Teaching and Learning Patterns in School Mathematics [electronic resource] : Psychological and Pedagogical Considerations /
Rivera, Ferdinand.author.
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.  .
Description based upon print version of record.
Copyright; Contents; Chapter 1: Introduction; Fundamental Notions of Patterning and Pattern Generalization; Two Basic Issues in Patterning Activity; Additional Considerations Regarding Patterns, Structures, and Generalization; Exploring a Complex Theory of Graded Pattern Generalization; Overviews of the Remaining Chapters; Chapter 2: Contexts of Generalization in School Mathematics; Abduction, Induction, and Deduction in Generalization; Abduction; Induction; Abduction and Induction Together; Deduction; Near and Far Generalization; Various Characterizations of Generalization
Reflections on and Implications for a Theory of Graded Pattern GeneralizationChapter 3: Types and Levels of Pattern Generalization; Kinds and Sources of Generalization; Types of Structures; Attending to, or Being Aware of, Structure; Modes of Representing Generalizations; Levels of Understanding Structure-Driven Algebraically Useful Generalizations; Toward a Theory of Graded Pattern Generalization; Chapter 4: A Theory of Graded Representations in Pattern Generalization; The Emergence of Structure; Alternative Theories to Emergence; A Parallel Distributed Processing View of (Semantic) Cognition
Implications of Parallel Distributed Processing on Pattern Generalization ProcessingChapter 5: Graded Pattern Generalization Processing of Elementary Students (Ages 6 Through 10 Years); Distinguishing Between Approximate and Exact Generalization: Changing Competencies in Number, Shape, and Figural Processing; Number; Shape; Figural Properties; On the (Presymbolic and Nonsymbolic 5) Algebraic Nature of Elementary Students' Embodied Generalizations: Gestures, Picture...; Arithmetical Numbers as Intuited and Tacit Variables in Algebraic Contexts
The Emergence of Function Understanding via Structural Incipient GeneralizationChapter 6: Graded Pattern Generalization Processing of Older Students and Adults (Ages 11 and Up); Object-Dependent Numerical Processing and Conversion; Object-Dependent Figural Processing and Conversion; Relationship-Dependent Numerical Processing and Conversion; Relationship-Dependent Figural Processing and Conversion; On The (Nonsymbolic) Algebraic Nature of Older Students' Pattern Generalizations: Variables as Situated, Induced, and Figur...; Functions as Generalizing Extensions
Chapter 7: Patterns and Graded Algebraic ThinkingRepresentations of Patterns in Three Contexts: Presymbolic, Nonsymbolic, and Symbolic Algebraic Generalizations; Patterns in the Elementary School Mathematics Curriculum: The Emergence of Presymbolic and Nonsymbolic Structural Generaliz...; Patterns in the Middle and High School Mathematics Curriculum and Beyond: The Emergence of Symbolic Algebraic Structures in...; The Graded Shape of Pattern Generalization; References; Author Index; Subject Index
Includes bibliographical references and index.
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.  .
Mathematics-Study and teaching .
Educational psychology.
Learning.
Instruction.
Curriculums (Courses of study).
Education-Curricula.
Teaching.
Mathematics Education.
Pedagogic Psychology.
Learning & Instruction.
Curriculum Studies.
Teaching and Teacher Education.