The document discusses teaching mathematics through problem solving using a three P's framework of purpose, process, and payoff. It emphasizes that important math concepts are best taught by engaging students in thinking about and developing problems of a reflective nature. The teacher's role is to pose thought-provoking problems and facilitate discussion of various solution methods rather than simply demonstrating examples. Effective problem solving lessons involve preparing students mentally, letting them work through problems with guidance, and conducting class discussions to justify and evaluate results.

1. Teaching Through Problem Solving Numeracy Barbara A. Guillory, M.A.Ed. National Board Certified Teacher

2. Three P’s Purpose: To provide a comprehensive framework for teaching mathematics through problem solving. Process: Discussions, reflections and small group interactions Payoff: Engage students thinking about and developing problems of a reflective nature

3. Important mathematics concepts and procedures can best be taught through problem solving, that is, tasks or problems can and should be posed that engage students in thinking about and developing the important mathematics they need to learn. Van de Walle (2001, p.40)

4. Why Teach Problem Solving? Problem Solving… Focuses attention on ideas and sense making. Develops “mathematical power”. Helps students develop confidence in doing mathematics Generates assessment data to make instructional decisions. Is a lot of FUN!!!

5. What is a Problem? A problem is any task or activity where the students have no prescribed or memorized methods or rules for accomplishing the results.

6. Features of a problem for Learning Mathematics It must begin where the students are It must clearly focus on the mathematics students are to learn It must require explanations for answers and methods

7. Problem Solving Group Dynamics Individual students Paris Groups Activities Hands-on materials Drawings Paper and pencil Mental math Calculators

8. Japan vs. United States Steps of a typical Japanese eight-grade mathematics lesson: Teacher poses a complex though-provoking problem. Students struggle with the problem Various students present ideas or solutions to the class. Class discusses the various solution methods. The teacher summarizes the class’ conclusions. Students practice similar problems Steps of typical American eight-grade mathematics lesson: Teacher instructs student in a concept or skill Teacher solves example problems with class Students practice on their own while the teacher assists individual students

9. 2 X 2 List two misconceptions of the teacher’s role in problem solving. List one responsibility of the teacher in creating a problem solving environment.

10. Developing a Problem Solving Lesson Before During After Getting Ready Get students mentally ready to work on the task Be sure all expectations for products are clear Students Work Let go Provide hints Listen carefully Observe and assess Classroom Discourse Accept student solutions with evaluation Conduct discussions as student justify and evaluate results and methods

11. Locating Problem Solving Tasks Your Textbook Using Traditional Textbooks Using Reform Programs Children’s Literature Other Resources

12. Teaching Tips and Questions Four Suggestions: Predict! Don’t hope. Be clear in your own mind about the purpose of the task or activity. Recognize that there is much more to a problem than the answer. Do not confuse open-ended problem solving with encouraging creativity.

13. Problem Solving Steps Understand the problem Devise a plan Carry out the plan Reflecting back on the process and outcome

14. Understanding the Problem Can you state the problem in your own words? What are you trying to find or do? What are the unknowns? What information can you obtain from the problem? What information is not needed or is missing?

15. Devising a Plan Look for a plan Look at a similar problem to determine if the same strategies can be used. Make a table Make a diagram Use guess and check Work backward

16. Carrying out the Plan Implement the strategies selected when devising a plan Check each step of your plan Write all necessary steps to your plan (keeping records)

17. Reflecting (Looking back) Check the results in the original problem Look for another method for solving the problem. Ask does your answer make sense, is it reasonable?

18. Building Attitudinal Goals Build in success Praise efforts and risk taking Listen to all students Provide special successes

19. “ When problem solving is interwoven with learning, students are learning mathematics by doing mathematics!”