Category Archives: teaching

Thoughts on Today

As promised, I was on daytime TV this week!  The clip is available for posterity at the Today Show website.

What’s interesting to me is the big discrepancy between the way this clip looked to a lot of my friends and colleagues and the way it felt actually to participate in it.  Many of my friends were disappointed that I didn’t say more, and wished the discussion had been at a higher mathematical level.

As for me, I walked out of the shoot feeling it had been a success.

Why the difference?

For one thing, Dr. Mrs. Q and I had been watching the show to get ready, and knew what to expect.  It was pretty clear that no serious math lecture was going to happen.  There was a planned question directed at me:  “How long will it take for someone to answer this question [generalized Fermat]?”  If that had happened, I had about 10 seconds planned in which I’d say “We don’t really know, and that’s what’s exciting, most of math remains a mystery even though we teach it in a way that makes it seem everything was settled centuries ago.”  And it would have been great to have said that!  But that would have been the absolute maximum amount of math possible to work into the segment.  And once you’re on the air, things move very quickly, and things are not very tightly tied to the cue cards.

Danica McKellar, who was on with me, handled the problem of content very intelligently; she understood perfectly well that it didn’t make sense to try to really explain a Diophantine question in the context of the show, so she made sure to tell viewers that they could read about it on her twitter feed, where she provided links to a full description.  That seems to me a totally sensible approach to conveying information about math on live national TV.  The thing we do in class is a great thing to do when you have an hour to talk to 200 people.  What you do when you have 10 seconds to talk to 2 million people is totally different.

What I wanted to accomplish on the show:

  • Give some sense that there still exist math problems we don’t know how to solve;
  • Demonstrate that mathematicians are not grubby almost-dead weirdos in robes, but normal people you might see on the street (or, in Danica’s case, even on the screen.)

Both of these seem like things you can totally do in 10 seconds, and things that are worth broadcasting to 2 million people if you get the chance.  I think we were only partially successful with the first goal, but did fine with the second.

There are a lot of different channels and I think that if we want to teach as much math as possible we have to broadcast on as many channels as we can get access to.  And each channel has its own rules.  My book is going to look really different from McKellar’s books, which in turn look really different from the Today show segment, and all three, of course, are drastically different from what we do in a classroom.  But every extra channel is a chance to transmit more math, or even just the mere idea that math is still happening.  The new Museum of Math in New York.  Sitcoms and movies and cop shows with mathy characters, even when the math is distorted or outright wrong.  Nim as an immunity challenge on Survivor.  Ubiquitous Sudoku.  I endorse it all!  If I knew a good way to set up a math booth at the Gathering of the Juggalos, I would totally do it.  (I was actually thinking of David Zureick-Brown for this, if he’s interested.)

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Writing seminar in mathematics

Tim Carmody, now a senior reporter at the Verge, taught a writing seminar at Penn called “Writing Seminar in Mathematics:  The Language of the Universe”:

For Galileo, the universe is written in the language of mathematics; for Descartes, the methods used in algebra and geometry teach us how to reason about anything, from philosophy to politics. Arguably, mathematics is fundamentally about writing–a set of rules that tell us what we are allowed to write and in what order. This seminar explores how mathematics’ emphasis on careful analysis, methodical argument, and logical proof can teach us to write in both scientific and nonscientific contexts. It will also examine the cultural and literary backgrounds of mathematical discoveries–the amazing, often funny stories behind the theorems scientists and engineers use every day. Our readings will come from philosophy and the history of science as well as Douglas Hofstadter’s wonderful book Gödel, Escher, Bach.

That sounds amazing!  I would love to teach a course like this, maybe someday as a FIG.  But probably not as a MOOC.  Perhaps an entirely new acronym is required.


Kato’s lecture notes are like a modernist novel about commutative algebra

Recorded and posted by U Chicago grad student Zev Chonoles.  What a strange and wonderful pleasure.

As we’ve seen, there is an analogy between Z and C[T]. In fact, the analogy between Z and Fp[T] is even stronger; for example the theory of zeta functions is very similar for Z and Fp[T]. We don’t know the true reason why they are so similar; perhaps they are children of the same parents. But we don’t know who their parents are; their parents are missing.


The class group is a bitter group and a sweet group. It is bitter because when it is non-trivial it
makes a mess. It is sweet because it makes things interesting.

There is a cake shop in Balmont, which is north of Chicago. The class group is the same as this
cake shop; it is a very nice cake shop.

I could go on but you should really just read these yourself.


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Historical textbook collection

I’m working in the math department library today and have gotten distracted by a collection of historical math textbooks that’s just gone on the shelves.

From College Mathematics:  A First Course (1940), by W. W. Elliott and E. Roy C. Miles:

The authors believe that college students who take only one year of mathematics should acquire a knowledge of the essentials of several of the traditional subjects.  From teaching experience, however, they are convinced that a better understanding is gained if these subjects are presented in the traditional order.  Students who take only one year of college mathematics are usually primarily interested in the natural sciences or in business administration.

The book covers algebra, trigonometry, Cartesian geometry, and calculus.  The definition of the derivative as a limit is given, but the epsilon-delta definition of limit is not.  Startling to think that science majors came to college never having taken algebra or analytic geometry.

Further back in time we get Milne’s Progressive Arithmetic, from 1906.  This copy was used by Maggie Rappel, of Reedsville, WI, and is dated January 15th, 1908.  Someone — Maggie or a later owner — wrote in the flyleaf, “Look on page 133.”

On the top of p .133 is written

Auh!  Shut up your gab you big lobster, you c?

You got me, Maggie!

I can’t tell what grades this book is intended for, but certainly a wide range; it starts with addition of single digits and ends with reduction of fractions to lowest terms.  What’s interesting is that the book doesn’t really fit our stereotype that math instruction in olden times was pure drill with no attention paid to conceptual instruction and explanation.  Here’s a problem from early in the book:

How many ones are 3 ones and 4 ones?  Write the sum of the ones under the ones.  How many tens are 6 tens and 2 tens?  Write the sum of the tens under the tens.  How do you read 8 tens and 7 ones?  What, then, is the sum of 24 and 63?  Tell what you did to find the sum.

From the introduction:

Yet the book is not merely a book of exercises.  Each new concept is carefully presented by questions designed to bring to the understanding of the pupil the ideas he should grasp, and then his knowledge is applied.  The formal statement of principles and definitions is, however, reserved for a later stage of the pupil’s progress.

Would these sentiments be so out of place in a contemporary “discovery” curriculum?

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Knuth, big-O calculus, implicit definitions (difficulty of)

Don Knuth says we should teach calculus without limits.

I would define the derivative by first defining what might be called a “strong derivative”: The function f has a strong derivative f'(x) at point x if


I think this underestimates the difficulty for novices of implicit definitions.  We’re quite used to them:  “f'(x) is the number such that bla bla, if such a number exists, and, by the way, if such a number exists it is unique.” Students are used to definitions that say, simply, “f'(x) is bla.”

Now I will admit that the usual limit definition has hidden within it an implicit definition of the above kind; but I think the notion of limit is “physical” enough that the implicitness is hidden from the eyes of the student who is willing to understand the derivative as “the number the slope of the chord approaches as the chord gets shorter and shorter.”

Another view — for many if not most calculus students, the definition of the derivative is a collection of formal rules, one for each type of “primitive” function (polynomials, trigonometric, exponential) together with a collection of combination rules (product rule, chain rule) which allow differentiation of arbitrary closed-form functions.  For these students, there is perhaps little difference between setting up “h goes to 0” foundations and “O(eps)” foundations.  Either set of foundations will be quickly forgotten.

The fact that implicit definitions are hard doesn’t mean we shouldn’t teach them to first-year college students, of course!  Knuth is right that the Landau notation is more likely to mesh with other things a calculus student is likely to encounter, simultaneously with calculus or in later years.  But Knuth seems to say that big-O calculus would be self-evidently easier and more intuitive, and I don’t think that’s evident at all.

Maybe we could get students over the hump of implicit definitions by means of Frost:

Home is the place where, when you have to go there,

They have to take you in.

(Though it’s not clear the implied uniqueness in this definition is fully justified.)

If I were going to change one thing about the standard calculus sequence, by the way, it would be to do much more Fourier series and much less Taylor series.

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Help me be a great Nim teacher

I’ll be at Marvelous Math Morning at CJ’s school this Saturday, playing Nim with kids ranging from K-5.  One simple goal is to teach them the winning strategy for the version of the game where there’s one pile and each player can draw 1 or 2 chips.  I’ve done that with CJ and he really liked it — and I think the idea of a perfect strategy is one of those truly deep mathematical concepts that even little kids can grasp.

But what else should I do?  What other Nims and Nimlikes should I teach these kids and what lessons should I try to impart thereby?

Update:  First two commenters both mentioned Tic-Tac-Toe.  At what age do kids typically learn how to play Tic-Tac-Toe and at what age have they learned a perfect strategy?  CJ is in kindergarten and has not seen this, or at least he hasn’t seen it from me.  I’ll ask him tonight.

Update:  Nim a success!  I played mostly one-pile, and the kids were definitely able to grasp pretty quickly the idea of winning and losing positions, and the goal of chasing the former and avoiding the latter.  I didn’t encounter anyone who’d played nim before.  I felt some math was transmitted.  Mission accomplished.

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The quotable Alfie Kohn

“When I talk to CEOs they often ask me, how can I get rid of the dead wood in my company?  I tell them the first question they have to ask is, why am I hiring live trees and then killing them?”

(Alfie Kohn)




Miscellaneous awesome

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Do mathematicians have “tiger mothers”? or: that’s funny, Norbert, you don’t look Chinese!

All the world — or at least all the world of parents of young kids — or at least all the world of educated parents of young kids who fret about their kids’ psychic and material well-being — is abuzz about Amy Chua’s article “Why Chinese Mothers are Superior,” which starts out:

A lot of people wonder how Chinese parents raise such stereotypically successful kids. They wonder what these parents do to produce so many math whizzes and music prodigies, what it’s like inside the family, and whether they could do it too. Well, I can tell them, because I’ve done it.

What follows is a cheerful recounting of Chua’s stern regimen with her daughters.  Here she is with 7-year-old Lulu, who was having trouble with a piano piece:

Back at the piano, Lulu made me pay. She punched, thrashed and kicked. She grabbed the music score and tore it to shreds. I taped the score back together and encased it in a plastic shield so that it could never be destroyed again. Then I hauled Lulu’s dollhouse to the car and told her I’d donate it to the Salvation Army piece by piece if she didn’t have “The Little White Donkey” perfect by the next day. When Lulu said, “I thought you were going to the Salvation Army, why are you still here?” I threatened her with no lunch, no dinner, no Christmas or Hanukkah presents, no birthday parties for two, three, four years. When she still kept playing it wrong, I told her she was purposely working herself into a frenzy because she was secretly afraid she couldn’t do it. I told her to stop being lazy, cowardly, self-indulgent and pathetic.

And her thoughts on grades:

If a Chinese child gets a B—which would never happen—there would first be a screaming, hair-tearing explosion. The devastated Chinese mother would then get dozens, maybe hundreds of practice tests and work through them with her child for as long as it takes to get the grade up to an A.

As it happens, I was just reading Reuben Hersh and Vera John-Steiner’s enjoyable new book Loving and Hating Mathematics, so of course I was reminded of Norbert Wiener’s childhood recollections of being trained in mathematics by his father:

He would begin the discussion in an easy, conversational tone.  This lasted exactly until I made the first mathematical mistake.  Then the gentle and loving father was replaced by the avenger of blood.  The first warning he gave of my unconscious delinquency was a very sharp and aspirated “What?”…. My lessons ended in a family scene.  Father was raging.  I was weeping and my mother did her best to defend me, although hers was a losing battle.

And afterwards:

Wiener’s student Norman Levinson wrote of his teacher, “Even forty years later when he became depressed and would reminisce about this period, his eyes would fill with tears as he described his feelings of humiliation as he recited his lessons before his exacting father.  Fortunately he also saw his father as a very lovable man and he was aware of how much like his father he himself was.”

Ann Hulbert in today’s Slate has more on Chua as the latter-day Leo Wiener.

I tend to think that getting strong in mathematics requires devoting a lot of time to it.  Hours a day on average, just like piano.  I certainly did that — but not because my parents forced or threatened or tantrummed me into it.  Chua leads off by suggesting that her method tends to produce “math whizzes.”  Is it true?  It goes against all my experience of how mathematics works.  But readers, I am curious — did any of you learn math like this?  Feel free to respond anonymously — I recognize this survey requires more self-revelation than most.

(Also, never fear, we’re not considering giving CJ and AB this treatment.  I’ve lived in an apartment with thin walls where I had to listen to a kid practice piano four hours a day, and friends, nothing would make me go back to that.)

Split-screen blackboard

From Andrew Gelman, an interesting pedagogical suggestion:

The split screen. One of the instructors was using the board in a clean and organized way, and this got me thinking of a new idea (not really new, but new to me) of using the blackboard as a split screen. Divide the board in half with a vertical line. 2 sticks of chalk: the instructor works on the left side of the board, the student on the right. On the top of each half of the split screen is a problem to work out. The two problems are similar but not identical. The instructor works out the solution on the left side while the student uses this as a template to solve the problem on the right.

Has anyone tried anything like this?  It sounds rather elegant to me.


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