The Unique Effects of Including History in College Algebra - The Modules (3)

D. Goodwin (Black Hills State University) and G. W. Hagerty (Black Hills State University) and S. Smith (Black Hills State University)

Education: Looks at the history of education, the changing society and how education requirements grow with the society, as shown by the following excerpt. “Early in the 18th century, … advancements in agriculture displaced many farm workers sending them to the cities to look for work…. [A]s cities grew, so did the demand for an elementary education…. By the end of the 19th century, … a free elementary educational system was established in the hopes that the majority of the population would obtain an eighth grade education…. The early 20th century saw the development of the airplane, the assembly line, the radio, telephones and much more … [causing] an increased push for every citizen to obtain a high school diploma…. In 1957 the Soviet Union launched Sputnik to begin the space race. The reaction … was to increase the math and science offered in the public high school…. [M]ath educators quickly produced a curriculum to include topics of precalculus, calculus and statistics in the high schools… There was also a push to produce more college graduates.” (Hagerty and Smith, 2006). As this is primarily a freshman course, the students should spend quality time reflecting on their career goals and their educational needs. The student should use this time to reconsider taking Trigonometry and Calculus.

Rational Functions: Looks at using rational functions as a means to approximate real events, such as the absorption rate of a medication into the blood stream along with the time for elimination and metabolism of the medication. This gave an opportunity to bring together many of the concepts discussed over the semester in one application. This allows the following question to be framed: “Are students learning College Algebra in order to develop mathematical concepts for use in their careers or are students learning College Algebra so that when someone else explains an event using mathematical concepts the student has a better chance of following the discussion and understanding their world better?”

Exponential Functions: Looks at the historical development of exponential functions as a means to support the biological sciences, as well as applications in business. Beyond the richness of the topic in applications, it introduces the ideas of calculus and how close the need is for calculus to move forward, as detailed in this excerpt: “Mathematical research was moving at a feverish pace during the 17th and 18th century… 'It would be fair to say that Johann Bernoulli began the study of the calculus of the exponential function in 1697' (The Number e, 2007). The problem of determining the value of \(\left(1+{\frac{1}{n}}\right)^n\) as \(n\) goes to infinity is important as it relates directly to the problem of continuous compounding of interest. It turns out that as \(n\) goes to infinity, \(\left(1+{\frac{1}{n}}\right)^n\) goes to \(e\). Hence we can write that \[\lim_{n\rightarrow\infty}\left(1+{\frac{1}{n}}\right)^n=e.\] "Once we have obtained the knowledge of exponential functions, we will have developed an understanding of mathematics that will put us on the boundary of all known mathematics around 1600 (or just a short 400 years ago). Furthermore, the types of problems which caused mathematicians to develop exponential functions were requiring additional mathematical tools to understand…. This new knowledge of exponential functions brings us to the doorstep of calculus.” (Hagerty and Smith, 2006).

Logarithmic Functions: By focusing on the work of Napier, this topic looks at the effort required and how much of today’s mathematics was developed out of someone’s desires to simplify and improve life. By focusing on the time span required to develop Napier's version of logarithms, including the time to create the original charts to make the concept practical, the students can gain an appreciation for the difficulty of the concept and realize that they too are going to need to put in effort to understand the concept.

In the first year of the use of the modules, it became apparent that the concepts needed to be made relevant to today. After the initial look at logarithms, students first common remark was, “If we don’t use logarithms for simplifying multiplication and division and slide rules are obsolete, why should we learn logs?” To solve this problem, every module was brought up to date by including a paragraph that included a discussion of how the concept is used today. One goal was to find a current application in as many career fields as possible.