Noel Dietz

Math G

MW 5:00 P.M.

 

TEXT BOOK REVIEW

            I have chosen to review the History of Mathematics: An Introduction, by David M. Burton, for this assignment.  Burton gives a full account of how mathematics has developed over the past five thousand years.  His narrative is basically a chronicle beginning with the origin of mathematics in the great civilizations of antiquity and progressing through the first decades of the twentieth century.

            A tremendous amount of detail has been provided regarding the lives of the people responsible for the early development of mathematics and it is really a historian’s bonanza.  It is quite obvious that the intellectual life of the mathematicians featured in this book towered over most of their contemporaries.  The book furnishes us with assorted problems of varying degrees of difficulty and these problems usually typify a particular historical period requiring the procedures of that period to complete the answers.  They appear to be an integral part of the text and, as one works with them, they learn some very interesting mathematics as well as some wonderful history.

            Burton has designed the book for college juniors and seniors and, as a result, it appears to be somewhat of a stretch for me compared to our regular text, Mathematical Ideas.  Even though some areas of it are a little beyond our current level of comprehension, it does give us a great preview of what we can expect if we choose to continue to the next level of mathematics.

            I really enjoyed the chapters on Pythagoras, Descartes and Newton and I believe they really reinforced the historical studies that I have already made on these mathematical giants.  The complete historical backgrounds of these intellectuals and others provide a perfect example of the difference between Burton’s book and Mathematical Ideas.  Our current text gives us more modern technical classroom support which I believe is what we most need at this particular period in our academic careers.  There are smaller references to past major mathematical personalities, but most of the chapters are devoted to the pure principles and standards for our level of understanding.  Another distinguishing feature is the treatment and emphasis on problem solving, which is used throughout our text.  Problem solving not only encourages us to think about how mathematics can be used, it also helps to prepare us for more advanced material in our future courses.

            The first half of the History of Mathematics is primarily devoted to specific individual developers of theories and although a lot of these theories are in use today, the applications cited for them do not apply to today’s demanding standards.  Burton waits until Chapter 8 to feature Descartes and Newton with the “Dawn of Modern Mathematics.”  Our text begins on Chapter 1 with the Art of Problem Solving By Inductive Reasoning, so I am much more comfortable with our text.

            However, in Burton’s book, the theory of sets is explained in 51 pages utilizing such great minds as Georg Cantor, Leopold Kronecker, Gotlobb Frege and David Gilbert.  Clear definitions, understandable theorems and specific proof for each theorem are used throughout this section of his book.  The detail supplied for this one area, appears to be much more clearly presented and perceptibly illustrated than the same material in our current text book.

            Conversely, Burton describes the basic concepts of algebra, including linear equations, in “Mathematics in Early Civilizations,” with no real illustrations of useful applications and no major effort is made to explain linear inequalities, polynomials and factoring.  Chapter 7 in our text covers those subjects perfectly and completely and further exemplifies the need for using modern technology and updated problems and exercises.  Mathematical Ideas has updated exercises that focus on real life data that make the work interesting and more relevant.  I believe that I may be the only history major in our class, so I really enjoyed the historical approach used by Burton to put early mathematics and mathematicians in proper perspective.  We have certainly learned this semester that most of our modern day mathematics evolved from the magnificent minds of ancient scholars.  Burton confirms this in one of his early chapters regarding Babylonian Mathematics: “When converted to modern algebraic notation, the Babylonian instructions amount to formulas equivalent to today’s modern rules.”

            I would choose the History of Mathematics: An Introduction, as a perfect companion to our text, Mathematical Ideas, but certainly not as a replacement.  I believe that it is essential to learn the historical significance of a subject along with its more modern reasoning and importance.  It is also critical that we both learn and retain modern mathematical concepts.  This requires that we learn critical thinking skills; to reason mathematically, to communicate mathematically, and to identify and solve mathematical problems.  This can best be achieved with modern text books focusing on interesting and appropriate applications of mathematics to help motivate the student.  Our present text book is a wonderful example of all of the above.  I will keep it always, not only as a valuable reference, but as a fine memento of one of the most interesting and rewarding classes of my college career.