Friday, May 13, 2011

Teaching Problem Solving, Part 3: Mathematical Habits of Mind

After working on some problems that I think are good, we'll discuss the two questions I posed in my earlier post.

1.  Why am I working on this problem instead of one of the other six?  Why am I working on this problem instead of watching tv, organizing my sportscaster bobblehead collection, or eating that chocolate santa that's still sitting on my kitchen counter?

2. What problem solving skills/mathematical habits of mind helped you make progress on a particular problem?

Unless there is outright mutiny, we may avoid sharing solutions all together.  Instead, I hope to really explore how we--experts--use mathematical habits of mind when solving unfamiliar problems and how this can be taught to students.  The plan is to explore the framework I am developing (here's where it all began) for mathematical habits of mind. The ones in bold are habits I believe could be helpful in solving one or more of the problems.

Mathematical Habits of Mind

1.  Pattern Sniff
A.          On the lookout for patterns
B.          Looking for and creating shortcuts
2.  Experiment, Guess and Conjecture
A.          Can begin to work on a problem independently
B.          Estimates
C.          Conjectures
D.          Healthy skepticism of experimental results
E.          Determines lower and upper bounds
F.           Looks at small or large cases to find and test conjectures
G.          Is thoughtful and purposeful about which case(s) to explore
H.          Keeps all but one variable fixed
I.            Varies parameters in regular and useful ways
J.            Works backwards (guesses at a solution and see if it makes sense)
3.  Organize and Simplify
A.          Records results in a useful way
B.          Process, solutions and answers are detailed and easy to follow
C.          Looks at information about the problem or solution in different ways
D.          Determine whether the problem can be broken up into simpler pieces
E.          Considers the form of data (deciding when, for example, 1+2 is more helpful than 3)
F.           Uses parity and other methods to simplify and classify cases
G.          Uses units of measurement to develop and check formulas
4.  Describe
A.          Verbal/visual articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions
B.          Written articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions
C.          Can explain both how and why
D.          Creates precise problems
E.          Invents notation and language when helpful
F.           Ensures that this invented notation and language is precise
5.  Tinker and Invent
A.          Creates variations
B.          Looks at simpler examples when necessary (change variables to numbers, change values, reduce or increase the number of conditions, etc)
C.          Looks at more complicated examples when necessary
D.          Creates extensions and generalizations
E.          Creates algorithms for doing things
F.           Looks at statements that are generally false to see when they are true
G.          Creates and alters rules of a game
H.          Creates axioms for a mathematical structure
I.            Invents new mathematical systems that are innovative, but not arbitrary
6.  Visualize
A.          Uses pictures to describe and solve problems
B.          Uses manipulatives to describe and solve problems
C.          Reasons about shapes
D.          Visualizes data
E.          Looks for symmetry
F.           Visualizes relationships (using tools such as Venn diagrams and graphs)
G.          Visualizes processes (using tools such as graphic organizers)
H.          Visualizes changes
I.            Visualizes calculations (such as doing arithmetic mentally)
7.  Strategize, Reason and Prove
A.          Moves from data driven conjectures to theory based conjectures
B.          Tests conjectures using thoughtful cases
C.          Proves conjectures using reasoning
D.          Looks for mistakes or holes in proofs
E.          Uses indirect reasoning or a counter-example
F.            Uses inductive proof
8.   Connect
A.          Articulates how different skills and concepts are related
B.          Applies old skills and concepts to new material
C.          Describes problems and solutions using multiple representations
D.          Finds and exploits similarities between problems (invariants, isomorphisms)
9.   Collaborate and Listen
A.          Respectful to others when they are talking
B.          Asks for clarification when necessary
C.          Challenges others in a respectful way when there is disagreement
D.          Participates
E.          Ensures that everyone else has the chance to participate
F.           Willing to ask questions when needed
G.          Willing to help others when needed
H.          Shares work in an equitable way
I.            Gives others the opportunity to have “aha” moments
10.          Contextualize, Reflect and Persevere
A.          Determines givens
B.          Eliminates unimportant information
C.          Makes and articulates reasonable assumptions
D.          Determines if answer is reasonable by looking at units, magnitudes, shape, limiting cases, etc.
E.          Determines if there are additional or easier explanations
F.           Continuously reflects on process
G.          Works on one problem for greater and greater lengths of time
H.          Spends more and more time stuck without giving up


  1. Sorry we didn't get to these questions. My group picked the squareness problem, and had a lot of fun with it. One person picked a subtractive measure, which didn't seem right to me (at first!), but we worked together to make her measure work. I'm intrigued by what we came up with and want to explore the differences between her measure and mine.

    I chose the squareness problem because I'd already been attracted to it when I read about it here. I liked creating my own measure of a quality that started out as a visual gestalt.

    The Habits of Mind we used included skepticism, testing conjectures using thoughtful cases, recording results in a useful way, and most of the collaboration skills.

    Thanks for leading a wonderful meeting. I'll write more about it at my blog.

  2. Avery, at a discussion at "Escape from the Textbook" Henri suggested we have an online group meeting with you, centered on this topic. This is of much interest to me, and I just forwarded this blog post to the Math Future and Natural Math email lists. I hope we can make it happen! Please contact me