Category Archives: teaching

Commutativity, with fractions

Talking to AB about multiplying rational numbers. She understands the commutativity of multiplication of integers perfectly well. But I had forgotten that commutativity in the rational setting is actually conceptually harder! That four sixes is six fours you can conceptualize by thinking of a rectangular array, or something equivalent to that. But the fact that seven halves is the same thing as seven divided by two doesn’t seem as “natural” to her. (Is that even an instance of commutativity? I think of the first as 7 x 1/2 and the second as 1/2 x 7.)

Teaching-track positions at Washington University: a testimonial

My Ph.D. student Silas Johnson is teaching at Washington University in St. Louis. This is a kind of job that’s getting more and more popular; teaching-focused, non-tenurable but also not on a limited term. I’m pretty interested in the nuts and bolts of how these jobs work, so I asked Silas to explain it to me. Take it away, Silas! The rest of this post comes from him.

Washington University in St. Louis is hiring two new Lecturers this year: one in math, and one in statistics. These are long-term teaching-track positions, meaning they’re intended to be permanent but do not come with the possibility of tenure. I’m currently a Lecturer here, and I enjoy it a lot. I get to teach a lot of interesting courses; so far, my 3-3 load has usually included two sections of calculus and one upper-division course, with the latter including everything from probability to number theory. I also support the department’s broader undergraduate teaching mission. For example, since arriving, I’ve worked on a project to streamline the course requirements for our math major, tried out new ideas for calculus recitations, and conducted teaching interviews for postdoctoral candidates. Most importantly, my colleagues have been wonderfully supportive as I adjust to the university, try out new teaching methods and techniques in my courses, and work on departmental projects.

These teaching-track faculty positions seem to be increasingly popular, though the nature and details of such positions vary from department to department. While non-tenured, my position is on a parallel promotion ladder from Lecturer to Senior Lecturer to Teaching Professor. This structure is fairly typical, though the titles vary (Assistant/Associate/Full Professor of Instruction is also common). In our case, the promotions also come with increased guarantees of job security.

Teaching-focused positions are sometimes stereotyped as a lesser option, perhaps even a backup plan for those who can’t find a research postdoc or tenure-track job. I disagree; I see this as a good path for mathematicians who, like me, have a genuine interest in teaching and want to make it the focus of their careers. Our department and college are clear about the value they place on teaching-track faculty, too; we vote in department meetings, serve on important committees, and are treated as equals in just about every way. (The only thing we can’t do is vote on tenure-track hiring and promotion.)

Overall, I really like it here. I’m happy with my decision to pursue a teaching career, and I’m glad there are other mathematicians out there who are interested in doing the same. I would encourage such people to apply for our position and others like it. If you’re a grad student, by the way, there are teaching-focused postdoc positions too!

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I really like talking with AB about arithmetic and her strategies for doing problems.  All this Common Core stuff about breaking up into hundreds and tens and ones that people like to make fun of?  That’s how she does things.  She can describe her process a lot more articulately than most grownups can, because it’s less automatic for her.  I learn a lot about how to teach math by watching her learn math.


I’m at Disney World with CJ, on a Pirates of the Caribbean-style ride, a car careening through a tunnel.  On the wall of the tunnel there are math posters, the kind you’d see in a high school classroom, about Pascal’s triangle, conic sections, etc.  And I feel sort of annoyed and depressed, because I know that Disney is going to make a big deal about how educational this ride is, but actually, nobody except me is looking at the posters, nobody who didn’t already know the math could get anything out of the posters, the way the car speeds down the track.

Please interpret and derive relevant policy prescriptions for math pedagogy in comments.

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Doctoral programs can have a strong influence on the weak-minded

Daniel Drezner:

First, I cannot stress enough the cult-like powers of a PhD program. Doctoral programs can have a strong influence on the weak-minded. Even if you’re pretty sure what you want going into a program, that can change as you’re surrounded by peers who want something different. You might think you’re strong-willed, but day after day of hearing how a top-tier research university position is the be-all, end-all of life can have strange effects on your psyche.

I really do feel this is something we handle well at Wisconsin.  Our Ph.D. graduates go on to a wide variety of positions, some in primarily teaching colleges, some in research institutions, some in industry, some in government.  We do not consider the North American research university the be-all and end-all of life.  We are not just trying to produce clones of ourselves.  We really do strive to help each of our students get the best job among the jobs they want to get.  

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Should we fire people who pick the wrong Final Four?

A thought experiment touched off by Cathy’s latest post on value-added modeling.

Suppose I’m in charge of a big financial firm and I made every trader who worked for me fill out an NCAA tournament bracket.  Then, every year, I fired the people whose brackets ended up in the lowest quintile.

This makes sense, right?  Successful prediction of college basketball games involves a lot of the same skills you want traders to have:  an ability to aggregate information about uncertain outcomes, fluency in quantitative reasoning, a certain degree of strategic thinking (what choices do you make if your objective is to minimize the probability that your bracket is in the bottom 20%?  What if your fellow traders are also following the same strategy…?)  You might even do a study that finds that firms whose traders did better at bracket prediction actually ended up with better returns 5 years later.  Even if the effect is small, that might add up to a lot of money.  Yes, the measure isn’t perfect, but why wouldn’t I want to fire the people who, on average, are likely to make less money for my firm?

And yet we wouldn’t do this, right?  Just because we think it would be obnoxious to fire people based on a measure predominantly not under their control.  At least we think this when it comes to high-paid financial professionals.  Somehow, when it comes to schoolteachers, we think about it differently.

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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.”


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.)

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 on October 3, 2013.

[2] Report on Remedial Education in the UW System: Demographics, Remedial Completion, Retention and Graduation, September 2009, accessed from on October 6, 2013.

[3] Findings of the Underprepared Learners Workgroup, accessed from 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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