Now in Print! The Heart of Calculus

The Heart of Calculus: Explorations and Applications
Philip M. Anselone and John W. Lee
List Price: $60.00 | MAA Member Price: $48.00

This book contains enrichment material for courses in first and second year calculus, differential equations, modeling, and introductory real analysis. It targets talented students who seek a deeper understanding of calculus and its applications. The book can be used in honors courses, undergraduate seminars, independent study, capstone courses taking a fresh look at calculus, and summer enrichment programs. The book develops topics from novel and/or unifying perspectives. Hence, it is also a valuable resource for graduate teaching assistants developing their academic and pedagogical skills and for seasoned veterans who appreciate fresh perspectives.

The explorations, problems, and projects in the book impart a deeper understanding of and facility with the mathematical reasoning that lies at the heart of calculus and conveys something of its beauty and depth. A high level of rigor is maintained. However, with few exceptions, proofs depend only on tools from calculus and earlier. Analytical arguments are carefully structured to avoid epsilons and deltas. Geometric and/or physical reasoning motivates challenging analytical discussions. Consequently, the presentation is friendly and accessible to students at various levels of mathematical maturity. Logical reasoning skills at the level of proof in Euclidean geometry suffice for a productive use of the book.

There are 16 chapters in the book, divided about equally between pure and applied mathematics. The first three chapters are on fundamentals of differential calculus and the last three are on the monumental discoveries of Newton and Kepler on celestial motion and gravitation. The intervening chapters present significant topics in pure and applied mathematics chosen for their intrinsic interest, historical influence, and continuing importance. There is great flexibility in the choice of which chapters to cover and the order of coverage because chapters are essentially independent of each other.

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William Satzer Reviews 101 Careers in Mathematics

William J. Satzer reviewed 101 Careers in Mathematics, 3rd Edition edited by Andy Sterrett as part of MAA Reviews. 

This is a wonderful book, potentially of great value to students and those who advise them. It has some frustrating gaps too, but in a way they also emphasize how useful it is and could be. In brief, this book presents a collection of profiles of people who have (or had) a career that involves some aspect of mathematics. Nearly all the people here have at least one degree in mathematics; the few exceptions have degrees in field like physics, operations research, or a statistics-related area. Short essays at the end of the book discuss the processes of interviewing and finding a job, and what it’s like to work in industry (or, more broadly, outside the academic community).

There are 25 new entries in this new edition that bring the total number of profiles to 146. The “101 Careers” of the title is best regarded as meaning “lots of careers”; even the first edition had more than 101 profiles. Counting careers is also a little funny: they don’t match up one-to-one with people. As many of the profiles demonstrate, many people have more than one career. Indeed it is increasingly uncommon for people to have a single career throughout their lives.

Read the full review here.

MAA Books Beat: Playing to Learn Game Theory

Written by Steve Kennedy, MAA Books Beat is a column written for MAA FOCUS. Playing to Learn Game Theory appears in the June/July 2014 issue.

Playing to Learn Game Theory

by Steve Kennedy

Suppose the graph depicted in figure 1 represents a network of communities. Two doctors are planning to set up practice in one of the nine communities. A practice will attract all the patients in every town closest to it (closest means in terms of number of graph edges joining the vertices). In the case of a distance tie, each doctor will attract half the patients in the town. Where should the doctors locate to ensure their best result? (They are not conspiring; each doctor is acting independently and in ignorance of the other’s choice.)

Of course, neither doctor, presumably desiring to maximize her practice, would choose one of the degree-two vertices. The other six vertices look equally valuable. Suppose Doctor A chooses vertex 2. Were Doctor B to choose any of vertices 3, 5, or 8, each doctor would get 4.5 towns’ worth of business. If Doctor B chooses vertex 6, she “wins” by gaining five towns to A’s four. (If she chooses vertex 7, the payoffs are reversed and A wins five to four.)

If we conceive of the competition as a game between the two doctors, then this game has no Nash equilibrium. Each doctor can guarantee herself only four-ninths of the available customers.

The example comes from the new MAA ebook, Game Theory through Examples by Erich Prisner. The Doctor Locator Game is one of scores of clever, rich, interesting games described and analyzed in this lively text. There are also scores of apps, linked to the etext, so that the reader can play the games, the best way to understand them. In the doctor game, the reader can, with two mouse clicks, choose vertices to represent the doctors’ choices and immediately see a color-coded graph giving the division and the count.

The example games introduce and exposit all the important concepts of game theory. The Doctor Locator Game, on a different graph, introduces the idea of a Nash equilibrium and explains how to find one. The obvious next question of their existence is answered by the above example.

New eBook: Game Theory through Examples

Game Theory through Examples

by Erich Prisner

Classroom Resource Materials

Game Theory through Examples is a thorough introduction to elementary game theory, covering finite games with complete information.

The core philosophy underlying this volume is that abstract concepts are best learned when encountered first (and repeatedly) in concrete settings. Thus, the essential ideas of game theory are here presented in the context of actual games, real games much more complex and rich than the typical toy examples. All the fundamental ideas are here: Nash equilibria, backward induction, elementary probability, imperfect information, extensive and normal form, mixed and behavioral strategies. The active-learning, example-driven approach makes the text suitable for a course taught through problem solving. Students will be thoroughly engaged by the extensive classroom exercises, compelling homework problems and nearly sixty projects in the text. Also available are approximately eighty Java applets and three dozen Excel spreadsheets in which students can play games and organize information in order to acquire a gut feeling to help in the analysis of the games. Mathematical exploration is a deep form of play, that maxim is embodied in this book.

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Bill Wood Reviews Exploring Advanced Euclidean Geometry with GeoGebra

Bill Woods reviews Exploring Advanced Euclidean Geometry with GeoGebra by Gerard Venema as part of MAA Reviews.

Exploring Advanced Euclidean Geometry with GeoGebra is written for an inquiry-based approach, with lots of exercises and just enough narrative and historical commentary to hold it all together. It is not the sort of book you read without some paper and probably a computer in front of you. What makes the book special is the inclusion of GeoGebra exercises (clearly identified with a *) to encourage experimentation. Exercises may ask students to construct a visualization of a theorem, verify results, and build examples and conjectures. Eventually the student gets to proving a theorem, but not before playing with the statement quite a bit.

The book focuses on “advanced” planar Euclidean geometry, which the author defines to mean anything developed after Euclid’s Elements. This makes it an excellent candidate text for a second course in Euclidean geometry using inquiry-based methods that minimize lecture and maximize student discovery. There is also much value to be mined as a supplement to other Euclidean geometry texts. The author suggests a structure in which this text is used as something of a lab manual rather than a primary text.

Even if it does not fit for course adoption, this book is worth any geometry teacher’s attention as way to reconnect with the learning experience they want for their students.

Read the full review here.

New in the MAA Store

101 Careers in Mathematics

3rd Edition
Andrew Sterret, Editor

This third edition of the immensely popular 101 Careers in Mathematics contains updates on the career paths of individuals profiled in the first and second editions, along with many new profiles. No career counselor should be without this valuable resource.

The authors of the essays in this volume describe a wide variety of careers for which a background in the mathematical sciences is useful. Each of the jobs presented shows real people in real jobs. Their individual histories demonstrate how the study of mathematics was useful in landing well-paying jobs in predictable places such as IBM, AT&T, and American Airlines, and in surprising places such as FedEx Corporation, L.L. Bean, and Perdue Farms, Inc. You will also learn about job opportunities in the Federal Government as well as exciting careers in the arts, sculpture, music, and television. There are really no limits to what you can do if you are well prepared in mathematics.

The degrees earned by the authors profiled here range from bachelor’s to master’s to PhD in approximately equal numbers. Most of the writers use the mathematical sciences on a daily basis in their work. Others rely on the general problem-solving skills acquired in mathematics as they deal with complex issues.

Order your copy today at the MAA Store. 

Coming Soon to the MAA eBooks Store

Game Theory through Examples
Erich Prisner
Classroom Resource Materials

This book is a thorough introduction to elementary game theory, covering finite games with complete information. The core philosophy underlying this volume is that abstract concepts are best learned when encountered first (and repeatedly) in concrete settings. Thus, the essential ideas of game theory are here presented in the context of actual games, real games much more complex and rich than the typical toy examples. All the fundamental ideas are here: Nash equilibria, backward induction, elementary probability, imperfect information, extensive and normal form, mixed and behavioral strategies. The active-learning, example-driven approach makes the text suitable for a course taught through problem solving. Students will be thoroughly engaged by the extensive classroom exercises, compelling homework problems, and nearly sixty projects in the text. Also available are approximately eighty Java applets and three dozen Excel spreadsheets in which students can play games and organize information in order to acquire a gut feeling to help in the analysis of the games. Mathematical exploration is a deep form of play; that maxim is embodied in this book.

Game Theory through Examples is a lively introduction to this appealing theory. Assuming only high school prerequisites makes the volume especially suitable for a liberal arts or general education spirit-of-mathematics course. It could also serve as the active-learning supplement to a more abstract text in an upper-division game theory course.

375 pp., 2014
eISBN: 978-1-61444-115-1
Catalog Code: GTE
Price not yet set

MAA Review of Beyond the Quadratic Formula

Mark Hunacek reviews Beyond the Quadratic Formula by Ron Irving as part of MAA Reviews.

"Combining mathematics and history, this text tells, in a way accessible to beginning students, the interesting story of how formulas came to be discovered for the roots of third and fourth degree polynomials, and why nobody will discover corresponding formulas for fifth (or higher) degree polynomials.

Specifically, the book begins with an introductory chapter on polynomials (treated informally as formal expressions rather than rigorously defined), and is followed by a chapter on quadratic equations, in which the familiar quadratic formula is derived from several different points of view. The study of cubic equations begins in the next chapter, which discusses Cardano’s formula. One of the more amusing aspects of Cardano’s formula is that even nice, simple numbers can wind up being represented by horrendous sums of cube roots of expressions involving square roots; lots of examples are provided in the book. "

Read the full review here.

To purchase the book, visit the MAA Store or the MAA eBooks Store.

Richard Wilders Reviews Math Made Visual

Richard Wilders reviews Math Made Visual: Creating Images for Understanding Mathematics by Claudi Alsina and Roger B. Nelson as part of MAA Reviews. "This wonderful book is a fitting edition to the Classroom Resource series. Indeed any teacher of mathematics at the high school level or above should have a copy. Let me take that a bit further: anyone with an interest in mathematics should have a copy! While I ended up reading it pretty much straight through, this is a wonderful reference book which can be consulted whenever one is stuck for a way to make a concept come to life or for an activity to get students involved in mathematics. The first two-thirds of the text consists of a wonderful set of examples of how visualization can aid understanding and inspire exploration. Each section ends with a set of challenges for the reader. These problems would make wonderful projects for pre-service high school teachers — many of them can be implemented in Geometer’s Sketchpad. The final section of the book consists of hints for solving these challenges. Sandwiched between these two sections is a short section providing suggestions as to how these ideas can be used on a classroom. While technology would certainly help, many of the hints involve simple paper folding and cutting. Geometer's Sketchpad would certainly suffice to create most all of the 2-dimensional figures." Read the full review here. Purchase the book today at the MAA eBooks Store.

Now in the MAA Store

Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem by Al Cuoco and Joseph J. Rotman Learning Modern Algebra aligns with the CBMS Mathematical Education of Teachers–II recommendations, in both content and practice. It emphasizes rings and fields over groups, and it makes explicit connections between the ideas of abstract algebra and the mathematics used by high school teachers. It provides opportunities for prospective and practicing teachers to experience mathematics for themselves, before the formalities are developed, and it is explicit about the mathematical habits of mind that lie beneath the definitions and theorems. This book is designed for prospective and practicing high school mathematics teachers, but it can serve as a text for standard abstract algebra courses as well. The presentation is organized historically: the Babylonians introduced Pythagorean triples to teach the Pythagorean theorem; these were classified by Diophantus, and eventually this led Fermat to conjecture his Last Theorem. The text shows how much of modern algebra arose in attempts to prove this; it also shows how other important themes in algebra arose from questions related to teaching. Indeed, modern algebra is a very useful tool for teachers, with deep connections to the actual content of high school mathematics, as well as to the mathematics teachers use in their profession that doesn't necessarily “end up on the blackboard.'' Order your copy today in the MAA Store! List Price: $60.00 MAA Member: $48.00

Beyond the Quadratic Formula by Ron Irving The quadratic formula for the solution of quadratic equations was discovered independently by scholars in many ancient cultures and is familiar to everyone. Less well known are formulas for solutions of cubic and quartic equations whose discovery was the high point of 16th century mathematics. Their study forms the heart of this book, as part of the broader theme that a polynomial’s coefficients can be used to obtain detailed information on its roots. The book is designed for self-study, with many results presented as exercises and some supplemented by outlines for solution. The intended audience includes in-service and prospective secondary mathematics teachers, high school students eager to go beyond the standard curriculum, undergraduates who desire an in-depth look at a topic they may have unwittingly skipped over, and the mathematically curious who wish to do some work to unlock the mysteries of this beautiful subject. Order your copy today in the MAA Store! List Price: $55.00 MAA Member: $44.00

New in the MAA eBooks Store

Order your copies today in the MAA eBooks Store.
	Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem by Al Cuoco and Joseph J. Rotman MAA Textbooks Series Learning Modern Algebra aligns with the CBMS Mathematical Education of Teachers-II recommendations, in both content and practice. It emphasizes rings and fields over groups, and it makes explicit connections between the ideas of abstract algebra and the mathematics used by high school teachers. It provides opportunities for prospective and practicing teachers to experience mathematics for themselves, before the formalities are developed, and it is explicit about the mathematical habits of mind that lie beneath the definitions and theorems. Learning Modern Algebra is designed for college students who want to teach mathematics in high school, but it can serve as a text for standard abstract algebra courses as well. (Print version coming soon to the MAA Store.)
	Beyond the Quadratic Formula by Ron Irving Classroom Resource Materials Series Beyond the Quadratic Formula is designed for self-study, with many results presented as exercises and some supplemented by outlines for solution. The intended audience includes in-service and prospective secondary mathematics teachers, high school students eager to go beyond the standard curriculum, undergraduates who desire an in-depth look at a topic they may have unwittingly skipped over, and the mathematically curious who wish to do some work to unlock the mysteries of this beautiful subject. (Print version coming soon to the MAA Store.)
	New Horizons in Geometry by Tom M. Apostol and Mamikon A. Mnatsakanian Dolciani Mathematical Expositions Series Apostol and Mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics.Using new and intuitively rich methods, they give beautifully illustrated proofs of results, the majority of which are new, and frequently develop extensions of familiar theorems that are often surprising and sometimes astounding. It is mathematical exposition of the highest order.

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Use the code 1689472568 to receive 10% off your book purchase in the MAA eBooks Store!

Two New CRM Books

The MAA has two new Classroom Resource Materials books! Paradoxes and Sophisms in Calculus By Sergiy Klymchuk and Susan Staples Paradoxes and Sophisms in Calculus offers a delightful supplementary resource to enhance the study of single variable calculus. By the word paradox the authors mean a surprising, unexpected, counter-intuitive statement that looks invalid, but in fact is true. The word sophism describes intentionally invalid reasoning that looks formally correct, but in fact contains a subtle mistake or flaw. In other words, a sophism is a false proof of an incorrect statement. A collection of over fifty paradoxes and sophisms showcases the subtleties of this subject and leads students to contemplate the underlying concepts. A number of the examples treat historically significant issues that arose in the development of calculus, while others more naturally challenge readers to understand common misconceptions. Sophisms and paradoxes from the areas of functions, limits, derivatives, integrals, sequences, and series are explored. The book could be useful for high school teachers and university faculty as a teaching resource; high school and college students as a learning resource; and a professional development resource for calculus instructors.

Exploring Advanced Euclidean Geometry  with GeoGebra By Gerard Venema This book provides an inquiry-based introduction to advanced Euclidean geometry. It utilizes dynamic geometry software, specifically GeoGebra, to explore the statements and proofs of many of the most interesting theorems in the subject. Topics covered include triangle centers, inscribed, circumscribed, and escribed circles, medial and orthic triangles, the nine-point circle, duality, and the theorems of Ceva and Menelaus, as well as numerous applications of those theorems. The final chapter explores constructions in the Poincaré disk model for hyperbolic geometry. The book can be used either as a computer laboratory manual to supplement an undergraduate course in geometry or as a stand-alone introduction to advanced topics in Euclidean geometry.  Both books are currently available in the MAA eBooks Store. Print copies will be available in the MAA Store shortly.