Mind

Skills of inquiring mind .... as defined by a maths teacher

Habits of mind

1. Pattern Sniff

A. On the lookout for patterns
“Ok. We’ve been working on this staircase problemand it seems that you can’t write perfect squares as a sum of consecutive whole numbers.”

B. On the lookout for shortcuts
“It would be nice if there were a faster way to do 57x34 than adding 57 to itself 34 times. Think we can find a way?”

2. Experiment, Guess and Conjecture

A. Can begin to work on a problem independently
“I’m not sure how to solve this problem, but I’m confident I can make some progress.”

B. Estimates
“Without doing any calculations, I’m guessing that it will take him 30 seconds to walk up the down escalator.”

C. Conjectures
“Based on my work, I think the following is true.”

D. Healthy skepticism of experimental results
“Boy, it sure seems like this 4, 2, 1 thing always repeats but we don’t have a proof yet.”

E. Determines lower and upper bounds
“I know it will take the people at least 10 minutes tocross the bridge because the 10 minute soldier has to cross the bridge. I also found a solution that takes 19 minutes so I know the final answer is somewhere between 10 and 19 minutes.”

F. Looks at small or large cases to find and test conjectures
“I made a table of the first 5 cases and I think I see a pattern. I’m going to see if this pattern holds for the 100^th case.”

G. Is thoughtful and purposeful about which case(s) to explore

H. Keeps all but one variable fixed

“So I’m exploring the equation y=mx+b and I’m wondering how the graph changes as m and b change. For now, I’m going to set m to 1 and just look at how the graph changes when I change b.”

I. Varies parameters in regular and useful ways

(Even/odd example)

J. Works backwards (guesses at a solution and see if it makes sense)

3. Organize & Simplify

A. Records results in a useful way
“I’m going to make a table.”

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
“I think I can solve this problem by solving these other 2 simpler problems.”

E. Considers the form of data (deciding when, for example, 1+2 is more helpful than 3)

“I’m going to leave my fraction as 6/36 because the 6 represents the number of ways you can roll a 7 with 2 standard dice and the 36 represents the total number of rolls.”

F. Uses parity and other methods to simplify and classify cases
“Next time we play 21 Nim I’m going to keep track of whether the running sum is a multiple of 3, one more than a multiple of 3, or 2 more than a multiple of 3.”

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

“The algorithm for dividing fractions is simple. Now I just need to work on making sense why this works.”

D. Creates precise problems

E. Invents notation and language when helpful
“For the sugar weighing problem, I don’t want to have to write out every solution in words so I’m going to let the symbol 3w~3s stand for the act of putting the 3 pound weight on one side of the balance scale, measuring out 3 pounds of sugar on the other side of the scale, and then setting aside the sugar.”

F. Ensures that this invented notation and language is precise
“I need to be careful that I am differentiating between sugar that I am measuring and sugar I am using as a weight.”

5. Tinker & Invent
A. Creates variations

B. Looks at simpler examples when necessary

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

“I see how this structure is made.”

D. Visualizes data

E. Looks for symmetry

F. Visualizes relationships (using tools such as Venn diagrams and graphs)

G. Vizualizes 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

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. Listen & Collaborate

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

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

Reference:

http://mathteacherorstudent.blogspot.com/2010/09/habits-of-mind.html