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Engaging (with) Mathematics and Learning to Teach

An Integrated Approach to Mathematics Preservice Education

Mathematics education research indicates the value of a meaning-making and problem-solving approach to the teaching mathematics in primary and lower secondary classrooms. Yet teachers, most of whom have not experienced such pedagogies in their own mathematics learning, often find it difficult to implement such approaches. Based on over twenty-five years in mathematics preservice education, this book is intended to support preservice tutors and their students in bridging this gap.The book takes six topics from the primary and lower secondary curriculum: place value number systems; the four rules of number; polygons, their properties and their symmetries; natural numbers including factors, multiples, powers and simple number theory; fractions, decimals and irrational numbers; and polyhedra. Each topic is located very briefly in the research literature and its place in or linked to the primary and lower secondary curriculum is discussed. Relevant mathematical activities follow, many of which can transfer directly from the university to the school classroom with very little adaptation. The final topic chapter is rather different. It deals with group theory, an aspect of mathematics which is related to primary and lower secondary mathematics structurally but not in terms of recognisable content.There is an emphasis throughout on the need to reflect on mathematical experience, to develop sensitivity and self-awareness and to promote an approach to the subject that is creative and inclusive.

Titel: Engaging (with) Mathematics and Learning to Teach
Untertitel: An Integrated Approach to Mathematics Preservice Education
Verfasser: Povey, Hilary
Dokumenttyp: Buch
Medientyp: Text
Erscheinungsdatum: 2017
Publikation in MIAMI: 21.12.2017
Datum der letzten Änderung: 21.12.2017
Verlag/Hrsg.: WTM-Verlag für wissenschaftliche Texte und Medien
Schlagwörter: Zahlentheorie; Polygon; Polyeder; Gruppentheorie; Mathematikunterricht; Grundschule; Sekundarstufe 1
Fachgebiete: Mathematik; Bildung und Erziehung
Rechtlicher Vermerk: © 2017 WTM – Verlag für wissenschaftliche Texte und Medien, Münster
Sprache: Englisch
Anmerkungen: Druckausgabe: Povey, Hilary: Engaging (with) Mathematics and Learning to Teach : An Integrated Approach to Mathematics Preservice Education. Münster : WTM, 2017. ISBN 978-3-95987-051-1
Format: PDF-Dokument
ISBN: 978-3-95987-052-8
URN: urn:nbn:de:hbz:6-10219398258
Permalink: https://nbn-resolving.org/urn:nbn:de:hbz:6-10219398258
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Inhalt:
CONTENTS
Acknowledgements ..... i
Links to software ..... ii
Introduction ..... 1
SECTION ONE
Introduction ..... 7
1. What we are about ..... 9
1.1. Our approach to mathematics subject knowledge for teaching ..... 10
1.2. Our pedagogical principles ..... 13
1.3. Troubling the account ..... 19
2. Doing mathematics ..... 22
2.1. “Big ideas” in mathematics ..... 23
2.2. Mathematical thinking ..... 25
2.3. Models of thinking ..... 28
SECTION TWO
Introduction ..... 33
1. Place value ..... 36
1.1. Number systems in history ..... 38
1.2. The role and meaning of zero ..... 41
1.3. Counting in other bases ..... 43
2. The four rules ..... 46
2.1. Alternative algorithms ..... 48
2.2. Problems in other number scripts ..... 50
2.3. Working in other bases ..... 51
3. Polygons ..... 53
3.1. Properties of quadrilaterals ..... 55
3.2. Semi-regular tiling patterns ..... 57
3.3. Triangles from folds ..... 59
3.4. The four centres of a triangle ..... 61
3.5. From circles to polygons ..... 62
3.6. Polygon symmetries ..... 63
3.7. Four tiles ..... 64
4. Natural numbers and beyond ..... 66
4.1. Patterns on grids ..... 67
4.2. Make 17 ..... 69
4.3. Flipping a coin ..... 69
4.4. A problem of power ..... 70
4.5. Diophantine equations ..... 72
5. Rational numbers and beyond ..... 74
5.1. Early Egyptian fractions ..... 75
5.2. Counting camels ..... 76
5.3. Bicimals and other numbers ..... 76
5.4. Beyond the rationals ..... 79
6. Polyhedra ..... 80
6.1. Spinning the cube ..... 81
6.2. Slicing the cube ..... 82
6.3. Filling the cube ..... 83
6.4. What are the Platonic solids? ..... 83
6.5. Symmetries of the Platonic solids ..... 84
6.6. The symmetries of a marked cube ..... 88
6.7. Semi-regular solids ..... 88
7. Group theory ..... 93
7.1. Dancing squares ..... 95
7.2. Isomorphism ..... 96
7.3. Four properties defining a group ..... 7
7.4. The Latin Square property ..... 98
7.5. Subgroups ..... 99
7.6. Cosets ..... 100
7.7. Lagrange’s theorem ..... 101
7.8. Revisiting the marked cube ..... 101
7.9. Cyclic groups and cyclic subgroups ..... 102
7.10. Two proofs ..... 103
7.11. Hexagon symmetries ..... 105
7.12. More about inverses ..... 106
7.13. Groups with six elements: how many are there? ..... 106
Conclusion ..... 111
References ..... 113
APPENDICES
Appendix 1 Mayan number puzzle ..... 125
Appendix 2 Symbol sequence cards ..... 126
Appendix 3 Grid multiplication puzzle ..... 128
Appendix 4 Squares, triangles and hexagons – from Smile 1734 ..... 130
Appendix 5 Proof that there are only eight semi-regular tilings ..... 131
Appendix 6 Proof for two tiles ..... 133