Brown math students, Daniel Katz has a deal for you. As a puzzle maven and a trained actor, Katz will make calculus more fun than you probably expect. And as long as you are paying attention, he’ll make sure the odds of learning are in your favor.
“In making puzzles, you want to create a positive experience for the solver,” said Katz, an alumnus of the Brown Graduate School who has returned to be the department’s first lecturer, a position that gives him responsibility for overseeing how calculus and much of undergraduate mathematics are taught. “When I design homework problems or exam problems I think in a very similar way. A friend of mine describes the art of writing puzzles as setting up an intellectual battle with the solver that you intend to lose. If you win this competition the customer is not satisfied. There is an art of giving students a problem where the tools they’ve learned in class will help them to solve it.”
That art requires thinking like the student. There should always be a logical thread they can follow if they bring their wits and their acquired skills.
“I construct the puzzle by solving it,” he said. “What’s one small piece of info I could come across as a solver that I could use? What’s the next thing I’d like to see?”
Katz’s goal is not necessarily to make math easy — students need to learn real skills and meet high standards — but to make it engaging. When he designs a problem, he can lead students to solve problems that are at least a little different from ones they’ve seen before. That helps prevent math class from turning into a droning exercise of rote regurgitation.
More fulfilling problem sets are only part of the promise. Katz is also well trained to hold an audience’s attention. When he graduated from MIT in 2003 he not only received a bachelor’s degree in math but also in theater arts.
“In describing teaching to people, more than once I have used the word ‘audience’ instead of ‘students,’” Katz said. “There are definitely differences — teaching is much more interactive — but there are the simple technical things of people being able to hear you and making the delivery interesting, and also reading the audience. When you are doing comedy, you can tell when the audience is not with you. If that happens you don’t just go on doing the exact same thing in the exact same way.”
Katz, a native of the Philadelphia area, came to Brown after MIT for graduate study in number theory. The work he did here sounds appropriately graduate level: Graph Theory and Recursive Towers of Function Fields earned him a master’s degree in 2005 and Sumfree Subsets in Cubes of Arbitrary Dimension was his Ph.D. dissertation under adviser Joseph Silverman in 2009. But in his first job as an assistant professor at Guilford College in Greensboro, N.C., Katz focused on a heavy undergraduate teaching load and on changing how young students view math.
“There is sort of an acceptance in the United States that it’s okay to not understand math,” Katz said. “Too often it leads to people as soon as they come into the math classroom just assuming this is not going to be for them.”
Katz realized that his professional passion to make math accessible melds perfectly with his personal passion for puzzles (he’s a regular at competitive puzzle events). Most undergraduates won’t have the tools or the inclination to ponder cubes of arbitrary dimension, but they can be familiar enough with Sudoku to become drawn in by deceptively complex questions such as, “How few numbers can be given at the beginning of a Sudoku puzzle that would still make it possible to find the solution?”
Katz has therefore begun to do research on the mathematics of puzzles. In January in Boston, for example, Katz delivered a whole lecture on a Sudoku variant called Futoshiki at the Joint Mathematical Meetings, a major national conference. Futoshiki adds the twist of inequalities to Sudoku and therefore presents some different mathematical questions.
Traditionally Futoshiki involves just one grid to fill in, but Katz likes to find new ways to tinker with the format, ensuring that like his homework sets, the exercise doesn’t just feel like another mundane matter of carrying out the same old techniques. (He designed a multigrid Olympic-themed Futoshiki that is available online.)
“I like this idea of adding an extra bit that changes the rules and changes the game,” he said.
At Brown he plans to continue to shake up the way undergraduate students sometimes view math.