Noel Dietz

Math G

MW 5:00 P.M.

TEXT BOOKREVIEW

            Ihave chosen to review the History of Mathematics: An Introduction, by DavidM. Burton, for this assignment. Burton gives a full account of how mathematics has developed over thepast five thousand years.  Hisnarrative is basically a chronicle beginning with the origin of mathematics inthe great civilizations of antiquity and progressing through the first decadesof the twentieth century.

            Atremendous amount of detail has been provided regarding the lives of the peopleresponsible for the early development of mathematics and it is really ahistorian’s bonanza.  It isquite obvious that the intellectual life of the mathematicians featured in thisbook towered over most of their contemporaries.  The book furnishes us with assorted problems of varyingdegrees of difficulty and these problems usually typify a particular historicalperiod requiring the procedures of that period to complete the answers.  They appear to be an integral part ofthe text and, as one works with them, they learn some very interestingmathematics as well as some wonderful history.

            Burtonhas designed the book for college juniors and seniors and, as a result, itappears to be somewhat of a stretch for me compared to our regular text, MathematicalIdeas.  Even though some areas of itare a little beyond our current level of comprehension, it does give us a greatpreview of what we can expect if we choose to continue to the next level ofmathematics.

            Ireally enjoyed the chapters on Pythagoras, Descartes and Newton and I believethey really reinforced the historical studies that I have already made on thesemathematical giants.  The completehistorical backgrounds of these intellectuals and others provide a perfectexample of the difference between Burton’s book and Mathematical Ideas.  Our current text gives us more moderntechnical classroom support which I believe is what we most need at thisparticular period in our academic careers.  There are smaller references to past major mathematicalpersonalities, but most of the chapters are devoted to the pure principles andstandards for our level of understanding. Another distinguishing feature is the treatment and emphasis on problemsolving, which is used throughout our text.  Problem solving not only encourages us to think about howmathematics can be used, it also helps to prepare us for more advanced materialin our future courses.

            Thefirst half of the History of Mathematics is primarily devoted tospecific individual developers of theories and although a lot of these theoriesare in use today, the applications cited for them do not apply to today’sdemanding standards.  Burton waitsuntil Chapter 8 to feature Descartes and Newton with the “Dawn of ModernMathematics.”  Our textbegins on Chapter 1 with the Art of Problem Solving By Inductive Reasoning, soI am much more comfortable with our text.

            However,in Burton’s book, the theory of sets is explained in 51 pages utilizingsuch great minds as Georg Cantor, Leopold Kronecker, Gotlobb Frege and DavidGilbert.  Clear definitions,understandable theorems and specific proof for each theorem are used throughoutthis section of his book.  Thedetail supplied for this one area, appears to be much more clearly presentedand 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 ofuseful applications and no major effort is made to explain linear inequalities,polynomials and factoring.  Chapter7 in our text covers those subjects perfectly and completely and furtherexemplifies the need for using modern technology and updated problems andexercises.  Mathematical Ideas has updatedexercises that focus on real life data that make the work interesting and morerelevant.  I believe that I may bethe only history major in our class, so I really enjoyed the historicalapproach used by Burton to put early mathematics and mathematicians in properperspective.  We have certainlylearned this semester that most of our modern day mathematics evolved from themagnificent minds of ancient scholars. Burton confirms this in one of his early chapters regarding BabylonianMathematics: “When converted to modern algebraic notation, the Babylonianinstructions amount to formulas equivalent to today’s modernrules.”

            Iwould choose the History of Mathematics: An Introduction, as aperfect companion to our text, Mathematical Ideas, butcertainly not as a replacement.  Ibelieve that it is essential to learn the historical significance of a subjectalong with its more modern reasoning and importance.  It is also critical that we both learn and retain modernmathematical concepts.  Thisrequires that we learn critical thinking skills; to reason mathematically, tocommunicate mathematically, and to identify and solve mathematicalproblems.  This can best beachieved with modern text books focusing on interesting and appropriateapplications of mathematics to help motivate the student.  Our present text book is a wonderfulexample of all of the above.  Iwill keep it always, not only as a valuable reference, but as a fine memento ofone of the most interesting and rewarding classes of my college career.