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What Makes Good Problem Solving. PREP Workshop, July 2003 Maria G. Fung. George Polya’s Framework. Understanding the problem Designing a plan (strategy) Carrying out the plan Looking back. Understanding the Problem. Get Familiar with Common Paradigms Ratios, percents

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What Makes Good Problem Solving

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What makes good problem solving l.jpg

What Makes Good Problem Solving

PREP Workshop, July 2003

Maria G. Fung


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George Polya’s Framework

  • Understanding the problem

  • Designing a plan (strategy)

  • Carrying out the plan

  • Looking back


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Understanding the Problem

  • Get Familiar with Common Paradigms

    • Ratios, percents

    • Counting/Enumeration

    • Patterns

    • Algebraic relationships

  • Find Key Information

    • Identify known quantities

    • Determine paradigms involved in problem

    • Draw pictures as appropriate


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Designing a Plan (Strategy)

  • 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


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Designing a Plan (Strategy)

  • More common strategies

    • Venn diagram

    • Finite differences

    • Change perspective

    • Sub-problems

    • Solve an easier related problem

    • Change focus


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Carrying Out the Plan

  • Homework Problems

    • Peer review grading

    • In-class discussions and presentations

  • Problems of the Week

    • Written and graded using Oregon Scoring Guide rubric

  • Scoring Practice with Oregon children’s work using the Guide


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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)


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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


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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

      • 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


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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?


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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?


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Problem Writing Unit

  • Students pick their best portfolio problems and one other problem they love

  • Groups discuss how to improve each problem

    • 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


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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


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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


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Online Math Mentoring Project

  • Opportunity to act as mentors to children in Fundamental (3-6) and Pre-algebra (5-8) 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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