Beauty Elementary Mathematics How Teachb: Beauty of Elementary Mathematics, the: and How to Teach It

Why is 2 times 3 equal to 3 times 2? One may think this is an axiom, but it has a proof, and a beautiful one at that. Elementary mathematics is as deep and as beautiful as higher mathematics. It includes some of the most important mathematical discoveries ever, for example the concept of the number, and the place-value method of representing numbers. We are so accustomed to this method, that we forget how clever and beautiful it is - resulting in its incredible efficacy. All this was a surprise for the author, a university professor of mathematics, when he went to teach in elementary school. He realized that good teaching of elementary mathematics requires understanding its fine points and conveying their beauty to the students. Sensing the beauty and understanding go hand in hand. The book outlines the material from kindergarten to grade 6 (with an excursion into algebra as well). It also discusses teaching principles, and their close relatives - thinking principles. Teachers and parents who imbue these principles are likely to convey the love of mathematics to the child. 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