Talking Habits of Mind: Part 1

I've been participating in a professional develop group since last year called Escape from the Textbook. The group is the brainchild of Henri Picciotto (and others?). While his webpage isn't flashy, definitely check it out if you're a math teacher. I recommend his site knowing that I may never see you again here.  Alas.

Henri and Kim Seashore (who's a math ed PhD student at Berkeley and would be awesome regardless of her current/future credentials) organized a conference last February at The Urban School in San Francisco where Henri teaches. I'm not sure if there will be a repeat this year as Henri and Kim now know how much work it is to organize a conference. Fortunately, a group of interested and available teachers (and pre-service teachers) have also been meeting a few times a year with the broad goal of talking and sharing things that typically can't be found in textbooks (or are seriously lacking).

We met this past Saturday morning. After an exiting dive into the uses of tangrams and Miri Math Geometry Tools fueled by the open ended question "how can these tools be used", we spent some time continuing the conversation from last year of how mathematical habits of mind can be explicitly taught in the classroom.

Tangrams: Picture taken from www.bcps.org

Miri Math Mirrors: Picture fromhttp://www.enasco.com/prod/images/products/71/VC121881l.jpg

We focused our work around the "Counting Diagonal" problem I found out about from Kate Nowak who blogged about this problem in August.  Here's the problem (I didn't alter it at all for the workshop, but have altered it a teeny bit here based on feedback and my own preferences):

Draw a rectangle on a square grid. An example 9 by 3 rectangle is drawn for you below. Draw one diagonal. How many squares does the diagonal pass through? Develop a rule to determine the number of squares a diagonal passes through for any rectangle of any size. 

Ideally you wouldn't even need the last sentence of the problem and students would go there on their own.  Anyway, it's a really nice problem and I recommend spending some time with it if it's unfamiliar.

On Saturday, I had people work on this problem in groups, but asked people to reflect on places where students might get stuck and habits of mind that could be helpful in order to move forward. In the end we highlighted different habits of mind that people used themselves. Here's a link to what we compiled as a group. If you really want to get down and dirty, you can compare this to the list I made myself of potentially useful habits before the meeting.

Some big picture takeaways:

• The ability to collect lots of data helps make this problem accessible.
• The ability to collect lots of data in many different formats can lead to a rich discussion of different ways to organize this data and how these different ways illuminate different patterns. I think too often this is where we scaffold (giving students the format to collect data), with the potential downside of eliminating this important mathematical thinking and the possibility for different representations that can be helpful in different ways
• This problem is a naturally good group task as cases could be divided among group members and there are a number of different solution paths that can be shared.
• Many people had insights about the relationship between related rectangles such as 2x3, 4x6, etc, an intermediate place in the problem where students can feel success.
• It's important to develop the habit of listening to and understanding other people's insights and methods, especially when they are different from your own.
• Lots of opportunities to predict, test your prediction, create conjectures, and prove these conjectures
• There are good opportunity for variations and extensions (one group that was familiar with the problem worked on the 3D equivalent). Someone else brought up the question of using other shapes or other paths. Students should never "be done."
• Tying good problems to specific content is really hard and society still values content more than habits of mind