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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.
|1||Toward Number Theory as a Conceptual Field||1|
|2||Coming to Terms with Division: Preservice Teachers' Understanding||15|
|3||Conceptions of Divisibility: Success and Understanding||41|
|4||Language of Number Theory: Metaphor and Rigor||83|
|5||Understanding Elementary Number Theory at the Undergraduate Level: A Semiotic Approach||97|
|6||Integrating Content and Process in Classroom Mathematics||117|
|7||Patterns of Thought and Prime Factorization||131|
|8||What Do Students Do with Conjectures? Preservice Teachers' Generalizations on a Number Theory Task||139|
|9||Generic Proofs in Number Theory||157|
|10||The Development of Mathematical Induction as a Proof Scheme: A Model for DNR-Based Instruction||185|
|11||Reflections on Mathematics Education Research Questions in Elementary Number Theory||213|
|About the Contributors||243|