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. 

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?

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.