Mathematics Education: Models and Processes

Mathematics Education: Models and Processes

Mathematics Education: Models and Processes

Mathematics Education: Models and Processes

Synopsis

To define better techniques of mathematics education, this book combines a knowledge of cognitive science with mathematics curriculum theory and research. The concept of the human reasoning process has been changed fundamentally by cognitive science in the last two decades. The role of memory retrieval, domain-specific and domain-general skills, analogy, and mental models is better understood now than previously. The authors believe that cognitive science provides the most accurate account thus far of the actual processes that people use in mathematics and offers the best potential for genuine increases in efficiency. As such, they suggest that a cognitive science approach enables constructivist ideas to be analyzed and further developed in the search for greater understanding of children's mathematical learning. Not simply an application of cognitive science, however, this book provides a new perspective on mathematics education by examining the nature of mathematical concepts and processes, how and why they are taught, why certain approaches appear more effective than others, and how children might be assisted to become more mathematically powerful. The authors use recent theories of analogy and knowledge representation -- combined with research on teaching practice -- to find ways of helping children form links and correspondences between different concepts, so as to overcome problems associated with fragmented knowledge. In so doing, they have capitalized on new insights into the values and limitations of using concrete teaching aids which can be analyzed in terms of analogy theory. In addition to addressing the role of understanding, the authors have analyzed skill acquisition models in terms of their implications for the development of mathematical competence. They place strong emphasis on the development of students' mathematical reasoning and problem solving skills to promote flexible use of knowledge. The book further demonstrates how children have a number of general problem solving skills at their disposal which they can apply independently to the solution of novel problems, resulting in the enhancement of their mathematical knowledge.

Excerpt

This book is intended primarily for those interested in education, especially mathematics education, and also for those concerned with cognitive development and cognitive science. Our purpose in writing the book was to define better techniques of mathematics education by combining a knowledge of cognitive science with mathematics curriculum theory and research. Our concept of the human reasoning process has been changed fundamentally by cognitive science in the last two decades. The roles of memory retrieval, domain-specific and domain-general skills, analogy, and mental models are better understood now than previously. We believe cognitive science provides the most accurate account so far of the actual processes that people use in mathematics, and offers the best potential for genuine increases in efficiency. As we indicate, a cognitive science approach enables constructivist ideas to be analyzed and further developed in our search for greater understanding of children's mathematical learning.

The book is not simply an application of cognitive science, however. We provide a new perspective on mathematics education by examining the nature of mathematical concepts and processes, how and why we teach them, why certain approaches appear to be more effective than others, and how we might assist children to become more mathematically powerful. We use theories of analogy and knowledge representation, combined with research on teaching practice, to help children form links and correspondences between different concepts, and between the same concept in different contexts, so as to overcome problems associated with fragmented knowledge. In so doing, we capitalize on new insights into the values and limitations of using concrete teaching aids that can be analyzed in terms of analogy theory.

In addition to addressing the role of understanding, we analyze skill acquisition models in terms of their implications for the development of mathematical competence. We place a strong emphasis on the development . . .

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