1. Good Teaching Through Mathematical Problem Solving

2. George Polya “Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. To achieve the higher aims I am talking about, there are some general tactics of problem solving—the right attitude for problem solving and ability to attack all kinds of problems, not only simple problems that can be solved with simple arithmetic, but more complicated problems of engineering, physics and so on, which will be further developed in the high school.” (California Mathematics Council).

3. What Is Problem Solving The process of applying previously acquired knowledge to new and unfamiliar situations

4. NCTM Problem Solving Standard Instructional programs from pre-K-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

5. Mathematics Standards of Learning for Virginia Public Schools – February 2009 Mathematical Problem Solving Students will apply mathematical concepts and skills and the relationships among them to solve problem situations of varying complexities. Students also will recognize and create problems from real-life data and situations within and outside mathematics and then apply appropriate strategies to find acceptable solutions. To accomplish this goal, students will need to develop a repertoire of skills and strategies for solving a variety of problem types. A major goal of the mathematics program is to help students become competent mathematical problem solvers.

6. VA SOL STANDARDS Problem solving has been integrated throughout the six content strands. The development of problem-solving skills should be a major goal of the mathematics program at every grade level. Instruction in the process of problem solving will need to be integrated early and continuously into each student’s mathematics education. Students must be helped to develop a wide range of skills and strategies for solving a variety of problem types.

7. Problems That Emphasize Understanding and Logical Thinking How can two fathers and two sons divide three automobiles among themselves with each receiving one?

8. Problems That Emphasize Understanding and Logical Thinking Some months have 30 days, some have 31. How many have 28 days?

9. The Handshake Problem The evening before classes begin at Knox College, a "Pumphandle" is held. Suppose a thousand faculty and students participate. If everyone shakes hands with everyone else once, how many handshakes occur during the Pumphandle? If you were the 500th person to enter the Pumphandle line, how many hands would you shake by the end of the evening?

10. 3-5 Minutes Try it by yourself…

11. 3-5 Minutes Discuss with a partner…

12. Problem Solving: A Head Scratching Dilemma What do you do when you first read a problem and have NO IDEA what it said? How do you make sense of an unfamiliar situation? How do you help your students to focus on thinking about the problem?

13. How did you approach the problem? Act it out Guess and check Compute Simplify Use a formula Write an equation Make a table Consider a simpler case Look for patterns Use drawings

14. These are called… Act it out Guess and check Compute Simplify Use a formula Write an equation Make a table Consider a simpler case Look for patterns Use drawings

15. Why are heuristics important? How do heuristics help problem solvers?

16. It helps to have an organized process.

17. Polya’s Problem Solving Process 1.Understand the problem 2.Devise a plan 3.Carry out the plan 4.Look back

18. Understand the Problem Reread the problem What are you trying to find out? What facts do you have? What is the problem asking? What do you know that is not stated in the problem? What questions do you have about the situation?

19. Understand the Problem Draw a picture to help you understand the problem. Try a specific case to help you learn about it. Is it possible to solve the problem? Is it a reasonable problem? Why or why not? Restate the problem in your own words

20. Devise a Plan Do you know a related problem? Think of a familiar problem where you have to answer a similar question. Here is another problem related to yours. Could you use it to help you? Try to solve a simpler problem. What strategies (heuristics) do you know?

21. Devise a Plan How can you organize the information given? What plan of attack will you use? Can you develop a pattern? Guess a solution or a first step. Can you write an equation? Display examples or cases in a chart or table.

22. Carry Out the Plan Use the strategy/heuristic you identified in the “Devise a Plan” phase Sometimes the original strategy you begin you leads to another Work the problem/solve

23. Look Back Did you answer the question(s) asked? Is your answer in the correct unit? Does your answer seem reasonable? Describe your solution process. Are there extensions/generalizations you can make? Could you have solved the same problem another way?

24. Look Back Could you have solved the same problem another way? Could you use the same method for another problem? What problems are similar to this one? Compare your solution to your guess or estimate.

25. Look Back What are the advantages & disadvantages of your method? What were the pitfalls in this problem? Verbalize the stumbling blocks you overcame to solve this problem. What solution strategies were most helpful in solving the problem?

26. Problems Exercises VS

27. Problems Exercises VS What do you think makes a problem different than an exercise? Let’s try to identify a few…

28. Is it a problem or exercise? How many squares are in the picture below?

29. Is it a problem or exercise? Tonya traveled 175 miles in 3.25 hours to go on vacation. What is the average speed she was traveling?

30. Is it a problem or exercise? Find 30% of 92.

31. Is it a problem or exercise? A Tale of the Cats The lucky cats on Stratton Street Had seven mice apiece to eat. The rest made do With only two; The total score Being twenty-four. How many cats ate mousie meat? This is a poem from the book Fun With Figures by J. A. H. Hunter (1965, Dover Publishing).

32. Problems VS Exercises Uses words and/or symbols Require prior knowledge of the topic Can use calculators to solve Challenging Solution is not immediately obvious Multiple ways to approach/solve Use higher level thinking skills Apply a strategy/ heuristic Math in a context Uses algorithms Not in a context Drill and practice Knowledge/ recall level reasoning and thinking

33. Making Problem Solving the Focus of School Mathematics Going beyond the textbook to create meaningful problem-solving situations Teaching problem-solving strategies and skills Helping students experience success in problem solving to motivate them to do more Modeling good problem solving behaviors for the students

34. Presented to you by the Tidewater Team for Improving Math Excellence Dr. Margi Mason mmmaso@wm.edummmaso@wm.edu Eric Shippee ewship@wm.eduewship@wm.edu