My grad school advisor keeps telling me that I need to write more about my thoughts and observations in the classroom for my action research project. I figure I might as well multi-task here and just blog about my research along the way.

We are now a month into the new school year, but I collected some data from them during the first week that I never had time to analyze and write about. There are some interesting (but not all that surprising) things that I found. First, I gave them a journal prompt that asked:

In math class, what is the role of the teacher and what is the role of the student?

Of my 72 students, 79% identified the teacher as the authority (status and epistemic), the student as passive recipient, and/or the role of school as knowledge transmission. Here are some of their responses:

Role of Teacher

All of the following are pieces of direct quotes from students:
Grade the work
Impart knowledge on the students
Teach the material
Share their knowledge
Teach clearly
Help solve problems and do them over and over again
Pass on knowledge
Teach math concepts so they are simple and easy to understand
Show examples of a problem
Teach students how to do the assigned work
Give and deliver information

Role of Student

All of the following are pieces of direct quotes from students:
Listen to the teacher
Take all the knowledge the teachers have to offer
Learn from the teacher
Take notes
Do worksheets
Pay attention
Study diligently
Be quiet
Absorb the knowledge
Absorb information
Learns what the teacher teaches

To be honest, this wasn't surprising...but it is alarming. It's alarming because the way in which we teach math inevitably (and implicitly) simultaneously teaches students things about themselves as mathematicians. Here is the evidence (responses from beginning of the year survey):
80% of my students think they can't do a math problem unless I tell them how to do it first...
85% think they need to memorize things...
and about half of them don't think they can create mathematical ideas, formulas, and rules.

All of this is further support that, as I cited in my research proposal (bold added for discussion here):

1. "our classrooms are the primary experiences from which students abstract both their definition of mathematics (Schoenfeld, 1994) and their sense of self as an active participant in the authoring of mathematics (Lawler, 2010)."

2. "Identity is a model for self-direction and, as a result, a possibility for mediating agency (Holland et al., 1998). Many students have established their identity as receivers of knowledge, with no active role in creating or critiquing mathematical claims. As a result, their sense of agency is surrendered. Research supports the view that such environments cause students to surrender their sense of thought and agency in order to comply with the procedural routines outlined by the teacher/authority figure (Boaler, 2000). Signs of this include negative attitudes towards math, lack of connected knowing, and the belief that mathematics is absorbed rather than created."

I'm interested in the idea of agency (mathematical and otherwise). I'm interested in the hidden curriculum in our classes and how it impacts students' definition of math, students' formation of self, the mediation (or perpetuation) of status/race/economic/power issues, and the recognition of their own ways of thinking and being mathematical in the world.
As I have posted about before, I really want to do a problem-based unit this year in which students attempt to answer the question, "How far away is the horizon line?" I'm getting ready to start the unit in about a week, so I have been thinking about it a lot lately. Mostly, I have been thinking a lot about the idea of a line that is tangent to a circle and how students might conceptualize that. The "visual" that I get in my head when I think about this horizon line question is this:
This morning I was sitting around the house with my girlfriend, and I decided to see what visual she might come up with and what ideas she might have about tangent lines. First, I asked her to draw the visual that comes to mind when she thinks about the horizon line problem. This is what she drew:
Pretty fantastic, right?!?! Certainly more artistic than my picture! It was interesting to me how differently two people might be thinking about the same scenario.

Then I asked her to draw a circle. Then I asked her to draw a line that touched that circle in only one point. She drew line #1 below (I added the numbering to make some distinctions here). Our conversation went something like this:

ME: Tell me why you decided to draw it that way.
HER: Well, cause it would only touch the circle in one point.
ME: What would happen if you continued your line?
HER: It would cross the circle on the other side.
ME: So would that work?
HER: I guess not....Well, I was thinking about this (draws line #2).
ME: Why did you decide not to draw that one?
HER: It just seems like it would touch the circle in more than one point.
ME: What if we zoomed in? (I drew the picture on the right)
HER: Hmmm...not sure. I still feel more certain that line #1 would only touch in one spot.

To me, this was really interesting. I wonder about how students think. I wonder about their mental models. And, mostly, I wonder how much we actually listen to them and respond to how THEY think. It can be tempting to tell students about a tangent line in the context of this problem, but that would be a missed opportunity for rich discussion. Perhaps more importantly, it would be imposing a way of thinking on them that is incompatible with how they are currently thinking. You hear a lot of people say that they don't like math. I wonder how much of that is due to the fact that they have learned that math doesn't care about their ideas, that math is always right, and that they need to learn to think more like math.

For one hour of my school day, I work with an amazing group of Seniors in a class I have called "Mathematical Thinking" (more about that in a later post, perhaps). Mostly, the class is a mixture of problem-based units and other miscellaneous open-ended puzzles, problems, and mathematical games.

Yesterday, we worked on a puzzle/game called Cartesian Chase. I played a few games against students to demonstrate the rules (we confined ourselves to a 3x7 rectangle) and then just let them play for while. Then, I had them stop and record anything they were noticing in terms of a strategy that seemed to be working. Then, they switched and played with new partners for a while longer. I stopped them again after a few games and had them record updated strategies. We ended with a class a few "undefeated" people playing each other. It quickly became apparent that there was a winning strategy at play.

In the process of all this, here is what I noticed:
- nearly ALL of the students were engaged and playing for the whole time
- students were having fun with each other
- we had a few early conjectures in place about what strategy might be best
- students uncovered structure in the problem, used it to win every time, & were able to clearly explain it
- after the game was "solved," a few students were curious: "what if we added another column?" and "what happens with other board sizes?"

I work hard to bring this same spirit of playfulness to other lessons. I work hard to make every day feel like a puzzle in our class. For some reason, I can never quite bridge that gap in the way I would like. I think I get pretty close most days, but for some reason "how many burger combinations are possible?" still feels more like a math problem and less like a puzzle to students. Maybe it has to do with our intent as teachers? Do we place too much emphasis on students "knowing" something specific by the end of the lesson? Could we set up the task better (slower?) so that it emerges as a puzzle? I have a lot of questions, but I do know that I value what students are learning about themselves as mathematicians and thinkers from a lesson just as much, if not more than, I value students knowing some piece of the thing we call "mathematics."
I should start this post by saying that I work in a school with "inclusive classrooms." Students are in the same class based on grade level alone, no tracking by "ability level" or any other metric. I suppose I should also say that I am doing an action research project on mathematical agency. Agency is a slippery word to define, but for the sake of simplicity for now let's say mathematical agency is defined by a positive self-concept as a mathematician (as one who believes in their ability to make sense of mathematical tasks and situations and to judge the validity of those responses). Lately, I have been thinking most closely about how the set-up and discussion of tasks in the classroom can affect student agency and power - for good and bad.

This all came about the other day when some of Brian Lawler's credential students were in observing my 10th grade class working on "Consecutive Sums." Basically, the prompt is: "explore consecutive sums and see what you discover" (where equations such as 1+2+3=6 and 7+8=15 are considered consecutive sums). Eventually, the conversation turned into trying to figure out which numbers can (or cannot) be written as consecutive sums. I put a table up on the board with the numbers 1-25 on it and asked groups to send people up to fill in the chart once they had found one. What ended up happening is that about 5 students dominated this part of the lesson while others sat and watched. I know there are plenty of suggestions about better ways to handle this particular part of the lesson, but I think the implications are greater than that.

I have read several articles and books for my action research (a couple good ones if you are interested) that outline an amazing vision for a classroom community in which students present ideas, challenge each other, and construct meaning together. Most times when I try this, one of two things happens:
1. I select and sequence student share outs so that certain voices are heard that are usually silenced. Mostly, because I have created the conversation, there isn't much to talk about and students seem disinterested. They aren't debating anything; they aren't solving things collaboratively. OR...
2. I'll select one or two pieces of work to get a conversation started and then step out of the way. This usually gets students talking and debating. The only problem is, it's usually no more than 10 students out of a class of 20-30.

I'm not sure I have any answers to this yet, or even that an answer exists that will work for all groups of students. But, I am really interested by the intricacies of the tasks we choose, by how we set up those tasks, by how we get students talking about those tasks, by how we conclude those tasks, and, especially, how ALL of those moves inevitably make a difference in what students are learning about themselves as capable mathematicians. This last bit, to me, is far more important than the "mathematics" that they learn.