Stephen R. Campbell

This study contributes to a growing body of research on teachers' content knowledge in mathematics. The domain under investigation was elementary number theory. Our main focus concerned the con- cept of divisibility and its relation to division, multiplication, prime and composite numbers, fac- torization, divisibility rules, and prime decomposition. We used a constructivist-orientedtheoretical framework for analyzing and interpreting data acquired in clinical interviews with preservice teach- ers. Participants' responses to questions and tasks indicated pervasive dispositions toward procedural attachments, even when some degree of conceptual understanding was evident. The results of this study provide a preliminary overview of cognitive structures in elementary number theory.

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.

... While there is agreement that multidisciplinary collaboration is needed, Zachary Stein and Kurt Fischer at Harvard University Graduate School in ... Having predicated her doctorate with many years of classroom experience, Kathryn Patten, Outreach Coordinator of the Educational ...

Campbell, S.R., Handscomb, K., Zaparyniuk, N, Sha, L., Cimen, O.A. Shipulina, O.V. (2009). Investigating image-based perception and reasoning in geometry. Paper presented at the American Educational Research Association Annual Meeting, San Diego, U.S.A.

Campbell, S.R., Li, M., Zaparyniuk, N. (2012). Facilitating world-wide communication and collaboration in mathematics education research using a virtual world. 36th Conference of International Group for the Psychology of Mathematics Education. Taiwan, ROC.

This paper explores how a young child (56 m) builds an understanding of the cardinality principle through communicative, touchscreen-based activities involving talk, gesture and body engagement working via multimodal, touchscreen interface using contemporary mobile technology. Drawing upon Nemirovsky's perceptuomotor integration theoretical lens and other foundational aspects of Husserlian phenomenology, we present an in-depth case study of a preschool child developing mathematical expertise and tool fluency using an iPad application called TouchCounts to operate with cardinal numbers. Overall, this study demonstrates that the one-on-one multimodal touch, sight and auditory feedback via a touchscreen device can serve to assist in a child's development of cardinality.

This study investigates procedural and conceptual aspects in preservice elementary school teacher's understanding of the Fundamental Theorem of Arithmetic. The data were collected by the means of a written questionnaire and individual interviews. The results suggest that the idea of the uniqueness of prime decomposition is very difficult to grasp. Participants' responses indicated, either implicity or explicity, that a possibility of alternative prime decompositions was often not overruled, and this influenced students' ability to make inferences regarding factors and divisors of natural numbers. Some pedagogical implications are discussed.

This study contributes to a growing body of research on teachers' content knowledge in mathematics. The domain under investigation was elementary number theory. Our main focus concerned the concept of divisibility and its relation to division, multiplication, prime and composite numbers, factorization, divisibility rules, and prime decomposition. We used a constructivist-oriented theoretical framework for analyzing and interpreting data acquired in clinical interviews with preservice teachers. Participants' responses to questions and tasks indicated pervasive dispositions toward procedural attachments, even when some degree of conceptual understanding was evident. The results of this study provide a preliminary overview of cognitive structures in elementary number theory.

Although representation and visualization are assumed to be at the core of understanding in Mathematics, history shows that visual impairments in general, and blindness in particular, are not irrevocable impediments to learning mathematics. In this paper, adopting Vigotsky’s mediation theory, we discuss how an undergraduate blind student demonstrates that lack of access to the visual field, does not impede his ability to visualize, but, rather, modifies it.

Most (under)graduate students with profound vision disability avoid taking mathematics courses at university level. In this paper, we argue that mathematical communication and learning are inherently multimodal and embodied; hence, sight disabled students are also able to conceptualize visuospatial information. Adapting Vygotsky's mediation theory, we show that lack of access to visual fields in an advance mathematics course does not obstruct a blind student's ability to visualize, but rather modifies it. We also argue proper precise tactile materials might empower blind students to better visualize mathematical functions.