Thanks to those who offered resources about this. I also got a riveting and satisfying response via email from my dear friend and mentor Rob Weiman about this and wanted to share it with you. Rob used to be my math coach, and is now getting his PhD at University of Delaware. Eat it up, people, this guys is amazing. Our facilitators here have done a great job with 2 and 3, but I'd love to see us do a bit more of 1 and 4.

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I think that when you are looking at cases of teaching in a group, video or otherwise, one thing to do is to set up very norms ahead of time about how we talk about the teaching. In situations where I have been with groups looking at video, the facilitators took great pains before showing the video to spell out very explicitly a few basic ideas:

1. These teachers have given us a great gift to learn from their practice. It is a privilege that we have through their generosity, we need to be thankful and respectful to them, and appreciate their making their practice public so that we all can learn, them included.

2. In general, when looking at video, the facilitators have not asked for general critiques, or evaluations of the teaching, but have asked specific questions about teaching moves. For instance, they would ask, what moves did the teacher make that pushed for justification? Or how did the problem advance, or inhibit sense-making for this group of students, or What moves did the teacher make that effected the ways students talked about the math?

3. Whenever people made comments in answer to these specific questions, they had to provide evidence to back up their claims, So if they would say something like, "when they said "good job", to Johnny, that really shut down the conversation." The facilitator would respond with, "Where is your evidence? How do you know that that shut down the conversation" The facilitator would also ask for the specific transcript line (if there was a transcript, and generally there was) so that everyone was working from he same instance of evidence.

4. Alternatives were presented as wonderments, not as "better" methods. If they were not, the facilitator would rephrase them. So if someone said, "If kids were in groups, this would have gone a lot better", the facilitator, might respond, we don't know how it would have gone. We can ask ourselves what might have happened if this had been a group activity, rather than an individual activity, but we have no evidence to support the claim that it would have gone better. One thing we could do, in this situation is to try it out and see how it goes.

In general, the ravaging of teachers comes from an ideological standpoint. i.e. I know what good teaching is, and this is not it. What we really want to encourage is an evidence-based orientation. We simply cannot say what is good or bad, we can say that this move at this particular point seems to have had this effect based on this evidence. And we can wonder what a different move may have produced. However, we simply cannot know what a different move may have produced because that different move did not happen. Our job is not to critique other teachers, or champion one particular mode of teaching, but to learn about teaching based on this example of practice, an example that we are privileged to witness through the generosity of this particular teacher.

If this is all happening in a class, the facilitator, or some other person in the class can make a big difference by saying these things over and over. At the very least, we can all imagine some instances of our teaching that could be ravaged, and the scariness of making our practice public, so a little empathy can really go a long way to changing the discussion.

As for teaching versus teacher distinction. Attached is a large review of the research literature about the effect of teaching on student learning. One of the authors is my advisor, who is pretty big on this distinction between teachers and teaching. Pages 377-378 of this review (it is part of a much larger book, it is not a 400 page review!) address this distinction specifically, and give evidence of its prevalence. Of larger interest, perhaps is the conclusion of the review, that we know remarkably little about how teaching effects learning. This review addresses some of the reasons why people who try to use evidence to support their claims find it so difficult to claim that specific teaching techniques are effective, and does attempt to say that despite the difficulties, there appear to be two big ideas, that if procedural fluency is the goal, then clear modeling, immediate practice with immediate feedback is effective, and if conceptual understanding is the goal, then struggle with meaningful mathematics is effective. This struggle also seems to help with procedural fluency.

The point here is that folks who are so ready to critique their fellow teachers should know that the best researchers in the business have had trouble making the claims they are so ready to use as blunt instruments to level the well-intentioned efforts of their colleagues, but that is just my take.

Other sources for this idea:

I. James Hiebert

- Hiebert's introduction to Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development (fondly known as "The Purple Book") by Mary K. Stein, Margaret Schwan Smith, Marjorie Henningsen and Edward A. Silver, is nice short (3 pages) description of this idea as well.

- Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world's teachers for improving education in the classroom. New York: Free Press.

This is a bit of a classic, it's main thesis is that the difference between the US and other countries is the teaching practices that are culturally embedded in the US mathematics classroom, and the lack of any kind of institutionalized structures to improve this instruction. Thus there is a ever-widening gap between the US and some other countries that have practices that are not only more effective, but also evolved and improve, sometimes by specific design.

II. Deborah Ball

Deborah Ball writes about teaching being an unnatural activity. Although this is not really about separating teachers from teaching, it does separate a teachers personality and skills in the adult world from the kinds of personal skills teachers must learn and cultivate as teachers. These personal skills may be seen as "teaching skills" rather than "teacher attributes" since these attributes simply are not the kind of attributes that people have naturally in the real adult world. (One example she gives is that cultural survival depends on people assuming shared meaning in most of our discourse, but teachers need to often drop that assumption. So, for instance, in math class, it is a very good move to ask a student what they mean by bigger, but that a guy in a bar talking about sports would soon find himself drinking alone if he asked what somebody meant by bigger, every time it came up in a discussion of the Jets and Giants offensive linemen.)

Another thing Ball writes about extensively is that we measure teacher knowledge through all these "proxies" rather than the knowledge that they need as teachers. So for instance, we look at how many college courses they took in math, or what their SAT score was, or whether they measured in math, rather than actually identifying and testing the specific knowledge they would need for teaching mathematics. Indeed, her whole research program, for which she has received huge recognition, is directed toward trying to identify and develop measures for the kind of mathematical knowledge specific to teaching mathematics. She calls this mathematical knowledge for teaching (MKT)

- Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers' mathematical knowledge: What knowledge matters and what evidence counts? In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 111-156). Charlotte, NC: Information Age Publishing.

This is another huge review, somewhat dry. However, there is a smal section at the beginning where she talks about how historically we have used proxy measures to determine how knowledgeable teachers are.

- Ball, D. & Forzani, F. (2009) The work of teaching and the challenge for teacher education, Journal of Teacher Education, 60(5), 497-511.

Her point about teaching as an unnatural act is on page two of this article. You might also want to check out her website, she has lots of stuff to read there. Not connected to this topic, but a nice read and the thing that launched her is:

- Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373-397.

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If you guys have difficulty finding any of these articles, let me know. I've digital files for most of them.

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