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Polya's Problem Solving Steps

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- 1. + Problem Solving Dr. Mary Pat Sjostrom
- 2. + Why teach problem solving? “Learning to solve problems is the principal reason for studying mathematics.” (NCSM, 1977) “Problem solving is not only a goal of learning mathematics but also a major means for doing so.” (NCTM, 2000) “High school students should have significant opportunities to develop a broad repertoire of problem-solving (or heuristic) strategies. They should have opportunities to formulate and refine problems because problems that occur in real settings do not often arrive neatly packaged. Students need experience in identifying problems and articulating them clearly enough to determine when they have arrived at solutions.” (NCTM, 2000)
- 3. + Problem or Exercise? A problemis a task for which one has no ready strategy for solving; there are no memorized or prescribed methods. Most of the “problems” posed in U.S. mathematics classrooms are really exercises. The purpose of an exercise (like the purpose of exercises done in a physical workout) is to develop skill with a particular method or algorithm. Exercise (like working out) is important, but it should not be the primary goal of the curriculum. Some research has found that as much as 90-95% of time spent in mathematics classroom is devoted to exercises – learning specific algorithms and practicing them through exercise.
- 4. + “Students learn from the kind of work they do in class.” (Hiebert, 1997) Practicing paper-and-pencil skills on worksheet exercises faster at doing these types of exercises. Watching the teacher demonstrate methods for solving problems better at imitating methods on similar problems. (Also may learn that they do not have the capability to think on their own.) build new relationships. In other words, construct understanding. Reflecting on how things work, how certain ideas or procedures are similar/different, how what they know relates to new situations
- 5. + Problem Solving can be taught Many in the United States think people are born problem solvers or they are not – they “got the math gene”! Schoenfeld examined expert and novice problem solvers and found that experts carry out Polya’s steps, reflect on and monitor their thinking, and move flexibly between the steps. Novices tended to jump to step 3; they grab some numbers and just try to work the problem.
- 6. + Problem Solving can be taught Japanese teachers routinely begin a lesson by posing a problem, allow students time to work on it, lead a discussion of student ideas, then have students go back to work. The lesson concludes with student presentations of their solutions. Then the teacher helps the students to summarize the mathematics and if possible generalize (e.g., develop a formula or articulate a concept).
- 7. + Role of the Teacher Teaching through problem solving requires a change in the role of the teacher, a shift in how the teacher thinks about teaching and learning. Students do most of the mathematics – it may seem like the teacher “isn’t teaching.” The teacher must select quality mathematical tasks that allow students to learn mathematics content as they figure out strategies and solutions. The teacher must plan and ask questions that encourage students to verify their solutions and reflect on the strategies they use. Listening is critical – the teacher listens to students! (Who talks most in your class? Who listens?) (Van de Walle, 2010)
- 8. + Polya, 1945 Devise a Plan Look Back Problem Solving Heuristics Understand the Problem Carry out the Plan
- 9. + Problem Solving Heuristics Understand What the Problem is known? What are the data? What are the conditions? What are you trying to find out?
- 10. + Problem Solving Heuristics Devise Have a Plan you solved a problem like this before? What strategies might work in this problem?
- 11. + Problem Solving Heuristics Carry out the Plan Check each step. Can you see that each step is correct? Can you prove that it is correct? If you aren’t making progress, feel free to return to steps 1 and/or 2.
- 12. + Problem Solving Heuristics Look Can Back you check the result? Does the answer make sense? Can you get the same answer a different way? Can you use the result, or the method, to solve a different problem? Is there more than one answer?
- 13. + Problem Solving Strategies Working Finding backwards a pattern Adopting Solving a different point of view a simpler problem Considering extreme cases Posamentier et al., 2010
- 14. + Problem Solving Strategies Making a drawing or diagram Intelligent guessing and testing Accounting for all possibilities Organizing data Logical reasoning Posamentier et al., 2010
- 15. + References Hiebert, J., et al. (1997). Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann. National Council of Supervisors of Mathematics. (1997). Position paper on basic mathematical skills. Retrieved Feb. 9, 2011, from http://www.mathedleadership.org/docs/resources/positionpapers/NCSMPositionPaper01_1977.pdf National Council of Teachers of Mathematics. (2000). Principles and standards of school mathematics. Reston, VA: NCTM. Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press. Van De Walle, J. A, Karp, K, & Bay-Williams, J. M.. (2010). Elementary and middle school mathematics (7th edition). Boston: Pearson Education.

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