Magnificent Mistakes in Mathematics

by Alfred S. Posamentier, Ignmar Lehmann

Two veteran math educators demonstrate how some "magnificent mistakes" had profound consequences for our understanding of mathematics' key concepts. In the nineteenth century, English mathematician William Shanks spent fifteen years calculating the value of pi, setting a record for the number of decimal places. Later, his calculation was reproduced using large wooden numerals to decorate the cupola of a hall in the Palais de la Decouverte in Paris. However, in 1946, with the aid of a mechanical desk calculator that ran for seventy hours, it was discovered that there was a mistake in the 528th decimal place. Today, supercomputers have determined the value of pi to trillions of decimal places. This is just one of the amusing and intriguing stories about mistakes in mathematics in this layperson's guide to mathematical principles. In another example, the authors show that when we "prove" that every triangle is isosceles, we are violating a concept not even known to Euclid - that of "betweenness." And if we disregard the time-honored Pythagorean theorem, this is a misuse of the concept of infinity. Even using correct procedures can sometimes lead to absurd - but enlightening - results.Requiring no more than high-school-level math competency, this playful excursion through the nuances of math will give you a better grasp of this fundamental, all-important science.

Genres: Nonfiction, Science, Mathematics

308 pages, Hardcover

First published January 1, 2013

About the author

Alfred S. Posamentier (born October 18, 1942) is among the most prominent American educators in the country and is a lead commentator on American math and science education, regularly contributing to The New York Times and other news publications. He has created original math and science curricula, emphasized the need for increased math and science funding, promulgated criteria by which to select math and science educators, advocated the importance of involving parents in K-12 math and science education, and provided myriad curricular solutions for teaching critical thinking in math.

Dr. Posamentier was a member of the New York State Education Commissioner’s Blue Ribbon Panel on the Math-A Regents Exams. He served on the Commissioner’s Mathematics Standards Committee, which redefined the Standards for New York State. And he currently serves on the New York City schools’ Chancellor’s Math Advisory Panel.

Posamentier earned a Ph.D. in mathematics education from Fordham University (1973), a Master’s degree in mathematics education from the City College of the City University of New York (1966) and an A.B. degree in mathematics from Hunter College of the City University of New York.

Posamentier was born in Manhattan in New York City, the son of Austrian immigrants. He has one daughter (Lisa, born in 1970), and one son (David, born in 1978). He resides in River Vale, New Jersey and is the current Dean of the School of Education and professor of mathematics education at Mercy College, New York. He was formerly professor of mathematics education and dean of the School of Education at The City College of the City University of New York, where he spent the previous 40 years.

Reviews

[Triscia Katerina · Rating 2 out of 5]

First chapter was pretty interesting, afterwards it got pretty painful to read. There wasn’t really anything insightful about this book. Honestly quite disappointed as the first chapter set a great tone.

[Deborah Cleaves · Rating 5 out of 5]

Challenging, fascinating.

[Ed · Rating 2 out of 5]

This book was a disappointment that did not live up to the advance hype. The hype is exemplified by the first paragraph of the blurb, which ends "Veteran math educators demonstrate how some 'magnificent mistakes' had profound consequences for our understanding of mathematics' key concepts." I did not find any of the mistakes leading to any worthwhile consequences. The Library Journal Review quotes Harold D. Shane, Mathematics, Emeritus, Baruch Coll., CUNY "Many of the errors are clever and instructive, some are repetitive and uninteresting; only a few should be designated as "magnificent." VERDICT This volume could be useful to a teacher of mathematics as a source of examples that can hammer home important concepts. Beyond that, it will have a limited readership." The following summary of the book is also from the Shane review.

510 Mathematics - a compendium of mathematical errors, arranged by topic into five chapters. Only the first, on errors made by eminent mathematicians, has much narrative. Here, we see number-theoretic conjectures that were based upon a few examples and a lot of wishful thinking. In Chapter Two, the authors address errors in arithmetic based mostly upon misunderstandings of notations. Chapter Three deals with algebraic errors primarily caused by inadvertent division by zero or the introduction of extraneous roots. In Chapter Four we have errors in geometric proofs, mostly arising from deliberately misleading diagrams. The final chapter deals with probability, showing mistakes made by confusing the concepts of conditionality, independence, and mutual exclusivity of events.

[Adrian Goh · Rating 1 out of 5]

Gave up reading it. It basically lists out all the possible mistakes a student can make in the classroom, including some mistakes that they are not likely to make. While there were some interesting historical facts about great mathematicians and their mistakes, the rest of the book covered mistakes that were nowhere near to magnificent.

[Mills College Library]

510 P8552 2013

[Q · Rating 2 out of 5]

It was generally confusing, but I did find a few interesting "mistakes", such as the Monty Hall problem and the Coin Flipping Trick.

[Wyl Young · Rating 1 out of 5]

Reading this is probably MY magnificent mistake in math, found it to be a tedious mediocrity.