Numbers are the backbones of mathematics. From 1 to infinity, numbers accompany and underlie the learning of mathematics and research. While perceived as familiar and understood, numbers present fascinating and often mysterious patterns, relationships and pedagogical issues. The Learning and Teaching of Number explores how mathematics education research has addressed issues related to the structure of numbers and number operations and provides a classroom context. It invites readers to explore less-travelled paths through a well-trodden terrain of number. This fascinating book combines mathematical content with pedagogical ideas and research results. Focusing on number, the book illustrates central ideas related to numbers via a variety of tasks at different levels of complexity. The Learning and Teaching of Number will allow the reader to examine and develop personal understanding of number sets and the relationships among them; enhance personal understanding of familiar topics associated with number operations; engage in a variety of tasks and strengthen personal problem-solving skills; enrich their repertoire of mathematical tasks and pedagogical actions; and consider research ideas and results related to teaching numbers, number operations and number relationships. This is a valuable resource for teacher education courses, graduate programs in mathematics education and professional development programs. Teacher trainers and maths teachers will find their personal understanding of numbers and relationships enriched and will draw connections between research and classroom pedagogy which will extend and enhance their teaching.

Numbers are the backbones of mathematics. From 1 to infinity, numbers accompany and underlie the learning of mathematics and research. While perceived as familiar and understood, numbers present fascinating and often mysterious patterns, relationships and pedagogical issues. The Learning and Teaching of Number explores how mathematics education research has addressed issues related to the structure of numbers and number operations and provides a classroom context. It invites readers to explore less-travelled paths through a well-trodden terrain of number. This fascinating book combines mathematical content with pedagogical ideas and research results. Focusing on number, the book illustrates central ideas related to numbers via a variety of tasks at different levels of complexity. The Learning and Teaching of Number will allow the reader to examine and develop personal understanding of number sets and the relationships among them; enhance personal understanding of familiar topics associated with number operations; engage in a variety of tasks and strengthen personal problem-solving skills; enrich their repertoire of mathematical tasks and pedagogical actions; and consider research ideas and results related to teaching numbers, number operations and number relationships. This is a valuable resource for teacher education courses, graduate programs in mathematics education and professional development programs. Teacher trainers and maths teachers will find their personal understanding of numbers and relationships enriched and will draw connections between research and classroom pedagogy which will extend and enhance their teaching.

This book offers multiple interconnected perspectives on the largely untapped potential of elementary number theory for mathematics education: its formal and cognitive nature, its relation to arithmetic and algebra, its accessibility, its utility and intrinsic merits, to name just a few. Its purpose is to promote explication and critical dialogue about these issues within the international mathematics education community. The studies comprise a variety of pedagogical and research orientations by an international group of researchers that, collectively, make a compelling case for the relevance and importance of number theory in mathematics education in both pre K-16 settings and mathematics teacher education. Topics variously engaged include: *understanding particular concepts related to numerical structure and number theory; *elaborating on the historical and psychological relevance of number theory in concept development; *attaining a smooth transition and extension from pattern recognition to formative principles; *appreciating the aesthetics of number structure; *exploring its suitability in terms of making connections leading to aha! insights and reaching toward the learner's affective domain; *reexamining previously constructed knowledge from a novel angle; *investigating connections between technique and theory; *utilizing computers and calculators as pedagogical tools; and *generally illuminating the role number theory concepts could play in developing mathematical knowledge and reasoning in students and teachers. Overall, the chapters of this book highlight number theory-related topics as a stepping-stone from arithmetic toward generalization and algebraic formalism, and as a means for providing intuitively grounded meanings of numbers, variables, functions, and proofs. Number Theory in Mathematics Education: Perspectives and Prospects is of interest to researchers, teacher educators, and students in the field of mathematics education, and is well suited as a text for upper-level mathematics education courses.

This book offers multiple interconnected perspectives on the largely untapped potential of elementary number theory for mathematics education: its formal and cognitive nature, its relation to arithmetic and algebra, its accessibility, its utility and intrinsic merits, to name just a few. Its purpose is to promote explication and critical dialogue about these issues within the international mathematics education community. The studies comprise a variety of pedagogical and research orientations by an international group of researchers that, collectively, make a compelling case for the relevance and importance of number theory in mathematics education in both pre K-16 settings and mathematics teacher education. Topics variously engaged include: *understanding particular concepts related to numerical structure and number theory; *elaborating on the historical and psychological relevance of number theory in concept development; *attaining a smooth transition and extension from pattern recognition to formative principles; *appreciating the aesthetics of number structure; *exploring its suitability in terms of making connections leading to aha! insights and reaching toward the learner's affective domain; *reexamining previously constructed knowledge from a novel angle; *investigating connections between technique and theory; *utilizing computers and calculators as pedagogical tools; and *generally illuminating the role number theory concepts could play in developing mathematical knowledge and reasoning in students and teachers. Overall, the chapters of this book highlight number theory-related topics as a stepping-stone from arithmetic toward generalization and algebraic formalism, and as a means for providing intuitively grounded meanings of numbers, variables, functions, and proofs. Number Theory in Mathematics Education: Perspectives and Prospects is of interest to researchers, teacher educators, and students in the field of mathematics education, and is well suited as a text for upper-level mathematics education courses.

This book explores the idea that mathematics educators and teachers are also problem solvers and learners, and as such they constantly experience mathematical and pedagogical disturbances. Accordingly, many original tasks and learning activities are results of personal mathematical and pedagogical disturbances of their designers, who then transpose these disturbances into learning opportunities for their students. This learning-transposition process is a cornerstone of mathematics teacher education as a lived, developing enterprise. Mathematical Encounters and Pedagogical Detours unfold the process and illustrate it by various examples. The book engages readers in original tasks, shares the results of task implementation and describes how these results inform the development of new tasks, which often intertwine mathematics and pedagogy. Most importantly, the book includes a dialogue between the authors based on the stories of their own learning, which triggers continuous exploration of learning opportunities for their students.

This book explores the idea that mathematics educators and teachers are also problem solvers and learners, and as such they constantly experience mathematical and pedagogical disturbances. Accordingly, many original tasks and learning activities are results of personal mathematical and pedagogical disturbances of their designers, who then transpose these disturbances into learning opportunities for their students. This learning-transposition process is a cornerstone of mathematics teacher education as a lived, developing enterprise. Mathematical Encounters and Pedagogical Detours unfold the process and illustrate it by various examples. The book engages readers in original tasks, shares the results of task implementation and describes how these results inform the development of new tasks, which often intertwine mathematics and pedagogy. Most importantly, the book includes a dialogue between the authors based on the stories of their own learning, which triggers continuous exploration of learning opportunities for their students.

This book shows how the practice of script writing can be used both as a pedagogical approach and as a research tool in mathematics education. It provides an opportunity for script-writers to articulate their mathematical arguments and/or their pedagogical approaches. It further provides researchers with a corpus of narratives that can be analyzed using a variety of theoretical perspectives.Various chapters argue for the use of dialogical method and highlight its benefits and special features. The chapters examine both "low tech" implementations as well as the use of a technological platform, LessonSketch. The chapters present results of and insights from several recent studies, which utilized scripting in mathematics education research and practice.

Lesson play is a novel construct in research and teachers' professional development in mathematics education. Lesson play refers to a lesson or part of a lesson presented in dialogue form-inspired in part by Lakatos's evocative Proofs and Refutations-featuring imagined interactions between a teacher and her/his students. We have been using and refining our use of this tool for a number of years and using it in a variety of situations involving mathematics thinking and learning. The goal of this proposed book is to offer a comprehensive survey of the affordances of the tool, the results of our studies-particularly in the area of pre-service teacher education, and the reasons that the tool offers such productive possibilities for both researchers and teacher educators.

Lesson play is a novel construct in research and teachers' professional development in mathematics education. Lesson play refers to a lesson or part of a lesson presented in dialogue form-inspired in part by Lakatos's evocative Proofs and Refutations-featuring imagined interactions between a teacher and her/his students. We have been using and refining our use of this tool for a number of years and using it in a variety of situations involving mathematics thinking and learning. The goal of this proposed book is to offer a comprehensive survey of the affordances of the tool, the results of our studies-particularly in the area of pre-service teacher education, and the reasons that the tool offers such productive possibilities for both researchers and teacher educators.

The idea of teachers Learning through Teaching (LTT) - when presented to a naive bystander - appears as an oxymoron. Are we not supposed to learn before we teach? After all, under the usual circumstances, learning is the task for those who are being taught, not of those who teach. However, this book is about the learning of teachers, not the learning of students. It is an ancient wisdom that the best way to "truly learn" something is to teach it to others. Nevertheless, once a teacher has taught a particular topic or concept and, consequently, "truly learned" it, what is left for this teacher to learn? As evident in this book, the experience of teaching presents teachers with an exciting opp- tunity for learning throughout their entire career. This means acquiring a "better" understanding of what is being taught, and, moreover, learning a variety of new things. What these new things may be and how they are learned is addressed in the collection of chapters in this volume. LTT is acknowledged by multiple researchers and mathematics educators. In the rst chapter, Leikin and Zazkis review literature that recognizes this phenomenon and stress that only a small number of studies attend systematically to LTT p- cesses. The authors in this volume purposefully analyze the teaching of mathematics as a source for teachers' own learning."

The idea of teachers Learning through Teaching (LTT) - when presented to a naive bystander - appears as an oxymoron. Are we not supposed to learn before we teach? After all, under the usual circumstances, learning is the task for those who are being taught, not of those who teach. However, this book is about the learning of teachers, not the learning of students. It is an ancient wisdom that the best way to "truly learn" something is to teach it to others. Nevertheless, once a teacher has taught a particular topic or concept and, consequently, "truly learned" it, what is left for this teacher to learn? As evident in this book, the experience of teaching presents teachers with an exciting opp- tunity for learning throughout their entire career. This means acquiring a "better" understanding of what is being taught, and, moreover, learning a variety of new things. What these new things may be and how they are learned is addressed in the collection of chapters in this volume. LTT is acknowledged by multiple researchers and mathematics educators. In the rst chapter, Leikin and Zazkis review literature that recognizes this phenomenon and stress that only a small number of studies attend systematically to LTT p- cesses. The authors in this volume purposefully analyze the teaching of mathematics as a source for teachers' own learning."

This book is grounded in the author's experiences of teaching mathematics for prospective elementary school teachers and conducting research on their understanding of mathematical concepts. It is a reflection on practice and an attempt to cope with a double challenge: that of a teacher, in helping prospective teachers make sense of mathematics, and that of a researcher, in an attempt to understand and describe the challenges faced by students. This work fits within the current community interest on teacher education and provides a novel focus, with both theoretical and practical considerations. The central claim in this book is that encounters with mathematical content by prospective elementary school teachers constitute relearning, rather than learning, of mathematics. The specific focus is on topics related to elementary number theory (e.g. divisibility, prime factorization), which is referred to as a "forgotten queen" (following Gauss' reference to number theory as a queen of mathematics). This is the content area that has not received significant attention in mathematics education research. The book can be summarized as an attempt to address the following questions: What is relearning of mathematical content and how is it similar to or different from learning? What are the examples of specific mathematical topics or concepts that require relearning? What pedagogical approaches can support relearning? The detailed analysis of research data and pedagogical approaches presented in the book are intertwined with stories of personal experiences of the author, which makes the reading not only intellectually stimulating but also enjoyable.

This book is grounded in the author's experiences of teaching mathematics for prospective elementary school teachers and conducting research on their understanding of mathematical concepts. It is a reflection on practice and an attempt to cope with a double challenge: that of a teacher, in helping prospective teachers make sense of mathematics, and that of a researcher, in an attempt to understand and describe the challenges faced by students. This work fits within the current community interest on teacher education and provides a novel focus, with both theoretical and practical considerations. The central claim in this book is that encounters with mathematical content by prospective elementary school teachers constitute relearning, rather than learning, of mathematics. The specific focus is on topics related to elementary number theory (e.g. divisibility, prime factorization), which is referred to as a "forgotten queen" (following Gauss' reference to number theory as a queen of mathematics). This is the content area that has not received significant attention in mathematics education research. The book can be summarized as an attempt to address the following questions: What is relearning of mathematical content and how is it similar to or different from learning? What are the examples of specific mathematical topics or concepts that require relearning? What pedagogical approaches can support relearning? The detailed analysis of research data and pedagogical approaches presented in the book are intertwined with stories of personal experiences of the author, which makes the reading not only intellectually stimulating but also enjoyable.

Number theory has been a perennial topic of inspiration and importance throughout the history of philosophy and mathematics. Despite this fact, surprisingly little attention has been given to research in learning and teaching number theory per se. This volume is an attempt to redress this matter and to serve as a launch point for further research in this area. Drawing on work from an international group of researchers in mathematics education, this volume is a collection of clinical and classroom-based studies in cognition and instruction on learning and teaching number theory. Although there are differences in emphases in theory, method, and focus area, these studies are bound through similar constructivist orientations and qualitative approaches toward research into undergraduate students' and preservice teachers' subject content and pedagogical content knowledge. Collectively, these studies draw on a variety of cognitive, linguistic, and pedagogical frameworks that focus on various approaches to problem solving, communicating, representing, connecting, and reasoning with topics of elementary number theory, and these in turn have practical implications for the classroom. Learning styles and teaching strategies investigated involve number theoretical vocabulary, concepts, procedures, and proof strategies ranging from divisors, multiples, and divisibility rules, to various theorems involving division, factorization, partitions, and mathematical induction.

Number theory has been a perennial topic of inspiration and importance throughout the history of philosophy and mathematics. Despite this fact, surprisingly little attention has been given to research in learning and teaching number theory per se. This volume is an attempt to redress this matter and to serve as a launch point for further research in this area. Drawing on work from an international group of researchers in mathematics education, this volume is a collection of clinical and classroom-based studies in cognition and instruction on learning and teaching number theory. Although there are differences in emphases in theory, method, and focus area, these studies are bound through similar constructivist orientations and qualitative approaches toward research into undergraduate students' and preservice teachers' subject content and pedagogical content knowledge.

Collectively, these studies draw on a variety of cognitive, linguistic, and pedagogical frameworks that focus on various approaches to problem solving, communicating, representing, connecting, and reasoning with topics of elementary number theory, and these in turn have practical implications for the classroom. Learning styles and teaching strategies investigated involve number theoretical vocabulary, concepts, procedures, and proof strategies ranging from divisors, multiples, and divisibility rules, to various theorems involving division, factorization, partitions, and mathematical induction.