New Strategies in Problem Solving: How to Develop Confident and Flexible Problem Solvers

1. New Strategies in Problem Solving: How to Develop Confident and Flexible Problem Solvers Incarnate Word Academy Elementary Level March4, 2011 Dr. Sarah Ives Assistant Professor of Mathematics Texas A&M University – Corpus Christi

2. Outline of Session • Overview of Problem Solving – • NCTM Problem Solving Standard • What is a problem? • Is there a process? • What are some strategies? • Problem Solving in Action • Planning for Instruction on Problem Solving • Questions and Further Discussions

3. NCTM Problem Solving Standard • The National Council of Teachers of Mathematics has included Problem Solvingas one of the process standards: • Instructional programs from pre-kindergarten through grade 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 NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Key Curriculum Press.

4. What is a Problem? Charles and Lester (1982) define a mathematical problem as a task for which: • The person confronting it wants or needs to find a solution; • The person has no readily available procedure for finding the solution; and • The person must make an attempt to find a solution. Charles, R. I., & Lester, F. K., Jr. (1982). Teaching problem solving: What, why, & how. Palo Alto, CA: Seymore. “Problem solving means engaging in a task for which the solution method is not known in advance” (NCTM, p 52)

5. The Problem-Solving Process Polya’s (1957) 4-step process: • Understand the problem – Provide time for students to identify the goal, what information is needed and what is extraneous, and detect any missing information; • Devise a plan to solve the problem – Students will use various strategies, have them discuss different ways to solve the same problem; • Implement a solution plan – encourage students to use their own ingenuity to develop a solution plan; and • Reflect on the problem – have students look back at the problem; they should be ready to explain and justify their solutions when asked. Polya, G. (1957). How to Solve It (2nd Ed.). New York: Doubleday.

6. Problem Solving Strategies Use Table or Chart Act out or Model problem Draw a picture Solve a simpler problem Find a Pattern Guess & Check Consider all Possibilities Work Backwards Changing Point of View Logical Reasoning Write Open Sentence

7. Problem Solving in Action Now it’s your turn! • Barnyard animals • Euler Squares • Pocket change • Candy bags • Rabbits & hutches • Party tables • Frogs & Lily pads Source: http://sci.tamucc.edu/~sives/1350/problem_solving11.html

8. Planning for Instruction on Problem Solving • Several Important Components: • Selecting appropriate tasks and materials, • Identifying sources of problems, • Clarifying the teacher’s role, • Organizing and implementing instruction, and • Changing the difficulty of problems.

9. Teacher’s Role Instead of: • Focusing on helping students “find an answer”, • Providing solution strategies, • Expecting specific responses, The teacher: • Is prepared to see where the students’ observations and questions may take them. • Encourages multiple approaches and allows time for communication and reflection about those strategies. • Is ready to ask questions that uncover students’ reasoning behind the process (Rigelman, 2007, p. 312). Rigelman, N. (2007). Fostering mathematical thinking and problem solving: The teacher’s role. Teaching Children Mathematics, 13(6), 308-314.

10. Organizing and Implementing Instruction • Classroom Climate • Open, supportive, encourage children to try different strategies • Grouping Children • Include individual, small-group, and whole-class problem-solving experiences • Allocating time • Problem solving should be an integral part of mathematics instruction, not ‘Friday’s only’ • Assessing children’s understanding • Ongoing assessment of understanding and problem-solving skills can be done by having students discuss and present solutions

11. Additional Resources • LINKS TO THE INTERNET • Problems of the Week: http://www.mathforum.org/pow/ Contains several weekly “Problems of the Week” as well as a mechanism to submit solutions electronically. Past “Problems of the Week” and solutions are also available. • Open-ended Math Problems: http://www.fi.edu/school/math2/ Contains open-ended math problems at several different levels of difficulty for middle school students. • Education Place’s Brain Teasers: http://www.eduplace.com/math/brain/ Contains math puzzles for Grades 3-8 as well as solution hints. Source: Bezuk, N., Cathcart, W., Pothier, Y., & Vance, J. (2011). Page 59.

12. Additional Resources • RESOURCES FOR TEACHERS • Reference Books: Problem Solving • Baroody, A. (1993). Problem Solving, Reasoning, and Communicating: Helping Children Think Mathematically. New York: Macmillan. • Charles, R., Lester, F., & O’Daffer, P. (1987). How to Evaluate Progress in Problem Solving. Reston, VA: National Council of Teachers of Mathematics. • O’Daffer, P. G. (1988). Problem Solving: Tips for Teachers. Reston, VA: National Council of Teachers of Mathematics. • Reys, B. (1982). Elementary School Mathematics: What Parents Should Know about Problem Solving. Reston, VA: National Council of Teachers of Mathematics. Source: Bezuk, N., Cathcart, W., Pothier, Y., & Vance, J. (2011). Page 59.

13. Questions, Comments? • Main Source: Bezuk, N., Cathcart, W., Pothier, Y., & Vance, J. (2011). Learning Mathematics in Elementary and Middle Schools: A Learner-Centered Approach (5th Ed.). Boston, MA: Pearson. • http://sci.tamucc.edu/~sives • Sarah.Ives@tamucc.edu • (361) 825-2151