**Making Sense of Complex Numbers**- I wrote a series of posts developing an argument against the idea that complex numbers were introduced for "whimsical" purposes. (this, then this) Teaching and reading about elementary school math helped push me to the conclusion that rotations might be the best way into imaginary numbers. At the highest level of sophistication, though, complex numbers are nothing more than points. So: start with rotations, end up with points.

**Feedback and Revision -**Before this year I was pretty pessimistic about feedback. If not pessimistic, then confused. I started making progress when I realized some potential drawbacks of immediate feedback. The next step was understanding how feedback could come in the form of questions and that feedback didn't even need to be individualized. (Though it could.) It was hard to square these conclusions with SBG-as-feedback, and I spent some time worrying about that. But I also ended up worrying about how we talk about feedback, and this lead to my series of posts about feedback and revision.

**Exponents as Numbers -**Past research made me very familiar with the sorts of mistakes that kids make with exponents, but I didn't really have a prescription. I still don't, but a few posts this year brought me closer. I argued -- and I don't know if I still agree with this -- that a sophisticated understanding of exponents is closer to seeing exponents as a number and not as an operation and certainly not as an abbreviation. How do we help students see the sorts of numbers that exponents represent? I think geometric series are key. I argued that we could use this to give kids a sense of what a "power" is. I then wrote about two lessons that show how I tried this in class last year. (here and here)

**Teaching Proof -**Much of my writing about proof this year has a focused thesis: proof is hard because geometric reasoning is hard,

*not*because logic is hard. In fact, to the extent that logical reasoning can be called an independent skill, children (even young children) already have it. To show this I asked parents to try an experiment with their kids and the results were clear. Rebecca's children can reason logically, so can these kids, and so can your's. This goes against a popular teacher account of why students struggle with proof, and it has implications for instruction. We need to spend less time teaching "everyday" logic and more time scaffolding geometric proof with the full range of proof activities. In general, I came to think that we need to get more specific about the proof-knowledge our kids are missing.

**Researchers and Teachers**

*-*Some of the most fun I had this year was reading and writing about

*From The Ivory Tower to the Schoolhouse*with Raymond. (Thanks, Raymond!) The book is all about the ways university research does and doesn't make its influence felt in classrooms, and our posts dug into these ideas. The trouble is that good ideas aren't always popular ones, and a lot has to do with the popularizer and the message. These sorts of concerns popped up when I wrote about feedback and generally caused me to be anxious about my own career.

***

This list is disparate. Does anything unify these concerns?

Besides for a pain-in-the-ass contrarian streak (but isn't every argument contrarian?) I think that my writing this year struggled mightily with the theory/practice divide. Teachers that I know (myself included) tend to seek activities and easily usable answers and resources. But on the topics that I've thought the most about -- proof, exponents, feedback, complex numbers -- I see the existing answers as inadequate. Teachers aren't theorists, though, and the way that we communicate is through easily usable activities and resources. (That is, sharing resources

*is*teacher discourse.)
What will it be: essays or resources? This next year I'd like to do a better job with both.