Book Reviews - March 2008 - Volume 108 (3)

How The Other Half Thinks: Adventures In Mathematical Reasoning

Author: Sherman Stein
McGraw-Hill, 2 Penn Plaza, New York, New York 10121
2001; 177 pages, Hardback $18.95

Reviewer: Daniel J. Schneck
Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219

So, do you think “a,b” and “c” are just the first three letters of the English alphabet? Think again! As the author of this book shows, “a's,” “b's,” and “c's,” sequentially strung together in various combinations and permutations, can be analyzed quite effectively to illustrate fundamental principles of mathematical reasoning - for example, how to deal with: (a) probability and statistics (Geometric probability in Chapter 1, “The Needle and the Noodle”; the limit process of Calculus in Chapters 2, “Win by Two,” and 4, “Slumps and Streaks”; and Topology in Chapter 3, “The Complete Triangle”); (b) combinatorics (Chapters 5, “Thrifty Strings,” and 8, “Twins”); (c) principles of reflection (Chapter 6, “Counting Ballots”); and (d) set theory (Chapter 7, “Infinity”).

The basis for the approach used to show “How The Other Half Thinks” is established right up front (in the Preface), where the point is made that, “Occasionally, in some difficult musical compositions there are beautiful, but easy parts - parts so simple a beginner could play them. So it is with mathematics as well.” That the author succeeds in living up to this approach is why, in my opinion, this delightful book belongs in the library of anyone who either deals professionally with, or just has a casual interest in the process of getting logically and, for the most part, intuitively, from point A to point B.

In eight well-written chapters, sandwiched in-between a Preface, Acknowledgments, an Epilogue, two Appendixes (Triangles, and Twins: A Supplement), Suggestions For Further Reading, and an Index, the author develops formulations that do, indeed, appeal to the inherent, intuitive attributes of the reader, to enable him or her to solve various mathematical problems. The problems all have in common systematic, logical approaches to pattern recognition ... and ... from such recognition, how one proceeds to draw conclusions that: (a) follow directly from the reasoning that led up to them (to avoid the trappings of non sequiturs); (b) are consistent with a given set of initial conditions (so that we all start reasoning from the same point); (c) satisfy constraints imposed on the a-b-c-string patterns (to avoid “changing horses in midstream”); and, (d) reasonably explain whatever additional observations can be made from these patterns (for example, extrapolations derived therefrom).

Throughout the book, the author relies on common sense and simple logic to analyze experimental results which, by inductive reasoning (i.e., going from the specific to the general) allows one to formulate theories to explain the experimental findings. The reader is then constantly asked to search for patterns that allow one to progress - from inductively reasoning his or her way to general principles - to deductively forecasting specific results, and extracting complexity from simplicity. In each case examined, the author makes the point (page 97) that, “we depended on just three important tools in the mathematical style of thinking: common sense, careful attention to details, and a willingness to be flexible.” Moreover (page 94), “The deeper we went into the original problem ... the more general our thinking became. Clutter gave way to clarity. The concrete was replaced by the abstract, and the abstractions turned out to be extremely useful.” [See, also, a related article that expands upon this concept of extracting essence from details: Schneck, Daniel J., “All God's Creations Got Rhythm!” American Laboratory, 32(#20):6-8; October, 2000].

Another of the many positive features of this book is the ample use of meaningful illustrations and diagrams to make a point. For example, the effective use of geometric equivalents to provide visual interpretations of analytical reasoning, animates that reasoning in the classic “picture-is-worth-a-thousand-words” sense. Furthermore, illustrations greatly help in discussing how to deal with permutations and combinations that are required to satisfy certain specific constraints - constraints such as: (a) forcing certain variables to be quantified according to specific criteria; (b) requiring that certain sequential combinations of variables can only appear once in any given string of a finite number of defined parameters (Chapter 5, “Thrifty Strings”); and/or, (c) insisting that a particular, well-defined relationship among certain variables - for example, that at all times under consideration, one variable is to be consistently larger than another one - must always hold (Chapter 6, “Counting Ballots”).

By far, the most challenging Chapter, even requiring a separate Appendix (B) for further elaboration, is Chapter 8, “Twins.” For me, it required several readings and a great deal of patient, rigorous contemplation. I am not registering this as a complaint, for there is very little that I did not like about this book. I do feel, however, that the purpose of the Epilogue would have been better served if it came at the beginning of the book, rather than at the end. This equivalent of a Coda (from the Latin, cauda, meaning, “tail”) - to again use a musical analogue - is a very nice summary of all of the major topics covered. Like the Coda of a musical composition, the Epilogue is a final passage, introduced after the completion of the essential parts of the work, which gives it a definite and satisfactory ending. However, in this case, I felt that the summary provided in the Epilogue - had the reader been alerted to it much earlier - would have given him or her an advanced sense of, and better “feel” for intent, as well as content. Speaking personally, it would have helped to put the chapters in proper perspective before, rather than after the fact, as I sequenced my way through the material. Also, although the author does cite a few real-life examples of how the principles discussed are actually applied, I would have liked to see some more practical applications of the lines of reasoning developed.

Be that as it may, I highly recommend this book to science and/or mathematics teachers because of its strong and effective emphasis on basic principles of logical reasoning, and because it lives up to its billing. We need much more of that in the classroom.

The Chemical Elements: A Historical Perspective

Author: Andrew Ede
Greenwood Heinemann, P.O. Box 6926, Portsmouth, NT 03802-6926
or Greenwood Publishing Group, P.O. Box 7247-7011, Philadelphia, PA 19170
2006; 183 pages, Hardback $65.00

Reviewer: Dorothy L. Gabel
Indiana University, Bloomington, IN 47408

This excellent book on the historic perspective of the chemical elements provides information not only on the history of the chemical elements, but also contains very useful appendices that should be of interest to chemistry students and their instructors at the high school through college levels. The book would also be useful for elementary teachers and middle school teachers who should have taken a chemistry or physical science course in their preparation for teaching science.

The four page “Introduction” gives an excellent overview of the historical development of chemistry and sets the tone for the ten chapters that follow. These are: Ancient Matter Theory, Greek Matter Theory from Thales to Aristotle Matter Theory in China, the Middle East, and India, Alchemy and the Concept of Elements, Corpuscles and Atoms, Bringing Order to Chaos, Making Elements Elemental, Seeking Order; The Periodic Table, The Atomic Elements, and The Elements Beyond 92.

This book provides a historical perspective of the history of chemistry from the viewpoint of the discovery of the elements from The Stone Age 2.5 to 2.6 million years ago to the present time with the creation of new radioactive elements through the bombardment of known elements with neutrons or other nuclear particles.

One of the nice features of the book is that every chapter contains interesting diagrams of reactions, tables, photographs, or sketches of equipment used by scientists over the ages. The book is divided into ten chapters plus a conclusion. This is followed by Appendix 1, an updated Periodic Table, Appendix 2 that contains the History of the Elements, Appendix 3, The Elements listed by Alphabetical order, their names, and the earliest date they were discovered, and Appendix 4, the Elements Listed by Date of Discovery.

Appendix 2, History of the Elements contains information that would be very useful to most chemistry teachers. It includes a section on Elements Known in Antiquity (Carbon, Gold, Silver, Copper, Sulfur, Tin, Lead, Mercury and Iron). For each of these elements, the symbol, atomic number, atomic weight, melting and boiling points, and atomic radius is given. In addition, a paragraph about the Latin name, mining or smelting of the element, and uses such as diamonds being made of carbon is included for the above eight elements. Another set of elements is included in a section called Dated Discoveries (In addition to the same information given in the previous set, the date of discovery, name of person who made the discovery, and uses of the element are given. This includes all of the remaining elements.

Also included in the book is a Glossary that contains words related to chemistry that may be unfamiliar to students, an excellent Annotated Bibliography that students interested in this topic might pursue, and a general bibliography on the history of chemistry and related topics. It also contains a comprehensive index.

Application for Teaching

Chemistry teachers might have each student in the class adopt a different element and become an expert on that element's composition in terms of particles making up the atoms, chemical reactions the element undergoes, and its physical appearance. At the end of the semester they could make a presentation to the class how the atoms that are members of the same family (vertical row on the periodic table) are alike or differ from one another in chemical and physical properties!

Recommendation

I highly recommend The Chemical Element as a resource for all high school chemistry teachers, and also to be included in secondary school libraries.

It is the best book on this topic of which I am aware.