An Expert Mathematician…

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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