1. Helping Children with Problem Solving CHAPTER 6 Tina Rye Sloan To accompany Helping Children Learn Math9e, Reys et al. ©2009 John Wiley & Sons

2. 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, 9 th Edition, © 2009

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

4. 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? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Problem Types (cont’d)

5. 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Problem Types (cont’d)

6. 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? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Problem Types (cont’d)

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

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

9. 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Factors for Success in Problem Solving

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

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

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

13. Problem-Solving Strategies Act It Out Make a Drawing or Diagram Look for a Pattern Construct a Table Guess and Check Work Backward Solve a Simpler or Similar Problem Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

14. 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Problem-Solving Strategies

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

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

17. Helping All Students with Problem Solving Manage Time Manage Classroom Routines Manage Student Needs Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

18. Interviewer: “Paco had 13 cookies. He ate 6 of them. How many cookies does Paco have left?” Meredith: Fifth Month of Kindergarten Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Student Interviews

19. 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 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Student Interviews (cont’d)

20. 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 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Student Interviews (cont’d)

21. 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 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Student Interviews (cont’d)

22. 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 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Student Interviews (cont’d)

23. 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 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Student Interviews (cont’d)

24. 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 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Student Interviews (cont’d)