I just picked a graduate program for a PhD in math education. I’ve never made such a big decision before or had to know myself and my interests so thoroughly. I choose a program I didn’t expect. A party school in the desert. Arizona State. And it’s scary because I chose based on my best guess of my academic interests instead of rankings, reputation or location. And that is a scary thing because it requires knowing myself and isn’t anything anyone can help me with. And if I pick incorrectly then the mistake is all mine and I’ll be surrounded with coeds in booty shorts wondering what exactly I’m doing in Tempe in the biggest school in the country. At least I’ll have my bike!

On the road to making this decision I visited a number of excellent schools full of excellent faculty. My favorite one hour visit was with Dr. Steffe at the University of Georgia. He started the meeting by asking me if I’d like some coffee. I was star struck: brilliant, famous math educator is buying me coffee. So brilliant and so courteous. He started the meeting by asking me about myself. I told him about teaching in Vegas an teaching in Seattle and how wildly different those experiences were. I spoke about how much an institution affects what happens inside my classroom. And then, because I knew that he had done groundbreaking work on student thinking in elementary school I spoke about the details of my practice. I told him about how I thought I’d done a good job teaching my students about why algebra worked and how to apply it and was shocked when so many people missed a simple integration question because they didn’t see how to simplify it into a polynomial. “I was so frustrated. I was so proud of all the work I’d done on algebra and felt like I’d identified the root cause of so many issues,” I told Dr. Steffe. “What I learned is that my students learned how to apply a rule in a limited number of examples of differentiation and were not able to extend that to integration. When I mentioned that we’d used a similar technique for differentiation they said I hadn’t told them they could think about algebra while integrating.”

Dr. Steffe pushed me with his questions. I can hardly remember all of them. I remember focusing on him as hard as I could trying to reflect and think as deeply as possible. I’m not used to having people ask questions about teaching that are too hard for me to answer easily. He spoke about fractions on a level that was honestly a challenge for me to follow. I understand fractions, but I’ve never really wondered about what basic arithmetic operations are needed to comprehend them. I’ve noticed: Kids don’t understand fraction. I’ve noticed that they mix up rules for multiplication and addition. But what are they really thinking when faced with a fraction? Often I would guess they are thinking about what rule we learned recently or what section they are in instead of trying to make sense of the math. In Calculus I got to the point of noticing that my student’s errors were largely algebraic but I didn’t really wonder what they were thinking about. I know that they don’t tend to think about everything they’ve ever learned. This seems like a foreign concept to them.

While reading some dissertations about rates of change and teaching Calculus I realized that I had no idea what my students were really THINKING of when I mentioned a rate. Reading the paper alerted me to teachers who could solve Calculus problems but didn’t understand rates and the connections between slopes, velocities and derivatives. They didn’t see Calculus as Slope and Sums like I did. I didn’t realize that my thought about Calculus are unusual for a high school teacher. After all that I thought I learned, I was still making assumptions that others thinking about math mirrored my own.

I asked Dr. Steffe if all the work he’d done to build a mental map of how students think about algebra and arithmetic had been extended to secondary mathematics. “That’s why I’m talking to you. Does this seem interesting? Are you curious?” He told me to come to Georgia to be the next Pat Thompson(a great math ed researcher focused on thinking and learning of Calculus and related topics who had been his advisee). This was really hard to turn down. I chose to work with Pat Thompson instead for reasons that had nothing to do with how impressed I was by Dr. Steffe. He sent me an email saying that I’d made a good choice which MADE my day. It was one thing to trust myself and wholly another to have someone with a lifetime in the field talk to me about my interests for an hour and confirm a really tough choice for me.

At the end of the hour I felt like my teaching had progressed. It was so obvious. What are students actually thinking? What kind of map do they have in their head? Why don’t I listen to their reasoning when they get a wrong answer? What can it tell me? What do their mistakes mean and what are they thinking when they make them? And now I wonder why I never learned about this in professional development. I’ve learned so many activities, organizational strategies, investment strategies, math problems, etc. But student thinking? We don’t talk about that. And shouldn’t we? Teaching without attending to student thinking is like doing Calculus with out knowing why the rules work. And I’m not sure how I missed that parallel.

The process of tracking abilities on different objectives and then tailoring instruction is related to student thinking. Visualizing errors they are likely to make and setting up situations to avoid those is also related to student thinking. These are procedures that one can follow without really understanding what it is all about. Now I’m not sure how to incorporate all these thoughts into class except to ask students what they are thinking about every day. I’m glad I have things under control enough to be able to do that.

Growing up as a teacher is such a hard balance from needing strategies to implement tomorrow and needing to really understand the discipline. I couldn’t extrapolate a year long plan and a plan for tomorrow based on the idea of “think about student thinking.” I needed help figuring out how to organize my desks and collect papers. I needed help setting up a gradebook and passing out books. I needed help in creating tests and powerpoints. How could I get down to the details of my student’s brains? I could see big problems. I knew that fractions were an issue. I knew that place value was important. I could see that students didn’t understand the big concepts of algebra because 2x+ 4 = 6 was not the same as 2x + 4 = 3x + 2. They were separate problems, with separate strategies. I think, more often than not I hate to admit, that I did all of these examples thinking the right things and maybe even saying the right things but struggling to communicate the right things. HOW in the WORLD do I actually get my students to think about why these require essentially the same concepts to solve? Just because I tell them why things work doesn’t mean they attend to that part of the instruction. How do I write questions that they can’t solve by memorization or recognition of certain structures? And how to I honor the place of recognition and memorization in math without making that the only thing we learn?

Do I want to build a map of student thinking or build a map of a school organization? I want to build a map. In fact, right now I should probably be building a map of the rest of the school year instead of blogging. But I don’t want to lose these moments when I’m immersed in books and classes. I don’t want to forget how much my intuition, reflection and experience taught me about myself, math and the world.

I’m a new CM getting ready to teach secondary math this fall, and I’ve found this post incredibly insightful. I think I have a ton of questions to ask you, but I’m not sure what they are right now. Could you point me to the dissertations/papers that you wrote about (paragraph 4)? I’m finishing a M.S. in electrical engineering this spring, and I was torn in attempting to decide between teaching math and science. I would love to read about teaching math and to talk with someone who has been doing it. Feel free to email me at tony.homan@gmail.com.

Thanks and I look forward to more of your posts.