Teaching and Learning Patterns in School Mathematics

This book draws on research findings on patterns in the last twenty years or so in order to argue for a theory of graded representations in pattern generalization. Pattern generalization encompasses the construction and justification of structures that give meanings to sequences of numerical and figural objects. While empirical studies conducted with different age-level groups have sufficiently demonstrated varying shifts in structural awareness and competence, which influence the eventual shape of an intended generalization, such shifts, however, are not necessarily permanent but parallel and graded, adaptive, and fundamentally distributed among a variety of cognitive and noncognitive sources that mutually influence each other. Thus, the emergence and complexity of the pattern generalization process cannot be reduced to a simple narrative of cognitive shifts from the arithmetic to the algebraic, from the recursive to the functional, from discerning details to perceiving properties, and so on and so forth. In this book, we pursue an alternative view of pattern generalization processing, that is, one that is not about permanent shifts or transition phases but graded and multimodal depending on individual learners' experiences with patterns and, especially, the manner in which they perceive, think about, and act on them.

A nonlinear graded perspective offers a much more robust and dynamic understanding of the similarities and differences in patterning competence since it is sensitive to, and acknowledges, the varying learning conditions and opportunities that shape generalization processing and representational conversion. Empirical evidence from a variety of sources will be provided to demonstrate this emergent perspective. Further, instructional implications commingle with research knowledge throughout the book, providing researchers andteachers with usable information that will help them cope with issues they may encounter when they use patterns to engage learners in generalization activity, which involves various aspects of abstract, quantitative, model-driven, structural, and regularity thinking.

Auteur
Dr. Rivera is interested in issues involving visualization and algebraic thinking, in particular, generalization. His overall research program at San Jose State University falls under the category of psychology of mathematics education, which involves studying learning and teaching conditions that influence curriculum and instruction. He recently concluded a three-year longitudinal study on pattern generalization at the middle school level. Results of his findings are reported in various peer-reviewed journals, book chapters, and conference proceedings. He has recently published a book, Toward a Visually-Oriented School Mathematics Curriculum, that is a volume in the Springer Mathematics Education Library series. This year, Dr. Rivera will be completing a one-year longitudinal research study in a Grade 2 class. He intends to follow the same kids through Grade 3 during the next school year. Dr. Rivera's research is supported by a Career grant from the National Science Foundation.

Résumé
This book synthesizes research findings on patterns in the last twenty years or so in order to argue for a theory of graded representations in pattern generalization. While research results drawn from investigations conducted with different age-level groups have sufficiently demonstrated varying shifts in structural awareness and competence, which influence the eventual shape of an intended generalization, such shifts, however, are not necessarily permanent due to other pertinent factors such as the complexity of patterning tasks. The book proposes an alternative view of pattern generalization, that is, one that is not about shifts or transition phases but graded depending on individual experiences with target patterns. The theory of graded representations involving pattern generalization offers a much more robust understanding of differences in patterning competence since it is sensitive to varying levels of entry into generalization. Empirical evidence will be provided to demonstrate this alternative view, which is drawn from the author's longitudinal work with elementary and middle school children, including several investigations conducted with preservice elementary majors. Two chapters of the book will be devoted to extending pattern generalization activity to arithmetic and algebraic learning of concepts and processes. The concluding chapter addresses the pedagogical significance of pattern learning in the school mathematics curriculum.

Contenu
-1 Introduction. -2 The Context of Generalization. -3 Types and Levels of Pattern Generalization. -4 The Theory of Graded Representations in Pattern Generalization. -5 Gestures, Words, and Incipient Generalizations. -6 Variables and Algebraic Generalizations. - 7 Patterns, Generalization, and Arithmetic Learning. -8 Patterns, Functional Thinking, and Algebraic Learning. -9 Pedagogical Issues and Implications. -10 Conclusions.