In 2010, Avery Pickford posted an entry on his edublog called “Habits of Mind”, which inspired me to think about the skills I think are important for secondary math students. Based on the lists of Avery and other edubloggers (it’s been 6 years, so I can’t remember the others), I compiled a list of my own, organizing skills by their location in the process of working on a problem. I call this list “An Expert Mathematician…” because I want students to internalize the commonalities they share with an expert of mathematics.

To have a multidimensional classroom, there needs to be many ways for students to be smart. “An Expert Mathematician…” is my list of ways to be smart.

**“An Expert Mathematician, …”**

** ****…when starting/playing with a problem, ..**

- … can restate a problem in her own words
- … states given information, eliminates unimportant information, and/or makes simplifying assumptions
- … can begin to work on a problem independently
- … explores with curiosity (and is thoughtful and purposeful about which case(s) to explore)
- … estimates efficiently
- … determines lower and upper bounds
- … makes the problem smaller
- … looks for patterns and/or shortcuts
- … is able to look at problem from completely different angle
- … makes conjectures

**…when working towards a solution, …**

- … organizes a plan of attack
- … finds patterns
- … is creative
- … extrapolates results from smaller problems
- … uses multiple representations to visualize relationships
- … creates models/diagrams/pictures or uses manipulatives
- … invents notation and language when helpful
- … organizes information
- … applies old skills and concepts to new problems
- … can work backwards to solve a problem
- … computes/solves efficiently
- … visualizes processes (e.g. graphic organizers)
- … can explain her thinking to others
- … reflects on the viability of current plan-of-attack and considers alternate plans/approaches

**…when solving, …**

- … generalizes patterns
- … creates algorithms
- … tests models
- … refines models
- … is thorough and exhaustive, covering all possibilities
- … is accurate with detail
- … checks reasonableness of answers
- … looks for additional or easier solution paths
- … finds multiple solutions
- … makes the problem bigger
- … discovers new problems (i.e. extensions)

**…when presenting a solution, …**

- … explains
*how* - … justifies
*why* - … displays results/diagrams/graphs/tables so they are clear, easy-to-understand (i.e. labeled, color-coded, etc)
- … focuses communication towards audience understanding (and draws on previous knowledge)
- … communicates clearly, concisely, and convincingly

**…when thinking/communicating mathematically, …**

- … formulates quality questions
- … asks questions that clarify misunderstandings
- … asks creative, outside-the-box questions
- … works on one problem for greater and greater lengths of time
- … spends more and more time stuck without giving up
- … finds beauty in mathematics
- … looks for mistakes or holes in proofs
- … connects different skills and concepts together

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