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The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition

Edward B. Burger, Michael Starbird

ISBN: 978-1-118-15659-9 December 2012 928 Pages

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WileyPLUS improves outcomes with robust practice problems and feedback, fosters engagement with course content and educational videos, and gives students the flexibility to increase confidence as they learn and prepare outside of class.


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.

Related Resources

  1. 1.     FUN AND GAMES

An Introduction to Rigorous Thought

1.1  Silly Stories, Each with a Moral

Conundrums that Evoke Techniques of Effective Thinking

1.2  Nudges

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



2.1  Counting

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


  1. 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


  1. 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?



5.1  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



6.1  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

6.4  Networking

Using Graphical Models to Find the Shortest, Closest, and Cheapest



7.1  Images

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



8.1  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



9.1  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



Applications of Rigorous Thinking

10.1        Great Expectations

Deciding How to Weigh the Unknown Future

10.2        Risk

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.
  • WileyPLUS, for The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition, provides a research based, online environment for effective teaching and learning. WileyPLUS builds students’ confidence because it takes the guesswork out of studying by providing students with a clear roadmap: what to do, how to do it, if they did it right.