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Strategies in Problem Solving: How to Develop Confident and Flexible Problem Solvers. ME by the SEa Conference June 17, 2011 Dr. Sarah Ives Assistant Professor of Mathematics Texas A&M University – Corpus Christi. Outline of Session. Overview of Problem Solving –

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strategies in problem solving how to develop confident and flexible problem solvers

Strategies in Problem Solving: How to Develop Confident and Flexible Problem Solvers

ME by the SEa Conference

June 17, 2011

Dr. Sarah Ives

Assistant Professor of Mathematics

Texas A&M University – Corpus Christi

outline of session
Outline of Session
  • Overview of Problem Solving –
    • NCTM Problem Solving Standard
    • What is a problem?
    • Is there a process?
    • What are some strategies?
  • 4-step Process to Problem Solving
  • Planning for Instruction on Problem Solving
  • Questions and Further Discussions
nctm problem solving standard
NCTM Problem Solving Standard
  • The National Council of Teachers of Mathematics has included Problem Solving as one of the process standards:
    • Instructional programs from pre-kindergarten through grade 12 should enable all students to –
      • Build new mathematical knowledge through problem solving
      • Solve problems that arise in mathematics and in other contexts
      • Apply and adapt a variety of appropriate strategies to solve problems
      • Monitor and reflect on the process of mathematical problem solving

NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Key Curriculum Press.

nctm problem solving standard4
NCTM Problem Solving Standard
  • “By learning problem solving in mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom in everyday life and in the workplace, being a good problem solver can lead to great advantages.” (NCTM, 2000, p. 52)
  • Students who can efficiently and accurately multiply but who cannot identify situations that call for multiplication are not well prepared. Unless students can solve problems, the facts, concepts, and procedures they know are of little use. (NCTM, 200, p. 182)
what is a problem
What is a Problem?

Charles and Lester (1982) define a mathematical problem as a task for which:

  • The person confronting it wants or needs to find a solution;
  • The person has no readily available procedure for finding the solution; and
  • The person must make an attempt to find a solution.

Charles, R. I., & Lester, F. K., Jr. (1982). Teaching problem solving: What, why, & how. Palo Alto, CA: Seymore.

“Problem solving means engaging in a task for which the solution method is not known in advance” (NCTM, 2000, p 52)

the problem solving process
The Problem-Solving Process

Polya’s (1957) 4-step process:

  • Understand the problem – Provide time for students to identify the goal, what information is needed and what is extraneous, and detect any missing information;
  • Devise a plan to solve the problem – Students will use various strategies, have them discuss different ways to solve the same problem;
  • Implement a solution plan – encourage students to use their own ingenuity to develop a solution plan; and
  • Reflect on the problem – have students look back at the problem; they should be ready to explain and justify their solutions when asked.

Polya, G. (1957). How to Solve It (2nd Ed.). New York: Doubleday.

1 understanding the problem
1. Understanding the Problem
  • Provide time for children to familiarize themselves with a problem; questions to ask include:
    • How would you tell the problem story?
    • Can someone describe the problem another way?
    • Did _______ give all the important information?
    • Is there something that needs to be added to what was said?
    • Could some information that ________ gave have been left out?
    • What will you know when you have solved the problem?
  • When children understand they are more likely to devote themselves to finding a solution

Bezuk, N., Cathcart, W., Pothier, Y., & Vance, J. (2011). Learning Mathematics in Elementary and Middle Schools: A Learner-Centered Approach (5th Ed.). Boston, MA: Pearson. (p. 43)

2 devising a plan
2. Devising a Plan
  • Different types of problems can have different problem solving strategies
  • Build in time for students to discuss possible solution strategies
  • A group may decide to try a common strategy or have each member try a different strategy
  • For a potentially difficult problem you could have a whole class discussion about solution strategies
  • Also encourage students to make estimates whenever appropriate
problem solving strategies
Problem Solving Strategies

Use Table or Chart

Act out or Model problem

Draw a picture

Solve a simpler problem

Find a Pattern

Guess & Check

Consider all Possibilities

Work Backwards

Changing Point of View

Logical Reasoning

Write Open Sentence

3 implementing a plan
3. Implementing a Plan
  • Once children have decided on a strategy/plan they should be given the freedom and encouraged to solve problems in different ways
  • Suggest students develop solution process in detail – no erasing ‘mistakes’
  • Compare to scientists experimenting for a cure, they wouldn’t destroy previous unsuccessful research efforts but use those results to inform future tries
4 reflecting on the problem
4. Reflecting on the Problem
  • Many times this step gets over-looked – it is important to focus on the process of problem solving and look back
  • Does the solution make sense?
  • Did we answer our problem?
  • Is the answer unique or could there be other solutions?
  • Also invite children to reflect on their solution process and to think of how others solved their problems
examples of problems
Examples of Problems
  • Barnyard animals
  • Euler Squares
  • Pocket change
  • Candy bags
  • Rabbits & hutches
  • Party tables
  • Frogs & Lily pads

Source: http://sci.tamucc.edu/~sives/1350/problem_solving11.html

Adapted from Young, E. 2011.http://sci.tamucc.edu/~eyoung/1350/problem_solving.html 

planning for instruction on problem solving
Planning for Instruction on Problem Solving
  • Several Important Components:
    • Selecting appropriate tasks and materials,
    • Identifying sources of problems,
    • Clarifying the teacher’s role,
    • Organizing and implementing instruction, and
    • Changing the difficulty of problems.
teacher s role
Teacher’s Role

Instead of:

  • Focusing on helping students “find an answer”,
  • Providing solution strategies,
  • Expecting specific responses,

The teacher:

  • Is prepared to see where the students’ observations and questions may take them.
  • Encourages multiple approaches and allows time for communication and reflection about those strategies.
  • Is ready to ask questions that uncover students’ reasoning behind the process (Rigelman, 2007, p. 312).

Rigelman, N. (2007). Fostering mathematical thinking and problem solving: The teacher’s role. Teaching Children Mathematics, 13(6), 308-314.

organizing and implementing instruction
Organizing and Implementing Instruction
  • Classroom Climate
    • Open, supportive, encourage children to try different strategies
  • Grouping Children
    • Include individual, small-group, and whole-class problem-solving experiences
  • Allocating time
    • Problem solving should be an integral part of mathematics instruction, not ‘Friday’s only’
  • Assessing children’s understanding
    • Ongoing assessment of understanding and problem-solving skills can be done by having students discuss and present solutions
annenberg online workshop on problem solving for grades 3 5
Annenberg Online Workshop on Problem Solving for grades 3-5
  • http://www.learner.org/courses/teachingmath/grades3_5/session_03/index.html
  • Example of Problem Solving in actual 4th grade classroom
    • “Sharing Division”
additional resources
Additional Resources
  • LINKS TO THE INTERNET
    • Problems of the Week: http://www.mathforum.org/pow/ Contains several weekly “Problems of the Week” as well as a mechanism to submit solutions electronically. Past “Problems of the Week” and solutions are also available.
    • Open-ended Math Problems: http://www.fi.edu/school/math2/ Contains open-ended math problems at several different levels of difficulty for middle school students.
    • Education Place’s Brain Teasers: http://www.eduplace.com/math/brain/ Contains math puzzles for Grades 3-8 as well as solution hints.

Source: Bezuk, N., Cathcart, W., Pothier, Y., & Vance, J. (2011). Page 59.

additional resources18
Additional Resources
  • RESOURCES FOR TEACHERS
    • Reference Books: Problem Solving
      • Baroody, A. (1993). Problem Solving, Reasoning, and Communicating: Helping Children Think Mathematically. New York: Macmillan.
      • Charles, R., Lester, F., & O’Daffer, P. (1987). How to Evaluate Progress in Problem Solving. Reston, VA: National Council of Teachers of Mathematics.
      • O’Daffer, P. G. (1988). Problem Solving: Tips for Teachers. Reston, VA: National Council of Teachers of Mathematics.
      • Reys, B. (1982). Elementary School Mathematics: What Parents Should Know about Problem Solving. Reston, VA: National Council of Teachers of Mathematics.

Source: Bezuk, N., Cathcart, W., Pothier, Y., & Vance, J. (2011). Page 59.

questions comments
Questions, Comments?
  • Bezuk, N., Cathcart, W., Pothier, Y., & Vance, J. (2011). Learning Mathematics in Elementary and Middle Schools: A Learner-Centered Approach (5th Ed.). Boston, MA: Pearson.
  • NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: Key Curriculum Press.
  • http://sci.tamucc.edu/~sives
  • Sarah.Ives@tamucc.edu
  • (361) 825-2151