Category Archives: teaching

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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March math link dump

  • If blood found at a crime scene contains a series of genetic markers found in about 1 in a million people, and if you search a database of genetic material from 300,000 people and find just one match, person X, for the blood at the scene, what is the probability that person X is innocent of the crime?  If you said “1 in a million” you might be a prosecutor.  If you said “1 in a million, and I’m barring any expert testimony that says otherwise” you might be a judge.
  • Good article in the New York Times about the challenge of teaching teachers to teach.  Deborah Ball of Michigan talks about what math teachers need:
  • Working with Hyman Bass, a mathematician at the University of Michigan, Ball began to theorize that while teaching math obviously required subject knowledge, the knowledge seemed to be something distinct from what she had learned in math class. It’s one thing to know that 307 minus 168 equals 139; it is another thing to be able understand why a third grader might think that 261 is the right answer. Mathematicians need to understand a problem only for themselves; math teachers need both to know the math and to know how 30 different minds might understand (or misunderstand) it. Then they need to take each mind from not getting it to mastery. And they need to do this in 45 minutes or less. This was neither pure content knowledge nor what educators call pedagogical knowledge, a set of facts independent of subject matter, like Lemov’s techniques. It was a different animal altogether. Ball named it Mathematical Knowledge for Teaching, or M.K.T. She theorized that it included everything from the “common” math understood by most adults to math that only teachers need to know, like which visual tools to use to represent fractions (sticks? blocks? a picture of a pizza?) or a sense of the everyday errors students tend to make when they start learning about negative numbers. At the heart of M.K.T., she thought, was an ability to step outside of your own head. “Teaching depends on what other people think,” Ball told me, “not what you think.”

    The idea that just knowing math was not enough to teach it seemed legitimate, but Ball wanted to test her theory. Working with Hill, the Harvard professor, and another colleague, she developed a multiple-choice test for teachers. The test included questions about common math, like whether zero is odd or even (it’s even), as well as questions evaluating the part of M.K.T. that is special to teachers. Hill then cross-referenced teachers’ results with their students’ test scores. The results were impressive: students whose teacher got an above-average M.K.T. score learned about three more weeks of material over the course of a year than those whose teacher had an average score, a boost equivalent to that of coming from a middle-class family rather than a working-class one. The finding is especially powerful given how few properties of teachers can be shown to directly affect student learning. Looking at data from New York City teachers in 2006 and 2007, a team of economists found many factors that did not predict whether their students learned successfully. One of two that were more promising: the teacher’s score on the M.K.T. test, which they took as part of a survey compiled for the study. (Another, slightly less powerful factor was the selectivity of the college a teacher attended as an undergraduate.)

    Ball also administered a similar test to a group of mathematicians, 60 percent of whom bombed on the same few key questions.

  • Thurston teams up with the House of Miyake for a Paris runway show loosely based on the fundamental 3-manifold geometries.  Thurston talks fashion:
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Getting that A, by means fair and foul

This week’s Capital Times leads with a story on grade inflation at UW-Madison.  I’m with ex-chancellor John Wiley on this:  “Grade inflation is one of those topics that initially seem clear and simple, but become murkier and more confusing the longer you think about them.”  I more or less stand by what I wrote about grade inflation in Slate in 2002.  The discussion on grade inflation has improved since then, actually:  I think people generally understand now that our moral standing doesn’t rest on whether our shorthand for “student did fine, showed they basically learned the material, is about average among classmates” is “B+” or “C.”  The Cap Times focuses on the more important question of whether different grading standards between departments creates weird incentives for undergraduates.

“I’m trying to get into medical school and it’s frustrating,” says Sheala M_____, a junior majoring in pharmacology and toxicology.  “I can work my butt off and come out of school with a 3.5 in my major, and a women’s study major going pre-med can come out with a 3.9 due to a much easier schedule. All of my courses have very strict policies — some where only 10 percent or 20 percent can get A’s.”

If you like statistics and large .pdf files you can look directly at the source of the article’s numbers: the registrar’s data for GPA in every department in Madison in 2008-2009, broken down by course number and class year.  For instance:  Sheala M_____ is required to take statistics, pathology, and biochem, which have average GPAs around 3.  (All give well above 20% A’s.)  The courses in her major, on the other hand, will be  in the pharmaceutical sciences department, where the average undergrad GPA is 3.43 and 46% of the grades are A.  The corresponding figures for women’s studies are 3.5 and 48%; not much of a thumb on the med school admission scales.  (Remember, the women’s studies pre-med has to take orgo too!)  That said:  I think the weird incentives are real and I think they’re bad.

Meanwhile, at my alma mater, Winston Churchill HS in Potomac, MD, up to 50 students may have broken into the school computer system and changed their grades.    The description of WCHS’s current reliance on computer-graded multiple-choice tests is sort of depressing.  But the worst part is I now have to stop making fun of my friends who went to high school with Blair Hornstine.

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Deforming Galois representations in Rwanda

I just now learned that my friend Ravi Ramakrishna from Cornell spent a sabbatical term last spring at the Kigali Institute of Science and Technology in Rwanda.  And he blogged his semester.  Good reading for anyone interested in math in the developing world, or who likes awesome pictures of gorillas and volcanoes.  Ravi made a side trip to Uganda with Teach and Tour Sojourners; seems like a nice program, though note that you pay your own way to the continent.

See also:  Dino Lorenzini’s notes on visiting math departments in Africa.

Note:  I don’t know if Ravi actually deformed any Galois representations while in Rwanda.  But come on, if you know the guy, you know he probably did.  He can’t leave those things alone.

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well, did we, okay did we determine that Moebius maps were like isometries or whatever?

From the Michigan Corpus of Academic Spoken English, a 16,000 word transcript of an undergraduate math study session.  In case you ever wanted to know what it really sounds like when students work on our homework.

S1: what if- what if A plus B, equals two times Y and C plus D equals two? [S3: yeah. ] it just has to be proportional so you can’t break it up… but if we have A and C being whatever, then let’s make them something that works.

S2: like one?

S1: let’s… like what if you made, A equal Z and C equal one or something.

S2: but they can’t equal whatever because in the bottom A over C has to equal Z.

S1: i know. [S2: okay ] you make it so that it works.

S2: so you want A to be equal to Z, and C to be equal to one.

S1: okay, so what if we do that…? well no then that gives us uh, Z in the Y equation. unless B equals like Y minus Z or something well it could be done… it’s gonna get complicated though… so if A equals Z,

S2: i think this sucks.

Continue reading

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