Books for Middle School Students
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Euclid: The Thirteen Books of The Elements Translated with introduction and commentary by Sir Thomas L. Heath
- "If your personal library could only have one book, this is the one. This is a translation in three volumes of the 2300-year-old Greek monument, a compilation by Euclid of the mathematical knowledge known to his day and containing his own contributions. Elements set mathematics in its course - the notion of axioms and proof and the logical ordering of theorems drawing upon prior ones and axioms to prove the new one. Elements will be still printing in some form long after the Parthenon has whithered away and much of today's books forgotten, for even the blemishes in this work - especially the tacit assumptions on which a few proof steps are based - helped to make mathematics the more perfect discipline that it is today. While Euclidean Geometry was placed on sounder axiomatic foundation in the nineteenth century, the small number of axioms chosen by Euclid serves not only as a manageable set for the beginner to prove the theorems, but also to illustrate the need to add more axioms and to learn the need for rigor in proof more than would be if the modern set of axioms were used from the outset. This is the greatest elementary textbook in mathematics that the beginner must take the first steps from, after learning to add, multiply and divide integers and fractions. Note that the book uses the word "axiom" for what is called "Common Notions,' and the real axioms in the modern sense are its five postulates. The young reader may advantageously begin the book right in the middle at Postulate 1, skim the commentaries and sail on to Proposition 1 and then go slowly through the proofs and the commentaries on the proofs. You might want to continue at least until Proposition 28. Skip to Proposition 47(Pythagoras' Theorem), take in Proposition 48 (the converse of Pythagoras' Theorem) which is the end of the first book of the Elements, and then go back to the beginning of the book and take in those commentaries that interest you until you reach Postulate 1. Thirteen books comprised the Elements, which this translation groups in three volumes. The first volume, containing the precious introduction by Sir Thomas Heath, and the books I and II of the Elements, would suffice for the beginning steps of a mathematical education. The advanced student who is passionate about mathematics may go on to Eudoxus' theory of irrationals (Book V) and Method of Exhaustion (Book XII), which are in volumes 2 and 3, respectively, of the present translation. Gems scattered through the volumes include the proof of the infinitude of primes, the irrationality of the square root of 2, the characterization of even perfect numbers and the description of regular polyhedra." Review by G. Rubin Thomas
"At the age of eleven, I began Euclid. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world."
- Bertrand Russell, Autobiography
Good and Evil by Richard Taylor - Prometheus Books (1999)
"Although Man is the crown of creation, it seems that by the 21st century A.D. he is still mostly an automaton that is so busy with the maintenance and well-being of itself and others of its kind that it gives little time to look at the big picture. And those few who have considered the questions over the millennia differ in their assessment or interpretation of the picture. The picture is a composite of the answers to questions the most important of which are these: What is good? What is good for man? What is a good life? Students in their early years do not think long on these questions. Yet, pondering these are as important as or more important than the pursuit of a high score in national math contests. In fact, having at least an understanding of the seriousness of these questions will make your life better no matter who you are or what you have achieved. Most people “feel” they have the answers to these questions. They don’t! The first part of this work is a brief history of the quest for the answers to these questions of ethics. The second part answers the big questions by combining the views of Protagoras and William James on morals and the good. This work is not pulp fiction, nor the latest hot seller, but a manual of living for the thoughtful person. It is the core of moral ethics laid out before you in clear and simple language and devoid of circular reasoning. The major reward in studying this work is a new objectivity in your interaction with fellow beings. A not too minor reward is a confident familiarity with the whole of western moral philosophy from its origins near the times of the Seven Sages of Greece all the way to the 20th century." Review by G. Rubin Thomas
The Republic by Plato - Translation by C.D.C. Reeves, Hackett Pulbishing Co. (2004)
- "Here is the iterative dialectic method of Socrates for attaining knowledge. Plato’s Republic is an essay on justice and it portrays the just man or the just state as nature or true morality would have them.
This is one of the most influential books in the world. Alfred North Whitehead, the great logician and mathematician, has said that the history of Western Philosophy is a footnote to Plato.
A glimpse of this is Christianity's substitution of what is in accordance with god for the Greek idea of what is good according to nature as explained in Plato's work .
This is Plato's most famous work and is said to be the basis of all Western Philosophy." Review by G. Rubin Thomas
The Art of Problem Solving by
Richard Rusczyk and Sandor Lehoczky - Two volumes: Vol. I ISBN 1-885875-01-0 Vol II ISBN 1-885875-03-7
"The relaxed conversational style of these two volumes employs a holistic approach to the teaching and learning of mathematics. The lively, happy, and caring tone, coupled with explanation of ideas that might be hidden from full view, rigor of treatment, and comprehensive coverage of all the basic topics make these volumes, perhaps, the best non-calculus school mathematics text books. The volumes are ideal for both schools and home-schooling. Those schools offering semester or year-long math strands in subjects such as Number Theory, Combinatorics, Graph Theory, or Probability could either omit the relevant chapters from the second volume or incorporate them in the course materials. The work is most suitable for students preparing for national math contests. Middleschool students who are profoundly gifted in mathematics should find it fun to self-study the books in a leisurely way over a period of a couple of years starting as early as their fifth grade. The work is valuable also for parents who are home-schooling their gifted children. Parents would find these volumes suitable teaching aids and almost a complete library for school math concepts including Limits." Review by Prof. George Rubin Thomas
The Parrot's Theorem by Denis Guedj - Translated from the French by Frank Wynne
- "This fantasy and adventure story possessing also literary beauty is really a mathematical novel providing an entertaining introduction to mathematical concepts appropriate for gifted students of middle school age." Review by G. Rubin Thomas
The Number Devil by Hans Magnus Enzenberger
- "This book consists of a loosely connected series of stories about a boy named Robert, an indifferent math
student who finds himself sharing his dreams with a "number devil." The number devil guides Robert through
some very deep mathematical terrain, exposing him to results that most people don't learn how to prove until
their second year in graduate school. The tour is lively and full of wonders, but it is very
short on explanations." Review by Dr. Laurens Gunnarsen
The Man Who Counted
by Malba Tahan
- "This book shares much with THE NUMBER DEVIL, consisting as it does of a similar series of loosely connected
stories about a single character, Beremiz Samir, The Man Who Counted. The level of the mathematical content
in this book is rather less elevated than in THE NUMBER DEVIL, but the story has an undeniable exotic appeal:
it purports to chronicle the life of a self-taught itinerant mathematician in 14th-century Iraq, and the
artful narrative echoes with the names, places and customs of that far-off land and age." Review provided by Dr. Laurens Gunnarsen
The Joy of Mathematics by Theoni Pappas
- "Unlike the first two titles above, it consists of many very brief entries, each presenting some curious little
mathematical fact, and most of these naturally suggest some sort of activity: paper-folding, drawing, cutting
and pasting, and so forth. There is no plot, no story, and no characters -- only
math in quick, small, surprising doses that is still not heavy
reading." Review by Dr. Laurens Gunnarsen
Mathematics of Choice: How to count without counting by Ivan Nivan
- "How many ways can six people be seated around a round table? Problems like this and more complex ones require a confident familiarity with permutations and combinations. Niven explains these and related concepts in a most lucid manner.
Treating also the generating polynomials, the Pigeon Hole principle, the Inclusion-Exclusion principle and the Principle of Mathematical Induction and other important concepts, this work is perhaps the classic text for beginners in combinatorial mathematics. Students who intend to write mathematical contests should have this in their personal library." Review by G. Rubin Thomas
A Mathematical Mosaic by Ravi Vakil
- "This book conveys the beauty and excitement of mathematics and contains many of the important elementary results that the future mathematician should master now. The proofs, the gentle nudging with appropriate problems, the historical notes and the short essays on the lives of eminent young mathematical contemporaries of the author make this one of the best books for the gifted middle school and high school student.
" Review by G. Rubin Thomas
What is Mathematics: An Elementary Approach to Ideas and Methods by Richard Courant and Herbert Robbins, Revised by Ian Stewart
- "A new edition of the 1941 classic portrait of the mathematical world. Offers insights into the fields of number theory, geometry, topology, functions and limits, and calculus. A chapter on recent developments includes Fermat's Last Theorem, the Four-Color Theorem, and more. Accessible to those with knowledge of basic high school mathematics..
" Review by Dr. Kathleen Zehender
Journey through Genius by William Dunham
- "This book offers a tour of twelve great theorems of mathematics, starting with the work of Hippocrates in 440 BC and proceeding through the centuries to the work of Georg Cantor in 1891. Each theorem is placed in historical and biographical context. The proofs can be understood by anyone with knowledge of basic high school mathematics." Review by Dr. Kathleen Zehender
Maths Challenge, Volumes 1 to 3, edited by Tony Gardiner, publisher: Oxford University Press
- "Each volume offers twenty or more challenges "which provide stimulating questions to help students think more deeply about basic mathematical ideas." A sampling of chapter topics from the last volume includes Pythagorean triples, divisibility rules for prime factors, the pigeonhole principle, and a history of pi." Review by Dr. Kathleen Zehender
The Moscow Mathematical Puzzles by Boris Kordemsky, edited by Martin Gardner (Dover)
- "A popular book of recreational mathematics from Russia." Review by Dr. Kathleen Zehender
A Mathematician's Apology by G.H. Hardy
- "Probably the greatest essay about mathematics and the practioners of that art by a leading mathematician from the first half of the 20th century and the discoverer and mentor of the Indian genius, Ramanujan." Review by G. Rubin Thomas
Littlewood's Miscellany Edited by Bela Bollabas
- "The music of mathematics fills the pages of this collection of musings by a great mathematician from the first half of the 20th century. Profoundly gifted middle and high school students should read the first two pages of the section "Mathematics with minimum raw material" and the last section "The mathematician's art of work." Review by G. Rubin Thomas
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Last updated - March 5, 2007