Al Cuoco on Wild About Math

Sol Lederman of Wild About Math interviewed Al Cuoco on his latest book Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem (co-author Joseph Rotman).

Listen to the interview here.

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Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem
by Al Cuoco and Joseph Rotman

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."

New MAA eBook

Illustrated Special Relativity Through Its Paradoxes
A Fusion of Linear Algebra, Graphics, and Reality
by John dePillis and José Wudka

Illustrated Special Relativity illustrates and resolves several apparent paradoxes of special relativity including the twin paradox and train-and-tunnel paradox. Assuming a minimum of technical prerequisites the authors introduce inertial frames and use them to explain a variety of phenomena: the nature of simultaneity, the proper way to add velocities, and why faster-than-light travel is impossible. Most of these explanations are contained in the resolution of apparent paradoxes, including some lesser-known ones: the pea-shooter paradox, the bug-and-rivet paradox, and the accommodating universe paradox. The explanation of time and length contraction is especially clear and illuminating.

At the outset of his seminal paper on special relativity, Einstein acknowledges the work of James Clerk Maxwell whose four equations unified the theories of electricity, optics, and magnetism. For this reason, the authors develop Maxwell’s equations which lead to a simple calculation for the frame-independent speed of electromagnetic waves in a vacuum. (Maxwell did not realize that light was a special case of electromagnetic waves.) Several chapters are devoted to experiments of Roemer, Fizeau, and de Sitter to measure the speed of light and the Michelson-Morley experiment abolishing the aether.

Throughout the exposition is thorough, but not overly technical, and often illustrated by cartoons. The volume might be suitable for a one-semester general-education introduction to special relativity. It is especially well-suited to self-study by interested laypersons or use as a supplement to a more traditional text.

Purchase your copy today in the MAA eBooks Store.

Beckenbach Book Prize 2014

The Beckenbach Book Prize, established in 1986, is the successor to the MAA Book Prize established in 1982. It is named for the late Edwin Beckenbach, a long-time leader in the publications program of the Association and a well-known professor of mathematics at the University of California at Los Angeles. The Prize of $2,500 is intended to recognize the author(s) of a distinguished, innovative book published by the MAA and to encourage the writing of such books. The award is not given on a regularly scheduled basis. To be considered for the Beckenbach Prize a book must have been published during the five years preceding the Award.

The 2014 Beckenbach Book Prize winner is:

A Historian Looks Back: The Calculus as Algebra  and Selected Writings by Judith Grabiner Judith Grabiner has written extensively on the history of mathematics. This collection, representing some of Grabiner's finest work, highlights the benefits of studying the development of mathematical ideas and the relationship between culture and mathematics. A large part of the book (Part I) is a welcome reprinting of Grabiner's The Calculus as Algebra: J.-L. Lagrange, 1736-1813 (1990), which focuses on Lagrange's pioneering effort to reduce the calculus to algebra. Ten articles (Part II) span a range of other mathematical topics, including widely held myths about the history of mathematics and the work of such mathematicians as Descartes, Newton, and Maclaurin. Six of these articles won awards from the MAA for expository excellence. This collection is an inspiring resource for courses on the history of mathematics. It reveals the creativity that has produced the mathematics we see as the finished product in textbooks.

The MAA will be honoring Judith Grabiner and her book at the 2014 Joint Mathematics Meetings in Baltimore. Learn more here.

MAA Review of Learning Modern Algebra

Mark Hunacek reviews Learning Modern Algebra by Al Cuoco and Joseph J. Rotman as part of MAA Reviews. This is an interesting, well-written book, in search of an appropriate course in which it could be used as a text. From the title, one would think that it was intended primarily as a text for an introductory abstract algebra course, but using it that way would require a fairly radical overhaul of the traditional syllabus of such a course. This is intentional: the authors make clear in the Preface to the book that they believe that this traditional syllabus (namely number theory, followed by groups and then rings) to be not only “totally inadequate for future teachers of high school mathematics” but also “unsatisfying for other mathematics students” as well. They propose that abstract algebra should be taught in two semesters: number theory and rings in the first, groups and linear algebra in the second. Even for such a course, however, this book would likely not be appropriate for both semesters; it covers a lot of number theory and ring theory, but very little group theory and linear algebra. (More about the specific contents later.) The primary intended audience of the book is future high school teachers. The authors take great pains to relate the material covered here to subjects that are taught in high school mathematics classes. And not just high school algebra classes: there is, for example, a fairly lengthy and quite detailed section on straightedge and compass constructions, including statements and (at least partial, and often full) proofs of many sophisticated results regarding impossible constructions. Read the full review here. To order your copy today, visit the MAA Store or the MAA eBooks Store.

Cuoco and Rotman on Learning Modern Algebra

Authors Al Cuoco and Joseph J. Rotman recap their experiences of writing Learning Modern Algebra: From Early Attempts to Prove Fermat's last Theorem. "About five years ago, I helped organize a conference at the University of Arizona focused on the mathematical preparation and professional development of secondary mathematics teachers. Consistent with the missions of the Institute for Mathematics and Education at UA and The Focus on Mathematics partnership in Boston (the sponsors of the event), we looked to invite an eclectic group of mathematicians, teachers, and mathematics educators. I had admired Joe Rotman’s writing for some time and had long wanted to hear more about his ideas on precollege education, ever since I visited Peter Braunfeld and colleagues at Urbana-Champaign, maybe 15 years ago. So we invited him and he accepted. Several of the presentations were, as Joe recounts in his post, attempts to make abstract algebra a more useful part of preparation for high school teachers. Like Joe, I was underwhelmed with much of what I heard. On the other hand, I had spent over two decades teaching high school, and I used ideas from number theory (especially Ireland-Rosen), algebra (of the Birkhoff Mac Lane variety), and other classic texts all the time to help bring some coherence and underlying structure to my high school courses. So, when Joe brought up the idea of a new text in this tradition, one that emphasizes rings and fields over groups, that puts experience before formality, and where abstract results emerge from concrete computations, I jumped at the chance to collaborate. And it’s been a very interesting collaboration. Yes, Joe made me crazy sometimes, but we gradually came to a common style and approach that got easier to negotiate as the chapters developed. Underneath, we really do share the same values, tastes, and dispositions. And I think part of the reason that we get along so well comes from the fact that we both enjoy a good calculation. By the way—I learned that it takes four times as long for two people to write a book as it does for one person. You can extrapolate what would happen with three authors." — Al Cuoco "About five years ago, I was invited to a meeting in Arizona about preparation of high school math teachers. I guess the reason for my being asked was a book I had written in the 90s, Journey into Mathematics, for a transition course between the usual first university calculus courses and the following math courses that take proofs seriously. The Arizona meeting was the first meeting of “educators” I had ever attended and, to tell the truth, I was quite the snob, sneering down my nose at guys who think they know how things ought to be taught. Well, I discovered that a lot of them are also very good mathematicians, and it might be worthwhile listening to them. One of the topics discussed was how abstract algebra courses designed for future high school math teachers affects what is actually taught in high schools. The standard of such a course is divided into three parts: number theory, group theory, and commutative ring theory. I was appalled by descriptions of how group theory was being shoe-horned into high schools, in the rare cases it is taught at all (but many other talks were pretty good). In an impromptu talk at the meeting, I began by saying that even though groups are my friends, in light of what I had been hearing, they should not be highlighted in high school. When I got home, I thought more about this, and I decided I could design an abstract algebra course for teachers that would be more useful than what is done now. But it’s been a long time since I’ve had any contact with high school math (except for sniffing at my daughter’s class when she was in a linear algebra course pretending to be Euclidean geometry). I met Al Cuoco in Arizona, and he knows high school curricula. I sent him an e-mail briefly describing my ideas, and asking whether he knew anyone knowledgeable about contemporary high school math who might share my ideas. He said, “What about me?” I was delighted, and our collaboration was born. What’s nice is that, in spite of disagreements along the way (I’m not sure how many times he said I drive him crazy), we essentially share the same values and also tolerate each other’s sense of humor. If anyone thinks we are too serious, we invite them to read the tale of Ricky the raccoon (p. 134). We hope that our work not only finds sympathetic readers, but that it can actually improve the way things are now done." — Joseph Rotman Interested in purchasing a copy? Order from the MAA Store or 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

Debbie Gochenaur Reviews Uncommon Mathematical Excursions

Debbie Gochenaur reviews Uncommon Mathematical Excursions: Polynomia and Related Realms by Dan Kalman as part of MAA Reviews. Offering an assortment of topics in the algebra, geometry, and calculus curricula, this book is intended as an enrichment for those familiar with these topics at the upper-secondary or introductory college math level. While the book may be geared especially for teachers who have taught their courses enough times to be thoroughly comfortable with the content, advanced students, as well as scientists and mathematicians in general, may find topics within this book intriguing. It has the potential to appeal to a broad audience. The book is meant to guide the readers in exploring ideas that are related to his/her core mathematical knowledge. The book is divided into three parts — The Province of Polynomia, Maxiministan, and The Calculusian Republic. Kalman ensures that the reader can make connections to known undergraduate mathematics and advanced secondary topics in the algebra, geometry, and calculus domains; he works to help the reader make extensions and perhaps understand more clearly the depth of mathematics in these seemingly elementary topics. Overall, the reader may be surprised by aspects of a particular topic that will lead to a greater understanding and appreciation for the mathematics. Read the full review here. Purchase your copy in the MAA Store.

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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