What Makes Good Problem
Solving
PREP Workshop, July 2003
Maria G. Fung
George Polya’s Framework
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Understanding the problem
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Designing a plan (strategy)
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Carrying out the plan
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Looking back
Understanding the Problem

Get Familiar with Common Paradigms
Ratios, percents
 Counting/Enumeration
 Patterns
 Algebraic relationships
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Find Key Information
Identify known quantities
 Determine paradigms involved in problem
 Draw pictures as appropriate
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Designing a Plan (Strategy)
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Common Strategies
Draw diagram
 Make exhaustive list
 Draw a logic matrix (objects vs properties so
we can identify/exclude possibilities)
 Guess and check
 Solve algebraic equation
 Work backwards

Designing a Plan (Strategy)

More common strategies
Venn diagram
 Finite differences
 Change perspective
 Sub-problems
 Solve an easier related problem
 Change focus

Carrying Out the Plan

Homework Problems
Peer review grading
 In-class discussions and presentations
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Problems of the Week

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Written and graded using Oregon Scoring
Guide rubric
Scoring Practice with Oregon children’s
work using the Guide
Carrying Out the Plan
Portfolio Problems
 Have students write summaries of
problem-solving exploration activities
(with open-ended problems)
 As with weightlifting, this skill is
developed by consistent practice (not by
watching)

Oregon Scoring Guide Rubric

Rubric for assessing problem solving
(scale of 1-6)
Conceptual Understanding
 Processes and Strategies
 Verification
 Communication
 Accuracy (scale of 1, 2, or 5)

Teaching and learning tool
 Proficiency in scoring children's work

Portfolio Problems
Problems that the students write using a
particular strategy and then solve
 Assessment of Portfolio Problems:

10 points per problem
 6 points for writing the problem

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4 points for writing an appropriate mathematically
interesting problem, following the correct strategy
2 points for clarity and good use of language
4 points for solving the problem correctly,
with complete explanations
Example of a Portfolio Problem

Lucky Lollipops (Original Version)
Logan the Leprechaun loves Lucky Lollipops.
He decides to increase his luck by eating one
lucky lollipop every day for 12 days, and also
by giving away 1 lollipop on the first day and,
for the other days, by giving away as many
lollipops as he had given on all the previous
days plus one more. How many lollipops total
did Logan give away? Extra question: If he
started giving away lollipops on a Tuesday,
how many had he given away at the end of the
day Sunday?
Math Forum Version of Portfolio
Problem in the Pre-Algebra POW

Lucky Lollipops - posted March 10, 2003
Logan the Leprechaun loves Lucky Lollipops. He
decides that he is going to give away his lollipops, and
to increase his luck he's thought of the following
routine:
I'll give away one lollipop on the first day and, for the
other days, I'll give away as many lollipops as I've
given on all the previous days plus one more.
How many lollipops did Logan give away on the 12th
day? on the 24th day?
Extra: Write an expression to generalize Logan's
routine. How many lollipops did Logan give away on
the nth day?
Problem Writing Unit

Students pick their best portfolio problems and
one other problem they love
 Groups discuss how to improve each problem

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Change context to a more interesting one for
children
Discuss how to modify problem to make it simpler
or more complex depending on level
What would happen if
Class presentations of “best” problems
 Resources for finding good word problems

What Makes a Good Problem
More than one step
 More than one method to solve
 Possibly more than one answer
 Clear language with no redundant
information
 Fun and relevant to children’s lives
 Develops, illustrates, or enhances an
important mathematical idea

Some Student-Generated
Responses to Getting Stuck
Consider another strategy
 Put problem aside for a while and come
back
 Try to explain the problem to a caring ear
 Build a model or draw a diagram or
picture
 Solve an easier related problem or
consider a sub-case

Online Math Mentoring Project
Opportunity to act as mentors to children
in Fundamental (3-6) and Pre-algebra (58) Problems at www.mathforum.org/pow
 Each student is assigned from 6-20
different replies to mentor
 Great experience in reading and
evaluating real solutions
 Practice at giving feedback and good
hints to children
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