Where does Pac-Man live?

Here’s a question I’ve always liked: what kind of world does Pac-Man on?

We know we live on a sphere, or course, or something close to it. But we often imagine our world on a rectangular map. What are the rules? Well, if you go out the left side, you come back on the right side. If you go to the top or bottom, you arrive a single point (the north or south pole). So following the same logic, what are the rules in Pac-Man’s world? If he goes out the left side or his world, he comes back in the right side. But—if he goes out the top, he comes back in the bottom. Our map represents a sphere. What shape is he living on?

You could also think of it like this: take a stretchy square of paper, glue the left side to the right side and the top to the bottom. What do you get?

It turns out that this shape is of great mathematical interest. See if you can figure it out. If you get stuck, there’s a nice video here.

The nightmare

The other day a friend related a math nightmare. She was in a prison-like compound. People guarded the exits, and they wouldn’t let her out until she solved a calculus problem. She couldn’t ever do it, and her teeth started to fall out.

Apparently this had been a recurring nightmare for her through high school. I wonder how many people have had dreams like this about math. Or any dreams where math played a starring role.

Mathematical Flimflam

Link: Mathematical Flimflam

One thing that’s always struck me is how mathematics has been interpreted by so many people in so many contexts as a direct conduit to God’s thoughts. This link is a modern manifestation of that—but sadly it’s a pretty shallow vision of what God’s thoughts look like. There are some pretty identities, but under deeper scrutiny they’re not particularly revelatory. This is a kind of mathematical flimflam—it looks like there’s something deep going on under the surface, but really it’s quite superficial, and has more to do with how we write numbers than what their inherent qualities are.

Then we have a segueway into assigning numbers to letters and interpreting words. There is no meaning here at all.

I worry that the inherent love for math that (I believe) we all have in us leaves us open to being drawn in by these kinds of vacuous mathematical runarounds when we don’t have authentic experience with math to compare it to. There are lots of ways to use mathematics to help a lie—it happens in religious thought, politics, economics. Being separated from our mathematical birthright leaves us open to being tricked. And plenty of people are ready to trick us.

Check out the comments as well. Very interesting responses, to my mind. Quite passionate. And a couple skeptics, who demonstrate why the numbers to letters argument doesn’t mean anything.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

G. H. Hardy (1877 – 1947)

This is an entertaining talk on education and creativity, though not specifically about math. There are some powerful stories near the end in particular.

One Manifesto

Link: One Manifesto

People have been writing about the failure of math education for a long, long time. For example:

S. K. Stein, Strength in Numbers, John Wiley & Sonse, 1996 

If you browse through The Mathematics Teacher, the main journal devoted to instruction in mathematics you will find constant lamentation, going back to its first volume in 1908, where one teacher wrote, “One of the most obvious facts about mathematics in our schools is a general dissatisfaction.” The tone in 1911 was even less cheery, “Our conference is charged with gloom. I have attended funerals, but I do not remember a more mournful occasion than this. We are failures and our students are not getting anything worthwhile.”

Year after year, the complaints in The Mathematics Teacher persist. I will skip ahead to 1958, when we read, “The traditional curriculum is meaningless, and by heading for abstract mathematics the modernists are moving further from reality.” … Still, in 1994, the University of Chicago School Mathematics Project complained, “The student today still encounters a variant of the elementary school curriculum designed for the pupil of a hundred years ago.”

I cribbed this tidbit from the website this title links to. I think the author of this site does a good job laying out a comprehensive manifesto—if you have some free time, check it out. There are plenty of specific anecdotes and quotes that are worth reading (and I’ll be putting my favorites of those up here from time to time). There are also lots of math puzzles. A warning though—most of them look pretty tricky. If you’re timid about math, this probably is not a good place to start. If you’re more experienced and looking for a challenge, it may be right up your alley.

To return to education for a moment, I quoted Stein above to make the point that there is virtually no time period we can point to where mathematics was generally well taught. Personally, I think that the problem is deeper than math education. The point is to teach students to think independently and creatively; everything else should support this goal. But the particular problem of mathematics is that most people don’t understand that it is a subject where creative and independent thought is possible. To quote the great mathematician and teacher G. Polya:

“A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.”

There are long standing arguments about how best to teach math. At this moment, they tend to be polarized between entrenched, immovable opponents. Both sides, of course, have plenty of valid points, and both sides, at some level, miss the whole picture. To be taught well, mathematics needs to be motivated. The reason to learn it is not because you’ll need to use it someday—it’s because it’s interesting now, and because you have questions now that you want to answer.

In my experience, there is a near universal interest in math. I’ve seen students who were classified as lost get joy out of the subject. But we have to get people—students, teachers, and the public—to do math in order to see that they actually enjoy it. Music you can listen to. Art you can look at. Math you have to do.

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.

J.H.Poincare (1854-1912)

Problem of the Week (or, ask another question)

Link: Problem of the Week (or, ask another question)

My old school, Saint Ann’s, in Brooklyn Heights, has an ongoing problem of the week competition for its middle schoolers. The problems tend to be fun, and require a little more thinking (and explanation) than most standard math problems, and the teachers have fun brainstorming new ones.

This one is pretty cute, and I think you can solve it if you want to. On the front of asking good questions though, I think there are some pretty interesting followup questions you could ask about this problem. The main one is, what happens to the ball after it hits the right side of the table? It will keep bouncing around the table, but will it ever go into the corners? Will it ever hit its starting position again?

Following these questions up, you can start to vary the size of the table and the angle and placement from which you shoot the ball in the first place. I suspect there’s a whole theory of how the ball bounces around the table that is elegant and satisfying. You just have to follow the initial question with a few more of your own.

Anyone else have any thoughts on good questions to ask about the billiard ball bouncing problem in the link?

After Tutoring

Just had a fun little tutoring session with a new student. (How could it surprise you that I tutor?) She strikes me, happily, as one of the many people who have a decent aptitude for math but, sadly, don’t like it much and don’t think they’re any good as it. How much I enjoy tutoring shocks me sometimes. The last problem got me genuinely excited: a neat geometric picture involving a stationary circle with another circle beside it shrinking away to nothing. You make a line from the top of the first circle to the place where the two circles intersect. The question, basically, is where does that line intersect the horizontal line through the center of both circles? We both looked at the picture and both of us thought, intuitively, that it should go off infinitely down the line.

But how to prove it? I suggested translating the problem from geometric to mathematical language. It involved quite a few steps, and wasn’t at all obvious how to do it at first, but she advanced from one to the next with little prompting from me, and we managed to state the whole thing as an algebra problem, where the answer would be given by a limit. Kind of astonishing that it’s possible to translate pictures into mathematics like this. We ran the limit, and with a little more work, found that the answer was: 4. A very finite number.

I have to say, this is one of the great things about math: you get surprised. Where did our intuition go wrong? What did the 4 have to do with the geometric picture (it was a graph, I should mention). I was immediately seized by a desire to start messing with the initial conditions to try to understand what made it be that number, and not another. And why not infinity? Something was wrong with our intuition, it seems. It’s a powerful thing to refine your intuition, and very satisfying.

The work of translating from one situation to another is also very key to the art of mathematics. Many people assume that all you need to do is solve problems, but before you can solve a problem you usually need to look at it from different perspectives. This algebraic/geometric correspondence is one of the richest in mathematics, in my opinion. It’s these connections between fields where the true insight lies.

A little example: have you ever noticed that as you go from one square number to another (1, 4, 9, 16, 25, 36, 49, etc.) you’re jumping an odd number each time (1, 3, 5, 7, 9, 13, etc.)? Looking at that arithmetically isn’t too enlightening: what does going from 3X3 from 4X4 have to do with the number 7?

But if you make it geometric, and think about a 3 by 3 array of dots going to a 4 by 4 array of dots, it becomes clear how to add those seven dots (in an L shape around the outside). Think about it a little more, and it becomes clear why it should be seven, and why it should always be odd.

And before you’ve known it, you have a proof of the purely arithmetic fact that the sum of the first n odd numbers is n^2. You can figure out that 1+3+5+7+9+…+91+93+95+97+99 = 2500 other ways (and I encourage it!), but the argument you get from playing with the square arrays sure is sweet.

Finding a good question

I just met with my advisor, Sandor Kovacs. Meeting with him is great, and I walk out excited to get to work. No exception today, but my work feels particularly slippery at the moment.

The reason is that I don’t yet have a good question. I’m narrowing in on it, but it keeps eluding me. There’s a truism in math that finding a good question is half the work. I estimate that locating mine is going to take me pretty much this entire academic year.

I’ll be teaching a course this summer called Turtles All the Way Down, in which I hope to teach, among other things, the art of asking questions. Finding your question, in or out of mathematics, can inspire your creative work for years. Yet we devote so little time to learning how to do it well.

The fun thing about math is that the classic questions which begin math usually feel like they should be easy. Doubling the square is a good example. If I give you a length and tell you to double it, it’s really easy to do it (just take two of those lengths and put them together). Likewise, if I give you a square (imagine a one inch by one inch square if you need a specific example), and tell you to double that—by which I mean double the area it take up—it seems like it should be just as easy: just double the sides of the square. But then you have a two inch by two inch square, and that’s four times as big as a one inch by one inch square (draw it!). So how do you actually get the square that’s twice as big?

A seemingly innocuous question. The answer isn’t terribly hard, but it’s far from obvious, and requires some insight. Once you see it, though, there’s that aha! feeling. (Plato brings it up in the Meno). It also points in a very interesting direction. In fact, if we follow this question as fully and honestly as we can, we end up travelling the path that leads to modern mathematics. My old colleague Paul Lockhart once remarked, after a very bright second grader he knew cried in frustration when he was trying to work out an implication of this problem, that these were the tears where mathematics begin.

Off to work!