# Learning and Teaching Number Theory: Research in Cognition and Instruction

## Synopsis

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.

## Excerpt

Stephen R. Campbell and Rina Zazkis

Since the beginning of the ancient Greek Pythagorean tradition over two and a half millennia ago, striving for a conceptual understanding of numbers and their properties, patterns, structures, and forms has constituted the heart, if not the soul, of mathematics and mathematical thinking. Today, a constellation of activities involving various operations, procedures, functions, relations, and applications associated with numbers occupy the main bulk of the K–12 mathematics curriculum. These activities are typically conducted under the auspices of arithmetic and algebra in various guises, such as counting and measuring with numbers, tabulating and graphing collections of numbers, and formulating and solving equations applied in less formal and more familiar “day-to-day,” “real-world,” situational contexts.

With all of the attention that has been given to informal meanings and familiar contexts in mathematics education these days, it seems that not too much consideration has been given to formal contexts concerned with properties and structures of number per se. Mathematical meaning is not just a matter of grounding concepts in familiar day-to-day real-world experiences. It is also a matter of developing the conceptual foundations for making clear and general abstract distinctions. A case in point has to do with understanding differences between whole numbers and rational numbers and the different kinds of procedures involved in operating with them. It is well known that learners have many procedural and semantic difficulties in this regard (e.g., Durkin & Shire, 1991; Greer, 1987; Mack, 1995; Silver, 1992). Teachers need to have and learners

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