The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition
December 2012, ©2013
Transform your mathematics course into an engaging and mind-opening experience for even your most math-phobic students. Now in its Fourth Edition, The Heart of Mathematics: An Invitation to Effective Thinking succeeds at reaching non-math, non-science-oriented majors, encouraging them to discover the mathematics inherent in the world around them. Infused with the authors' humor and enthusiasm throughout, The Heart of Mathematics introduces students to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking.
- 1. FUN AND GAMES
An Introduction to Rigorous Thought
1.1 Silly Stories, Each with a Moral
Conundrums that Evoke Techniques of Effective Thinking
Leading Questions and Hints for Resolving the Stories
1.3 The Punch Lines
Solutions and Further Commentary
1.4 From Play to Power
Discovering Strategies of Thought for Life
- 2. NUMBER CONTEMPLATION
How the Pigeonhole Principle Leads to Precision Through Estimation
2.2 Numerical Patterns in Nature
Discovering the Beauty of the Fibonacci Numbers
2.3 Prime Cuts of Numbers
How the Prime Numbers Are the Building Blocks of All Natural Numbers
2.4 Crazy Clocks and Checking Out Bars
Cyclical Clock Arithmetic and Bar Codes
2.5 Public Secret Codes and How to Become a Spy
Encrypting Information Using Modular Arithmetic and Primes
2.6 The Irrational Side of Numbers
Are There Numbers Beyond Fractions?
2.7 Get Real
The Point of Decimals and Pinpointing Numbers on the Real Line
- 3. INFINITY
3.1 Beyond Numbers
What Does Inﬁnity Mean?
3.2 Comparing the Infinite
Pairing Up Collections via a One-to-One Correspondence
3.3 The Missing Member
Georg Cantor Answers: Are Some Inﬁnities Larger than Others?
3.4 Travels Toward the Stratosphere of Infinities
The Power Set and the Question of an Inﬁnite Galaxy of Inﬁnities
3.5 Straightening Up the Circle
Exploring the Inﬁnite Within Geometrical Objects
- 4. GEOMETRIC GEMS
4.1 Pythagoras and His Hypotenuse
How a Puzzle Leads to the Proof of One of the Gems of Mathematics
4.2 A View of an Art Gallery
Using Computational Geometry to Place Security Cameras in Museums
4.3 The Sexiest Rectangle
Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle
4.4 Soothing Symmetry and Spinning Pinwheels
Can a Floor Be Tiled Without Any Repeating Pattern?
4.5 The Platonic Solids Turn Amorous
Discovering the Symmetry and Interconnections Among the Platonic Solids
4.6 The Shape of Reality?
How Straight Lines Can Bend in Non-Euclidean Geometries
4.7 The Fourth Dimension
Can You See It?
- 5. CONTORTIONS OF SPACE
5.1 Rubber Sheet Geometry
Discovering the Topological Idea of Equivalence by Distortion
5.2 The Band That Wouldn’t Stop Playing
Experimenting with the Möbius Band and Klein Bottle
5.3 Knots and Links
Untangling Ropes and Rings
5.4 Fixed Points, Hot Loops and Rainy Days
How the Certainty of Fixed Points Implies Certain Weather Phenomena
- 6. MODELING OUR WORLD THROUGH GRAPHS
6.1 Circuit Training
From the Königsberg Bridge Puzzle to Graphs
6.2 Feeling Edgy?
Exploring Relationships Among Vertices, Edges, and Faces
6.3 Plane Old Graphs
Drawing in the Plane and Coloring Maps
Using Graphical Models to Find the Shortest, Closest, and Cheapest
- 7. FRACTALS AND CHAOS
Viewing a Gallery of Fractals
7.2 The Infinitely Detailed Beauty of Fractals
How to Create Works of Inﬁnite Intricacy Through Repeated Processes
7.3 Between Dimensions
Can the Dimensions of Fractals Fall Through the Cracks?
7.4 Mysterious Art of Imaginary Fractals
Creating Julia and Mandelbrot Sets by Stepping Out in the Complex Plane
7.5 They Dynamics of Change
Can Change Be Modeled by Repeated Applications of Simple Processes?
7.6 Predetermined Chaos
How Repeated Simple Processes Result in Utter Chaos
- 8. TAMING UNCERTAINTY
8.1 Chance Surprises
Some Scenarios Involving Chance That Confound Our Intuition
8.2 Predicting the Future in an Uncertain World
How to Measure Uncertainty Using the Idea of Probability
8.3 Random Thoughts
Are Coincidences as Truly Amazing as They First Appear?
8.4 Down for the Count
Systematically Counting All Possible Outcomes
8.5 Drizzling, Defending, and Doctoring
Probability in Our World and Our Lives
- 9. MEANING FROM DATA
9.1 Stumbling Through a Minefield of Data
Inspiring Statistical Concepts Through Pitfalls
9.2 Getting Your Data to Shape Up
Organizing, Describing, and Summarizing Data
9.3 Looking at Super Models
Mathematically Described Distributions
9.4 Go Figure
Making Inferences from Data
9.5 War, Sports, and Tigers
Statistics Throughout Our Lives
10. DECIDING WISELY
Applications of Rigorous Thinking
10.1 Great Expectations
Deciding How to Weigh the Unknown Future
Deciding Personal and Public Policy
10.3 Money Matters
Deciding Between Faring Well and Welfare
10.4 Peril at the Polls
Deciding Who Actually Wins an Election
10.5 Cutting Cake for Greedy People
Deciding How to Slice Up Scarce Resources
- “For the Algebra Lover” Mindscapes in each section. These problems were added for students who want to practice their computational skills and for professors who prefer to integrate more algebra into this course.
- Chapter 6 on graph theory.
- Accessible, student-friendly writing style engages students to read and understand the concepts and encourages critical thinking.
- “Life Lessons” offer effective methods of thinking that students will retain and apply beyond the classroom.
- End-of-section Mindscape activities for the development of application, problem solving, and argumentation skills.
- Activities that encourage collaborative learning and group work like the hands-on manipulative kit that directs students to model their thinking and actively explore the world around them.
- “A Look Back” provide a review of concepts taught in that section
- Technology Icon located throughout the book indicates that there’s an interactive activity available online (e-manipulatives)
- Visual techniques that inspire students to model their thinking and actively explore the world around them.
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