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Summary
Summary
With profound implications for classroom practice, this text examines the significance of children's understanding and learning of mathematical notations in their development as mathematics learners. Using a series of interviews and in-depth conversations with kindergarten and elementary school children, the author investigates young children's understanding of different mathematical notations, including written numbers and the written number system, commas and periods in numbers, notations for fractions, data tables, number lines, and graphs. Logically organized according to the ages of the children, with each chapter focusing on one to three children and one aspect of notations, this seminal work discusses key concepts, such as:
The relationship between conceptual understanding and notations. The interaction between children's inventions and conventional notations. How children appropriate conventional notations and transform them to make sense of them, and how the classroom culture can encourage, foster, and take advantage of children's invented notations.Author Notes
Bárbara M. Brizuela is an assistant professor of education at Tufts University. She currently works with children and teachers in public schools in the Boston area.
Reviews 1
Choice Review
Brizuela (Tufts Univ.) holds a doctorate in education from Harvard. Her book draws on data from her dissertation as well as recent work with D. Carrahar and A. Schliemann, including work on TERC, a nonprofit education research and development organization, and the Tufts Algebra in Early Mathematics Project. Brizuela assumes a Piagetian lens in reflecting on her observations of K-3 students' learning of and from mathematical notations. Her assertions that "there is a constant interaction between mathematical notations and conceptual understanding and that there is a similar interplay between invented and conventional mathematical notations" are supported by excerpts from interviews with children using and coming to understand the use of "periods" and commas in numbers, graphs and tables, and number lines. Brizuela argues that conventions are not merely transmitted but involve active construction and meaning making on the part of learners. She also notes the ways in which children's struggles with notation mirror those in the historical development of mathematical notation. Readers of Symbolizing, Modeling and Tool Use in Mathematics Education, ed. by K.P. Gravemeijer, will find this book of interest. The clear writing, focus, and theoretical frame make Brizuela's book a valuable contribution. ^BSumming Up: Recommended. Upper-division undergraduates and above. A. O. Graeber University of Maryland College Park
Table of Contents
| Foreword | p. vii |
| Acknowledgments | p. ix |
| 1. Overview | p. 1 |
| Research Focus | p. 3 |
| Connections to History of Mathematics and Notational Systems | p. 6 |
| A Definition for Notations | p. 7 |
| Organization of the Book | p. 9 |
| 2. George: Written Numbers and the Written Number System | p. 11 |
| George's Fine-Motor Skills and Understanding of the Number System | p. 13 |
| George's Use of Dummy Numbers | p. 16 |
| The Role of Relative Position in George's Ideas About Written Numbers | p. 21 |
| Reflections | p. 26 |
| 3. Paula: "Capital Numbers" | p. 27 |
| Making Sense of Conventions | p. 28 |
| Capital Numbers: Paula's Invented Tool | p. 31 |
| Inventions | p. 35 |
| Conventions | p. 37 |
| Reflections | p. 38 |
| 4. Thomas: Commas and Periods in Numbers | p. 42 |
| Thomas's Developing Understanding of Periods and Commas in Numbers | p. 43 |
| Reflections | p. 51 |
| 5. Sara: Notations for Fractions That Help Her "Think of Something" | p. 55 |
| Solving Algebraic Problems During Class | p. 58 |
| Solving Algebraic Problems During an Interview | p. 63 |
| Reflections | p. 65 |
| 6. Jennifer and Her Peers: Data Tables and Additive Relations | p. 67 |
| Second Grade: Looking at Children's Self-Designed Tables | p. 71 |
| Jennifer in Third Grade | p. 77 |
| Reflections | p. 80 |
| 7. Jennifer, Nathan, and Jeffrey: Relationships Among Different Mathematical Notations | p. 81 |
| Details of the Study | p. 82 |
| The Problem Presented to the Children | p. 83 |
| The Children's Reactions to the Problem | p. 84 |
| The Children's Notations | p. 87 |
| The Children's Interpretations of Jennifer's Graph | p. 91 |
| Reflections | p. 98 |
| 8. Final Reflections | p. 100 |
| References | p. 107 |
| Index | p. 117 |
| About the Author | p. 125 |
