Książki

The studies presented in this book should be of interest to anybody concerned with the teaching of arithmetic to young children or with cognitive development in general. The 'eaching experiment· was carried out with half a dozen children entering first grade over two years in biweekly sessions. Methodologically the authors' research is original. It is a longitudinal but not a naturalistic study, since the experimenter-teachers directed their interaction with each individual child with a view to his or her possible progress. It is experimental in the sense that two groups of subjects were selected according to criteria derived from an earlier study (Steffe, von Glasersfeld, Richards & Cobb, 1983) and that the problems proposed were comparable, though far from identical across the subjects; but unlike more rigid and shorter "learning" or ''training" studies it does not include pre-and posttests, or predetermined procedures. Theoretically, the authors subscribe to Piagefs constructivism: numbers are made by children, not found (as they may find some pretty rocks, for example) or accepted from adults (as they may accept and use a toy). The authors interpret changes in the children's counting behaviors in terms of constructivist concepts such as assimilation, accommodation, and reflective abstraction, and certain excerpts from protocols provide on-line examples of such processes at work. They also subscribe to Vygotsky's proposal for teachers '0 utilize the zone of proximal development and to lead the child to what he (can) not yet do· (1965, p. 104).

Construction of Arithmetical Meanings and Strategies

I: On the Construction of the Counting Scheme.- Children's Counting.- The Counting Types.- Perceptual Unit Items.- Figural Unit Items.- Motor Unit Items.- Verbal Unit Items.- Abstract Unit Items.- Ontogenetic Analysis.- Stages.- Adaptation.- Counting as a Scheme.- The First Part of the Counting Scheme.- The Third Part of the Counting Scheme.- Other Sources of Numerosity.- Perceptual Mechanisms.- Spatial Patterns.- Meaning Theory.- Reflection and Abstraction.- II: The Construction of Motor Unit Items: Brenda, Tarus, and James.- 1. Brenda.- The Perceptual Period.- The Motor Period.- Discussion of Brenda's Case Study.- The Perceptual Period.- The Motor Period.- 2. Tarus.- The Perceptual Period.- The Motor Period.- Discusion of Tarus's Case Study.- The Perceptual Period.- The Motor Period.- 3. James.- The Perceptual Period.- The Motor Period.- Discussion of James's Case Study.- The Perceptual Period.- The Motor Period.- Perspectives on the Three Case Studies.- Period Criterion.- The Incorporation and Invariant Sequence Criteria.- The Reorganization Criterion.- III: The Construction of Verbal Unit Items: Brenda, Tarus, and James.- 1. Brenda.- Discussion of Brenda's Case Study.- 2. Tarus.- Discussion of Tarus's Case Study.- 3. James.- Discussion of James's Case Study.- Perspectives on the Case Studies.- The Verbal Period as a Subperiod in the Figurative Stage.- Counting-on.- IV: The Construction of Abstract Unit Items: Tyrone, Scenetra, and Jason.- 4. Tyrone.- The Motor Period.- The Abstract Period.- Discussion of Tryone's Case Study.- 5. Scenetra.- The Motor Period.- The Verbal Period.- The Abstract Period.- Discussion of Scenetra's Case Study.- 6. Jason.- The Motor Period.- Creating Verbal Unit Items.- The Abstract Period.- Discussion of Jason's Case Study.- Perspectives on the Case Studies.- Stages.- Incorportation Criterion.- Transition to the Abstract Period.- The Reorganization of Counting.- V: Lexical and Syntactical Meanings: Brenda, Tarus, and James.- 1. Brenda.- The Perceptual Period.- The Motor Period.- The Verbal Period.- Discussion of Brenda's Case Study.- The Perceptual Stage.- The Figurative Stage.- 2. Tarus.- The Perceptual Period.- The Motor Period.- The Verbal Period.- Discussion of Tarus's Case Study.- The Perceptual Stage.- The Figurative Stage.- 3. James.- The Perceptual Period.- The Motor Period.- The Verbal Period.- Discussion of James's Case Study.- The Perceptual Stage.- The Figurative Stage.- Perspectives on the Case Studies.- The Perceptual Stage.- Finger Patterns.- The Figurative Stage.- Mobile Finger Patterns.- Sophisticated Finger Patterns.- Spatio-Auditory Patterns.- Dual Meanings of Number Words.- Counting as the Meaning of Number Words.- Summary of the Types of Preconcepts and Concepts.- Meanings of "Ten".- Ten as an Enactive Concept.- Ten as a Countable Figural Unit.- Ten as a Countable Motor Unit.- Adding Schemes.- The Perceptual Stage.- The Figurative Stage.- Comments on Prenumerical Children.- VI: Lexical and Syntactical Meanings: Tyrone, Scenetra, and Jason.- Systems of Integration.- Integrations.- Sequential Integration Operations.- Progressive Integration Operations.- Part-Whole Operations.- 4. Tyrone.- The Emergence of the Integration Operation.- The Period of Sequential Integration Operations.- The Period of Progressive Integration Operations.- The Period of Part-Whole Operations.- Discussion of Tyrone's Case Study.- The Emergence of the Integration Operation.- The Period of Sequential Integration Operations.- The Period of Progressive Integration Operations.- The Period of Part-Whole Operations.- Unit Types of the Unit of Ten.- 5. Scenetra.- Recognition and Re-Presentation of Patterns.- The Emergence of the Integration Operation.- The Period of Sequential Integration operations.- The Period of Progressive Integreation Operations.- Discussion of Scenetra's Case Study.- The Emergence of the Integratoin Operation.- The Period of Sequential Integration Operations.- The Period of Progressive Integration Operations.- Unit Types of the Unit of Ten.- 6. Jason.- Recognition and Re-Presentation of Patterns.- The Emergence of The Integration Operation.- The Period of Sequential Integration Operations.- The Period of Progressive integration Operations.- The Period of Part-Whole Operations.- Discussion of Jason's Case Study.- The Emergence of the Integration Operation.- The Period of Sequential Integration Operations.- The Period of Progressive Integration Operations.- The Period of Part-Whole Operations.- Unit Types of the Unit of Ten.- Perspectives on the Case Studies.- The Emergence of the Integration Operation.- Numerical Patterns.- Number Sequences.- Stages in the Construction of the Numerical Counting Scheme.- Piaget's Invariant Sequence and Incorporation Criteria.- The Reorganization Criterion.- Units of One.- The Unit of One in Sequential Integration Operations.- The Unit of One in Progressive Integration Operations.- The Unit of One in Part-Whole Operations.- Units of Ten.- The Stage of Sequential Integration Operations.- The Stage of Progressive Integration Operations.- The Stage of Part-Whole Operations.- Other Perspectives.- VII: Strategies for Finding Sums and Differences: Brenda, Tarus, and James.- Brenda.- Independent Solutions.- Number Word Coordinations.- Tarus.- Independent Solutions.- Number Word Coordinations.- James.- Independent Solutions.- Number Word Coordinations.- Perspectives on the Case Studies.- Number Facts.- VIII: Strategies for Finding Sums and Differences: Tyrone, Scenetra, and Jason.- Sequential Integration Operations.- Jason.- Tyrone.- Scenetra.- Discussion: Sequential Integration Operations.- Progressive Integration Operations.- Jason.- Tyrone.- Scenetra.- Discussion: Progressive Integration Operations.- Part-Whole Operations.- Jason.- Tyrone.- Perspective on the Case Studies.- Arithmetical Context.- Thinking Strategies and Integration Operations.- Thinking Strategies and the Basic Facts.- Thinking Strategies and the Construction of Part-Whole Operations.- Goals for Teaching Thinking Strategies.- IX: Modifications of the Counting Scheme.- Predicting Modifications of the Counting Scheme.- Mathematical Learning.- The Perceptual Stage.- Temporary Modifications.- Procedural Accommodations.- Engendering Accommodations.- Isolated Procedural Accommodations.- The Figurative Stage.- Procedural Accommodations.- Temporary Modifications.- Retrospective Accommodations.- Re-presentation and Review of Prior Activity.- The Figurative Stage: Tyrone, Scenetra, and Jason.- Procedural Engendering Accommodations.- Temporary Modifications.- Metamorphic Accommodations.- Stages in the Construction of Part-Whole Operations.- Sequential Integration Operations.- Procedural Accommodations.- Engendering Accommodations.- Progressive Integration Operations.- Internal Reorganizations.- Part-Whole Operations.- Phylogenetic Perspectives.- Zones of Potential Development in Retrospect.- Figurative Stage.- Sequential Integration Operations.- Progressive Integration Operations.- Part-Whole Operations.- Final Comments.- References.