Why Teach Open-Ended Problem Solving - To help young people be better problem solvers is to prepare them not only to think mathematically but to approach life's challenges with confidence in their problem-solving ability. The thinking and skills required for mathematical problem solving transfer to other areas of life. The writers of the groundbreaking report Everybody Counts: A Report to the Nation on the Future of Mathematics Education put it this way:
Experience with mathematical modes of thought builds mathematical power—a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. Mathematics empowers us to understand better the information-laden world in which we live (National Research Council, 1989). Learning mathematics by grappling with open-ended and challenging problems accommodates diverse learning styles. The active and varied nature of problem solving helps students with diverse learning styles to develop and demonstrate mathematical understanding (Moyer, Cai, & Grampp, 1997). Traditional teaching approaches involving rote learning and teacher-centered instructional strategies often do not meet the learning needs of many students who may be active learners or require multiple entrances into the curriculum.
Learning through open-ended problem solving helps students to develop understanding that is flexible, that can be adapted to new situations and used to learn new things (Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human, 1997).
“Things learned with understanding are the most useful things to know in a changing and unpredictable world,” explains Hiebert and colleagues. Yet, usefulness is not the only reason to learn with understanding. To learn with understanding is to also grapple intellectually with mathematics as a subject. “When we memorize rules for moving symbols around on paper we may be learning something, but we are not learning mathematics,” says Hiebert. “Knowing a subject means getting inside it and seeing how things work, how things are related to each other, and why they work like they do.”
When students encounter mathematical ideas that interest and challenge them in an openended problem solving context, they are more likely to experience the kinds of internal rewards that keep them engaged, says Hiebert (Hiebert et al., 1997). Students who must resort to memorizing will lack understanding and will likely feel little sense of satisfaction, perhaps withdrawing from learning altogether. In fact, he says, evidence suggests that if students memorize and practice procedures repeatedly in a rote fashion, it's difficult for them to go back later and gain a deeper understanding of the mathematical concepts underlying those procedures. Researchers Jerry Becker and Shigeru Shimada (1997) concur: “Lessons based on solving open-ended problems as a central theme have a rich potential for improving teaching and learning.”
Recognizing the centrality of problem solving to mathematics learning, education leaders have made it a focal point of standards reform for the past two decades. In the spring of 2000, the National Council of Teachers of Mathematics renewed its commitment to problem solving when it published Principles and Standards for School Mathematics (NCTM, 2000), an update of the council's 1989 statement about standards for teaching and learning mathematics, Curriculum and Evaluation Standards for School Mathematics. Like that seminal work, the council's updated standards identify problem solving as an essential component of math learning for all grade levels.
Furthermore, many states have adopted content and performance standards and assessments based on the NCTM standards that include an emphasis on problem solving. While the Northwest states are at different stages in the process of adopting standards and developing assessment systems, most are addressing the importance of teaching and assessing reasoning, communicating, making connections, and applying knowledge to problem situations—key tenets of problem solving.
Experience with mathematical modes of thought builds mathematical power—a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. Mathematics empowers us to understand better the information-laden world in which we live (National Research Council, 1989). Learning mathematics by grappling with open-ended and challenging problems accommodates diverse learning styles. The active and varied nature of problem solving helps students with diverse learning styles to develop and demonstrate mathematical understanding (Moyer, Cai, & Grampp, 1997). Traditional teaching approaches involving rote learning and teacher-centered instructional strategies often do not meet the learning needs of many students who may be active learners or require multiple entrances into the curriculum.
Learning through open-ended problem solving helps students to develop understanding that is flexible, that can be adapted to new situations and used to learn new things (Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human, 1997).
“Things learned with understanding are the most useful things to know in a changing and unpredictable world,” explains Hiebert and colleagues. Yet, usefulness is not the only reason to learn with understanding. To learn with understanding is to also grapple intellectually with mathematics as a subject. “When we memorize rules for moving symbols around on paper we may be learning something, but we are not learning mathematics,” says Hiebert. “Knowing a subject means getting inside it and seeing how things work, how things are related to each other, and why they work like they do.”
When students encounter mathematical ideas that interest and challenge them in an openended problem solving context, they are more likely to experience the kinds of internal rewards that keep them engaged, says Hiebert (Hiebert et al., 1997). Students who must resort to memorizing will lack understanding and will likely feel little sense of satisfaction, perhaps withdrawing from learning altogether. In fact, he says, evidence suggests that if students memorize and practice procedures repeatedly in a rote fashion, it's difficult for them to go back later and gain a deeper understanding of the mathematical concepts underlying those procedures. Researchers Jerry Becker and Shigeru Shimada (1997) concur: “Lessons based on solving open-ended problems as a central theme have a rich potential for improving teaching and learning.”
Recognizing the centrality of problem solving to mathematics learning, education leaders have made it a focal point of standards reform for the past two decades. In the spring of 2000, the National Council of Teachers of Mathematics renewed its commitment to problem solving when it published Principles and Standards for School Mathematics (NCTM, 2000), an update of the council's 1989 statement about standards for teaching and learning mathematics, Curriculum and Evaluation Standards for School Mathematics. Like that seminal work, the council's updated standards identify problem solving as an essential component of math learning for all grade levels.
Furthermore, many states have adopted content and performance standards and assessments based on the NCTM standards that include an emphasis on problem solving. While the Northwest states are at different stages in the process of adopting standards and developing assessment systems, most are addressing the importance of teaching and assessing reasoning, communicating, making connections, and applying knowledge to problem situations—key tenets of problem solving.
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