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# A Math Lesson in Three Acts

(With apologies to Dan Meyer)

I’ve often had a gut feeling that we actually invest life into math questions that grab us. Here’ s a question I like:

Question: How does a bishop behave on a torus chess board?

Many people, reading this question, might be turned off if the words are unfamiliar (bishops are chess pieces that move diagonally; a torus is what you get if you glue opposite sides of a rectangle together: like your computer screen in a game (for example, some Pac-Man versions) where you can move off the right side of the screen and come back in the left side, and ditto for the top and bottom). However, if the language isn’t in your way, or if you get past it (try http://www.youtube.com/watch?v=0H5_h-RB0T8 for example), there’s an interesting question here.

But even if you’re familiar with the language, the question may turn you off. It’s vague, after all: what does “behave” mean? “What do they want me to do here?” we can imagine the student crying.

Well, there’s no “they,” and I don’t want you to do anything. I’m just trying to share a question that is, for me, the launching point of a strange and exciting story.

So maybe I should just tell the story as it progresses for me.

Act 1: Stasis and disequilibrium.

A bishop sits on a chess board.

Casually, it looks off the right side of the board and sees that it is connected to the left side, and that the top is connected to the bottom. The bishop is the same (it moves diagonally), but there’s more to its world than it thought.

Not only that, the bishop is capable of different things in this different context. Suddenly the walls that prevented it from continuing its movement are gone. It progresses on the diagonal moving down and left to the bottom of the board and then through it. The old boundaries are gone! In fact, the diagonal is a kind of circle now, for when the bishop continues on its path, it wraps back to where it started. The world is different, the rules have changed.

How many squares can the bishop reach now? Which ones are they? What if it starts on a different square? Something in me wants to know.

If something in you does too, then stop reading this post, put your computer to sleep, and try to answer these questions.

Act 2: Escalating crises.

Once our character is out of its comfortable initial world, it is faced with a series of escalating crises. If we’re reading a novel, these come from the writer; here, they come from us. Doing math is partly a matter of writing the story as you go. This is one of the joys of the subject: you have more authorial control here than you can imagine. It’s also one of the difficulties: you can’t be a passive member of the audience. You make the bishop and the board happen. This is why you have to actually do the work if you want to experience the story in its fullest form.

If you have, maybe you know that the bishop can always reach the same number of squares wherever it starts (14 squares to be precise, counting the one it starts on). Then there are deeper questions: why is this true? The bishop is getting a sense of its power in these new settings.

There are the immediate questions I have, like–why 14 squares? How many would it hit on a smaller board, like a 6 by 6 board, or a 4 by 4 board? But these changes of scenery are precisely the new crises for our main character (the Bishop! Perhaps it’s because I played chess a lot when I was younger that I identify with chess pieces as characters. I like to chalk it up to my tremendous wellsprings of empathy). Suddenly, our question has widened: what exactly are the bishops powers when the board changes size?

And that’s just the beginning of the trouble! Once the board starts changing sizes, we have to consider the possibility that it changes shape too. What if the bishop lives on a rectangle? How many squares can it hit if it lives on a 4 by 6 rectangle for example?

At this point, our hero is starting to doubt facts about itself that it always took for granted. Nothing seems stable. Fortunately, we know some things about bishops for sure. For one thing, bishops, since they move diagonally, always stay on one color of the chess board: either they’re on the white squares or the black squares.

But, wait! Even the most cherished assumptions can be false! For on this board, the bishop moves from white squares to black squares. In fact, if you trace its diagonal, it’s actually attacking every square on the board!

Here, in the diagram to the left, you can see the beginning of the bishop’s move. On some board’s, the bishop is more powerful than anyone has ever dared imagine.

That may be a bit melodramatic, but I do feel a quickening of my pulse as the story continues. And the questions draw me in: when does the bishop hit every square? Is there any way to know how many squares it will hit?

Act 3: Climax and Resolution

The denouement comes with what some people call the aha moment, that instant of revelation where you see what’s really important when it comes to the bishop’s powers and all the other clutter and confusion falls away. As anyone who has felt it knows, it’s a magical moment. There are a lot of us suffering a lot of frustration because it’s so beautiful when that moment comes.

That’s why I’m not going to tell you the answer. Because with math, the most powerful endings are the ones you write yourself. And once you resolve the issues for yourself, the resolution is meaningful in the deepest kind of way.

Postscript: A New Stasis

Once you solve the problem, you experience some kind of mathematical version of catharsis, and what you’ve learned, both the facts and the arc of the story, settle into you and inform how you see and what you know. Bishops look different to you now, as do chessboards. But for the moment, things are settled.

But even this new stasis is deceptive. Didn’t we start with stasis? All we need is some new interruption, some idea to throw things into disarray again. Suddenly the world we know is ripe with possibility, ready to spin into disequilibrium as our curiosity nudges it.

Of course, maybe you don’t care for this problem. There are probably a lot of novels you won’t like either. But experiencing real stories, and real mathematical inquiries, is one of the most enriching thing you can do for yourself. Maybe this one doesn’t grab you, but one will, and when it does, you can be thankful that you are a human being, and human beings, for some miraculous reason, get to live their lives engaged by stories, by compelling questions, and by mathematics.

1. Bryan Meyer

This is a beautiful post! In particular, I loved your closing remark that “human beings, for some miraculous reason, get to live their lives engaged by stories, by compelling questions, and by mathematics.” I strongly believe that humans have a natural tendency to be intrigued by puzzling situations, make sense of our world, and remedy cognitive conflict (disequilibrium). In this way, teaching and learning mathematics is enjoyable because it appeals to our natural intuition. I think your “three acts” here could/should serve as a lesson format for all teaching (particularly in math). I wonder if Dan based his on a similar philosophy or if he agrees with this?

I have written recently about similar ideas (http://www.doingmathematics.com/2/post/2012/03/constructivist-learning-in-action.html) and have been trying to figure out how to make this type of cognitive resolution a daily part of my class. It isn’t always easy but I feel the attempt has made my teaching measurably better. Thanks again for a great post!

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Dan

Thanks, Bryan!

I like how you let the students make a prediction that reality proved wrong–thanks for sharing the link. The idea of disequilibrium is one that I return to over and over, and that is central in teaching, in my opinion. It’s one of the great motivators.

I think Dan and I are both students of stories and other things that engage people. We don’t like standing in front of a class and being boring. I think we’ve come to a lot of similar conclusions, but we also have a lot of differences, which are probably in large part a matter of who we teach and where, and what constraints are on us. Dan likes to hook his students with good visuals, for example, and uses real world examples. While I absolutely agree with his emphasis on leading with a great, concise question, I tend to favor pure mathematical questions, and don’t care so much about visuals. I don’t think either way is fundamentally better: utility is the final test, and my guess is that both of our methods work well in the situations where we teach.