I chose to review The Nature Of Mathematics by Karl J. Smith for this assignment, which is to compare and contrast our current Math G textbook, Mathematical Ideas, with a similar text. I chose to review The Nature of Mathematics because it seemed the title hinted that the book might reveal math’s mysteries and it’s progression over time, in a more descriptive fashion. Similar to the way taking an Anatomy class clears up some of the mysteries of how the human body works, it’s nature of being. One would hope that all math books would offer the same teachings, but beyond the teachings, math books should provide an interesting array of samplings that will hold the attention of the reader. If the text is able to do this the reader will be able to more readily retain the jest of the text, which ultimately is to learn mathematics. A text needs to spike interest and captivate its reader. A text needs to provide the instructor with a basis for his/her teachings, and the students with a reference in which to guide them, above and beyond the classroom. Let us compare and contrast the strengths and weaknesses of these two texts.

As I began to compare and contrast the two texts, I noticed a basic difference, Smith’s book was smaller in size and did not seem as heavy as Mathematical Ideas. As I eagerly compared the topics between the two textbooks I became aware of the fact that Smith’s text had fewer chapters, eleven to be exact, versus our current texts fourteen. Not only did Smith’s text have a smaller topic list but it seemed to be missing a few of the topics our current text covers. For instance, Smith’s text did not include the majority of logic topics that Mathematical Ideas has. Furthermore, our current text devotes two separate chapters to Sets and Logic, while The Nature of Mathematics combines the two topics into one chapter. Smith’s text has a chapter dedicated solely to the nature of computers. Within the chapter, there are various topics, such as microcomputer applications, hardware, software, and an explanation on a couple of computer programs. In contrast, Mathematical Ideas has a few sections devoted to computer mathematics, focusing on computer application base systems, specifically the binary, octal, and hexadecimal systems. Similarly, The Nature of Mathematics combines three chapters that Mathematical Ideas goes over in more detail. In chapter four of Smith’s text, he includes early numeration systems, prime numbers, natural numbers, rational numbers, as well as real numbers. Our current text goes over historical numeration systems and clock arithmetic in chapter four, then teaches prime and composite numbers in chapter five, and goes over real numbers and rational numbers in chapter six.

Based on the above information, I feel that our current text, Mathematical Ideas, offers a better topic list for our class. Now we will explore the areas of each text that offers the best explanation of various individual topics.

The area that I felt The Nature of Mathematics explained better than our current text is Probability. The chapter starts with a clear and simple definition of the probabilistic model. There is also an interesting historical note on the actual origin of the theory of probability. The text states that the theory stemmed from a gentleman in France (in the 17^th century) who wanted to know the probability of winning at gambling so he could adjust his stakes. I guess times have not changed too much as far as gambling is concerned, even though hundreds of years have past. Our current text also goes into quite a bit of detail about the origin of this theory, which includes a quote by Pascal, one of the phenomenal men who helped the gambler figure out his problem. Unfortunately, I found Pascal’s quote to be a bit distracting because I had to read it a couple of times to be able to completely understand it. Interestingly enough, both textbooks use the coin toss question as their first example of probability. One of the first "rules" I came upon in Smith’s textbook was "Relative Frequency", where the experiment is repeated, let’s say, "x" times, and an event occurs "y" times, then "y/x" is called the relative frequency of the event. This rule is used for exactly as is stated above, a simple and easy rule to understand. In Mathematical Ideas, the text explains the same problem in a way that I found harder to understand. Although I did understand our current text’s explanation of this problem, it took me longer to read as the text throw in about seven different terms (such as "empirically" and "fair") related to this particular problem, whereas The Nature of Mathematics used only one. Smith goes onto explain the many terms that Mathematical Ideas listed in the coin toss problem in more detail, using different problems to examine each of the topics individually. I believe that because Smith’s text has 73 pages in the Probability chapter versus the 48 pages in our current text, Smith was able to spend a greater amount of time on more experiments and examples, some of which are used in our current text also. Even though I felt The Nature of Mathematics explained Probability better than our current text, I still felt that our text did a fine job as well.

The area I felt that Mathematical Ideas, our current text, explains better than the opposing text is Statistics. Our current text spends 74 pages explaining statistics versus Smith’s 42 pages. I believe that this simple fact allows a better exchange of information and allows the reader to delve deeper into the subject, which leads to improved understanding. Mathematical Ideas starts it’s statistic chapter by showing an example of some statistical data inserted into a chart. This visual example is instrumental in helping me to better understand the statistical information. The chapter proceeds by providing information on how and why statistics are used in various ways in daily life. For instance, the reader is informed about how the government uses statistics for censuses and population trends. In The Nature of Mathematics, Smith uses the same type of charts and graphs that our current text uses, but the later text uses a wider variety. In Smith’s text you will find, bar graphs, line graphs, and histograms. In our current text you will find all the graphs mentioned above plus frequency polygons and comparison line graphs. The charts and graphs are visually stimulating, consisting of multiple colors. They are very easy to understand and follow. As I compared the calculation of standard deviation in each text, it was very plain that our current text gave a more detailed explanation by not only giving the formula, but including a detailed list of steps that one needs to follow to be successful while using the formula. This by far is easier to understand and provides the reader with a more effective tool than Smith’s text. The following is a short list of some of the information I was not able to find in The Nature of Mathematics that was included in our text: Chebyshev’s Theorem, the x-score, box plot, extrapolation, and more. All of the above reasons have helped me to determine why I chose statistics as the area that Mathematical Ideas explains better than the opposing text.

One of the most interesting aspects of the opposing text is the timelines that are included at the beginning of each chapter. The timelines follow the years through 3000 BC to 1999. These timelines give us information on interesting mathematical accomplishments as well as other historical happenings, like the announcement that smallpox was declared extinct in 1980 (good thing to know in case a person is ever on Who wants to be a Millionaire or Jeopardy). Other significant timeline offerings include the fact that in 1640 Desaegues was studying projective geometry, types of measurement used by the Babylonians and Egyptians, and profiles of important people in the study of mathematics. As I searched through Smith’s text, I stumbled upon a mathematician named Srinivasa Ramanujan (1887-1920) who died young (33years old) but was so well self-taught in mathematics that he became the first Indian to become a Fellow of the Royal Society because of his brilliance. I found this young man interesting. I also noticed a few names that I recognized specifically because I have taken this class. One of the names happened to be Alan M. Turning (1912-1954) who discovered the Turing machine. I recognized his name because when I was working on the video review assignment, he was profiled for his work on developing modern day computers.

In The Nature of Mathematics there are even cartoon (funnies) that were used, to visually express inequalities, like having a bird versus a hippo on a teeter-tooter (see saw). All of the examples above helped to make the book more visually stimulating. As I approached each chapter, the interesting facts and timelines encouraged me to want to learn more about the subject that was being explained. I have to say that for me when I have a picture or some sort of timeline that I call recall in my mind, it helps me learn more thoroughly and helps me retain the information more easily.

Visually, Smith’s book used only three colors to adorn its pages: black, white, and orange. Mathematical Ideas uses red to point out it’s examples and tests, yellow to highlight rules and definitions, and blue to give variance between black numbers, lines, and graphs. Consequently, there are photos and drawings in each text.

In conclusion, I would have to say that I would choose our current text versus The Nature of Mathematics because of several reasons. These reasons include: that our text is more up to date, our text contains a greater amount of information while thoroughly explaining each section, and our text utilizes a wide variety of examples as teaching tools. I am a reader and lover of books, so it is no wonder that I enjoyed the timelines and variety of interesting tidbits that Smith’s book provided, however, this is a math class, and as such, we need a text that will provide us with the best teaching tool, and that is in my opinion, Mathematical Ideas.