The Beauty of Elementary Mathematics: And How to Teach It

Contents
About the Author
Acknowledgments
Introduction
What’s in the Book
Part 1 Thinking Mathematics
	Beauty
	What Is Mathematics?
	Mathematics and the Humanities
Part 2 Didactics
	Layer upon Layer
	How Do You Catch a Cloud and Pin It Down?
	Three Stages: Concrete, Pictorial, Abstract
	Pace
	Thou Shalt Not Skip
	Words: The Cement of Thought
	Class Discussion
	Naming
	The Personal Whiteboard
	Laughter = Understanding
Part 3 Kindergarten
	Why?
	The First Operation: Naming a Unit
	Directions
	Before and After, Smaller and Bigger
	Equations in Kindergarten
Part 4 More Teaching Principles
	Reversal
	My First Equations: Using Reversal
	Role Reversal
	Conservation and the Commutativity of Addition
	There Is No Such Thing as “Too Simple”
Part 5 Operations in Search of Meaning
	Meaning before Calculation
	The Curriculum in Arithmetic
	Addition and Subtraction: Dynamic and Static
	Multiplication = Counting
	Distribution: Open the Box before Multiplying, or After?
	4 × 3 = 3 × 4 Is Not an Axiom
	Two Types of Division
	The Ultimate Crime: Dividing by Zero
	Remainder: When You Cannot Split
	Parity
	Isomorphism
	Average
Part 6 There Shalt Be Order: The Decimal System
	Collecting Tens
	A Lesson on the Decimal System
	The Second Most Useful Mathematical Invention (After the Number)
	Why Didn’t the Romans Have a “Zero” Symbol?
	Divisibility by 3
Part 7 The Checkered Notebook: Calculation
	The Calculator Debate
	In the Country of One-Fingered Creatures
	A Ten Is Born
	Why Are Math Notebooks Checkered?
	Should the Addition and Multiplication Tables Be Memorized?
	A Lesson on the Addition Table
	Vertical Multiplication
	Short Division
Part 8 More Teaching Principles
	Guessing
	Divide and Conquer
	Approximately
	Extreme Cases
	“Do, Then Listen”, or “Listen, Then Do”?
	The Archimedean Fulcrum of Mathematical Education
Part 9 Fractions
	What Is 5/8 of 240 — or, What Is a Fraction?
	A “Must” Lesson
	Fractions from Venus (Shapes), Division from Mars (Numbers)
	The Culprit: The Pizza Model
	Juggling One Ball
	Teaching Fractions Together with Division
	Why Is the Fraction Line a Sign of Division?
	Multiplying Fractions
	Dividing Fractions
	Expansion
	Addition and Subtraction: The Common Denominator
	Smallest Common Denominator
	Comparing Fractions
	Mixed Numbers
Part 10 Decimal Fractions
	Conversion
	Infinite Decimal Fractions
	From Infinite Decimal Fractions to Simple Fractions
	Addition and Subtraction of Decimal Fractions
	Popeye’s Secret
	A Famous Fraction: Percent
Part 11 Educational Revolutions
	Whole Words
	The New Math
	Investigation and the Math Wars
Part 12 Proportion
	The Seven Sisters
	The Two Meanings of Direct Proportion
	Per Unit: The Wisdom of km/Hour
	Inverse Rate, Pool Problems, and Parallel Resistors
Part 13 Geometry
	The Amazing Greeks
	The Most Beautiful Proof?
	An Axiom, or a Theorem? The Story of the Fifth Axiom
	Main Points
	Geometry vs. Number Theory: Which Is More Complex?
	First Lesson: Points and Lines
	Definable vs. Undefinable
	Angles
	The Right Angle
	Opposite Angles
	More on Parallel Lines: Infinity and Transitivity
	Corresponding and Alternate Angles
Part 14 Congruence and Similarity
	The Triangle Inequality
	Congruent Triangles
	Parallels and Similarity
	Auxiliary Constructions
	External Angles
	Hierarchy and Converse Theorems
	Symmetry
Part 15 The Ruler and the Compass
	Another Stroke of Genius of the Greeks
	Impossible Constructions
Part 16 Length, Area, and Volume
	Units of Length
	Square Centimeter and the Area of a Rectangle
	Pythagoras’ Theorem
	The Area of the Triangle and the Volume of the Pyramid
	One Example Suffices!
	The Cavalieri Principle
Part 17 The Magic Circle
	Geometric Locus
	The Perpendicular Bisector
	Angles in the Circle
	π
	The Area of the Circle: Is Earth Flat After All?
	The Area of the Sphere
	The Volume of the Ball
	Beyond the Firmament: Middle School Mathematics
	Epilogue: The Infinitude of Mathematics
Index