Category Archives: teaching

Show your work

Here’s another comment on that New York Times piece:

“mystery number game …. ‘I’m thinking of a mystery number, and when I multiply it by 2 and add 7, I get 29; what’s the mystery number?’ “

See, that’s what I mean, the ubiquitous Common Core approach to math teaching these days wouldn’t allow for either “games” or “mystery”: they would insist that your son provide a descriptive narrative of his thought process that explains how he got his answer, they would insist on him drawing some matrix or diagram to show who that process is represented pictorially.

And your son would be graded on his ability to provide this narrative and draw this diagram of his thought process, not on his ability to get the right answer (which in child prodigies and genius, by definition, is out of the ordinary, probably indescribable).

Actually, I do often ask CJ to talk out his process after we do a mystery number.  I share with the commenter the worry of slipping into a classroom regime where students are graded on their ability to recite the “correct” process.  But in general, I think asking about process is great.  For one thing, I learn a lot about how arithmetic facility develops in the mind.  I asked CJ the other night how many candies he could buy if each one cost 7 cents and he had a dollar.  He got the right answer, 14, not instantly but after a little thought.  I asked him how he got 14 and he said, “Three 7s is 21, and five 21s is a dollar and five cents, so 15 candies is a little too much, so it must be 14.”

How would you have done it?

Good student bad student good professor bad professor

From an essay called “Why Your Professors Suck”:

Good student: “When will the midterm be?”
Me: “Why do you care?”
Good student: “Um… I’d like to be able to plan when I should study for it.”
Me: “Oh, okay. I don’t know when it’s going to be.”
Good student: “Um… Okay. What’s it going to cover?”
Me: “I’m not sure, but it’ll be really great!”
Good student: “That’s good, I guess. Can you be more specific?”
Me: “Not really. But why do you care?”
Good student: “Well, you’re the professor!”
Me: “I am? That’s odd. You know, I got mostly Cs and Ds in college. Maybe you shouldn’t be listening to me.”
Good student: “But you do have a PhD, right?”
Me: “Sure, but any jerk can get a PhD. Just think about all your professors. It can’t be that hard!”

The author presents this as a special delivery of some much-needed real-world wisdom to the boringly conformist “good student.”  But I think it comes off as free-floating nastiness directed at a kid asking a perfectly reasonable question.  Discuss.

Update:  Actually, I think what follows this exchange makes it even a little worse:

This sends my “good” students into conniption fits. My cynical students enjoy watching these interactions.

Basically, I think I like my cynical bad students more than my good students because the good students are wrong and the cynical bad students are right.

So yeah — it’s not just pure nastiness, it’s served with a charming helping of “humiliate the disfavored student in public while the favored students look on and enjoy.”

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Wisconsin and the Common Core math standards

I have been inexcusably out of touch with the controvery in Wisconsin about the adoption of the Common Core state standards for mathematics.  I present without comment the text of a letter that’s circulating in support of the CCSSM, which I know has the support of many UW-Madison faculty members with kids in Wisconsin public schools.  All discussion (of CCSSM in general or the points made in this letter) very welcome.

(Related:  Ed Frenkel supports CCSSM in the Wall Street Journal.)

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To whom it may concern,

We the undersigned, faculty members in mathematics, science and engineering at institutions of higher education in Wisconsin, wish to state our strong support for Wisconsin’s adoption of the Common Core State Standards for Mathematics (CCSSM).  In particular, we want to emphasize the high level of mathematical rigor exemplified by these standards.  The following points seem to us to be important:

  • We know that what we have been doing in the past does not work.  Nationwide, over 40% of first-year college students require remedial coursework in either English or mathematics.[1] For many of these students, completing their remedial mathematics (that is to say, high school mathematics) requirement will be a significant challenge on their path to their chosen college degree.  The situation in Wisconsin mirrors the national one.  Over the University of Wisconsin system as a whole, 21.3% of all entering freshmen in the fall of 2009 required remedial education in mathematics.[2]  Over the Wisconsin Technical College System, the mathematics remediation figure is closer to 40%.[3]
  • The CCSSM set a high, but realistic, level of expectations for all students.  It is unrealistic, and unnecessary, to expect all students to master calculus (for example) in high school.  That would be the “one size fits all” approach that is often brought up as an argument against the Common Core.  Instead, the CCSSM attempts to identify a coherent set of mathematical topics of which it can be reasonably be said that they are essential for students’ future success in our increasingly technological and data-driven society.  “College and career ready,” yes, but also life and citizenship ready.
  • It is easy to point to a certain favorite topic and say that the Common Core delays discussion of that topic, or places it in a grade level higher than it has been taught previously.  It is also dangerous.  There is no merit in placing a topic at a grade level where students are unable to do more than repeat procedures without understanding or reasoning.  (One example would be the all-too-frequent expectation that students compute means and medians of sets of numbers, with no significant connection to context, and no discussion of when it would make sense to use one rather than the other.)  It is necessary to look at any set of standards as a coherent whole, and ask whether students who meet all expectations of the standards have been held to a sufficiently high level.
  • Any set of standards is a floor, not a ceiling.  Any local school district, school or individual teacher may set expectations beyond the standards, if they choose to do so.  There are certainly many students who will need more mathematics in high school than is required by the CCSSM: Science, Technology, Engineering or Mathematics (STEM)-intending students, or students who hope to attend an elite college or university, are two obvious groups.  These students should indeed take more mathematics, and opportunities should be made available for them to do so. The standards question, however, is whether all students should be required to learn more mathematics than is in the CCSSM; our answer is “no.”
  • Even for talented students, the rush to learn advanced topics and procedures should not come at the expense of students’ deeper understanding of the mathematical content being covered. Talented students also need quality guidance; they should not be rushed thoughtlessly for the sake of advancement.
  • There are undoubtedly some professional mathematicians, scientists and engineers who claim that the CCSSM are insufficiently rigorous; it is our understanding that they are a small minority.

We entreat you to keep Wisconsin in the group of States that are adopting the CCSSM.  We see the consequences of failed educational policies in our classrooms every day, and we only have the well being of our students in mind. The CCSSM is the right balance: already far higher than our previous State standards but not beyond what one can expect from a majority of students.

 


[1] Beyond the Rhetoric: Improving College Readiness Through Coherent State Policy, accessed from http://www.highereducation.org/reports/college_readiness/gap.shtml on October 3, 2013.

[2] Report on Remedial Education in the UW System: Demographics, Remedial Completion, Retention and Graduation, September 2009, accessed from http://www.uwsa.edu/opar/reports/remediation.pdf on October 6, 2013.

[3] Findings of the Underprepared Learners Workgroup, accessed from http://systemattic.wtcsystem.edu/system_initiatives/prepared_learners/Findings.pdf on October 6, 2013.

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.

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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.

Or:

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

f(x+\epsilon)=f(x)+f'(x)\epsilon+O(\epsilon^2)

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)

 

 

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