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
I experiment methodically and systematically
I make small changes to look for change and permanence
I use charts and other methods to organize information
I organize my work in a way that can be referenced by others
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
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.
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.
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?).
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!!!