I read an interesting blog post from Without Geometry Life is Pointless regarding the notion of problem solving and thinking.
The list, which references the works of many authors is a good example of techniques used in many disciplines. I've personally used many of these in Marketing, Analytics and Computer Science, but suspect these will be useful in any applied context.
In particular, I find the order of these elements are important. Start with bounding, then work backwards, and develop tests to validate your findings.
You'll notice that none of the entries is "read up or Google answers". Thinking in abstract is often times best done entirely without external influence. Its astounding how often you'll find holes and opportunities in subjects that are well documented.
1. Pattern Sniff
A.
On the lookout for patterns
“Ok. We’ve been working on this
staircase problem and 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 to
cross 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 100th 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
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
c 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
