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Semiotics in Mathematics Education

Epistemology, History, Classroom, and Culture


Edited by Luis Radford, Gert Schubring and Falk Seeger

Current interest in semiotics is undoubtedly related to our increasing awareness that our manners of thinking and acting in our world are deeply indebted to a variety of signs and sign systems (language included) that surround us.
Since mathematics is something that we accomplish through written, oral, bodily and other signs, semiotics appears well suited to furthering our understanding of the mathematical processes of thinking, symbolizing and communicating. Resorting to different semiotic perspectives (e. g., Peirce’s, Vygotsky’s, Saussure’s), the authors of this book deal with questions about the teaching and learning of mathematics as well as the history and epistemology of the discipline. Mathematics discourse and thinking and the technologically-mediated self of mathematical cultural practices are examined through key concepts such as metaphor, intentionality, gestures, interaction, sign-use, and meaning.
The cover picture comes from Jacob Leupold’s (1727) Theatrum Arithmetico-Geometrico. It conveys the cultural, historical, and embodied aspects of mathematical thinking variously emphasized by the contributors of this book.

Edited by Gerald Kulm

This book presents a coherent collection of research studies on teacher knowledge and its relation to instruction and learning in middle-grades mathematics. The authors provide comprehensive literature reviews on specific components of mathematics knowledge for teaching that have been found to be important for effective instruction. Based on the analysis of video data collected over a six-year project, the chapters present new and accessible research on the learning of fractions, early concepts of algebra, and basic statistics and probability.

The three sections of the book contain chapters that address research on the development of mathematics knowledge for teaching at the undergraduate level, instructional practices of middle-grades teachers, and the implications of teacher knowledge of mathematics for student learning. The chapters are written by members of a research team led by the Editor that has been working for the past six years to develop practical and useful theories and findings on variables that affect teaching and learning of middle grades mathematics.

Mathematics knowledge for teaching is a topic of great current interest. This book is a valuable resource for mathematics education researchers, graduate students, and teacher educators. In addition, professional developers and school district supervisor and curriculum leaders will find the concrete examples of effective teaching strategies useful for teacher workshops.

Edited by Erkki Pehkonen, Maija Ahtee and Jari Lavonen

The Finnish students’success in the first PISA 2000 evaluation was a surprise to most of the Finns, and even people working in teacher education and educational administration had difficulties to believe that this situation would continue. Finland’s second success in the next PISA 2003 comparison has been very pleasing for teachers and teacher educators, and for education policymakers. The good results on the second time waked us to think seriously on possible reasons for the success. Several international journalists and expert delegations from different countries have asked these reasons while visiting in Finland. Since we had no commonly acceptable explanation to students’success, we decided at the University of Helsinki to put together a book “How Finns Learn Mathematics and Science?”, in order to give a commonly acceptable explanation to our students’success in the international PISA evaluations. The book tries to explain the Finnish teacher education and school system as well as Finnish children’s learning environment at the level of the comprehensive school, and thus give explanations for the Finnish PISA success. The book is a joint enterprise of Finnish teacher educators. The explanations for success given by altogether 40 authors can be classified into three groups: Teacher and teacher education, school and curriculum, and other factors, like the use of ICT and a developmental project LUMA. The main result is that there is not one clear explanation, although research-based teacher education seems to have some influence. But the true explanation may be a combination of several factors.

How should I know?

Preservice Teachers' Images of Knowing (by Heart ) in Mathematics and Science


Kathleen T. Nolan

Elementary preservice teachers’school experiences of mathematics and science have shaped their images of knowing, including what counts as knowledge and what it means to know (in) mathematics and science. In this book, preservice teachers’ voices challenge the hegemony of official everyday narratives relating to these images.
The book is written as a parody of a physical science textbook on the topic of light, presenting a kaleidoscope of elementary preservice teachers’ narratives of knowing (in) mathematics and science. These narratives are tied together by the metaphorical thread of the properties of light, but also held apart by the tensions and contradictions with/in such a critical epistemological exploration. Through a postmodern lens, the only grand narrative that could be imag(in)ed for this text is one in which the personal lived experience narratives of the participants mingle and interweave to create a sort of kaleidoscope of narratives. With each turn of a kaleidoscope, light’s reflection engenders new patterns and emergent designs. The narratives of this research text highlight patterns of exclusion, gendered messages, binary oppositions, and the particle nature and shadowy texture of knowing (in) mathematics and science. The presentation format of the book emphasizes the reflexive and polyphonic nature of the research design, illustrated through layers of spoken text with/in performative text with/in metaphorical text.
The metaphor of a kaleidoscope is an empowering possibility for a critical narrative written to both engage and provoke the reader into imag(in)ing a critical journey toward possibilities for a different “knowing by heart” in mathematics and science and for appreciating lived experience narratives with/in teacher education.

Key Works in Radical Constructivism

(edited by Marie Larochelle)


Ernst von Glasersfeld

Key Works on Radical Constructivism brings together a number of essays by Ernst von Glasersfeld that illustrate the application of a radical constructivist way of thinking in the areas of education, language, theory of knowledge, and the analysis of a few concepts that are indispensable in almost everything we think and do. Ernst von Glasersfeld’s work opens a window on how we know what we know. The present work grew out of a desire to make more accessible this line of thought, to highlight its originality and consistency, and to illustrate its fecundity in the domains of cognition and learning.
The first three parts of this book contain texts by Glasersfeld that outline the constructivist approach and explicate the frequently drastic reconceptualizations he has suggested. Both the last part and the postscript consist of commentaries by Edith Ackermann, Jacques Désautels, Gérard Fourez, Leslie P. Steffe and Kenneth Tobin, scholars in the fields that Glasersfeld has been concerned with. They examine a number of critical aspects pertaining to (radical) constructivism’s current and future development, often tracing out paths that warrant further exploration and reflection, in particular concerning the sociopolitical dimension of knowledge.
Key Works on Radical Constructivism is intended as a reference book for researchers, educators, and students of education—and for anyone interested in grasping, or deepening their grasp of, radical constructivism’s tenets, ambitions and concerns. Readers will discover in this collection of firsthand contributions the contours of a bold, contemporary debate about a most compelling current of thought.

Mathematisation and Demathematisation

Social, Philosophical and Educational Ramifications

Edited by Uwe Gellert and Eva Jablonka

In this volume scholars from diverse strands of research have contributed their perspectives on a process of mathematisation, which renders social, economical or political relationships increasingly formal. At the same time, mathematical skills lose their importance as they become replaced by diverse technological tools; a process of demathematisation takes place. The computerization of financial transactions, calculation of taxes and fees, comparison of prices as well as orientation by means of GPS, visualisation of complex data and electronic voting systems—all these mathematical technologies increasingly penetrate the lifestyle of consumers. What are the perils and promises of this development? Who is in charge, who is affected, who is excluded?
A common concern of all the authors of this volume is an attempt to draw attention to issues related to the formatting power of mathematics and to its role as implicit knowledge, which results in a process of demathematisation. This process, having once received considerable attention, is now threatened to be eclipsed by the proliferation of a discussion of school mathematics, which shows a tendency of cutting off its own philosophical and political roots. Taken together, the contributions reveal a rather complex picture: They draw attention to the importance of clarifying epistemological, societal and ideological issues as a prerequisite for a discussion of curriculum.

Notational Knowledge

Developmental and Historical Perspectives

Edited by Eva Teubal, Julie Dockrell and Liliana Tolchinsky

Permanent external representations in the form of drawings, maps, musical scores, figures, graphs, writing, numerals, hallmarks and signatures are part of our daily landscape and permeate most social activities almost from the moment we are born. This book is about humans’ appropriation, understanding and use of external representations.
The authors, all established researchers, present first hand research in the domain of notational knowledge. They reflect on the peculiar features and representational mechanisms of notational systems based on cultural conventions such as musical notation, graphs, writing, numerals and mathematical notation as well as on unique notations that children create in new situations. There are two chapter clusters in the book. The first cluster considers these systems from a historical perspective. Authors focus on the characteristics of these systems in different cultures and at different times and analyze the ways in which notation systems evolve and transform our social interactions, our ideas about language and about other domains of knowledge. The second cluster of chapters takes a developmental perspective. In these chapters the authors focus on the individual appropriation of these systems and highlight the interest for studying permanent external representation as a domain of human development. In particular, authors explore the ways in which notation systems are acquired, the extent to which children are sensitive to their distinguishing constraints and to the particular contents they come to represent and question the future of notations. Both the historical and the developmental perspectives are crucial for understanding the relations between culture and cognition. We can learn about the human mind through analyzing the social processes of invention and transformations of cultural artifacts and also through the individual and social process of appropriation of the cultural artifacts already created.
A common theme in the book is that permanent external representations are not just instruments for expressing given information or tools for communication, they are objects to think with. They not only keep record of existing knowledge but are themselves instrumental in the creation of new knowledge. There are conceptual, linguistic and esthetic distinctions that may be unattainable without notational means.
The book will be useful for students of psychology, philosophy, linguistics and education and for every one interested in understanding ways in which knowledge is generated, recorded and scrutinized.
Although there are other volumes on writing, literacy, and numeracy, and some chapters are available in other volumes on the history of writing, mathematics or musical notation, the present proposal is unique and timely for the range of notational systems it embraces, for including both an historical and a developmental perspective and for the number of theoretical frameworks it discusses.

Stepping Stones for the 21st Century

Australasian Mathematics Education Research

Edited by Gilah C. Leder and Helen Forgasz

Over the years a number of "must read" articles and book chapters have appeared—work that has formed the foundational stepping stones of mathematics education research for the 21st century. Twelve such seminal articles have been reproduced in this book. Each is accompanied by two independent appraisals of the longer term impact of the work within and beyond the mathematics education research community. Collectively these writings cover a wide range of topics and provide a broad overview of the outstanding contributions of Australasian mathematics education research prior to 2000.

Theorems in School

From History, Epistemology and Cognition to Classroom Practice


Edited by Paulo Boero

During the last decade, a revaluation of proof and proving within mathematics curricula was recommended; great emphasis was put on the need of developing proof-related skills since the beginning of primary school.
This book, addressing mathematics educators, teacher-trainers and teachers, is published as a contribution to the endeavour of renewing the teaching of proof (and theorems) on the basis of historical-epistemological, cognitive and didactical considerations. Authors come from eight countries and different research traditions: this fact offers a broad scientific and cultural perspective.
In this book, the historical and epistemological dimensions are dealt with by authors who look at specific research results in the history and epistemology of mathematics with an eye to crucial issues related to educational choices. Two papers deal with the relationships between curriculum choices concerning proof (and the related implicit or explicit epistemological assumptions and historical traditions) in two different school systems, and the teaching and learning of proof there.
The cognitive dimension is important in order to avoid that the didactical choices do not fit the needs and the potentialities of learners. Our choice was to firstly deal with the features of reasoning related to proof, mainly concerning the relationships between argumentation and proof.
The second part of this book concentrates on some crucial cognitive and didactical aspects of the development of proof from the early approach in primary school, to high school and university. We will show how suitable didactical proposals within appropriate educational contexts can match the great (yet, underestimated!) young students’ potentialities in approaching theorems and theories.

Margaret Walshaw

Education has a long tradition of opening itself up to new ideas and new ideas are what Working with Foucault in Education is all about. The book introduces readers to the scholarly work of Michel Foucault at a level that it neither too demanding not too superficial. It demonstrates to students, educators, scholars and policy makers, alike, how those ideas might be useful in understanding people and processes in education. This new line of investigation creates an awareness of the merits and weaknesses of contemporary theoretical frameworks and the impact these have on the production of educational knowledge.
Working with Foucault in Education engages readers in selected aspects of education. Its ten chapters take a thematic approach and include vignettes that explore issues relating to curriculum development, learning to teach, classroom learning and teaching, as well as research in contemporary society. These explorations allow readers to develop a new attitude towards education. The reason this is possible is that Foucault provides a language and the tools to deconstruct as well as shift thinking about familiar concepts. They also provide the means for readers to participate in educational criticism and to play a role in educational change.