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Problem solving is one of the process standards clearly discussed in the Principles and Standards of School Mathematics. (1) The Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, 2006, NCTM, also clearly emphasizes problem solving.
So one might ask, how often should teachers have students “do” Problem Solving? The answer is not as simple as it may seem. As stated in the Principles and Standards, “Problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program.” (2)
Problem solving should be embedded within the majority of classroom mathematics instruction, indeed problem solving should serve as the foundation for mathematics instruction. Asking students to identify their understandings, communicate their thinking, utilize a variety of methods and reflect on a wide variety of solution strategies should be elements contained in every math lesson.
Ever since the first Trends in International Mathematics and Science Study (TIMSS) results were published in 1995, there has been a call to reform US mathematics programs to match more closely the model for mathematics instruction used in Japan. Original TIMSS data showed that U.S. lessons tended to be structured into two sections. First the teacher reviews the answers to the previous assignment, then s/he demonstrates or explains how to solve a sample problem. Next, students apply that same method to multiple practice problems, some of which are to be completed as homework. In contrast, Japanese teachers believe that problem solving is the standard approach for all mathematics instruction. The Japanese lessons start with a problem situation, as students independently approach possibilities for solutions and or strategies. Students then work with others to carry out a possible solution, while the teacher monitors approaches and completeness of strategies. The teacher then purposefully selects solutions to be shared with the entire class that will lead to a focused discussion of the intended mathematical concepts, with connections to previous |
learning and in ways that will connect to future learning of related concepts. Students are encouraged to analyze all of the solution strategies that are presented and determine which one is most effective. Deep understanding of a mathematical concept comes about in the process of problem solving when students invent their own solutions and/or methods. Japanese teachers focus a great deal of their planning on anticipating how they think their students will attempt to solve one complex task so they can pre-plan how to help students discover misconceptions and maximize the conceptual understanding that can occur. (3)
Most of the National Science Foundation-Grant curricula have incorporated the focus on teaching mathematics through problem solving, and those curricula have been used in many U.S. schools. The 2007 TIMSS study results have recently been published, and they show that there has been improvement in math instruction in the U.S. The average mathematics scores for both fourth and eighth-grade U.S. students were about 2 to 3 percent higher than in the first international test in 1995. That small but significant gain substantially boosted the U.S ranking into the top 20% of all countries that participated in the evaluation. Surely, some of this gain is from the focus on teaching mathematics through problem solving.
The Making Sense of Problem Solving materials have been written on the foundation that problem solving is an essential vehicle for mathematics instruction. In addition, the MSPS materials help teachers teach students how to be problem solvers.
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1 Principles and Standards of School Mathematics 2000, National Council of the Teachers of Mathematics (NCTM), Reston, VA.
2 Principles and Standards of School Mathematics 2000, NCTM, p. 52.
3 Shimizu, Y.,2003, Problem Solving as a Vehicle for Teaching Mathematics: A Japanese Perspective, in F. K Lester, Jr., and R. I Charles, eds., Teaching Mathematics through Problem Solving: Prekindergarten – Grade 6, NCTM, Reston, VA, p. 205-214.
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