Lessons in math don’t have to be so boring. Physicist Frank Wilczek on logic puzzles, games of chance and other ways to entice students.
You might not expect to find challenging mathematics on supermarket magazine stands, but it is there in abundance. New collections of cerebral puzzles are always coming out (including a weekly feature in this section). Many, such as Sudoku, involve pattern recognition. Others, such as Kakuro and Kenken, bring in simple arithmetic. Still others smuggle in graph theory and topology. My favorites are logic puzzles.
Though never labeled as such, all of these puzzles involve the same kinds of thinking as formal mathematics. Yet many people think of “math” as something scary and of puzzles as something fun. The reason for this paradox is that they’ve been misled about what mathematics is. Their main exposure to something with that name, in school, is often an off-putting ritual featuring memorization and mindless replication of useless abstractions.
Can we do better? An idea from economics, “revealed preferences,” may be helpful. To understand what people enjoy, look at what they choose. Those racks at the supermarket are telling us something important.
Whole magazines are devoted to logic puzzles, which come graded in difficulty from one star (suitable for beginners) to five stars (fiendishly difficult). The situations that they describe vary widely, from the everyday to the surreal. You might, for example, have lists of characters, presents and holidays, and the problem will be to figure out, from a bunch of clues, who gave what to whom, when.
If you consider the clues as axioms and the solution as a theorem, you’ll recognize that these puzzles embody the same logical structure as Euclid’s geometry. But they are easily digestible miniatures, self-contained and attuned to the human taste for narrative. In a wonderful variant called logic art, you deduce instructions for filling in a grid that ultimately produces a picture.
The branch of mathematics that governs logic puzzles is called propositional calculus. It is fundamental not only for mathematics but also for computer science. It’s a fascinating and open-ended problem to program a computer to solve them. (When my daughter Mira was in high school, we played around at this. Our programs got to the point that they could solve the four-star problems in a few minutes on a 2000 vintage laptop, but we didn’t do as well with the five stars.) Recreational logic problems can be a gateway, leading to a serious commitment to thinking and programming.
Another revealed mathematical preference, this time in geometry, comes to us from the Italian Renaissance. Around 1413, Filippo Brunelleschi discovered perspective—the art and science of capturing, in a drawing, the proportions of how things actually look. Contemporary artists including Masaccio, Donatello and da Vinci took up Brunelleschi’s constructions enthusiastically. Within a few decades, they created masterpieces that people have enjoyed and admired ever since.
Perspective introduced a new kind of geometry, called projective geometry, into mathematics. The concepts of projective geometry permeate the most vibrant, advanced parts of contemporary mathematics and computer graphics. Yet they can be introduced, following Brunelleschi, in rules for drawing that allow students to create splendid, convincing town squares and buildings within minutes.
I myself only learned these techniques recently, and it’s been a magical experience to play with them. To me, it is a no-brainer that this experience should be a very early part of the geometry curriculum. It’s another way into serious thinking and programming—and, of course, into art.
We know that people like games of chance and gambling. These lead naturally into adventures in probability and statistics, which can be tested in entertaining experiments. And these adventures scale up. Just a few steps take us into hot developments in big data.
I should admit that some very important branches of mathematics don’t have immediate entertainment value, at least for most people. Linear algebra, for example, is the language of quantum physics. Learning it is an essential part of understanding how the physical world works. Yet the early parts of linear algebra are quite dull and abstract. One must have patience to persevere until the more advanced, and spectacularly beautiful, parts of the subject open to view.
—Dr. Wilczek, winner of the 2004 Nobel Prize in physics, is a professor at the Massachusetts Institute of Technology and the author of “A Beautiful Question: Finding Nature’s Deep Design.”
Still, the more addictive parts of math are a good place for students to start. Once they’re hooked, they’ll be ready for the harder stuff.