I've been doing a lot of thinking about what would improve outcomes, on a large scale, for the truly underserved kids from around the U.S. I've had the good fortune to know many truly great teachers, and to have been part of a department that developed several of them (with more in the pipeline!). One of my students once compared being in one of my classes to watching a perfect game in baseball, and--except for the "watching" part--I have to say I've had few nicer compliments. But I've come to think that the emphasis on great teaching in our nation's current dialogue about the importance of education is at best unhelpful and at worst counterproductive. We don't need more great teachers.

Visiting China, I was struck by how rigorous the mathematics is at the nation's best schools, but also--and I was really surprised by this--how decidedly vanilla the teaching is. A typical classroom--at a top school--looks like this:

Notice: fifty or so students listening to a lecture by one teacher at the front, on a chalk board (no technology), and--this doesn't come through on the still photo--little to no actual student input. On this day, students were going over questions on a practice test for their regional end-of-year exams (already an activity I'm not sure I'd spend time on, certainly not with

*me*doing the presenting), and for forty or so minutes, literally all that happened was that students were told how to solve problems they had gotten incorrect, and marked the correct methods in their test booklets. But the problems were something else--deep, challenging, multidimensional. For example, the problem below (from the same class's papers) asks about what happens when a trapezoid is folded into a solid:
It's almost inconceivable that even an honors geometry class in the U.S. would ask a question this complicated.

The disjunction here poses a real question: how is it that Chinese students get to the level where they can do and appreciate such challenging problems, without getting excellent instruction? And after a lot of asking and thinking and soul-searching, I think I have the answer: there are very few bad teachers. The typical middle-to-strong Chinese student gets, so far as I can tell, much more

*consistent*instruction than a similar student in the U.S. Ask any student at my school--selective, very high-level--and you'll get the story of fifth grade, when they didn't really do any math in math class. (Or sixth grade, or third grade, or whatever.) So far as I can tell, this almost never happens in China. In China, there are some good teachers, a lot of mediocre teachers, but almost no bad teachers. Go into any Chinese math class on a given day, and my guess is, you'll see kids getting fair-to-middling teacher-led instruction in mathematics that is reasonably clear and factually correct. I'm claiming that Chinese students--unlike my students--don't get told that "zero isn't even or odd, it's special" or that you can't subtract 7 from 3, or that "there's a formula for solving polynomials of degree five or higher, we just haven't found it yet." And then this instruction is supported by a consistent experience of solving rich problems on homework and on tests. Eleven consecutive years of this kind of solid, not particularly imaginative teaching produces literally tens thousands of students who can tackle very challenging math problems--which is the point.
I've observed a lot of teachers and tried my hardest to help the teachers in my department improve their own practice, as I'm always trying to improve mine. But I don't know what to tell a teacher to make them into another Peter, or John, or Ray, or Natalie, or even me. I'm not sure it's possible to communicate to one person how they can become a great teacher, because one of the things about truly great teaching is that it's idiosyncratic: what John does has influenced me, but I can no more do John's teaching than I can do Groucho Marx's repartee.

On the other hand, I think it's possible, and probably not even that difficult, to delineate what it takes to be a reasonable, middle-of-the-road math teacher who produces a solid year of growth in the vast majority of his or her students. If we could have more of those, we wouldn't have to play catch-up--which is hard even for terrific teachers, not to mention the mediocre ones. Almost all of our students would, like the Chinese, finish eighth grade with some working knowledge of algebra and geometry--not just a collection of area formulas jumbled together--and the ability to tackle multi-step problems. In high school, kids finishing trig would actually know enough trigonometry to apply it in precalculus and calculus, because they weren't spending trig relearning facts about functions, equations, and geometry that they should have learned a year or more previously.

I'll finish this ~~rant~~ post with a brief list of items I'm looking for in the next generation of mediocre teachers. The expectations may not strike you as very high: but imagine what would happen if we could really expect them every day, every year from kindergarten through 12th grade.

- Except for testing days, each day's class has an objective: something students are to know, understand, or be able to do that they didn't know or understand, or weren't able to do nearly as well the previous day. Content is not simply repeated from year to year or even day to day.
- The day's objective is clearly related to overall course goals, to local and national standards, and to what the students already know.
- Assessment is frequent and individual: at least a couple of times per week, students' work is collected (or assessed in class) individually to find out what they know, to give them feedback on what they need to improve, and to adjust instruction. Assessment tasks are nontrivial, especially on formative assessments.
- The mathematics presented each day is correct.
- The mathematics is presented each day in a reasonably logical order. When asked, a teacher can explain the motivation for each step, not just what the step is.
- The time allocated to mathematics is spent actually doing mathematics, not graduation practice, watching a non-math movie, or taking a break. (I don't make these up, but please don't ask me to name names.)
- The time allocated to mathematics is spent with the students either (a) doing mathematics, (b) listening to brief explanations about how to do mathematics, or (c) asking each other or the teacher questions about mathematics. ["Will this be on the test?" is not a question about mathematics.]

I'm sure there are more ... leave them in the comments.