| Description |
1 online resource (xiii, 378 pages) : illustrations |
| Contents |
TABLE OF CONTENTS; PREFACE; ACKNOWLEDGEMENTS; CHAPTER 1: THE CONCEPT OF INVARIANCE AND CHANGE: THEORETICAL BACKGROUND; UNDERSTANDING PHENOMENA FROM THE ASPECT OF INVARIANCE AND CHANGE; The Concept of Invariance and Change; Change and Invariance from the Epistemological Perspective; Invariance and Change in the Development of Science; Invariance and Change in the Development of Mathematical Knowledge; THE CONCEPT OF INVARIANCE AND CHANGE IN THE MATHEMATICAL KNOWLEDGE OF STUDENTS; The Cognitive-Developmental Aspect of Invariance and Change; On Mathematics Instruction and Learning in School |
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Invariance and Change as an Educational ConceptInvariance and Change as Concepts for Mathematics Teachers; THE BASIC INTERPLAY BETWEEN INVARIANCE AND CHANGE; Different Types Invariance; The Taxonomy of Invariance and Change; SOME INTRODUCTORY ACTIVITIES IN INVARIANCE AND CHANGE; What Is Invariant? What Changes? An Investigative Activity about Freefall Durations; "Can This Change?" Three Experiments with Paper; "Different or the Same?" Optical Illusions; NOTES; REFERENCES; CHAPTER 2: INVARIANT QUANTITIES -- WHAT IS INVARIANT AND WHAT CHANGES? |
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INTRODUCTION: UNDERSTANDING THE INVARIANCE OF QUANTITY AS A BASIS FOR QUANTITATIVE THINKINGInvariance of a Quantity: Assembling a Finite Set from Subsets; Generalization of the Principle of Invariance of Quantity: "Large Numbers"; Invariance of Quantity in Terms of Arithmetic Operations; Aspects of Equality, Invariance, and Change When Partitioning Quantities; Generalization of Partitioning: Non-Integers and the Number of Components; Summary of the Activities in Chapter 2; ACTIVITY 2.1: DIVIDING DOLLS BETWEEN TWO CHILDREN |
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MATHEMATIC AND DIDACTIC ANALYSIS OF ACTIVITY 2.1: PARTITIONING A SET INTO TWO SUBSETS: POSING PROBLEMS AND PARTITION METHODSSummary of the Roles of the Concepts of Change and Invariance inActivity 2.1: Description of Ways to Split a Set into Two Subsets; ACTIVITY 2.2: HOW TO SPLIT A FRACTION. ALMOST LIKE ANCIENT EGYPT; MATHEMATIC AND DIDACTIC ANALYSIS OF ACTIVITY 2.2: INVARIANCE OF QUANTITY AND SPLITTING OF UNIT FRACTIONS; ACTIVITY 2.3: THEY ARE ALL EQUAL, BUT & MATHEMATIC AND DIDACTIC ANALYSIS OF ACTIVITY 2.3: FROM EQUAL ADDENDS TO CONSECUTIVE ADDENDS; Summary and Comments for Question 4 |
| Summary |
What is the connection between finding the amount of acid needed to reach the desired concentration of a chemical solution, checking divisibility by a two-digit prime number, and maintaining the perimeter of a polygon while reducing its area? The simple answer is the title of this book. The world is an interplay of variation and constancy -- a medley of differences and similarities -- and this change and invariance is, largely, a language of science and mathematics. This book proposes a unique approach for developing mathematical insight through the perspective of change and invariance as it applies to the properties of numbers and shapes |
| Bibliography |
Includes bibliographical references |
| Notes |
Print version record |
| Subject |
Symmetry (Mathematics)
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Mathematics -- Study and teaching.
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Mathematical models.
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Algebra.
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Geometry, Algebraic.
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Mathematics -- Philosophy.
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mathematical models.
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algebra.
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MATHEMATICS -- Geometry -- General.
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Algebra
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Geometry, Algebraic
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Mathematical models
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Mathematics -- Philosophy
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Mathematics -- Study and teaching
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Symmetry (Mathematics)
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| Form |
Electronic book
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| Author |
Ilany, Bat-Sheva, author
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| ISBN |
9789463006996 |
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9463006990 |
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