helping children with problem solving chapter 6 tina rye sloan to accompany helping children learn...

Helping Children withProblem Solving

CHAPTER 6

Tina Rye SloanTo accompany Helping Children Learn Math9e, Reys et al.

©2009 John Wiley & Sons

Focus Questions• What is the difference between solving problems and

practicing exercises?

• What does it mean to teach math through problem solving? What “signposts” for teaching guide this approach?

• What types of problems can be used in teaching through problem solving?

• What strategies for problem solving are helpful for elementary students?

• Why is looking back such an important phase in problem solving? What questions should students learn to ask themselves when they are solving problems and reflecting on their solutions?

Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009

Problem Types

•Problem-involves a situation in which the solution route is not immediately obvious

•Exercise-a situation in which the solution route is obvious

•Routine problem-the application of a mathematical procedure in the same way it was learned

•Non-routine problem-the choice of mathematical procedures is not obvious

Which terms are synonyms?


Are these problems or exercises?15 rows of stamps. 8 stamps in each row. How many stamps?

24 packs of baseball cards.

8 cards in a package.

How many baseball cards?


Problem Types (cont’d)

Try this. Is this a problem for you?

Use the numerals 0,1,2,3,4,5,6,7,8 to form a 3 by 3 square. The sum of the numbers in every row is 12. The sum of the numbers in every column is 12.



•Begin with the digits 1, 2 ,3, 4 ,5, 6, 7, 8, and 9. Use each digit at least once and form three four-digit numbers with the sum of 9636.

• ___ ___ ___ ___ +___ ___ ___ ___ + ___ ___ ___ ___ = 9636

Try this. Is this a problem for you?



Signposts for Teaching Mathematics Through Problem Solving

Signpost 1: Allow Mathematics to Be Problematic for Students

Signpost 2: Focus on the Methods Used to Solve Problems

Signpost 3: Tell the Right Things at the Right Time


Factors for Success in Problem Solving

• Instruction should build on what children already know.

•Engaging children in problem solving should not be postponed until after they have “mastered” computational skills.

•Children should be taught a variety of problem-solving strategies to draw from.


• Children’s problem-solving achievements are related to their developmental level. Thus, they need problems at appropriate levels of difficulty.

•Factors which contribute to children’s difficulties with problem solving include knowledge, beliefs and affects, control, and sociocultural factors.


Factors for Success in Problem Solving

Choosing Appropriate Problems

•Consider including problems that: ▫Ask students to represent a mathematical idea in

various ways.▫Ask students to investigate a numeric or geometric

concept.▫Require students to estimate, or to decide on the

degree of accuracy required, or to apply mathematics to practical situations.


Choosing Appropriate Problems

•Consider including problems that: ▫Ask students to conceptualize very large or very small

numbers.▫Ask students to use logic, to reason, to test conjectures,

or to gauge the reasonableness of information.▫Ask students to perform multiple steps, or use more

than one strategy.▫Are open-ended.


Polya’s Model of Problem Solving

1. Understand the problem.

2. Devise a plan for solving it.

3. Carry out your plan.

4. Look back to examine your solution.


Problem-Solving Strategies

Act It OutMake a Drawing or Diagram

Look for a PatternConstruct a TableGuess and CheckWork Backward

Solve a Simpler or Similar Problem


The Horse Problem

A man buys a horse for $60, sells it for $70, buys it back for $80, and sells it for $90. How much does the man make or lose in the horse trading business?

Describe your strategy.


Problem-Solving Strategies

The Importance of Looking Back

•Some of the most important learning that results from problem solving occurs after the problem has been solved, when students look back at the problem, at the solution, and at how they found the solution.


The Importance of Looking Back

•Look back at the problem.•Look back at the answer.•Look back at the solution process.•Look back at one’s own thinking.


Helping All Students with Problem Solving

•Manage Time•Manage Classroom Routines•Manage Student Needs


Interviewer: “Paco had 13 cookies. He ate 6 of them. How many cookies does Paco have left?”

Meredith: Fifth Month of Kindergarten


Student Interviews

Interviewer: “Carla has 7 candies. How many more candies does she need so that she will have 11 candies to share with her friends?” Allan: Fifth Month of First Grade


Student Interviews (cont’d)

Interviewer: “Nineteen children are going to the circus. Five children can ride in each car. How many cars will be needed to get all 19 children to the circus?”

Clint: Fifth Month of Second Grade



Interviewer: “Robin has 3 packages of gum. There are 6 pieces of gum in each package. How many pieces of gum does Robin have altogether?”

Bill: Third Month of Third Grade



Interviewer: “Tad had 15 guppies. He put 3 guppies in each jar. How many jars did Tad put guppies in?”

Darla: Third Month of Third Grade



Interviewer: “Tad had 15 guppies. He put 3 guppies in each jar. How many jars did Tad put guppies in?” Ellen: Third Month of Third Grade



Interviewer: “Nineteen children are taking a mini-bus to the zoo. They will have to sit either 2 or 3 to a seat. The bus has 7 seats. How many children will have to sit three to a seat, and how many can sit two to a seat?”

Allison: Fifth Month of Fourth Grade



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