A friend/co-worker, Andrew, and I hosted a workshop at Greater San Diego Math Council's Annual Conference this past weekend. We titled the workshop "Rich Mathematics Through Project-Based Learning," but rather than attempt to give a "here is how you do PBL" workshop (which fails to acknowledge that there are many ways to attempt PBL) we decided to use our own efforts at project-based learning as a lens through which participants, including ourselves, could examine the role of the teacher in fostering mathematical understanding. We each presented one approach to PBL, along with some student work and a dilemma that each approach seemed to pose about the role of the teacher in facilitating (or not?) the direction of student work/learning.

## Our Two Approaches

Andrew and I decided to present two very different approaches to planning and implementing projects in our classes. I don't want to try and articulate Andrew's position too much on my own (maybe if he drops by he can leave some thoughts in the comments), but his basic premise was that a "content-based" project might be structured something like this:

Andrew's dilemma here was, given that he is attempting to move students from A to B, "what is the role of the teacher in facilitating that progression?" This led participants to attempt to deconstruct the process and put in steps that might direct student learning. Contrasting that, I have been interested in experimenting around with a more "open-ended" approach to projects. I visualize it like this:

This approach brings an entirely different dilemma, yet still one that largely revolves around the role of the teacher. Specifically, how does a teacher manage such divergent outcomes and how can the teacher facilitate mathematical connections/understandings for students?

## A Closer Look at the Open-Ended Approach

We took an in-depth look at a specific example that I have used recently with students. First, I posed the following task for students:

Students played with that in groups for a day or so and we made some conclusions as a class. Then, I asked them to brainstorm as many questions as they could that they might be interested in pursuing based on the initial task. Here is what they came up with:

1. Can we create a rule/formula for the maximum number of squares based on the number of lines used?

2. How many triangles/rectangles/etc. can be created using only 12 lines?

3. Is there a difference between even and odd numbers of lines?

4. How many shapes can you create with 12 lines?

5. What would a graph of maximum number of squares vs. lines used look like? Linear? Exponential? Other?

6. What if by "lines" we meant "toothpicks" or "unit lines?"

I passed out some student work samples for participants to take a look at, with the prompt(s):

*Create as many squares as possible using only 12 lines*Students played with that in groups for a day or so and we made some conclusions as a class. Then, I asked them to brainstorm as many questions as they could that they might be interested in pursuing based on the initial task. Here is what they came up with:

1. Can we create a rule/formula for the maximum number of squares based on the number of lines used?

2. How many triangles/rectangles/etc. can be created using only 12 lines?

3. Is there a difference between even and odd numbers of lines?

4. How many shapes can you create with 12 lines?

5. What would a graph of maximum number of squares vs. lines used look like? Linear? Exponential? Other?

6. What if by "lines" we meant "toothpicks" or "unit lines?"

I passed out some student work samples for participants to take a look at, with the prompt(s):

*What do you notice?*

What do you wonder about?

What evidence of the Common Core Practice Standards do you see in student work?What do you wonder about?

What evidence of the Common Core Practice Standards do you see in student work?

## Looking for Rule/Formula (Question #1)

## Lines as "toothpicks" (Question #6)

## Even vs. Odd # Lines (Question #3)

## How many triangles (Question #2)

I'm still left with LOTS of questions about what all this means and what my role is in all of this work with students. What are the benefits of this divergence? What are the consequences? How do I help them make connections within and across each others' work? Do I push students reach certain conclusions with their work that I know are still out there? Do I let them just end where they end? If I did push, would they really own the outcome? Would they "know" it? Those last two questions have been nagging at me a lot recently (perhaps more on that in a later post?).

I do think, however, that I am committed to continuing to try figure it out with my students. It is the closest I have come to truly freeing them to think for themselves, follow their own curiosities, make their own conclusions, and be honest with themselves about what is still left unanswered (versus trying to convince me that they know something that they think I want them to know). I mentioned in our workshop that our answers to some of the big questions (what is math? what is the purpose of math education? what is a project? what is the role of the teacher?) determine a lot of the small things we do in our class every day. I'm trying to let myself live in a state of constant re-evaluation with those questions. I think we need to in order to be fully present in the craft of teaching.

I do think, however, that I am committed to continuing to try figure it out with my students. It is the closest I have come to truly freeing them to think for themselves, follow their own curiosities, make their own conclusions, and be honest with themselves about what is still left unanswered (versus trying to convince me that they know something that they think I want them to know). I mentioned in our workshop that our answers to some of the big questions (what is math? what is the purpose of math education? what is a project? what is the role of the teacher?) determine a lot of the small things we do in our class every day. I'm trying to let myself live in a state of constant re-evaluation with those questions. I think we need to in order to be fully present in the craft of teaching.