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This paper was written as an assignment for Ian Walton's Math G -Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.

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This paper was submitted by Britney Maracchini in Fall 2001 Math G at Mission College.

If you use material from this paper, please acknowledge it.

Britney Maracchini

December 5, 2001

Math G

M/W 5-7pm

Textbook Review Assignment

I chose #5, Burton’s History of Math , to compare to our current textbook, Mathematical Ideas by Miller, Heeren and Hornsby. While there was considerable overlap in subject material between the two books, the entire approach was very different.

Beginning with the Table of Contents of the books, it is easy to see that much of the same material is covered. Burton, for example, begins by covering the early number systems and symbols, arithmetic and geometry of the early Egyptians, and the first and second Alexandrian School.

Beginning in the sixth chapter, Burton takes his readers through the First Awakening, 14th and 15^th century Europe, the Dawn of modern math and Probability Theory. Finally, Burton deals with the Renaissance (Fermat, Euler, and Gauss), non-Euclidean Geometry and Set Theory. His approach, then, appears to be from a geographical and historical perspective. Burton goes to great length to give a detailed biography of all those associated with early development of mathematics and the cultural activities surrounding them at the time.

Miller et al. also began with the Egyptian and Babylonian cultures’ development of mathematics, but instead of approaching it chronologically and geographically, represented it from the standpoint that these cultures practiced inductive reasoning. After exploring inductive and deductive reasoning, strategies for problem solving are introduced. While Miller et. al., too, present the great mathematicians of the time, they spend far less time in intricate detail. Instead, they allow a side bar blurb to credit the minds behinds the ideas.

After the overall concept of problem solving is discussed, Miller et. al. then go back and pick up discussion of symbols and terminology before approaching logic. Only then do they feel comfortable that a strong enough foundation has been laid for them to go back in time and discuss numeration and mathematical systems.

Miller et. al. continue to discuss many of the same concepts as Burton did, such as probability, Euclidean and Non-Euclidean geometry, infinity and cardinality. However, the overall strategy in presenting materials appears to be building in concept and complexity rather than building in time.

It’s difficult to say whether Burton presented a better or worse topic list for our class because so much of it was the same subject matter. However, because of Miller et.al. approached the material in a building context, I think our current textbook is more relevant to contemporary topics.

Of all the topics discussed in both books, I think Burton did a better job of explaining basic set theory than did Miller et. al. He shows finite sets and several theorems associated with finite sets and unions. He then introduces infinite sets and graphically displays arrays in such a way that the concepts are crystal clear. He finishes off the discussion with cardinality.

Miller et. al., on the other hand, seem to assume the reader is already familiar with basic set theory. They offer a one page explanation of the symbols and terminology, followed by definitions and 6 examples. Little time is allowed to absorb and apply the concepts introduced.

Probability, however, was addressed much better in our current textbook than in Burton’s book. Burton writes about everyone even remotely involved in the origins of probability, including Pascal, Fermat, Gravat, Cardan, and Fra Luca. He portrays their lives and times and even presents some of their wrong thinking that eventually lead to probability theory. By the time Burton introduces probability theory as it is known today, I was totally confused.

Miller et. al. instead present a brief overview of the concept of probability. They then address its basic concepts, followed by 11 real world examples. By the time I had completed the 11 examples, I had a solid base from which to approach the concepts associated with properties of probability.

I would definitely not chose Burton’s book over our current text. This is an excellent book if you are interested in teaching and learning about the history of mathematics and the mathematicians behind the theories. To teach and learn mathematical concept, Burton’s book is too confusing in language and history. He writes in the genre of the mathematicians’ time, so words appear to be misspelled and misleading. Burton’s thrust appears to be with the mathematicians, not the math. By the time he has completed his account of the mathematician, the concept is lost. If the goal of this class is to teach and have us learn mathematical applications, our current text is far superior.