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.

Ryan’s Seminar

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 ${f(z)}$ is continuous,” or “it clearly follows that ${g(z)}$ is never ${0}$ 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 ${f(z)}$ continuous? Why indeed was ${g(z)}$ never ${0}$?

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

Learning Methods

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 ${f(z)}$ is an entire function. Then, the range of ${f(z)}$ is either the whole complex plane or the plane minus a single point.

The function ${f(z) = \exp(z)}$ shows the theorem is best. A very beautiful theorem.

Open Problems

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?

$\displaystyle \S;$

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.

1. March 18, 2010 4:21 pm

I tend to find that when learning is easy, it doesn’t stick as well. It’s just a bunch of facts in one ear, and then out the other.

Anybody can (eventually) learn to manipulate symbols using a rule set, but until you can do more with it, you really don’t understand it. One version of ‘more’ is in explain things other to others. Another version is in predicting similarities.

Classes I remember (twenty years later):

– A small group discussion combinatorical analysis. Size and interaction where great
– Ancient ruler and compass proofs (a whole class on them). We had to prove things using only what was available 200o years ago.
– Group/Ring theory, but because the textbook talked a lot about people, history and practical applications.

Recently I was reading a book on Number Theory that intermixed the theorems and formulas with their history. That combination makes it more real, more compelling, I think. The state of our knowledge was driven from its history, so it is a richer learning experience to understand it in that context.

March 18, 2010 4:24 pm

Everything I’ve read on the topic indicates that there’s a lot of solid evidence showing that an “interactive” approach where students are forced to debate, teach, and work amongst themselves is a much more powerful teaching method than lectures. One recent talk I watched addressing this is this one by Harvard physics professor Eric Mazur.

Anecdotally, I totally agree. I learn OK from lectures if I ask a lot of questions, but working with other people also trying to learn the material is much more effective.

Whether your professor’s specific approach is the very best way of doing so or not seems probably secondary to the fact that at least he escaped from the lecture-and-Q&A paradigm.

March 18, 2010 6:31 pm

I very much enjoyed that talk – thanks for linking!

I especially liked the quantitative data he had.

March 19, 2010 2:26 am

Very interesting talk, thank you for the link. Do you know if somebody has looked into this idea specifically for mathematics or algorithms teaching?

March 18, 2010 4:31 pm

Fascinating Story! I did not think that was possible.

Just very recently I read the famous essay “Lockhart’s Lament” which is also very stimulating on the topic of math education (it is the first google result).

Personally, I think it is better to cover LESS material, but to teach more mathematical maturity. You can always learn what you need, but you need to build your math “muscles” first.

Also, it seems like he was using the “Moore Method” which was populartized in Texas.

However, there might be some practical drawbacks against this type of thing. Perhaps it only works with small classes of very advanced students?

March 18, 2010 6:26 pm

Picard’s theorem needs the assumption that f is non-constant… =)

March 18, 2010 7:59 pm

Sorry, good point.

March 19, 2010 11:44 am

This is likely the most minor point ever made on this high-level blog but it seems to me that a more interesting way to phrase Picard’s theorem is to move the “non-constant” part into the conclusion: The range of an entire function is the whole plane, a single point, or the plane minus a single point.

March 18, 2010 6:44 pm

If you are teaching to “smart” people then Halmos gives this as an example in his auto-math-ography.

As I recall, it starts with him handing out a list of theorems that they needed to prove for Linear Algebra, and with him writing out the definition of the vector space. That was it.

They struggled. The first two weeks, they could barely muster the barest of proofs (“there is a zero”, etc.) He was guiding all the way (throwing in an occasional definition) but they had to do ALL the hard lifting themselves.

Tells you something (about Halmos and Chicago in those days.)

Anyway, in the complex analysis class, there’s a straightforward “philosophical” question that shows whether or not you understand the subject. Pity it’s never actually asked in any textbook.

A real function could’ve 16 derivatives but not the 17th whereas a complex function, if it has one, is infinitely differentiable.

Why?

6. March 18, 2010 7:06 pm

This sounds very similar to the Moore method. I’d be interested if you think there are some important differences between Ryan’s approach and a modified Moore method class.

One of my professors at UT Austin, Dr. Cline gave a talk this past year on using the Moore method in computer science classes. He had taken a class from Paul Halmos on the method and described the effect on him personally:
Pros:
A lifelong mastery of the material
Supreme Confidence
Cons:
Guilt/consternation associated with reading other people’s work
non-reliance upon books or journals or colleagues

7. March 18, 2010 7:21 pm

Picard’s Little Theorem is named after Charles Émile Picard. And he mostly used his name Émile rather than Charles, for example on his papers. By the way, in French at least, he is exclusively known as Émile Picard, membre of the Académie Française and of the Académie des Sciences.

March 18, 2010 8:00 pm

Thanks for this comment. It is still a great theorem by any name.

March 18, 2010 7:38 pm

All this while I thought I was one of the eight folks who had gone through the random student presenter model in a class 🙂 For me it was my first computational complexity course. Instead of using up real estate here, I’ll just put in a link to my related comment on Muthu’s blog.

9. March 18, 2010 8:45 pm

I took topology as a graduate student from a student of Moore, using the Moore method. I hated it. Education at the graduate level should consist both of knowing how to carry out research in the subject (how to understand the subject and prove theorems in the case of math) and also how it fits within the world of science. We got nothing of the latter, and they way we did the work we learned to understand a pitiful little bit of topology, although the part we understood we well pretty well. I learned more from my other professors in grad school, but rarely did they explain anything about the subject’s relation with the rest of math.

By the way, I was an assistant professor in the math dept at Case when you were there.

March 18, 2010 9:47 pm

Great story about the quarterback professor. It would be unthinkable in the current days, as you rightly say.

About the teaching style, I think I would not be upset with this style of teaching at a seminar course. If the professor announces that this is going to be his format, and as long as the students know what to expect, I would not be upset. Of course, it is a great way which prepares the student to learn the material well enough, to become intellectually mature and present on the spot, taking care of all the details. It would improve both their presentation skills and their knowledge of the subject.

But, I would not be happy if the course was announced as a conventional complex analysis class. The students would expect to learn more material and would not have a clue that this is the format. Also, I feel that this kind of a course would be more suited towards the student who were in their senior year, or grad students. Still it is a fun system, no one can escape because you are randomly chosen. Unless you are absent throughout, this can be taken care of by picking random names from the class roll rather than picking random names from the students present.

Maybe for a 4 month semester, a good experiment would be to teach the core material in 3 months conventionally, and have this system for the last month, for some optional topics.

March 18, 2010 10:26 pm

wow, really impressive story! i guess he played before they used the wonderlic —
http://en.wikipedia.org/wiki/Wonderlic_Test
i heard somewhere that scouts actually don’t like it when a player scores too high on the test. not sure if that’s true.

March 19, 2010 8:31 am

:The game has gotten very technical, the pay is too great, and the stakes are too high …

You’re referring to academic computer science here, right?

13. March 19, 2010 8:33 am

I think that more mathematical education should be based around discussion and interaction; sometimes a simple conversation can spark off a thought process that is more valuable to one’s comprehension of a subject than hours of lectures, and often I find that it is only when I attempt to explain something that I realise I don’t actually understand it as well as I thought.

I also think that teachers of abstract mathematics should be sure to include plenty of concrete applications and examples. Having recently begun research, I am finding that I am having to relearn basic things I thought I knew about, for example, group theory, as I have had very little experience in applying what I’ve learnt in lectures, and the practice is very different from the theory!

14. March 19, 2010 9:37 am

Truly fascinating… Picard’s theory is still taking me a while to digest..

15. March 20, 2010 4:31 pm

That thin monograph — Ahlfors, by any chance?

16. March 22, 2010 1:59 pm

Thank you SO MUCH for posting this story! Frank Ryan is my college roommate’s Dad (college was 30 years ago) and one hell of a nice guy. He was director of athletics at Yale (while his son was at Harvard), and I knew he had a Ph.D. in mathematics, but I never knew he’d been a pro football player and a college professor at the same time. WOW! I’m just in awe.