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The Unique Effects of Including History in College Algebra - Mathematics Is a Human Endeavor

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

While the students’ academic performance improved through the initial revision of College Algebra to include a stronger technology component, the students were still seeing mathematics as a hard and time-consuming subject that was not something they, personally, could excel at. The students believed that a “good math student” should be able to understand every concept immediately and pass tests without effort. Even though performance had improved, the beliefs the students held about “good math students” made it very difficult for them to believe that they could be successful in more difficult courses, such as Calculus. Thus, when writing the historical modules, it was important to show students that mathematics was not developed in a day and that even the greatest mathematicians did not understand mathematical concepts without effort.

When the historical modules were written, careful thought was given to include the length of time (in some cases centuries) that it had taken mathematicians to understand concepts. One goal was to convince students that it is okay to not understand all topics immediately from a lecture. The students were shown that some topics continued to mystify the entire mathematical community for hundreds of years. Through the modules, mathematicians (“good math students”) were portrayed as people who were fascinated and personally invested in the concepts they were studying and therefore invested a sufficient amount of effort and time to learn as much as they could about the concepts they were studying. In changing the students’ perceptions of what it takes to be successful as a mathematician, it was expected that the students would realize that they could succeed in Calculus.

To accomplish this goal, historical modules focused on the development of particularly stubborn College Algebra concepts, those that had taken many centuries to fully understand. For example, Quadratic Equations were studied in part for the practical use of finding areas of quadrilaterals. Egyptian rope stretchers circa 1650 BC often needed to find the area of quadrilaterals, although they probably were not able to solve quadratic equations (The Ahmes Papyrus, 2007). The parabola was also studied by the Greeks circa 400 BCE. These two concepts were united together (as they are taught today) with the creation of a coordinate system by René Descartes in 1637 - a coordinate system that would later evolve into the Cartesian coordinate system (Berlinghoff & Gouvea, 2002). This historical development graphically illustrates to the students that the concepts, now usually explained in a single lecture, were actually developed over centuries. The hope is that students would realize understanding concepts is the most important part of mathematics, no matter how long it takes to learn the concepts.

It is important for students to focus on the idea that mathematicians struggle together to solve important problems. As an example, here is an excerpt from the historical module on logarithms. “In 1594, Napier published his results on how large multiplication and division problems could be transformed into addition and subtraction problems.... John Napier’s logarithmic numbers were discovered through geometry and thus, were not of the simplest form. In 1615, Henry Briggs, a geometry professor at Oxford, visited Napier and discussed using a base ten system. This was agreed upon and the movement towards modern-day logarithms had begun … in the 380 years between the discovery of logarithms and the invention of the calculator, many uses for logarithms have been discovered. Hence, the importance of logarithms has increased” (Hagerty and Smith, 2006). As a connection to real world problems, the importance of logarithms in chemistry and related fields was included.

D. Goodwin (Black Hills State University) and G. W. Hagerty (Black Hills State University) and S. Smith (Black Hills State University), "The Unique Effects of Including History in College Algebra - Mathematics Is a Human Endeavor," Convergence (June 2010), DOI:10.4169/loci002530