Gerald L. Alexanderson
The Random Walks of George Polya
Series: Spectrum
In the first half of this charming book Gerald Alexanderson, presents a insightful portrait of George Polya, the great teacher. In the second half of the book, Alexanderson assembles eight papers that describe Polya's contribution to various fields.
George Polya enjoyed the esteem of the mathematical community not only for his deep and influential contributions in a variety of mathematical fields, but also for his groundbreaking work in the teaching of mathematics. His standing in the latter area could rest solely on his having written one of the most widely read books in mathematics, the still-popular How to Solve It. Various other books were almost equally influential, however: Mathematical Discovery and Mathematics and Plausible Reasoning, to name but two.
In addition to his championing problem-solving, he contributed to mathematics important results in complex and real analysis, inequalities, mathematical physics, combinatorics, probability theory, number theory and geometry. He coined the phrases "random walk" and "central limit theorem" and gave to mathematics the Polya Enumeration Theorem, along with many other ideas used widely today.
The present work attempts to describe how such versatility came about and, along the way, tells some stories about mathematics and mathematicians that, we hope, will enlighten and entertain.
The list of articles about Polya's work include:
Polya's Work in Probability by K. L. Chung
Polya's Work in Analysis, R. P. Boas
Comments on Number Theory,, D. H. Lehmer
Polya's Geometry, by Doris Schattschneider
Polya's Enumeration Theorem, by R.C. Read
Polya's Contributions in Mathematical Physics, M. M. Schiffer
George Polya and Mathematics Education, Alan H. Schoenfeld
Polya's Influence - References to his Work
300 pp., Paperbound, 1999
ISBN -0-88385-528-3
Isabella Bashmakova and Galina Smirnova
The Beginnings and Evolution of Algebra
Translated from the Russian by Abe Shenitzer with the editorial assistance of David A. Cox
Series: Dolciani Mathematical Expositions-23
How did algebra arise? What are its subject matter and methods? How have they changed in the process of its evolution? These natural questions are answered in the present book by its authors, I.G. Bashmakova and G.S. Smirnova, in an authoritative and compelling way.
There is hardly a branch of mathematics whose evolution has undergone as many surprising metamorphoses as has algebra, and these metamorphoses are described by the authors with vividness and clarity. The special merit of the book is that it corrects the widespread view that up to the 1830s the mainspring of the development of algebra was the investigation and solution of determinate algebraic equations, and especially their solution by radicals. The authors show that this viewpoint is one-sided and gives a distorted view of of its evolution. Specifically, they show that the role of indeterminate equation in the evolution of algebra was no less important than that of determinate equations.
A word about the authors. I.G. Bashmakova is a renowned authority on Diophantine analysis. G.S. Smirnova, her former doctoral student and collaborator, has made significant contributions to the history of algebra in Western Europe in the 16th century.
Contents:
1. Elements of algebra in ancient Babylonia;
2. Ancient Greek "geometric algebra";
3. The birth of literal algebra;
4. Algebra in the Middle Ages in the Arabic East and in Europe;
5. The first achievements of algebra in Europe;
6. Algebra in the 17th and 18th century;
7. Problems of number theory and the bnirth of commutative algebra;
8. Linear and commutative algebra;
9. Conclusion.
160 pp., Paperbound, 1999
ISBN 0-88385-329-9
Bonnie Gold, Sandra Keith, and William Marion, Editors
Assessment Practices in Undergraduate Mathematics
Series: MAA Notes-49
This book, a collection of assessment practices that have been tried by more than 100 contributors in mathematics at a wide variety of schools, attempts to offer the mathematics teacher suggestions from an insider's perspective. The book is not formulaic: no author claims to have "the answer," and many of the projects reported on are still in progress. On the other hand, the articles provide a wealth of suggestions from creative, energetic and concerned individuals who have had the courage to experiment and to critique their own efforts. Without doubt, the reader will find in these pages encouragement to experiment on his or her own, to find assessment methods which are personally meaningful.
Techniques offered in this book range from brief ten-minute classroom exercises and examples of alternative testing, group work and assignments, to examples of how departments may measure the placement of students into courses, the effectiveness of the major, and the quantitative literacy of their graduating students. Teachers beleaguered by formal end-of-term teacher evaluation forms, will find a variety of alternative assessment techniques that provides ways in which the quality of teaching can be better examined.
The book is unique among assessment books in representing the point of view of mathematicians exploring and examining methods of learning in their field.
350 pp., Paperbound, 1999
ISBN 0-88385-161-X
Frank Morgan
The Math Chat Book
Don't miss the chance to win $1,000...If you buy The Math Chat Book, and solve the puzzle found on the title page of the book, you could be a winner! The best correct response which stands for a year (or maybe earlier) wins $1000 award. No particular mathematical training is required.
Series: Spectrum
An ideal gift for anyone who enjoys the fun of mathematics--buy a copy for a favorite teacher, friend, or for yourself...
This book shows that mathematics can be fun for everyone. It grew out of Frank Morgan's live, call-in Math Chat TV show and biweekly Math Chat column in The Christian Science Monitor. The questions, comments, and even the answers come largely from the callers and readers themselves.
Why does the new year start earlier in Europe?
Why is the Fourth of July on a different day of the week each year?
How can you be elected President with just 22% of the vote?
Can a computer have free will?
Didn't some kid find a mistake on the SATs?
Do airplanes get lighter as passengers eat lunch?
College students make important progress on the still open "Double Bubble Conjecture."
One youngster asks, "If I live for 6000 years, how many days will that be?" His first answer is (6000 years)(365 days/year) = 2,190,000 days. That is not quite right: it overlooks leap years. An older student takes leap years into account, adds 1500 leap year days, and comes up with 2,191,500 days. The answer is still not quite right.
Every hundred years we skip a leap year (the year 1900, although divisible by four, was not a leap year), so we subtract 60 days to get 2,191,440. The answer is still not quite right. Every four hundred years we put the leap year back in (2000 will be a leap year), so we add back 15 days to get 2,191,455, the final answer.
This book makes no attempt to fit any mold. Although written by a research mathematician, it goes where the callers and readers take it, over a wide range of topics and levels. Almost anyone paging through it will find something of interest. It is time for everyone to see how much fun mathematics can be.
124 pp., Paperbound, 1999
ISBN 0-88385-530-5
William Dunham
Euler
The Master of Us All
Series: Dolciani Mathematical Expositions-22
New from William Dunham, award-winning author of Journey through Genius: The Great Theorems of Mathematics, and The Mathematical Universe....
Mathematician William Dunham has written a superb book about the life and amazing achievements of one of the greatest matheamticians of all time. Unlike earlier writings about Euler, Professor Dunham gives crystal clear accounts of how Euler ingeniously proved his most significant results, and how later experts have stood on Euler's broad shoulders. Such a book has long been overdue. It will not need to be done again for a long long time.--Martin Gardner
William Dunham has done it again! IN "Euler: The Master of Us All", he has produced a masterful portrait of one of the most fertile mathematicians of all time. With Dunham's beautiful clarity and wit, we can follow with amazement Euler's strokes of genius which laid the groundwork for most of the mathematics we have today. --Ron Graham, Chief Scientist AT&T
Without question, Leonhard Euler (1707-1783) ranks among history's greatest mathematicians. Over six decades of unmatched productivity, and despite a visual impairment that grew ever worse, charted the course of mathematics throughout the eighteenth century and beyond. His reputation is captured in Laplace's famous admonition, "Read Euler, read Euler. He is the master of us all."
Written for the mathematically literate reader, this book provides a glimpse of Euler in action. Following an introductory biographical sketch are chapters describing his contributions to eight different topics--number theory, logarithms, infinite series, analytic number theory, complex variables, algebra, geometry, and combinatorics. Each chapter begins with a prologue to establish the historical context and then proceeds to a detailed consideration of one or more Eulerian theorems on the subject at hand. Each chapter concludes with an epilogue surveying subsequent developments or addressing related questions that remain unanswered to this day. At the end of the book is a brief outline of Euler's collected works, the monumental Opera Omnia, whose publication has consumed virtually all of the twentieth century.
In all, the book contains three dozen proofs from this remarkable individual. Yet this is merely the tip of the scholarly iceberg, for Euler produced over 30,000 pages of pure and applied mathematics during his lifetime. Euler: The Master of Us All samples the work of a mathematician whose influence, industry, and ingenuity are of the very highest order.
192 pp., Paperbound
ISBN-0-88385-328-0