Word problems have been a staple of mathematics instruction for centuries, yet the rationale for their use has remained largely unexamined. A range of findings have shown how students consistently answer them in ways that fail to take account of the reality of the situations described. This monograph reports on studies carried out to investigate this "suspension of sense-making" in answering word problems. In Part One, a wide range of examples documenting the strength of the phenomenon is reviewed. Initial surprise at the findings was replaced by a conviction that the explanation lies in the culture of the mathematics classroom, specifically the rules implicitly governing the nature and interpretation of the word problem genre. This theoretical shift is reflected in Part Two. A detailed analysis of the way in which word problems are currently taught in typical mathematical classrooms is followed by reviews of design experiments illustrating how, by immersing students in a fundamentally changed learning environment, they can acquire what the authors consider to be more appropriate conceptions about, and strategies for doing, word problems. Part Three turns to a wider discussion of theoretical issues, a further analysis of the features of the educational system considered responsible for outcomes detrimental to many students' understanding and conception of mathematics, and suggestions for rethinking the role of word problems within the curriculum.
This book presents the reader with a comprehensive overview of the major findings of the recent research on the illusion of linearity. It discusses: how the illusion of linearity appears in diverse domains of mathematics and science; what are the crucial psychological, mathematical, and educational factors being responsible for the occurrence and persistence of the phenomenon; and how the illusion of linearity can be remedied.
This book examines the kinds of transitions that have been studied in mathematics education research. It defines transition as a process of change, and describes learning in an educational context as a transition process. The book focuses on research in the area of mathematics education, and starts out with a literature review, describing the epistemological, cognitive, institutional and sociocultural perspectives on transition. It then looks at the research questions posed in the studies and their link with transition, and examines the theoretical approaches and methods used. It explores whether the research conducted has led to the identification of continuous processes, successive steps, or discontinuities. It answers the question of whether there are difficulties attached to the discontinuities identified, and if so, whether the research proposes means to reduce the gap – to create a transition. The book concludes with directions for future research on transitions in mathematics education.
Within an increasingly multimedia focused society, the use of external representations in learning, teaching and communication has increased dramatically. Whether in the classroom, university or workplace, there is a growing requirement to use and interpret a large variety of external representational forms and tools for knowledge acquisition, problem solving, and to communicate with others. Use of Representations in Reasoning and Problem Solving brings together contributions from some of the world’s leading researchers in educational and instructional psychology, instructional design, and mathematics and science education to document the role which external representations play in our understanding, learning and communication. Traditional research has focused on the distinction between verbal and non-verbal representations, and the way they are processed, encoded and stored by different cognitive systems. The contributions here challenge these research findings and address the ambiguity about how these two cognitive systems interact, arguing that the classical distinction between textual and pictorial representations has become less prominent. The contributions in this book explore: how we can theorise the relationship between processing internal and external representations what perceptual and cognitive restraints can affect the use of external representations how individual differences affect the use of external representations how we can combine external representations to maximise their impact how we can adapt representational tools for individual differences. Using empirical research findings to take a fresh look at the processes which take place when learning via external representations, this book is essential reading for all those undertaking postgraduate study and research in the fields of educational and instructional psychology, instructional design and mathematics and science education.
This book examines the kinds of transitions that have been studied in mathematics education research. It defines transition as a process of change, and describes learning in an educational context as a transition process. The book focuses on research in the area of mathematics education, and starts out with a literature review, describing the epistemological, cognitive, institutional and sociocultural perspectives on transition. It then looks at the research questions posed in the studies and their link with transition, and examines the theoretical approaches and methods used. It explores whether the research conducted has led to the identification of continuous processes, successive steps, or discontinuities. It answers the question of whether there are difficulties attached to the discontinuities identified, and if so, whether the research proposes means to reduce the gap – to create a transition. The book concludes with directions for future research on transitions in mathematics education.
This book presents the reader with a comprehensive overview of the major findings of the recent research on the illusion of linearity. It discusses: how the illusion of linearity appears in diverse domains of mathematics and science; what are the crucial psychological, mathematical, and educational factors being responsible for the occurrence and persistence of the phenomenon; and how the illusion of linearity can be remedied.
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