Saturday, March 22, 2014

Guest Blog: She likes math, but she hates math homework

Emma, 6th grade, likes math (does math circle voluntarily on Saturdays, for example), but hates math homework.  When I asked her why, she sent me the following well-thought-out response--unedited by me (okay, I deleted two commas).  Math teachers, take note!
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As a 6th grader, here are the reasons I hate math homework, and solutions to the problems:

1) Quantity- Homework is supposed to be for us to practice what we've learned. So if you know the material, then it's a waste of time to do 20 similar problems when all you need to prove that you know the material is 4 or 5. And if you don't know, then you'll practice incorrectly, and it really won't help you to learn how to graph inequalities if you're repeating incorrect steps for any number of problems greater than that needed to prove that you know the material.

2) Quality- Having similar problems doesn't help anyone. Sure, if you want to memorize steps, it may be helpful, but in order to understand the math behind the equation and prove why you need to take the steps, you should have a variety of problems. Because in order to understand how to use a method, you should take the time to practice with different types of problems, or else you run across a slightly different problem and you don't know how to do it. 

3) Comprehension- Having problems to solve like "x+2=5. x=?" doesn't help you learn math. That's arithmetic. Arithmetic is something that you can do with a calculator. Math is knowing why x=3. And just showing your work doesn't help, because again, you're just showing your calculations, which again, could be done with a calculator. What helps is asking, "Why does x=3?" Because then you have to look up from your calculator and think about what makes the problem work.

4) Grading (as a class)- When homework is being graded as a class, then you should again focus on the logic hidden behind the problem. Ask a kid to come up to the board and prove why their answer is correct (having fewer problems would also prove to be helpful here). When you go over them by just stating the answer, it doesn't help the kids who got the answers wrong. Unless they just made an arithmetic error, they still don't get why it works, still don't understand the material, and now only know that they were wrong, which doesn't help. And as teachers, your job is not to make kids pass or fail, but to get them to learn something. 

Sunday, March 16, 2014

Three Parables of Teaching

A peril of teaching for a while--and thinking about teaching--is that everything you read becomes filtered through the question "What does this say about teaching and learning?"  But here are three short pieces that more-or-less deliberately engage issues of teaching and learning in somewhat parabolic (if not elliptic) ways.

"Shooting an Elephant," by George Orwell, captures the experience of being a new teacher (whether new to the profession, or new to a school) confronted by a discipline problem and a class or hallway full of students what you're going to do about it.  I think it also reminds us that we're never really as prepared as we think we are, that first time, and that the chances of things ending well--for the elephant surely, but also for us--are slim indeed.

Sideways Stories from Wayside School, by Louis Sachar, is full of wonderful and off-the-wall tales and fables, but it's the first one that has stuck with me:  Joe is held back during recess by Mrs. Jewl because he can't count properly.  When Joe counts a set of objects, the numbers come out in any old random order, but he always gets the number of objects right.  The (very Wittgensteinian) irony is that whenever Mrs. Jewl tries to explain to him how to count the "right" way, he does exactly what she does, and comes up with the wrong answer.  Ever since my mentor Steve Starr read me this story, I've tried to listen more and worry more about whether the student has what appears to be a robust way of getting the right answer than whether he or she is doing it my way.

Finally, Philip Roth's "The Conversion of the Jews" is the hilarious and deeply sad story of Ozzie, a young pre-teen Jewish boy in 1950's New Jersey who has a history of asking his rabbi the wrong--and by that I mean "the hard"--questions.  The story's precipitating incident is an argument in which his rabbi asserts the existence of a historical Jesus but says that he couldn't have been the Son of God as the New Testament describes because, as Ozzie quotes the rabbi, "‘The only way a woman can have a baby is to have intercourse with a man."  Ozzie asks "if He could make ail that in six days, and He could pick the six days he wanted right out of nowhere, why couldn’t He let a woman have a baby without having intercourse."  And the rabbi refuses to give a consistent answer, leading Ozzie to burst out "You don't know anything about God!" What strikes me here is that Ozzie demands only two things of his rabbi: intellectual honesty, and a modicum of kindness.  And really, for a teacher, is it reasonable to expect anything less?

Sunday, March 9, 2014

It's not how big your class is, it's what you do with it

Big news over the last couple of weeks--besides the nascent testing rebellion going on in CPS and other districts--has been the publication of Diane Schanzenbach's paper, "Does Class Size Matter?" by the National Education Policy Center.  Among this paper's key findings:
  • Old studies claiming zero or negative correlations between small classes and achievement relied on faulty meta-analysis of published data.
  • The STAR experiment in Tennessee, which was a randomized trial, showed a 0.15-0.20 standard deviation gain from assignment to classes of 13-17 rather than 22-25, with higher gains in African-American and low-income subgroups.
  • Teachers of smaller classes are able to (and, in the STAR case, did) use a variety of individualizing strategies, including tracking individual achievement, differentiating instruction, and making personal connections with students.   
  • Contrary to popular belief, these effects were larger with more experienced teachers.
  • Although the STAR study is the most comprehensive randomized study in the US, its findings are backed by other studies that managed to control for other variables in the process.
These findings are summarized as simply "All else being equal, increasing class sizes will harm student outcomes."

But I'm still a skeptic, mostly because of the phrase "all else being equal."  In particular:
  • The STAR gain of 0.15 sd requires reducing classes to about half of what they currently are in Chicago, which would require--roughly--doubling the number of teachers.  CPS currently has 22,000 teachers, and it's hard for me to imagine that the district would be able to shazam up anywhere near 22,000 additional teachers without dredging the bottom of the applicant pool.  But would these new, bottom-of-the-pool teachers actually improve outcomes?
  • The reason why you have to get down to 13-17 students per class to see the payoff is that once the denominator of an expression is large, reducing it a little doesn't increase the quotient very much.  For example, if a class lasts 50 minutes and has 30 students, each student gets at most 1 minute and 40 seconds of "air time" or attention.  If the same class has only 25 students, each student gains 20 seconds of "air time" or attention--which isn't very much. So in the real world of the class size reductions that are plausible in the short term, you're not going to see much payoff.
  • The main pedagogical advantages the small-class-size teachers had over the the regular-class-size teachers in the STAR survey were all things all teachers should be doing anyway: monitoring what individual students are doing and learning, giving students second or third opportunities to learn material they didn't get the first time, and making personal/emotional connections with their students. It's easier to do those things in smaller classes, sure, but they're hardly small-class-only techniques. In fact, studies have shown that weaker teachers placed in small classes take no more advantage of these techniques than they did in larger classes. So the STAR study suggests that teachers who are using these techniques move further than teachers who don't, which I kind of feel like we already knew.
  • In a world of finite resources, smaller classes are a trade-off.  In the US, we trade smaller classes than teachers get in other countries for more of them:  the standard load is five classes of 28-30 students (in Chicago) or 40-odd students (in much of California).  So obviously it's better to have five classes of 17 students than five classes of 28 or 40 (for one thing, you have many fewer students to keep track of).  But what if you traded back, having three classes of 45 or so students instead of five classes of 25?  You'd get an additional two prep periods a day to plan,conference with other teachers, and analyze (grade + reflect on) assessments.  That time would allow you to better pace the next day's lesson--which you could plan that day, rather than having to do a week's worth on Sunday just to be able to keep up during the week--and to learn more about teaching.  And you wouldn't have to stay up until midnight to do that.  The situation I'm describing is pretty close to what they have in China: two sections of 50 students, with most of the day devoted to planning and preparation.1
  • Back to the issue of air time:  how would getting an additional ten or twenty seconds of verbal feedback each day compare to getting written feedback on individual work every single day, which is what many Chinese classes offer?
  • Another trade-off we make is teacher quality.  Just as hiring 22,000 teachers would reduce quality, it seems reasonable that if we were willing to live with much larger classes, we might be able to increase overall teacher quality.  That's what Finland did when they first started turning their educational system around (although now class sizes are back down to about 20, which is the stated average in CPS elementary schools, as a concession to eliminating class tracking).
So what's the moral?  Better teaching clearly produces better outcomes.  Smaller class sizes have costs.  (At a district average of about $80-100K per teacher, even just an extra 10,000 teachers runs to about a billion dollars annually, which seems like a lot to pay for a five percentile point gain on tests.)  Expanding class sizes without increasing total student loads might have substantial benefits to teachers and students: increasing opportunities to work together and to plan and assess better and more frequently, with more and more individualized feedback.  Why are we talking about keeping "all other things equal" when they so rarely are?
Because veteran teachers know: it's not just how big your class is; it's what you do with it that matters.
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1.  "But China and Finland are so much more homogeneous, it's easier to teach big classes of those students!" I hear you cry.  Well, in the eight-county Chicago area, more than 50% of African-American students are in classes that are over 95% African-American, and over 25% of Latino students are in similarly segregated schools. (WBEZ)  So while the CPS system is much more diverse than China's or Finland's, it's not obvious that its classrooms are.

Saturday, February 22, 2014

Bayes's Theorem and Hiring Your Way To Great Teachers

Bayes' Theorem is the most important statistics theorem that nobody knows.  In essence, it says that when computing the probability of an event occurring, you must take into account the information you already know about what theoretical outcomes are actually possible.  In mathematical terms:  the probability of A given that B has occurred equals the probability that both occur, divided by the probability that B occurs (with or without A).

Bayes's theorem has wide applications in statistical inference, but today we're going to talk about one that's of crucial concern to anyone trying to improve teaching in a particular school:  hiring teachers.  Suppose that your interview protocol allows you to identify satisfactory-or-better teachers with 80% accuracy (if you have such a protocol, tell me!).  And suppose that 80% of the teachers in the applicant pool are satisfactory-or-better.  Then if you interview 100 teachers, here's what happens:

  • 80 teachers are satisfactory-or-better.  Of these, your interview protocol says that 80%, or 64, really are satisfactory-or-better, and 20%, or 16, it rates as unsatisfactory.
  • 20 teachers are unsatisfactory.  Of these, your interview protocol says that 80%, or 16, are unsatisfactory, and 20%, or 4, are inaccurately rated as satisfactory-or-better.
What's the takeaway?  Well, there are 68 teachers rated as satisfactory-or-better, but of these, 4 are actually unsatisfactory.  Thus if you hire one of the 68 teachers, you have about a 94% chance of getting a satisfactory-or-better teacher.

So far, the odds sound pretty good.  But those odds are highly dependent on the assumptions we made: that 80% of the candidates were satisfactory-or-better, and that your protocol helps you tell good from bad 80% of the time.  In my experience, neither of these things is necessarily true:
  • Many of the satisfactory-or-better teachers are pretty happy where they are, and aren't looking for new jobs.  In my experience, the applicant pool for Chicago Public Schools jobs is more like 40% satisfactory-or-better.  (Note: I'm not, not, not saying that only 40% of CPS teachers are satisfactory.  There are lots of satisfactory-or-better CPS teachers--but in my experience, many of them are committed to the schools where they teach.  What I'm saying is that when I was interviewing applicants for jobs at my school, only 40% of the applicants were satisfactory-or-better.)
  • Teachers can have great credentials and interview well without being great in the classroom.  You can catch that with demo lessons (I've known of candidates who hit home runs in the interview only to totally whiff the demo lesson), but even that won't tell you how well they relate to students and parents over the long-term, how well they collaborate with colleagues, and how committed they really are to improving their practice.  So my guess is that interview protocols are less than 80% reliable, say 70%.
Running that same thought-experiment with our revised assumptions yields very different results:
  • Only 40 applicants are satisfactory-or-better, and our protocol identifies 70%, or 28, of them as such.  12 applicants are incorrectly identified as unsatisfactory.
  • 60 applicants are unsatisfactory, and our protocol identifies 70%, or 42, of them as such.  18 applicants are incorrectly identified as satisfactory.
Thus 46 applicants are identified as satisfactory-or-better, of whom only 28 really are satisfactory.  So the probability that a given applicant who passes the interview process is actually satisfactory is 28/46, or about 61%.  Under these conditions, then, if you hire five candidates, only three will probably work out.  And you'll be stuck with two whom you sort of wish you didn't hire.  

What are some conclusions we can draw?
  • The usefulness of hiring procedures has as much to do with the overall quality of the applicant pool as it does with the theoretical reliability of the procedure itself.  If the pool has lots of unsatisfactory teachers, even a good test will end up with many of the apparently-satisfactory teachers being actually unsatisfactory.
  • If you don't have a great applicant pool, firing a teacher who isn't working out will only result in a substantial improvement 60% of the time.  That's better than nothing, but a whole lot less than the "Just fire all the bad teachers" voices usually let on.
  • If lots of bad teachers are suddenly fired, the applicant pool will get worse, both because the fired teachers are now in it, and because lots of people are trying to hire the good ones (remember that 3/5 of the teachers we hire, even under the pessimistic assumption, are good teachers).  And then as we've seen, hiring procedures become less effective at securing satisfactory teachers for jobs.  So as a system-wide policy, "fire the bad teachers" is unlikely to produce substantial improvements for a large fraction (probably more than half) of the kids in the system.
So it's unlikely that we can hire--or fire--our way to great teachers.  We need to take the teachers we already have and develop them instead.

Saturday, February 15, 2014

Teaching Mathematical "Grit": A Dialogue

This week, the CPAM listserv has been bubbling with discussion about how to teach grit in mathematics; unsurprisingly, both of us have pretty strong opinions on this subject.  So here was our part of the dialogue:

John:  During class, I have them work on one challenging problem at a time. They work. I walk around and listen. They are encouraged to try it themselves first, then discuss their work with their neighbors. I do not give hints or show methods to solve the problem. They know that they have been given a problem, not an exercise. That is, I expect that it will take time to solve it, it is related to what we are working on but is not a copy of other problems they have been asked to work on. I do not ask them to do the problem, but insist that they work on it. Every fifteen seconds, or so, I walk by and observe progress. If you want them to become persistent you must provide situations where persistence is the only way. At some point during the class, we discuss solutions that various students have proposed. If it is a worthy problem, one that requires persistence, there will be several ways to approach it. When the problem has been solved, they have learned the content for tonight's homework. That is the intent of the problem, to teach the new lesson by having them figure out what the next thing is. Then I give them another problem. I never give them a worksheet nor do I give notes. We spend class time working on worthy problems. They learn the content, they learn how to solve problems, they learn "grit" and most of all, they love it.

It takes a while for them to get used to the concept that I will not "show them how to do the problem before I ask them to do it." but once they do, they are actually learning math.

PJ.:  One lesson I learned about this is that it's hard to learn more than one thing at a time.  So a good problem to use to teach grit is one in which the actual skills embedded in the problem aren't that difficult, but maybe they have to think through a lot of different possibilities, go down a blind alley, or something.  It helps a lot if the problem has someplace fairly obvious to start (even if it's ultimately the wrong place) and if there's some obvious way to check that your answer is correct.  Many math contest problems require a lot of grit but fail on these two counts, because there's basically only one thing to do, and the "problem" is seeing what that one thing is.  On the other hand, problems like the camel & bananas problem (one version + solution here), or the farmer with the broken eggs (look here), or the locker problem (look here) can be good places to start.

In terms of classroom practice, I'm not as hands-off as John, but I think it's really important NOT to scaffold the problem, especially in problem-specific ways.  Scaffolding the problem by asking leading questions just leads the students, and teaches them that they needed your help to do the problem, which isn't what you want them to learn.  Following Polya and my friend Doug, I have a series of nudges I tend to give in these situations, and I choose the problem with those nudges in mind.  (That is, I'm thinking about which general strategies I want kids to apply.)  There aren't many of these nudges, and I use the questions a lot, to the point that I can start asking students "What do you think I'm going to tell you to do?"  A not-totally-exhaustive list, in no particular order:
  • Try a simpler case.
  • Try several simpler cases, and look for a pattern.
  • What are the conditions?  Can you state ______ as a mathematical sentence?
  • What's the unknown?
  • Look for congruent triangles/similar triangles. [I teach a lot of geometry]
  • Try chasing angles. [Same]
  • Reduce the problem to finding a point. [Same]
  • Try dropping one condition and satisfying the others.
  • Can you satisfy even one of these conditions?
  • What's a related problem?  Can you transform this problem into that one?
  • Use algebra
John:  I think it is more what we choose to emphasize when we try to get other people to understand what we are trying to do. I believe teachers tell kids too much so I emphasize that I do not tell them anything. The things you listed are certainly questions I ask my students , as well as things like, can you prove it, can you do it another way, does your answer make sense, does your answer agree with the answers others got, perhaps you should draw a bigger picture.....

[After this, another teacher wrote in about Carol Dweck's work on mindset, and fourth teacher said that she regularly assigns her kids hard problems to work over a week or two; students don't ask her for help unless they're "totally stuck".  I wrote back the following pair of replies:]

P.J.: Two brief extra thoughts:
  1. It turns out that kids actually respond when they are taught the science behind cognitive development -- that in itself can be a way to change their mindset -- which is maybe surprising?
  2. I actually encourage my students to see me relatively early on (before they're totally stuck), because I worry that their peers will give them too much help.  But I have a small enough roster that I don't have to worry about being overwhelmed.

Wednesday, February 12, 2014

δ, ε, and Mathematical Thinking

So a few of you have expressed interest (surprise, concern on my behalf...) that I'm doing formal limits with my class.  To be fair, the class is very advanced: only one of the fifteen students has yet to take Calculus, and almost all have spent at least one summer doing math.  But it's still a scary prospect.  So here's a report from day 1.

Why?  Why teach the formal definition?  And how do you motivate it?  We began this unit by doing an empirical investigation of iterates of the function f(x) = rx(1 - x) for 0 ≤ x ≤ 1 in the context of rabbit populations: if x represent this year's population density in a particular warren, f(x) represents next year's population density in that warren.  Students quickly discovered that a variety of long-term behaviors are possible (convergence to a single limit, oscillations between 2 or 4 points, and apparently "random" behavior that we couldn't quite nail down), and depend mostly on r.  So then as we want to refine our ideas and start writing proofs about limits, we realized that we needed a formal definition--otherwise, as I put it in class, "we're not doing mathematics, we're doing what those people across the hall [the science teachers] do."


So that's the class's official motivation.  But why slog through this? A traditional answer is this: if you're going to go on in proof-based mathematics, you need to be able to write proofs using the formal definition of limit.  A less-traditional answer is that we're going to need some formal properties of limits, continuous functions, and sequences in this very class, and this is our first opportunity to start working those muscles.  But the least-traditional, and most important answer is this: I'm using limits to teach mathematical thinking.


What?  A fundamental mathematical activity is defining, and it's very hard.  A good mathematical definition describes a phenomenon precisely, excluding everything else.  In lower-level classes, I tell students that, unlike in English, a mathematical definition is like a poem: every word is crucial in exactly the place that it is.  When possible, we actually analyze those poems: why do we define a trapezoid as "a quadrilateral with at least one pair of parallel sides", or an isosceles trapezoid as "A quadrilateral with at least one pair of parallel sides, such that the two angles formed by one of the parallel sides with the other two sides are congruent"?  We drop conditions, try to draw things that "break" the revised definition, argue about the merits of an inclusive versus exclusive definition (what theorems about parallelograms follow from our definition of trapezoid?).


But it's rare that students get to construct their own definition, and so that's exactly what we did yesterday.  We iterated through four stabs ("It's getting pretty violent in here!" quipped one student).  Each time we started by taking the fuzzy new idea and rephrasing it in mathematical language.  For example, when a student proposed "The terms in the sequence get closer to the limit," we rephrased as "|xnL| decreases."   But each time, one student or another would come up with an objection: "Look, we're saying that these terms are getting closer to 1733 1/3 ... but they're also getting closer to 2000, 2100, or anything bigger than that!"  Then we rephrased: "What do you mean by closer?" "As small as you want."  "Okay, then, so how do we say that mathematically?" "Smaller than any number."  "Okay, then we're going to have to name that any number...."


Who?  It helped that a few students had done the formal definition in a class the previous year, but I found an interesting way to handicap them:  I told them they could only give two kinds of contributions, "genuine questions" and "counterexamples" in response to other students' proposals.  That restriction didn't totally quiet them (although it did, somewhat) -- it forced them to think through what they had learned last year, and to apply the underlying ideas to the definition we were working with.  At least two key ideas (and one major counterexample) were found by students who had never studied limits formally.


How?  I've already said that I hamstrung students who had already studied the topic in a way that made them think mathematically without taking the work away from other students.  But I made a few other crucial decisions that really made this half-hour of discussion go well:



  • We started with a rich set of concrete examples--a bunch of sequences with a variety of long-term behaviors--on which we could draw as we worked on our definition.  This context put the meat of the activity--asking whether our current stab included the things we wanted to include, and excluded the things we wanted to exclude--within just about every student's grasp.
  • We started with sequences.  In the past, I've found that limits of sequences are much easier for students to get going on than the limit of a function at a point.  Sequences are simpler: the values are discrete, they only go in one direction, and the definition only involves one set of absolute values, not two.  Moreover, there are lots of relatively interesting (nonconstant) sequences whose N's can be explicitly calculated from their ε's.  Conceptually, too, there's a natural quality to "we're wondering what happens as time goes on" that "we're wondering what happens when x gets close to, but not exactly equal to, a" seems to lack.
Well?  Did it work?  I'm not sure yet.  The kids were engaged and spent half an hour discussing definitions before we settled on what we called the "working draft" we'd use until further notice.  Most of the students contributed to the discussion at least once, so that's something.  And when we went through a proof, together, the kids seemed to follow.  But I'll know more tomorrow, after kids try one on their own.  I can tell you this, though: it was way more engaging for everyone, way more interesting, than starting class by writing the definition on the board and then slogging through a bunch of examples.

Sunday, February 9, 2014

A Mathematical Adventure Underway!

My apologies for the relative silence.  We're back and running, somehow.

Today's entry is relatively brief but pretty exciting.  For the last two summers, I've taught a Chaos "Maxi" course at HCSSiM:  three weeks, 2.5 hours every morning and 3 hours of problems most every evening, with 15 of my best teenage friends.  The course was based on the Chaos half of my "KAM-Geometry" course that I teach at Walter Payton, but I sussed it up with more analysis and other theoretical math, and of course I cared a little less about students "getting" everything (so long as they weren't totally lost).  What a pair of crazy adventures those courses were!


Our annual "Toga Day" (do mathematics while dressed like Ancient Greek mathematicians)


Batman comes to help us prove the completeness of Fractal space under the house-dwarf metric


A board of conjectures

This spring, I'm teaching the high school version of the course -- but I've decided it won't be a high school version.  Instead, we'll do as much of the sussed-up, proof-heavy, analysis stuff as we can: because it's cool, because it makes the course math (instead of "here's another cool observation, wonder why it happens?"), and because I think the kids can do it.

We do the formal definition of limit tomorrow ... I'll let you know how it goes!