The Unique Effects of Including History in College Algebra - Introduction

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

Can student attitudes about mathematics be affected by knowing that RenĂ© Descartes, in the 17th century, united many centuries worth of algebraic and geometrical knowledge by developing the foundations for analytic geometry and creating a coordinate system that would later evolve into the Cartesian coordinate system of today (Berlinghoff & Gouvea, 2002)? What achievement and retention benefits are there in exposing students to the historical development of logarithms? Does teaching the origin of the mathematical term parabola affect students’ understanding of the concept and their ability to communicate mathematically about parabolas? Can historical ideas about complex numbers and Boolean algebra be used to show students that mathematics is useful for solving the real world problems of today? This article will discuss what Black Hills State University (BHSU) has learned about the effects of including historical modules in College Algebra on student attitudes, beliefs, understanding and mathematical communication. The evidence collected shows that the inclusion of historical modules caused positive changes in mathematical communication, student achievement and attitudes while the use of other strategies to improve College Algebra, such as technology incorporation, left these particular areas unchanged.

In 2002, BHSU mathematics faculty began examining its College Algebra course, which is taken by roughly 360 students each fall and about 180 students each spring. Fall semester students became the primary focus, as initial surveys indicated that a greater increase in Calculus enrollment was feasible with the students enrolling in College Algebra in the fall. Approximately 20% of fall College Algebra students desire to major in a discipline that requires enrollment in either Calculus or Survey of Calculus for graduation. In 2002, only 1% of College Algebra students continued on to take Calculus or Survey of Calculus courses. Students were surveyed to find out why they chose to take or avoid Calculus, and the majority of students who did not go on felt that they were not strong enough in mathematics to continue. From examining these results, BHSU faculty decided it was a realistic to set a goal that 8% of fall semester students successfully complete Calculus or Survey of Calculus.

Over the next two years, the College Algebra course was strengthened by adding appropriate technology to the course (Hagerty and Smith, 2005). Passing rates increased by about 10%. Students who desired to major in fields that require Calculus were now primarily earning As and Bs in the redesigned course. Even with this performance improvement, there was not a significant increase in the number of students continuing on to Calculus. Despite increased grades and passing rates, one thing had not changed: the attitudes of the students about continuing to take additional math courses. The students held onto the idea that their skills were weak because of the effort they needed to be successful. The students said they were concerned that, even with the success they were having in College Algebra, they would not have success in a class as difficult as Calculus. Furthermore, the additional effort of being successful was not worthwhile, the students said.

Given the growing body of literature (Bruckheimer & Arcavi, 2000; Heiede, 1996; Johnson, 1994; Kleiner, 1996; Siu, 2000; Rickey, 1996; Smith, 1996; Swetz, 2000) theorizing that embedding history within mathematics courses can change students' attitudes about themselves as problem-solvers and about the nature of mathematics, including the history of mathematics in the College Algebra course seemed to be a natural fit. Starting in 2004, historical modules were written for each week of the College Algebra course. Each week the historical topic was directly related to the math skills discussed during the week, with the idea of developing students' understanding that:

  1. Mathematics is a human endeavor,
  2. Mathematics is fluid discipline and mathematics knowledge changes from one generation to the next, and
  3. Mathematics is useful for solving today's real world problems.

This paper will briefly discuss the modules and then focus on each of these three reasons for including history in College Algebra. The paper will also discuss using history to improve mathematics communication. While improving mathematical communication was not one of the original purposes in including history, it was found that including history had a positive effect on the students' mathematical communication skills. The paper will close with a discussion of the observed effects of embedding mathematics history in the College Algebra courses, including an additional 10% gain in passing rates for the course.