# The Unique Effects of Including History in College Algebra

## Introduction

Can student attitudes about mathematics be affected by knowing that René 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.

## The Modules (1)

Each module was designed to improve students’ understanding and desire to continue mathematics, with the goal of increasing enrollment in Calculus. During the discussion phase of elevating the role of history of mathematics, it was decided to move beyond the level of history seen in even the most progressive of textbooks. Each week, a significant essay was written about both the development of mathematics as a tool to improve society, as well as the need for each generation to improve their mathematics skills to ensure the further advancement of society. The modules helped motivate the student on the topic of the week, but also was connected with previous lessons so that each lesson increased understanding of the effort required to be successful in mathematics, and also gave the students a greater appreciation for the need of mathematics in their lives.

The modules tell solid stories at manageable lengths. Students were asked to read two to three pages of mathematics history each week. Every teacher of every section of College Algebra implemented the historical modules beginning in 2005. The first nine history lessons are outlined below.

Introductory Lesson: Introduces the term "algebra" and its Arabic origin. It then briefly gives a timeline of the movement through Europe, including the addition of the Cartesian coordinate system, the introduction of the function nomenclature by Leibniz, and the popularization by Euler of the symbolic notation $$f(x)$$ for a function (The Function Concept, 2007). The goal was to introduce both the longevity of the subject and the amount of effort required to invent the subject as it is known today.

Quadratics and Parabolas: Looks at the development of quadratic equations (as an application of finding areas for quadrilaterals) and parabolas. The goal is to further extend the notion of the fluidity of mathematics and the time span required to develop today’s mathematics. Also, this module introduces the need for vocabulary and the process in which vocabulary was developed.

## The Modules (2)

Introduction to Polynomials: Looks at the efforts put forth in finding zeros of polynomials and includes a brief introduction to the lives of Niels Henrik Abel and Evariste Galois. This historical module excerpt illustrates the difficulties Abel and Galois had in breaking into the mathematical circles of their time. “Niels Henrik Abel, at the age of sixteen, proved that a general formula for solving a quintic (fifth degree) polynomial did not exist…. However, since he was largely self taught, leading mathematicians in Paris, such as Cauchy, largely ignored him… Evariste Galois had equally important discoveries. At sixteen, Galois had the desire to enter the most prestigious engineering school of the day, the École Polytechnique… [W]hen Galois submitted a paper to the school as part of the admission process, Cauchy lost the paper. He attended another school for the purpose of training to become a teacher. However, he kept his mathematical studies up and submitted a second paper to the École Polytechnique. This paper also appears to have been lost.” (Hagerty and Smith, 2006).

Polynomials: Looks at theoretical methods to help find zeros of polynomials. The module looks at Horner’s method and how information traveled in eras prior to modern-day technology. It includes a discussion of the difficulty of crediting the correct civilization with the development of a topic as it is believed that Horner did not develop the method credited to him; in fact, the Ancient Chinese knew of this method (Eves, 1992).

Technology: Looks at methods to use technology to find zeros of polynomials, and discusses the rapid changes in technology. The goal is to have the students take a look at when the Internet was developed and realize that instant messages were not always possible. The students need to realize that their parents enjoyed “Pong” and “Pacman” and their grandparents had the radio. Thus, the students need to revaluate the question “My parents didn’t need math, why do I?”

## The Modules (3)

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.

## Mathematics Is a Human Endeavor

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.

## Mathematics Is a Fluid Discipline

Mathematics is a dynamic field that renews and updates itself with each generation. College Algebra texts of 50 years ago are substantially different from the College Algebra texts of today. One important difference is the advent of various technologies such as calculators and computer algebra systems. The use of technology not only assists in solving problems, but has ultimately changed the focus of the instruction from algorithmic manipulations to conceptual understanding. It is important to share with students the historical development of technology and the influence of technology on the discipline of mathematics. When students truly understand mathematics concepts and how technology can be effectively used, perhaps they will be more likely to believe that mathematics is a useful and important subject to study. In order to stress these important connections, a lesson was written on the history and development of technology that was used with a calculator laboratory lesson to find x-intercepts of polynomials.

With this focus, it was expected that the students would realize that mathematics education that was acceptable for their parents’ generation may no longer be acceptable for them. Instead, today’s students need to look closely at the changing world and determine what mathematical ideas, skills and understandings are going to be important not only for their first job, but also for success throughout all the years of their employment.

## Mathematics for Today's Real World Problems

Mathematics is useful for solving today’s real world problems (NCTM, 2000). Because the redesigned College Algebra had an increased focus on real world applications, history was used to strengthen this connection by focusing on the real world problems that spawned the need to explore today’s College Algebra concepts. Modules were included that detailed how the Egyptians (in surveying the floods each year) developed methods to measure the area of quadrilaterals, how logarithms were developed as a means to save time multiplying and dividing large numbers, and how probability was secretly applied to the ancient games of gambling in order to make the most profitable choices. In the historical class discussions, real world applications for which the concept was originally developed were tied to today’s real world applications of the concept.

Using the historical development of complex numbers and Boolean algebra, students learned the idea that sometimes in the history of mathematics, concepts that were explored on a strictly pure, abstract level come to be extremely useful for real world applications long after they were originally developed. The usage of both complex numbers and Boolean algebra in electronics and computers occurred more than a century after their initial development. Students were posed these questions: “What would the world look like today if neither concept was developed?” and “Could both electronics and computers be at the stages they are today if early developers had not understood that complex numbers or Boolean algebra were the solutions they were looking for?”

## Using History Develops Mathematical Communication

Developing mathematical communication was not a primary reason for adding history to the College Algebra course. However, the inclusion of the historical modules appears to have been beneficial to communication. When a student is struggling with communication, instead of just trying to fit the words by rote into the mathematical procedures being taught, incorporation of the historical development of mathematical terms helps students make sense of mathematical terminology. This provides a much stronger basis for mathematical terms to be understood and used in communication.

Napier coined the word logarithm from the Greek words logos (ratio) and arithmos (number) to describe the math that he was working on (Eves, 1992). Based on his methods, this wordsmithing made perfect sense. However, today, the process used in logarithms has greatly changed as easier methods have been found. Through history, the cognitive structures associated with mathematical terms have been strengthened, allowing students to more easily communicate effectively. It also appears that this process of including the historical development of terms has allowed the student to relax and made it easier to incorporate new terms and concepts into their intellectual structures. While the majority of the lessons did not focus specifically on terminology, informally it was found that including the history lessons increased the usage of correct mathematical terminology during classroom discussions.

## Findings (1)

Over the last three years at BHSU, great strides have been made in improving student outcomes in College Algebra with the inclusion of history in the course. The same results theorized by many (Bruckheimer & Arcavi, 2000; Heiede, 1996; Johnson, 1994; Kleiner, 1996; Man-Keung, 2000; Rickey, 1996; Smith, 1996; Swetz, 2000) about the effects of inclusion of the historical development of concepts in mathematics courses can now be seen in the College Algebra course.

Figure 1 shows the changes in the percentage of students enrolled in the fall College Algebra course who continued on to enroll in Trigonometry. The dotted line shows the goal of 8%. The solid line shows the growth in the percentage of the students enrolling in Trigonometry. The number of students enrolled in College Algebra in the fall has consistently remained around 360 students. The number enrolling in Trigonometry has grown from 5 students in 2003 and 2004 to 20 students in 2007.

Figure 1. Percent of students enrolling in Trigonometry from 2003-2007

The increase started in 2005, the year the history modules were introduced. Besides quadrupling the number of students enrolling in Trigonometry, average student achievement in Trigonometry also increased by approximately 6-7%, from about a 70% overall average to an overall average of 76-77% for the course. Furthermore, the Trigonometry instructor reports that this class has doubled in size, attendance has increased, there is an improved success rate and a greater percentage of students are making their way to Calculus. In the coming semesters, as more students who were exposed to the revised College Algebra course matriculate into Calculus, increased enrollment and achievement is expected in Calculus.

There has been significant improvement on test problems where mathematical vocabulary and notation have created difficulty for students in the past. One example is that when students were asked to find the inverse function of $$f(x)=\frac{x+2}{2-x},$$ students would pick the reciprocal about 30% of the time prior to the use of history in the College Algebra course. After the introduction of history to the course, students are picking the reciprocal only about 15% of the time.

## Findings (2)

Most of the student attitude evaluations were informal. In a recent conversation with a student who is taking College Algebra at the request of her job, the student informed us that she greatly appreciated the historical readings. The readings gave her a greater understanding of why mathematics was important and gave her “non-mathematical” explanations (her definition of non-mathematical can be interpreted as not algorithmic) as to what was going on. She felt that the historical backgrounds made it easier for her to understand and learn the mathematical processes. This type of attitude seems to have grown in the students over time and as they realize mathematics is more than just a set of facts and rules to be memorized. In informal discussions with Trigonometry students who had taken College Algebra with the historical modules, students credited the history modules with assisting them to reflect on the mathematics needed for their careers, and the modules gave the students a better understanding for effort required for their success. Furthermore, the history modules combined with other changes appear to have also affected the students’ realization that they needed to understand the concepts as well as just complete the assigned work.

An approximate 10% increase in the College Algebra passing rate since 2005 has come since the addition of the historical modules. Today, over 70% of BHSU’s College Algebra students pass the course with a C or better.  The College Algebra faculty have maintained a strong stance against grade inflation and have worked diligently to ensure that the 20% increase in passing rates over the course of the reform was not caused by a reduction in the quality of the course.  One outside measurement tool has been the CAAP test (CAAP, 2004). The state mandates that all students take the CAAP test at the end of their sophomore year of college.  Analysis of this information gives strong statistical data that grade inflation is not occurring.  Furthermore, for students whose only math course while attending BHSU is College Algebra, CAAP  scores have improved by a statistically significant 10.9% since 2005 (p < .01).

The faculty at Black Hills State University believe that the historical aspect of College Algebra is vital for the student to realize both the need for additional mathematics courses and the effort required to be successful. The inclusion of history has also had some affects on instructors. College Algebra instructors are seeing their former students returning during office hours to discuss mathematical history. During the first month of the current semester, one instructor has had the opportunity to discuss concepts of Boolean Algebra and the Fibonacci sequence with former students. These discussions help ignite the fires in College Algebra instructors and make them want to reach even more students. The faculty feels that including historical development of mathematics is of key importance and believe that the full benefits of history inclusion have yet to be reached. At this time, the faculty is looking for means to increase the use of history. While much has been done to improve the College Algebra course, it seems as though there may be some effects that can only be accomplished through the effective inclusion of the historical development of mathematical concepts.

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