Read This!

The MAA Online book review column

Briefly Noted

February 2006

William Playfair (1759-1823) is generally recognized as the originator of now common statistical graphs such as line graphs for time series, pie charts, and bar graphs. But his books have not been generally available, and as Wainer and Spence start their preface: “…William Playfair’s Atlas is like the Bible: an ancient and revered book that is often cited but rarely read.” In addition to the Atlas, they also include Playfair’s The Statistical Breviary; shewing, on a Principle Entirely New, the Resources of Every State and Kingdom in Europe; Illustrated with Stained Copper-plate Charts, Representing the Physical Powers of each distinct Nation with Ease and Perspicuity.

Wainer and Spence have done us a great service by making Playfair’s two books available again. In their valuable 35-page introduction, they place Playfair’s life in the context of the times and also give us an historical account of his life. They discuss his graphs in light of how we today evaluate aspects of statistical graphs, and everyone interested in the theory of statistical graphs should read this introduction.

Most of the Atlas contains presentations and discussions of line graphs for time series, showing imports and exports between England and other European countries. He argues forcefully for the use of graphs instead of tables. The book is delightful reading, and the following quote, arguing for the use of graphs instead of tables, from his Introduction gives a strong flavor of the book (modernizing most of his f’s into s’s):

As the eye is the best judge of proportion, being able to estimate it with more quickness and accuracy than any other of our organs, it follows, that wherever relative quantities are in question, a gradual increase or decrease of any revenue, receipt, or expenditure, of money, or other value, is to be stated, this mode of representing it is peculiarly applicable; it gives a simple, accurate and permanent idea, by giving form and shape to a number of separate ideas, which are otherwise abstract and unconnected. In a numerical table there are as many distinct ideas given, and to be remembered, as there are sums, the order of progression, therefore, of those sums are also to be re-collected by another effort of memory, while this mode unites proportion, progression, and quantity, all under one simple impression of vision, and consequently one act of memory.

Nobody said it better during the next two hundred years.

[Gudmund Iversen; posted to MAA Reviews 1/27/2006]

Classics in Mathematics Education Research is an excellent read for any mathematician transitioning into the field of mathematics education or for someone who is interested in learning more about this field. It is also a first-rate introductory book for graduate students pursuing doctoral degrees in mathematics education. The book is composed of seventeen research articles that reflect and have influenced mathematics education research in the past thirty years.

The authors of the research articles are prominent mathematics education researchers whose works have greatly contributed to the field of mathematics education. Their research displays the importance of learning with understanding and demonstrates the integration of teaching and learning mathematics research. Furthermore, through these researchers the role of quantitative and qualitative studies has been solidified. Many of the articles can be classified into one of the following categories:

• Ethnomathematics: Role of cultural practices in learning mathematics,
  
• Gender issues,
  
• Meta-cognition: Emergence of cognitive perspective for studying student thinking and problem-solving, and
  
• Action Research: The role of teachers conceptions in research in teaching and learning.

The mathematical topics addressed in the articles range from Brownell's article on "The Place of Meaning in the Teaching of Arithmetic" to Tall and Vinner's article entitled "Concept Image and Concept Definition in Mathematics with Particular Reference to Limits and Continuity." There are also articles dealing with the number concept in the middle grades and geometry. Each article is preceded by a "Perspective," introducing the cited work and its relevance to mathematics education research.

Classics in Mathematics Education Research will give you a little taste of what constitutes important and meaningful research in the field of mathematics education! Invest in this book, if you want to learn more about the cultivation of mathematics education research.

[Hortensia Soto-Johnson; posted to MAA Reviews 12/28/2005]

In 1973, Wayne Roberts became Minnesota Coordinator for the American High School Mathematics Examination (AHSME).  He noticed that Minnesota students did not do as well as those from other parts of the country.  During a 1978 sabbatical leave in Massachusetts, he visited a small high school whose students consistently did very well on the AHSME.  The reason, he concluded, was that the Massachusetts students participated in a mathematics league.  On his return to Minnesota, he caused one to be created there and this book surveys its first twenty-five years.

In the Minnesota league, teams of fifteen students have meets, five or so a season, involving around eight schools.  There is a variety of events, with problems on various topics and at various levels of difficulty.  In recent years the number of teams participating has ranged between 160 and 180.  At the end of the year there is a statewide Math Bowl for league winners.

It works.  Minnesota's AHSME scores went up, and its students go on to more and better mathematical accomplishments than they had before the league existed.  The league also has the large advantage of regularly bringing together groups of mathematics students and teachers.

There should be more mathematics leagues.  They take money, though not a lot, and someone like Wayne Roberts to make them go, which may be more difficult to arrange than financing.

The book is a compendium of this and that: pictures, lists of winners, sample problems (quick — what's the largest prime factor of (25!)³ - (24!)³?), how the league has operated, and so on.  The section "Our Guiding Philosophy" deserves to be widely read.  Anyone interested in mathematics competitions will find this book useful.

[Underwood Dudley; posted to MAA Reviews 01/19/2006]

Absolutely, read this one! Making Transcendence Transparent is one of those books that stand out from the crowd because the authors have put a lot of good work into it, and plenty of imagination and creativity. It is witty, funny at times, highly entertaining, very readable and interesting to both the casual and advanced reader.

As the title and subtitle explain, the book is an introduction to transcendental number theory. As anyone who has already studied the subject knows, the theorems in this area of mathematics are rather involved and usually quite obscure at first glance. In Making Transcendence Transparent the authors try (and succeed) to penetrate this "darkness" by building the intuition of the reader and providing clear expositions of the idea of the proof before presenting the actual proof.

The book is partitioned into 9 chapters and an appendix, covering the following topics (among others): the basic theory, mostly definitions, of rational, irrational, transcendental and algebraic numbers; Liouville's theorem and Liouville's numbers, Roth's theorem; polynomial vanishing and the transcendence of e; the Lindemann-Weierstrass theorem; Siegel's lemma; the Gelfond-Schneider theorem; Mahler's classification of transcendental numbers; the Weierstrass P-function and periods and transcendence in function fields.

As mentioned in the first paragraph, the book is beautifully written and very creatively put together (for example, each chapter is named after a well-known real number... except the last chapter, which is named after a well-known transcendental function). The text helps us understand the concepts by building a very strong intuition and also motivates the concepts from a historic point of view. Furthermore, the topics selected are interesting and provide a broad view of the subject. My only objection: not enough problems. However there are suggested problems along the chapters which the authors call challenges, some of them for a good reason. Conclusion: read this one!

[Álvaro Lozano-Robledo; posted to MAA Reviews 01/04/2006]

Publication Data

Classics in Mathematics Education Research, ed. by Thomas P. Carpenter, John A. Dossey, and Julie L. Koehler. National Council of Teachers of Mathematics, 2004. Paperback, 226 pp., $35.95. ISBN 0-87353-565-0.

Minnesota Math League XXV, by A. Wayne Roberts. Beaver's Pond Press, 2005. Paperback, 242 pp., $25.00. ISBN 1-59298-111-9.

Making Transcendence Transparent: An Intuitive Approach to Classical Transcendental Number Theory, by Edward B. Burger and Robert Tubbs. Springer, 2004. Hardcover, 258 pp., $39.95. ISBN 0-387-21444-5.

Gudmund R. Iversen holds a PhD in statistics from Harvard University and is Professor Emeritus of Statistics at Swarthmore College where he taught statistics for many years

Hortensia "Tensia" Soto-Johnson is an Assistant Professor at the University of Northern Colorado, where she teaches elementary and secondary pre-service teachers. She also mentors mathematics education doctoral candidates. In her spare time, Tensia enjoys reading, practicing yoga, and most importantly spending time with her husband Roger and their son Miguel.

Underwood Dudley has retired from DePauw University and is now living in Florida.

Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
• May 1999: Books on math and physics, both popular and technical.
• July 1999: Various kinds of geometry, and classics in applied mathematics.
• August 1999: What's happening, Dirichlet, and Cryptonomicon.
• October 1999: Graph theory sources, John von Neumann, and Noncommutative Geometry.
• November 1999: Number theory in nine chapters, mathematics at work, and Jacques Hadamard.
• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
• March 2000: Cardano and Manifold: Time.
• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.
• August 2000: acrostics, field theory, categories, and partial differential equations.
• September 2000: excursions, history, and some high-level expository writing.
• November 2000: a festival of math videos and the Fourier-analytic proof of quadratic reciprocity.
• December 2000: Analysis problems, mathematical mysteries, and industrial mathematics.
• January 2001: Euler, Kolmogorov, and Honsberger.
• March 2001: history of mathematics, in three flavors: Harriot, prime numbers, and geometrical exactness.
• May 2001: three very different "applied mathematics" books.
• August 2001: introductions to algebraic number theory and to topology, and problem books.
• November 2001: Wilbur Knorr, John H. Conway, and vector analysis.
• February 2002: biological time, celestial mechanics, and things rarely talked about.
• March 2002: history of recent mathematics and differential geometry.
• June 2002: mathematics in other cultures, Archimedes, probabilistic thinking, and Dover's Phoenix.
• August 2002: Euclid's Elements, Hurewicz's Lectures, and Olympiads 2001.
• October 2002: Algebra from A to Z, Problems in Probability, and a really old trigonometry textbook.
• December, 2002: group representation theory, Irish mathematicians, and integration.
• January, 2003: careers, logic, dueling idiots, and differential geometry.
• March, 2003: the Putnam, the Liber Abaci, and the Millennial Conference.
• April, 2003: analysis, reprints, and Rithmomachia.
• May, 2003: conversation starters, dissections, and Fibonacci numbers.
• July, 2003: mathematical circles, the Scottish Café, and the zeta function.
• September, 2003: history — an encyclopedia, historiography, and algebra.
• October, 2003: selected papers and second editions.
• December, 2003: calculus, relativity, and biology.
• February, 2004: cryptography, generating functions, Wallis, and Stirling.
• March, 2004: differential equations, fundamental groups, reprints, and spherical buildings.
• April, 2004: Ramanujan graphs, time, and Medieval Islam.
• May, 2004: History: of Greek mathematics, of mechanics, of tables, of Reverend Bayes.
• June, 2004: The Grothendieck-Serre correspondence, mathematical biology, olympiad problem books, and units.
• July, 2004: Groups, Probability, Olympiads, and Isaac Newton.
• October, 2004: Klein, dogs, rivers, physics, geometry.
• November, 2004: Karl Pearson, chessboard problems, John Wallis, and contemporary art.
• January, 2005: Abel's legacy, reductive groups, curriculum foundations, and irresistible integrals.
• February, 2005: Differential equations, graphs, optimization, algorithms... and arithmetic on function fields of characteristic p.
• March, 2005: William Rowan Hamilton, π, Adolph Zeising, and Jesuits.
• April, 2005: differential equations, abstract algebra, and two kinds of number theory.
• May, 2005: algebra, chance, variations, and Chinese.
• Late May, 2005: motives, historical sources, Lebesgue, Cauchy.
• July, 2005: calculus, control theory, instabilities, biostatistics.
• August, 2005: difference equations, Galois theory, algebra, martingales in finance.
• October, 2005: 4-manifolds, geometry, Turing, and statistics in sports.
• December, 2005: lattices, irrational numbers, topology, and linear models.
• January, 2006: algebra, geometry, bubbles, jaws, Greek mathematics, and quotations.

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.