X Marks the Spot : The Lost Inheritance of Mathematics

**__X Marks the Spot is__** written from the point of view of the users of mathematics. Since the beginning, mathematical concepts and techniques (such as arithmetic and geometry) were created as tools with a particular purpose like counting sheep and measuring land areas. Understanding those purposes leads to a greater understanding of why mathematics developed as it did. Later mathematical concepts came from a process of abstracting and generalizing earlier mathematics. This process of abstraction is very powerful, but often comes at the price of intuition and understanding. This book strives to give a guided tour of the development of various branches of mathematics (and what they’re used for) that will give the reader this intuitive understanding. Features * Treats mathematical techniques as tools, and areas of mathematics as the result of abstracting and generalizing earlier mathematical tools * Written in a relaxed conversational and occasionally humorous style making it easy to follow even when discussing esoterica. * Unravels how mathematicians think, demystifying math and connecting it to the ways non-mathematicians think and connecting math to people’s lives * Discusses how math education can be improved in order to prevent future generations from being turned off by math. Cover Half Title Title Page Copyright Page Table of Contents List of Figures Preface Authors Chapter 1 Why This Book? Part I The Roots of Mathematics Chapter 2 Sticks and Stones The Unnumbered World Stones: Counting Sticks: Measuring Multiplying the Stick Tricks with Sticks Rate of Change Units Multiply Like Rabbits The volume? Infinity The Numbered World Chapter 3 Abstraction, Mistrust, and Laziness Abstraction Mistrust Premises Methods of Reasoning in Proofs Kinds of Proof Elegance Laziness Putting the Three Together Chapter 4 Algebra, Geometry, Analysis: The Mathematical Mindsets Algebra Here it is: Value Abstraction. Let’s bring him out again, x the unknown, Variable Extraordinaire! Geometry Sticks Are Lines When Are Shapes the Same? Shapes as Tools Analysis Building Machines Function Behavior and Unknown Functions Logic Logic in Math Logic and Meaning Digging up the Roots Part II Theory in Practice Chapter 5 Analytic Geometry Planes and Space, Numbers by Numbers by Numbers Function is Shape Parametric Graphing The Fault Is Not in Our Stars Curves and Surfaces Coordinate Systems More Dimensions than You Can Shake a Stick at Chapter 6 Calculus: Motion and Size The Derivative Limits Derivative Redux (Reduced, That Is) When Can We Differentiate? Function Approximation and Taylor Series Vectors Integral Calculus The Fundamental Theorem of Calculus Logarithm and Exponential The Absence and Presence of Calculus Chapter 7 The Language of Motion Linear Digression Chapter 8 Sound, Notes, and Harmonics Chapter 9 Probability and Statistics Cards and Randomness Combinatorics: Counting Without Counting Counting Cards Probability Distributions: Stick the Chances Stochastic Processes Statistics Chapter 10 Other Geometries: Not So Straight, These Sticks The Universe Is Bent Shortest Distance Between Two Points Metric Spaces Nature Spiky in Coast and Leaf Why Are They Called Fractals? Iteration and Chaos Chapter 11 Algebra and the Rise of Abstraction Construction and Its Limits Imaginary and Complex Numbers Algebraic Structures: Abstraction Spreads Wide Its Arms Morphisms: Preservatives Added and Multiplied Algebraic View of Geometry: Slouching Toward Topology Category Theory Part III Toolkit of the Theoretical Universe Chapter 12 The Smith and the Knight How the User Sees the Tool The Maker’s View of the Tool Beauty in the Use of Tools Beauty in the Making of Tools: Artists and Artisans Mathematics as Toolkit Using the Toolkit Expanding the Toolkit Chapter 13 Building the Theoretical Universe Fluids Electricity and Magnetism Chapter 14 Computers Logic Embodied Binary Arithmetic Embodied Computer Programming Modular Programming, Procedures, and Functions Speed, Memory, Bandwidth, Price, Size, and Efficiency Human Thought and Computer “Thought” Computer Use in Math and Science Chapter 15 The Theoretical Universe of Modern Physics: Toolkit Included Relativity General Relativity Quantum Mechanics Quantum Field Theory Chapter 16 Math Education and Math in Education Welcome to Math Problem World How to Teach It Math in Education Reclaiming the Lost A Stone’s Throw Index "X Marks the Spot is written from the point of view of the users of mathematics. Since the beginning, mathematical concepts and techniques (such as arithmetic and geometry) were created as tools with a particular purpose like counting sheep and measuring land areas. Understanding those purposes leads to a greater understanding of why mathematics developed as it did. Later mathematical concepts came from a process of abstracting and generalizing earlier mathematics. This process of abstraction is very powerful, but often comes at the price of intuition and understanding. This book strives to give a guided tour of the development of various branches of mathematics (and what they're used for) that will give the reader this intuitive understanding. Features: Treats mathematical techniques as tools, and areas of mathematics as the result of abstracting and generalizing earlier mathematical tools; Written in a relaxed conversational and occasionally humorous style making it easy to follow even when discussing esoterica; Unravels how mathematicians think, demystifying math and connecting it to the ways non-mathematicians think and connecting math to people's lives; Discusses how math education can be improved in order to prevent future generations from being turned off by math"-- Provided by publisher