There are a few themes in math education that are particularly interesting to me…habits of mind is one of them. They are the building blocks of "doing mathematics." The more I work with students, the more convinced I have become that they are also the strongest indicators of success in school mathematics. They are the things that good mathematicians of any age do on a regular basis. I seen a few collections of mathematical habits and there are certainly some great resources already (here, here, here, and here to start). After collecting and distilling, here is what I have so far:

Look for Patterns
Start Small
Be Systematic
Take Apart & Put Back Together
Conjecture and Test
Stay Organized
Describe and Articulate
Seek Why and Prove
Be Confident, Patient, and Persistent
Collaborate and Listen


I particularly like the last one because I get an amazing mental image of Vanilla Ice with his well-maintained flat top haircut and shiny "Hammer pants" ("STOP…collaborate and listen…Ice is back with a brand new invention…"). 
I have been toying with a feedback system for each that would look something like this:
Throughout the year we would keep track of how students progress in their development and implementation of each of the habits. Part of me hates to simplify and categorize the habits in this concrete way. The reality is that they are much more fluid and interconnected. At the same time, I think this helps students get specific feedback and helps them set specific goals for improvement. On that note, there are a few things I am wondering about and would like feedback on:

1. Is this a good list of habits? Are there ones that I am missing? Are there ones that are unnecessary? 
2. Suggestions for feedback systems? I hate to bring it up…but should this translate to a grade?
3. How can we rename the habits in a way that is more student friendly?

UPDATE 3/31

See my post here for an updated version of the habits.
 


Comments

02/14/2012 10:59am

These are what I have listed on my course descriptions this year:


• Persevering and accepting mistakes as part of the learning process
• Thinking and acting flexibly
• Communicating with clarity and precision
• Collaborating effectively
• Thinking creatively
• Applying logical reasoning and problem solving

I see now that my own list is missing something important to me: Posing Problems

I like your starter list, and my only initial comment is that I wonder what the difference between "stay organized" and "be systematic" is?

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02/14/2012 11:22am

To me, staying organized includes things like making a chart, recording work in an orderly way, and using other organizational techniques IN your work. Being systematic is a way of DOING your work. The "Consecutive Sums" problem comes to mind. Some students will experiment randomly. Mathematicians will experiment systematically:

1+2
2+3
3+4
4+5

or

1+2
1+2+3
1+2+3+4

…you get the idea. Systematic work makes it easier to spot patterns and structure within those patterns.

I have tried to create a list that takes the mystery out of good problem solving for students. I asked some students to make a flow chart of their problem solving recently and some looked like: "read the problem" --> "identify important info" --> "solve problem"…
WOW…pump the brakes! "Solve problem?!?" Clearly, they don't know (or haven't become aware of) how to solve a problem. I suggest we give them habits to help with that.

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02/17/2012 11:41am

Look for examples, counter-examples, non-examples, and border cases.

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LSquared
03/11/2012 5:37pm

I like your list! It seems to me that the "right" list depends on your students. I really like that your list is easy for high school students to understand. I particularly like your first three--having them as three separate steps, I think, makes each one seem easier to figure out how to do.

In its own way it's more a physical science thing than a math thing, but I think conjecturing and testing is a really key place for high school students to grow. There are two key things going on there--you have to make a conjecture that's specific enough that it can be tested, and once you have made a specific conjecture, you have to realize that you're not done yet, and you still need to test and look for reasons to support your conjecture.

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03/21/2012 1:07pm

Bryan, this is a great start for a mathematical habits of mind. I do like the feedback system, and I would consider having student self-evaluate from time to time, or identifying a particular mathematical habit of mind for a pupil to focus on for a period of time.

I like that this gets to the heart of what mathematics teaching and learning is and should be. The only addition I would have is something to promote creativity and risk-taking. But it could certainly be folded into some of the other HoM.

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03/21/2012 1:39pm

@LSquared Thanks for the feedback! I agree that there may not be a "right" list. I think the way in which I/we define the habits for students will definitely have some impact on how successful they are. For instance, if there is lots of overlap (i.e. is "start small" the same as "take apart and put back together"??) or if they are defined in a way that is ambiguous for high school students they won't be as effective. I want to try to focus on improving this list as much as possible before next year.

@Geoff I really like the self-evaluation idea and the idea of having students set personal goals based around the habits. I have been thinking about establishing "levels" for each habit so that students can have concrete evidence of how to improve but I'm skeptical of trying to separate all of these into levels (not sure they are that concrete). Thoughts?

I also agree that this is what mathematics teaching and learning should be. It is difficult to implement well in a year (without a cohesive 4 year or K-12 program) but this is at least a start.

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Jeffrey Paules
03/29/2012 8:14am

@LSquared I agree that having it in language high schoolers can understand. That's one area I think the Common Core standards fail, at least here in Oregon. Our older state standards were a lot easier for students to interpret.

vickie Woodlief
03/26/2012 3:07pm

This is so awesome:)

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Cindy Farmer
03/27/2012 2:26am

Check out this link to the 8 Common Core Standards of Mathematical Practice...habits of mind of mathematically proficient students. Your list sounds very similar. I really like the feedback system. I am a lead teacher at an IB school. We are working to vertically align our Approaches to Learning in a similar way where we could give feedback as to a child's growth in particular areas of learning habits (things such as "collaboration", "information literacy", "organization", "communication"...) I think developing one's metacognative capacity is important in any discipline.

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03/29/2012 8:30am

@Cindy and @Jeffrey
I have also updated my list and included more detailed descriptions here:
http://www.doingmathematics.com/2/post/2012/03/habits-of-a-mathematician-take-two.html

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