Usiskin / Zalman The Classification of Quadrilaterals

A Study in Definition
1. Auflage 2008
ISBN: 978-1-60752-600-1
Verlag: Information Age Publishing
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 125 Seiten

Reihe: Research in Mathematics Education

This monograph reports on an analysis of a small part of the mathematics curriculum, the definitions given to quadrilaterals. This kind of research, which we call microcurricular analysis, is often undertaken by those who create curriculum, but it is not usually done systematically and it is rarely published. Many terms in mathematics education can be found to have different definitions in mathematics books. Among these are “natural number,” “parallel lines” and “congruent triangles,” “trapezoid” and “isosceles trapezoid,” the formal definitions of the trigonometric functions and absolute value, and implicit definitions of the arithmetic operations addition, subtraction, multiplication, and division. Yet many teachers and students do not realize there is a choice of definitions for mathematical terms. And even those who realize there is a choice may not know who decides which definition of any mathematical term is better, and under what criteria. Finally, rarely are the mathematical implications of various choices discussed. As a result, many students misuse and otherwise do not understand the role of definition in mathematics. We have chosen in this monograph to examine a bit of mathematics for its definitions: the quadrilaterals. We do so because there is some disagreement in the definitions and, consequently, in the ways in which quadrilaterals are classified and relate to each other. The issues underlying these differences have engaged students, teachers, mathematics educators, and mathematicians. There have been several articles and a number of essays on the definitions and classification of quadrilaterals. But primarily we chose this specific area of definition in mathematics because it demonstrates how broad mathematical issues revolving around definitions become reflected in curricular materials. While we were undertaking this research, we found that the area of quadrilaterals supplied grist for broader and richer discussions than we had first anticipated. The intended audience includes curriculum developers, researchers, teachers, teacher trainers, and anyone interested in language and its use.

Autoren/Hrsg.

Usiskin, Zalman

Usiskin,

Zalman,

Inhaltsverzeichnis

1;Cover;1
2;Contents;5
3;Preface;9
3.1;Why Quadrilaterals?;9
3.2;Writing and Acknowledgements;10
3.3;Audiences;11
4;Introduction;13
5;Chapter 1. Definitions in Mathematics;17
5.1;An introductory example—isosceles triangles;19
5.2;Hierarchies, hierarchical chains, and partitions;22
6;Chapter 2. Quadrilaterals;25
6.1;Quadrilaterals and quadrangles;25
6.2;Quadrilaterals as polygons;25
6.3;Types of quadrilaterals;33
6.4;Quadrilaterals in Euclid;35
7;Chapter 3. Parallelograms;37
7.1;Showing that definitions are equivalent;39
7.2;The hierarchy underneath parallelograms;42
8;Chapter 4. Trapezoids;43
8.1;Choosing between exclusive and inclusive definitions;45
8.2;Trapezoids on the coordinate plane;48
9;Chapter 5. Rectangles;49
9.1;Analysis of defining conditions for rectangles;50
9.2;Why are so many different defining conditions in use?;53
9.3;The possible influence of non-Euclidean geometries;54
10;Chapter 6. Isosceles Trapezoids;57
10.1;Symmetry as a defining condition;58
10.2;Should rectangles be isosceles trapezoids?;61
11;Chapter 7. Kites;65
11.1;Equivalent defining conditions for kites;68
11.2;Isosceles triangles as degenerate kites;68
12;Chapter 8. Rhombuses (Rhombi);71
13;Chapter 9. Squares;75
13.1;Defining quadrilaterals using only the hierarchy;76
14;Chapter 10. Cyclic Quadrilaterals;79
15;Chapter 11. Properties of Quadrilaterals Exhibited in Hierarchies;85
15.1;Types of quadrilaterals as intersections of other types;86
15.2;Hierarchical properties of angles;88
15.3;Hierarchical properties of sides;88
15.4;Hierarchical properties of diagonals;89
15.5;Hierarchical properties of symmetry;91
15.6;Hierarchical positions on the coordinate plane;92
16;Bibliography;99
17;Appendix A: Definitions of Quadrilaterals;101
17.1;1. Quadrilateral;102
17.2;2. Polygon;103
17.3;3. Parallelogram;104
17.4;4. Trapezoids;104
17.5;5. Rectangle;104
17.6;6. Isosceles trapezoid;105
17.7;7. Kites;105
17.8;8. Rhombuses;105
17.9;9. Squares;106
17.10;10. Cyclic quadrilaterals;106
18;Appendix B: Individual Geometry Texts and the Definitions Found in Them;107
19;Appendix C: Bibliography of Texts Examined;115