereas older students are comfortable with describing things algebraically (like x+2 or x2).
  • I use the scoring system +1 for right answers and -2 for wrong answers because I want to discourage wild guessing. (Recall that as a sub, my goal is primarily to keep a class under control-- wild guessing descends in to chaos rather quickly.)
  • When I play this game, I announce that I'm writing down a rule (not a function) because most of my students don't know what a function is.
  • Examples of functions that I would use include x+1, 2x, 2x+5, x2, x(x-1), 2x, x/2, 1/x, 10 - x...
I love this game. One of my favorite things is that since it's not a part of the curriculum, the students that "aren't good at math" actually do quite well. This is because the psychological hurtles they have when facing math class aren't in place when they're just playing some dumb game the sub made up.