Math from Three to Seven

Math from Three to Seven

The Story of a Mathematical Circle for Preschoolers Alexander K. Zvonkin

Contents

Foreword to the American Edition

vii

Introduction

1

A Parent's Journal

1

Why a circle? Why a journal?

2

Should journals be edited?

4

A novice's reflections on pre-school math

5

Opinions

8

A Short History of Our Circle

9

Acknowledgments

12

Two Disclaimers

13

Chapter 1. The First Session: Narrative and Reflections

15

A session in action

15

Piaget's phenomena: reality or illusion?

24

Why read psychology books?

28

Why do we need theories?

32

Chapter 2. The Boys' Math Circle, Year One

35

Session 21. The M?obius band

35

Session 22. What is bigger, the whole or its part?

37

Session 23. The Tower of Hanoi

41

Session 24. A bit of topology

45

Session 25. The boy in the elevator

47

Session 26. Intersecting classes

48

Session 27. Pegboard quadrilaterals

50

Session 28. We start probability theory

52

Session 29. Total failure

54

Session 30. Pouring water

56

Session 31. Probability theory, again

60

Session 32. Diplomas

62

A few more problems

63

How to draw a cube?

67

iii

iv

CONTENTS

Chapter 3.

Children and

5 2

:

The

Story

of

One

Problem

69

A combinatorial puzzle

70

Equivalent problems

71

Denoting...

74

Proof

76

Physics and logic

79

Chapter 4. The Boys' Math Circle, Year Two

83

Session 33. Geometric similarity

83

Session 34. An uneventful session

86

Session 34. Almost calculating probabilities

87

Session 36. A game of three dice

90

Session 37. How many rectangles?

92

Session 38. Losing my grip

94

Session 39. Back on track

96

A short excursion into the past.

97

The programming language Kid

100

Session 40. First encounter with Dienes blocks

105

Session 41. More of the same (robots and Dienes blocks)

107

Session 42. Snowflakes

109

Session 43. On certain properties of addition

111

Session 44. Magic squares

115

Session 45. Generalized chains

117

Session 46. Isomorphic problems

119

Session 47. The end of the story about

5 2

.

120

Session 48. True and false statements.

122

A bit of programming, just with Dima

123

Session 49. Thinking about symbols

125

Session 50. A double anniversary

128

Session 51. Which path is longer?

129

Session 52. Breaking a code

131

Session 53. A genealogical tree

132

Session 54. The end of the school year

134

Chapter 5. Notation, abstraction, mathematics, and language

137

Symbols for words

137

"Simplified" notation?

139

Each person has more than one type of intelligence

141

Teaching mathematics as a native tongue

144

Chapter 6. The Boys' Math Circle, Year Three

149

Session 55. Logical problems

149

Session 56. Construction foreman

152

Session 57. Who is booter, Gobr or Stoon?

154

Session 58. Floor plans

158

A long hiatus

160

CONTENTS

v

Session 59. What does the other person see?

164

Session 60. Reflection

167

Session 61. How do you add invisible numbers?

169

Session 62. Which room is larger?

172

Session 63. Reason versus chance

173

Session 64. We battle against the odds, again

176

Session 65. Homeomorphism

180

Session 66. Topology

183

Session 67. Four colors

184

Miscellaneous jokes, conversations, and puzzles

185

Chapter 7. The Boys' Math Circle, Final Six Months

195

Session 68. Calendar conundrum

195

Session 69. Oral puzzles

197

Session 70. More programming

200

Session 71. Classroom puzzles . . . almost

203

Session 72. Subprograms

205

Session 73. Odd numbers and squares

208

Session 74. The geometry of numbers

210

Session 75. The Mayans

212

Session 76. All things must end, sometime

214

Chapter 8. At Home and in School

217

Mathematical discussions, with sad digressions about school

217

First graders

234

Chapter 9. The Girls' Math Circle, Year One

239

Introduction

239

Session 1. Piaget's phenomena, again

246

Session 2. Princes and princesses

251

Session 3. How many differences?

253

Session 4. Building from diagrams

256

Session 5. Permutations

259

Session 6. The boy's morning

261

Session 7. Play trumps science

262

Session 8. Between two mirrors

264

Session 9. In the courtyard

266

Session 10. Bi-colored cubes

270

Session 11. Fives

271

Chapter 10. The Girls' Math Circle, Year Two

273

Session 12. Something's amiss with probability theory

273

Session 13. Intersecting classes again

275

Session 14. The Tower of Hanoi

277

Session 15. Towers of equal height

278

Session 16. Turning 90

280

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