From the Back Cover
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 and teachers 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.
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.
