### On the Job

My friend and co-worker Maura once pointed out that it's not very often that we get to see our friends and family actually performing their jobs, so she posted a picture up on her blog of herself in the middle of teaching. I write all about my experiences teaching, but what does it actually look like? Here's a picture:

I like this photo because it captures a number of my teaching values:

- Kids learn more from engaging with each other's ideas. The girl at the board will learn more by orally explaining her thinking. The kids in the class will learn more from thinking about how other people see it rather than just how I, the teacher, sees it.
- Kids engaging with each other's ideas builds not just content knowledge, but mathematical habits of mind. I always want kids evaluating the reasonableness of other's ideas, articulating their reasoning, making connections between different ways of seeing, taking intellectual risks, testing out ideas, and so on. It's a lot harder for kids to develop these habits if the teacher does all the talking. These habits are what real mathematicians do and what real mathematicians will tell you makes them successful.
- The class, not the teacher, should be the source of ideas. Especially the beginning of the year kids will often complain, "Just tell us the answer!" I tell them that I already know the answer, so now it's their job to figure it out. The buy-in and learning increases when the intellectual authority of a class is shifted from the teacher to the class. I want the idea to be that none of us may know how to do it on our own, but we can use each other to come to the answer together. Furthermore, there's no reason why my ways of thinking are more valid than the many reasons they bring up. Just today, for example, a group of kids in one class came up with a way of finding the area of a trapezoid that I had never seen or thought of. If I had just lectured them on the formulas that I'm familiar with, none of that would have come out. Now, not only can they learn from the different methods, their understanding will be be deepened by looking for the connections between the methods.
- Kids should be physically positioned in a way that reflects the expectations and values of the class. I put the kids in groups all the time because I want them using each other as resources all the time. Even though this picture is of a whole class discussion with one person at the front (at least for now; more came up to the board later), I often pause class discussions for students to consult their team. The only time I put kids in rows is when they take an individual test.
- Maybe you can't read the problem on the board (click to enlarge), but it demands important things from students.
- There are a lot of access points to the problem and lots of correct ways of answering. I value multiple methods and ways of seeing, so I have to use problems that allow for these all to come out. (Full disclosure: I did not create this problem. I am really good at stealing the right stuff from the right teachers).
- The problem demands justification. I tell kids all the time that the answer itself is much less important than the "how do you know" piece. Justification is a cornerstone of mathematics, so it should be a cornerstone of my class.
- The discussion of the problem could go in a many different directions. With this specific problem, some ideas that have come up over in different classes include: why base and height have to be perpendicular; what "not drawn to scale" means; why the diagonal of a rectangle is longer than its sides; differences in the definitions of parallelograms and rectangles; why the area formulas for rectangles and parallelograms are identical; how many specific examples you need before you can make a conclusion; and many more. Depending on the class and what feels important to them, the problem allows for many different roads the discussion could take. Similarly, the open-endedness allows me in the teacher role to push on things that I know a given class needs.

- There is AP US History mess all over the board from the teacher I share a room with. I hate sharing a room (not because of that teacher, but because I want my own space)
- Come on, I never have that level of rapt attention from 9th graders. It would be nice, but they're 14 years old.

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