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# Teaching as Problem Solving

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Apply Polya's heuristics for problem solving to the "problem" of teaching.

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### Teaching as Problem Solving

1. 1. + Problem Solving Dr. Mary Pat Sjostrom Bear with me… this is relevant to this module.
2. 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)
3. 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. 4. + Devise a Plan Look Back Problem Solving Heuristics Understand the Problem Carry out the Plan Polya, 1945 Remember these?
5. 5. Can we view teaching as + problem solving?
6. 6. + Devise a Plan Look Back Teaching as problem solving Understand the Problem Carry out the Plan We can approach and solve teaching as we approach and solve a mathematics problem: heuristically, rather than algorithmically. How do Polya’s heuristics apply to teaching?
7. 7. + Understand the Problem  What are your goals for students?  What concept do you want students to understand?  What is the Big Idea of the mathematics?  What are the real world connections?  What do you expect students to already know about this concept?  What misconceptions might students have?  Where would you like student thinking to be at the end of this lesson?
8. 8. + Devise a plan  How do you usually teach this lesson?  What problem (or what coherent sequence of problems) can you pose that will engage students in developing this mathematical concept?  What resources are available to you? For the students?  How do you expect students to respond to __________?  How will you know students understand?
9. 9. + Carry out the plan  Teach the lesson.  Listen to student thinking.  Students should do the mathematics. (You don’t need the practice, they do!)  Monitor and reflect on your teaching.  You may have to modify the plan as the lesson progresses.
10. 10. + Look Back  How do you think the lesson went?  What worked? What didn’t work?  Did students learn what you wanted them to learn? How do you know?  What would you do differently if you could teach the lesson again?  Based on what happened today, what will you do when the class meets again?
11. 11. + 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. Posamentier, A. S., Smith, B., &Stepelman, J. (2010). Teaching secondary mathematics: Techniques and enrichment units (8th Edition). Upper Saddle River, NJ: Merrill Prentice Hall. Van De Walle, J. A, Karp, K, & Bay-Williams, J. M.. (2010). Elementary and middle school mathematics (7th edition). Boston: Pearson Education.