## Our Two Approaches

## A Closer Look at the Open-Ended Approach

*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 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.