This volume is a result of mathematicians, cognitive scientists, mathematics educators, and classroom teachers combining their efforts to help address issues of importance to classroom instruction in mathematics. In so doing, the contributors provide a general introduction to fundamental ideas in cognitive science, plus an overview of cognitive theory and its direct implications for mathematics education. A practical, no-nonsense attempt to bring recent research within reach for practicing teachers, this book also raises many issues for cognitive researchers to consider.

An accessible introduction to some of the cognitive issues important for thinking and learning in scientific or other complex domains (such as mathematics, physics, chemistry, engineering, or expository writing), with practical educational applications and implementation methods. Many students find it difficult to learn the kind of knowledge and thinking required by college or high school courses in mathematics, science, or other complex domains. Thus they often emerge with significant misconceptions, fragmented knowledge, and inadequate problem-solving skills. Most instructors or textbook authors approach their teaching efforts with a good knowledge of their field of expertise but little awareness of the underlying thought processes and kinds of knowledge required for learning in scientific domains. In this book, Frederick Reif presents an accessible coherent introduction to some of the cognitive issues important for thinking and learning in scientific or other complex domains (such as mathematics, science, physics, chemistry, biology, engineering, or expository writing). Reif, whose experience teaching physics at the University of California led him to explore the relevance of cognitive science to education, examines with some care the kinds of knowledge and thought processes needed for good performance; discusses the difficulties faced by students trying to deal with unfamiliar scientific domains; describes some explicit teaching methods that can help students learn the requisite knowledge and thinking skills; and indicates how such methods can be implemented by instructors or textbook authors. Writing from a practically applied rather than predominantly theoretical perspective, Reif shows how findings from recent research in cognitive science can be applied to education. He discusses cognitive issues related to the kind of knowledge and thinking skills that are needed for science or mathematics courses in high school or colleges and that are essential prerequisites for more advanced intellectual performance. In particular, he argues that a better understanding of the underlying cognitive mechanisms should help to achieve a more scientific approach to science education.

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.

The fifth volume in the Mathematical Cognition and Learning series focuses on informal learning environments and other parental influences on numerical cognitive development and formal instructional interventions for improving mathematics learning and performance. The chapters cover the use of numerical play and games for improving foundational number knowledge as well as school math performance, the link between early math abilities and the approximate number system, and how families can help improve the early development of math skills. The book goes on to examine learning trajectories in early mathematics, the role of mathematical language in acquiring numeracy skills, evidence-based assessments of early math skills, approaches for intensifying early mathematics interventions, the use of analogies in mathematics instruction, schema-based diagrams for teaching ratios and proportions, the role of cognitive processes in treating mathematical learning difficulties, and addresses issues associated with intervention fadeout. Identifies the relative influence of school and family on math learning Discusses the efficacy of numerical play for improvement in math Features learning trajectories in math Examines the role of math language in numeracy skills Includes assessments of math skills Explores the role of cognition in treating math-based learning difficulties

The last decade has seen a rapid growth in our understanding of the cognitive systems that underlie mathematical learning and performance, and an increased recognition of the importance of this topic. This book showcases international research on the most important cognitive issues that affect mathematical performance across a wide age range, from early childhood to adulthood. The book considers the foundational competencies of nonsymbolic and symbolic number processing before discussing arithmetic, conceptual understanding, individual differences and dyscalculia, algebra, number systems, reasoning and higher-level mathematics such as formal proof. Drawing on diverse methodology from behavioural experiments to brain imaging, each chapter discusses key theories and empirical findings and introduces key tasks used by researchers. The final chapter discusses challenges facing the future development of the field of mathematical cognition and reviews a set of open questions that mathematical cognition researchers should address to move the field forward. This book is ideal for undergraduate or graduate students of psychology, education, cognitive sciences, cognitive neuroscience and other academic and clinical audiences including mathematics educators and educational psychologists.

Didactics of Mathematics as a Scientific Discipline describes the state of the art in a new branch of science. Starting from a general perspective on the didactics of mathematics, the 30 original contributions to the book, drawn from 10 different countries, go on to identify certain subdisciplines and suggest an overall structure or `topology' of the field. The book is divided into eight sections: (1) Preparing Mathematics for Students; (2) Teacher Education and Research on Teaching; (3) Interaction in the Classroom; (4) Technology and Mathematics Education; (5) Psychology of Mathematical Thinking; (6) Differential Didactics; (7) History and Epistemology of Mathematics and Mathematics Education; (8) Cultural Framing of Teaching and Learning Mathematics. Didactics of Mathematics as a Scientific Discipline is required reading for all researchers into the didactics of mathematics, and contains surveys and a variety of stimulating reflections which make it extremely useful for mathematics educators and teacher trainers interested in the theory of their practice. Future and practising teachers of mathematics will find much to interest them in relation to their daily work, especially as it relates to the teaching of different age groups and ability ranges. The book is also recommended to researchers in neighbouring disciplines, such as mathematics itself, general education, educational psychology and cognitive science.