Integrating Problem Solving
Problem solving should be incorporated into lessons throughout the year, not taught as a separate unit or lesson. Fully integrating problem solving in the classroom gives students "the chance to solidify and extend what they know and … can stimulate mathematics learning" (National Council of Teachers of Mathematics2000, p. 52)."
A problem-solving activity must ask students to determine a way to get from what is known to what is sought. If students have already been given ways to solve the problem, it is not a problem, but practice. A true problem requires students to use prior learnings in new ways and contexts. Problem solving requires and builds depth of conceptual understanding and student engagement.
Standards related to problem solving proposed by the NCTM indicate that instructional programs from Kindergarten through Grade 12 should enable all students to …
… build new mathematical knowledge through problem solving |
Good problems give students the chance to solidify and extend what they know and, when well chosen, can stimulate
mathematics learning. |
… solve problems that arise in mathematics and in other contexts |
Good problem solvers tend naturally to analyze situations carefully in mathematical terms and to pose problems based on situations they see. They first consider simple cases before trying something more complicated, yet they will readily consider a more sophisticated analysis. |
… apply and adapt a variety of appropriate strategies to solve problems |
As with any other component of the mathematical tool kit, strategies must receive instructional attention if students are expected to learn them. In the lower grades, teachers can help children express, categorize, and compare their strategies. Opportunities to use strategies must be embedded naturally in the curriculum across the content areas. By the time students reach the middle grades, they should be skilled at recognizing when various strategies are appropriate to use and should be capable of deciding when and how to use them. |
… monitor and reflect on the process of mathematical problem solving |
Effective problem solvers constantly monitor and adjust what they are doing. They make sure they understand the problem. If a problem is written down, they read it carefully; if it is told to them orally, they ask questions until they understand it. Effective problem solvers plan frequently. They periodically take stock of their progress to see whether they seem to be on the right track. If they decide they are not making progress, they stop to consider alternatives and do not hesitate to take a completely different approach. |
Adapted with permission from Principles and Standards for School Mathematics (pp. 52, 53, 54), copyright 2000 by the National Council of Teachers of Mathematics.
Roles of Teacher and Student
The teacher and the students have distinct roles in the development of problem solving strategies, for example:
The teacher … |
The student … |
- allows the students' to explore where their observations and questions may take them
- encourages multiple approaches
- allows time for communication and reflection about strategies
- asks questions that uncover students' thinking
- presses for the students' reasoning behind the process.
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- explores the problem
- from the exploration, develops models and methods of thinking about the problem
- from these models and methods, develops his or her reasoning and proves his or her thinking to be reasonable and valid
- discusses his or her reasoning and solutions
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Adapted from Nicole R. Rigelman, "Fostering Mathematical Thinking and Problem Solving: The Teacher's Role." Adapted with permission from Teaching Children Mathematics (13, 6, February 2007, p. 312), copyright 2007 by the National Council of Teachers of Mathematics. All rights reserved.
Mathematical Problem Solving Process
The process of problem solving can be viewed as a continuous cycle that involves exploring,
developing models and methods, proving models and methods, and discussing reasoning and solutions. These actions may occur simultaneously or in the order listed, depending on the individual and/or the problem he or she is solving.
Adapted from Nicole R. M. Rigelman, Teaching Mathematical Problem Solving in the Context of Oregon’s Educational Reform (dissertation, Portland State University, 2002), p. 184, as cited in Nicole R. Rigelman, "Fostering Mathematical Thinking and Problem Solving: The Teacher's Role." Adapted with permission from Teaching Children Mathematics (13, 6, February 2007, pp. 312, 313), copyright 2007 by the National Council of Teachers of Mathematics. All rights reserved. |
