How does the analysis I gave of the worm pattern (here) apply to this pattern?

Wait, I was going to answer this question in a post, but then I realized that I need to take a step back. How do I judge the success of an analysis of student thinking? What would it mean for the analysis to apply to this pattern? What should a “theory” do here?

Ultimately, I want a theory that helps me answer teaching questions. (I don’t need a theory that speaks to psychologists or math ed researchers, or explains phenomena more broadly.) So, what are the teaching questions that I might have to answer about this problem?

**What is this problem good for? **With an abundance of problems on the internet and in books, it becomes necessary to have some way of choosing the ones that are best for my kids. Since I’m in the business of helping my kids learn more math, and since I often am looking for problems that fit into a particular sequence of ideas, I really need to know what math a problem can help my children learn. (Citing a standard is unhelpful here. A standard tells me nothing about where a problem falls in the development of learning that standard.) Further, there’s a whole website of visual patterns. How do I know which ones to choose for my class?
**What should I do if my students are stuck on this problem? **Should I let them struggle? Should I have a “mathematical chant” that I repeat over and over (e.g. “Pick another step to find”)? Should I show them how the pattern grows?
**What should we talk about in the beginning/end of class? **Mathematical conversations need to be purposeful — otherwise they sprawl and frustrate my students. What could we talk about?
**Will my students find this easy? Hard? **I don’t want to walk into class and be surprised by the difficulty, since that will just screw with everything else that I planned.

In turn, I think these questions boil down to just two concerns:

- What are the mathematical goals I could have for a lesson that uses this problem? [This would guide my selection of the problem, and tell me what we can talk about. It also gives me a gauge on the difficulty of the problem.]
- What sorts of feedback/hints would be helpful to give in/after class, if somebody needs them? [What if my kids are stuck?]

Any theory about how kids think about these problems, then, needs to answer to the need for mathematical goals and feedback.

Am I missing anything? Err, of course I’m missing something. Any thoughts on what I’m missing, or where I went wrong here?

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