The Quarterback and the Professor
How an NFL quarterback taught complex analysis
Frank Ryan is not just a theorist, but is also a mathematician who specialized in complex analysis. He got his Ph.D from Rice University on “A Characterization of the Set of Asymptotic Values of a Function Holomorphic in the Unit Disc,” in 1965.
Today I would like to talk about how we learn, and how we teach.
I have a story to tell about Ryan—I just shared it the other day with Alan Kay—and he insisted I had to post on it. Right away. So here it is Alan.
Ryan was not only a professor at Case Western Reserve, where I was an undergraduate in the ’60’s, but he was at the same time the starting quarterback for the Cleveland Browns. The starting quarterback for a NFL football team, and a professor with a full teaching load. During the football season he taught his class early in the week, and then on Sundays he was behind the center taking the ball. Handing it off, throwing passes, and getting sacked—like any other quarterback in the league. He was one of the best quarterbacks of his day, and had great successes. For example, he appeared in three straight Pro Bowls.
I still remember listening to him explain a fine point in complex analysis on Tuesday, and then watching him on TV getting tackled, on Sunday. It was hard to believe, even though I knew it was the same person, the player being taken hard to the ground was my professor. The player being tackled knew how to throw a perfect spiral 40 yards, and also could go to the board and prove Picard’s Little Theorem. Amazing.
Today, I believe there is no chance a quarterback—or any player—on an NFL team would want to or be allowed to be a full time professor during the season. The game has gotten very technical, the pay is too great, and the stakes are too high for any team to allow this to happen. But, back when I was taking complex analysis it happened. Really.
I still recall wincing when he got sacked during one tough game. Then, a few days later I saw him in class, with his arm in a sling and his speech slightly slurred. I assumed the slurring came from pain killers he was taking for the shoulder injury. Sling or not we pushed on, deeper into the beautiful structure of complex analysis.
As an undergraduate I took a seminar with Ryan on complex analysis. This was one of strangest classes I ever had in mathematics, and probably one of the best. It was a small group, about eight of us, taking his class on advanced topics in complex analysis. The course was based on a thin monograph, but Ryan did zero lecturing. Instead, at the beginning of each class he ran the following protocol:
- He would shuffle up a deck of playing cards, and we all would gather around a table.
- He then would deal out the cards one at a time face up on the table: we each got the cards landing in front of us.
- There were two bad cards: the Queen of Spades and the Queen of Diamonds.
Once these two cards were dealt, the class really began. The player who was “lucky” enough to get the Queen of Spades went to the blackboard and was expected to start explaining from exactly where we stopped at the end of the last class. You were allowed to use the book or notes and you typically explained the proof of some theorem.
After half the class was over, it was the other “lucky” person—who got the Queen of Diamonds—to take over from the first student.
Sounds not too hard, but it was a killer. The main problem was the thin book’s concept of a proof was not a detailed proof, but at best a high level sketch. Proofs were filled with phrases like: “it is easy to see that is continuous,” or “it clearly follows that is never in the unit disk.” The person at the board would say these phrases, but Ryan would usually jump in with a simple “why?” Why indeed was continuous? Why indeed was never ?
Sometimes the student at the board could answer and we moved on to the next point, but often they got stuck. The rest of us could help and make suggestions—of course we were usually lost too. The class might stay on a single question for the entire first half of class. If this happened, then the next student would have to get up and try to convince Ryan why it was true.
The student who was up second had an interesting prediction problem. They had 45 minutes to prepare for their turn, but they had no idea where the first student would get to in their 45 minutes. I remember being in this position—half listening to the class while trying to guess where I would have to start explaining.
The cards, the Queen of Spades and the Queen of Diamonds, were picked as the “bad” cards for a reason. The first is, of course, the worst card to get in the game of hearts: the player who is stuck with this card gets 13 points. The second is based on the original movie “The Manchurian Candidate”. In the movie this card is used to trigger Laurence Harvey to follow orders without question. One of the great movies, in my opinion
I sometimes wonder how we learn and how we should teach. A while ago I posted on EEE and still think about this—the Educational Extinct Event.
I hated Ryan’s class. One consequence of the way the class was organized was I learned relatively little material. In a more conventional class I think I would have learned more theorems, more proofs, more concrete facts from complex analysis.
I loved Ryan’s class. The class taught me to think on my feet—literally. I learned how to read a proof and find the “gaps” I needed to fill in myself. I may have learned relatively little complex analysis, but I learned a great deal about mathematics in general.
By the way Picard’s Little Theorem, named for Charles Picard, is:
Theorem: Suppose is an entire function. Then, the range of is either the whole complex plane or the plane minus a single point.
The function shows the theorem is best. A very beautiful theorem.
The main open problem is this: what is the best way to present mathematical material? I am especially interested in hearing what you think about Ryan’s method. Did you have some similar experience? Should we teach more classes in this way? Or is it better to cover lots of material?
I am currently at the major meeting of the ASL—the Association of Symbolic Logic. I will give you an update on what it is like at a logic meeting. The reason I am here is to chair a session held in honor of the Gödel lecturer, who this year is our own Sasha Razborov.