I have written about these "habits of a mathematician" before, but the list is changing, improving, and evolving so I wanted to share my most recent thoughts. Mostly, this list has grown out of a desire to show students that "doing mathematics" is not about trying to find an answer that exists at the back of a book (or elsewhere); doing mathematics is about creating, exploring, inventing, and authoring. I owe a lot to these people for their work before mine in helping shape this list:

Look for Patterns

I am always on the hunt for patterns and regularity
I bring a skeptical eye to pattern recognition
I look for reasons why a pattern exists

Solve a Simpler Problem

I start with small cases
I "chunk" things into small sub problems
I take things apart and put them back together

Be Systematic

I experiment methodically and systematically
I make small changes to look for change and permanence

Stay Organized

I use charts and other methods to organize information
I organize my work in a way that can be referenced by others

Visualize

I draw pictures and diagrams to represent a situation
I invent notation or representations to facilitate exploration

Experiment through Conjectures

I ask "what if..."
I invent numbers to test relationships
I use specialized cases to test my ideas
I modify conjectures to continue/deepen exploration

Be Confident, Patient, and Persistent

I am willing to take risks
I will explore even when I'm unsure about where to start
I do not give up easily

Collaborate and Listen

I listen carefully to and respect the ideas of others
I effectively use strengths of people around me
I ask people for their thoughts when they seem hesitant
I don't dominate a conversation

Seek Why and Prove

I seek to understand why things are the way they are
I look for clues that help me understand why
I create logical arguments that prove my ideas

Generalize

I look to understand ALL cases
I invent notation that helps me generalize
I algebraically represent the structure I see


I'm most interested in how a curriculum that is based on these might help students in "doing" their own mathematics. It's not enough to simply slap these up on wall and reference them often (although that might be a start). As I see it, there are some serious implications on curriculum, pedagogy, and assessment that go with these.

Curriculum

Most importantly, if we expect students to create their own mathematics we need to offer them the opportunity to do that. I think the best response to this is to center curriculum around rich, engaging, interesting problems (in short, a problem-based curriculum). This can still be done within a traditional, standards-based classroom but it definitely would suggest relying less on textbooks as an instructional tool. Here are two examples of what I see as possible problem-based units.

Assessment

At the end of each unit (or possibly periodically through the unit) students will select work(s) that demonstrate their use/growth in relation to these habits. They will fill out a reflection sheet (coming soon) and then submit that work to their portfolio. Three times a semester, students will evaluate their portfolio and will receive evaluations from a peer and myself (and possibly a parent/guardian?).

Pedagogy

In the past, I have prompted students to implement these habits (both explicitly and implicitly). The more I think about this, I'm inclined to think this is doing them a disservice for two reasons: 1) I'm telling them what to think or encouraging them to understand in my way and 2) they aren't the ones creating mathematics. Essentially, all I should do is help introduce the habits, provide students with opportunities to do mathematics, and help give them feedback as they grow and develop.

As always, a work in progress...feedback welcome!!!
 


Comments

Jeff d.
03/27/2012 3:56pm

I am starting to get on board with your portfolio idea. Having the students select the work that demonstrates those HoMs is a great idea as part of a larger assessment model.
I wonder what kind of scaffolding would be needed to help them understand what the habits look like and how they could demonstrate them. Many students already use many of the habits and I wonder what it would take to help them recognize that.

Reply
03/29/2012 9:38am

@Jeff Love your thoughts here! Part of what I struggle with is that this is a LOT to do in one year. Ideally, I think it would be great to have these integrated as part of a high school (or better yet) K-12 curriculum. The Common Core process standards seem to be a step in the right direction, but I'm skeptical about how they will actually impact mathematics teaching and learning.

My attempt in scaffolding their introduction has been to select problems early on that I suspect will elicit these habits and then, as a class, identify/name/outline them.

Mostly, I want students to see that this is "doing mathematics;" that it is about the thinking and sense-making. I like the portfolio assessment because it gives students a tangible way to show themselves and others that they ARE being mathematical in a variety of different ways. The assessment focuses on what students CAN do as opposed to pointing out "flaws" (as a teacher percceives them) in their reasoning.

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Sarah
08/26/2012 8:53pm

Something to think about ... I have copied your "ideals" and started to play with the formatting. I will have to compare with how it fits in to our "mathematical processes" (Ontario curriculum) and do some tweaking, but I think that these ideas need to find a place in my classroom.

Reply
03/07/2013 8:35am

Am I the only one that caught the Vanilla Ice reference?

Reply
03/04/2014 11:52am

Wow! Very happy to read this. I've been playing around with a similar set of HOM for a while but am still struggling to authentically integrate them into day-to-day classroom practice.

I think I have two problems: 1) facilitating the activities so that all pupils are making real progress and developing sound mathematical reasoning skills as well as covering the curriculum 2) The official tests we use to assess pupils do not test many of these skills, and so I feel I'm cheating them when, at the end of the year, they are leveled not according to their progress in HOM but according to a written test.

Here's what I started with in terms of habits: http://wp.me/p46ycg-9O any help here would be greatly appreciated. (BTW: very inspired by the quality of the problem solving activities you're blogging about - would you mind me asking where your main sources for these activities are?)

Reply
Bryan
03/06/2014 9:10am

Hi Nyima,

Thanks for stopping by and sharing your work. The graphics you have created around your habits of mind are beautiful and it is obvious that having them at your core is raising some very interesting questions about teaching and learning mathematics.

I pull activities from a variety of different places (and make some up myself) but I have found the Interactive Mathematics Program textbooks to be my "go-to" source for activities and units. With the right classroom culture and some slight modifications when necessary, their material is really quite powerful. You might also check out some of the stuff from Marc Driscoll (Fostering Algebraic/Geometric Thinking); it is lighter on the number of activities but very much relates to your questions about teaching through habits of mind.

As far as your other two inquiries:
1) I have found PLANNING to be of huge importance, even when teachers have good materials. I like using the "Thinking Through a Lesson" protocol as a planning guide for myself. Maybe you will find that useful as well?
http://www.teachinla.com/vpss/documents/course_info/3b-support.pdf

2) I really don't think this is an either/or dilemma. In Jo Boaler's recent summer course titled "How to Learn Math," she talked about how students from two very different schools reacted to standardized tests. In the school with an inquiry orientation to mathematics (in fact, I think they used the IMP texts) students said things like "When we see a question we don't know how to do, we just try to figure it out. That's what we always do in math class." In schools with more directed or procedural forms of instruction, students said things like "When I see a question I don't know how to do, I panic." I found that pretty interesting.

Thanks again for your comments. Keep in touch.

Bryan

Reply
03/10/2014 2:18pm

Thanks so much for those pointers - I ordered "Fostering Algebraic Thinking". I started to look at the Interactive Mathematics Program materials but thought that it might be too much for my year 6-8 kids (grade 5-7).

1) The TTLP is excellent and I'm going to begin trying it out this week. It's a good reflection however: I'm recognising that, when pupils get stuck, I still tend to guide them to how I would solve it. I think planning questions before-hand will help with this.

2) Good point - need to get my inquiry based learning more embedded and of better quality. It's too half-hearted right now...

(Only one of the HOM graphics are mine, I just added text and colour!)

Thanks again for your help, much appreciated : )




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