A modeling-based approach to college algebra.

Abstract

Virginia Commonwealth University (VCU) is engaged in reforming college algebra, a course that is a gateway to the college experience for many students. The details of a modeling-based approach to college algebra are outlined in this article. An assessment of factors related to the attitude toward mathematics comparing students in this modeling-based approach with students in a traditional course was conducted. The results of the assessment, withdrawal rates of the courses, and student comments from focus groups are discussed.

Introduction

College algebra is a gateway course to the college experience for many students. Consequently, successfully completing the course is an important first step to obtaining a college degree. In most colleges and universities the curriculum was designed over fifty years ago and focuses on preparing students for calculus--a course which fewer than 10% of college algebra students will take (Small, 2002). This and other factors including that students have seen this material in high school and a low passing rate have a negative impact on the attitudes of students in college algebra. The attitudes developed in the course have an influence on student willingness to take other mathematics courses and on their opinion of mathematics after finishing college.

The Committee on the Undergraduate Program in Mathematics (CUPM) is a group within the Mathematical Association of America who issue guidelines every decade for developing the ideal curriculum for undergraduate mathematics courses. In their most recent publication, CUPM (2004) advocates major changes to college algebra. In particular, the course should focus on content that is applicable to other academic disciplines, provide detailed study of a small number of topics, and help students develop the ability to communicate numerical ideas orally and in writing. In a series of discussions at the Algebra Initiative Colloquium (Lacampagne, 1995), mathematicians and educators reviewed all aspects of the algebra curriculum. One recommendation that emerged was that in order to prepare students for the ever-changing needs of business and industry, the curriculum "should be grounded in problem solving that reflects real-world situations and offers a variety of methods of solution" (p. 12). A college algebra course designed with these recommendations in mind has a focus on mathematical modeling. The intent is that students who complete such a course will be able to use mathematical modeling coupled with algebra skills to analyze various problems and recognize the usefulness of applying the process to real-world situations. Along the way, the educational experiences in which the students are engaged will have a significant influence on their attitudes toward mathematics.

In two studies (Fox & West, 2001; Oty et. al, 2000) researchers looked at student attitudes after completing modeling-based college algebra courses. One study emphasized application problems and group projects (Fox & West, 2001). Most students wrote positive comments about their experiences in their project summaries. The negative comments reflected students' frustration with the challenge they faced through the project experiences to become critical thinkers of mathematics. The authors felt that all comments revealed that the assignments satisfied their purpose of preparing students to work with real-world problems and most of the students were positive about the experience. In a similar course featuring science-based applications, students completed a Leikert scale survey about their attitudes at the end of the course (Oty et. al, 2000). With respect to mathematics, 80% of the students in the modeling-based course expressed that their attitude had improved during the class while 55% of students in a traditional college algebra course had a similar change in attitude. Based on these studies, it appears that modeling-based college algebra experiences are a positive influence on students' attitudes.

Method

With the recommendations of mathematicians and educators in mind (Lacampagne, 1995; CUPM, 2004) we piloted a modeling-based college algebra course and compared the attitudes of students in the new course with the attitudes of students in a traditional college algebra course. Specifically, the question posed in this study is: Do students in a modeling-based course have better attitudes toward mathematics than students in a traditional college algebra course? Students selected a section of college algebra based on personal preference and section availability. While random assignment was not possible, students did not know which sections would be taught through the modeling-based approach before the first day of class. Both approaches had similar grade level distributions with 80% of the students enrolled being freshmen. The modeling-based sections served as the treatment group while the traditional sections served as the control group. The research question was answered by analyzing data from three assessment measures: (1) student responses to the questions on the Fennema-Sherman Mathematics Attitudes scales, (2) the percentage of students who withdrew from college algebra in Fall 2004, and (3) the comments of focus group participants who completed the modeling-based course. Course outlines followed by details of the evaluation are presented below.

Modeling-based College Algebra

The modeling approach to college algebra incorporates extensive use of graphing calculators and collaborative group work to help students develop mathematical models and use them to solve a variety of problems. The mathematical skills covered in the course are those necessary to successfully complete the modeling process. As a result, many skills covered in a traditional college algebra course like rationalizing the denominator of functions or solving quadratic equations by completing the square are not covered in the course. The required textbook is Contemporary College Algebra: Data, Functions, and Modeling (Small, 2003). Using a combination of technology and algebra skills, students study a variety of applications. They begin by plotting a series of data points to analyze the pattern. Then they compare the data to the characteristics and shapes found in their library of functions (linear, quadratic, exponential, etc.) which is developed and added to throughout the semester. Through a series of approximations, an algebraic model emerges that is used to determine other data points and predict the data's future behavior.

Course sections are limited to 35 students with two teaching assistants available for every class. Small class size and extra help are important to student success in this course which centers on student exploration and contains very little traditional lecture. The course's weekly meetings consist of two 75 minute sessions in a regular classroom and one 50 minute computer lab session. A typical 75 minute session begins with students presenting homework problems on the blackboard as a review of prior material and a lead-in to the instructor's introduction of new material. Most class time is spent with students working collaboratively to develop mathematical models and applying them to problems posed. The formal lecture lasts 20 minutes or less and occurs intermittently throughout the collaborative work. A short lecture serves as an introduction to the day's topic and other lecture/discussion opportunities result from student comments or questions. The computer lab class period provides an opportunity to delve further into the big concepts covered by the class. Students work on problems that require a longer amount of time and review concepts by working problems included in the software package that accompanies the textbook (Small, 2003). Students are given homework assignments at the end of each class period on the day's material.

Grades are based on a variety of tasks. Four tests (three 75 minute exams and one comprehensive final) are given throughout the semester. The emphasis of each test is on modeling and applications with only 30% of each test devoted to skill problems. Homework, in class group assignments, quizzes, and two group projects are also included in the final grade. For the projects, students have three weeks to gather data, apply a model to the data, and write up their results in a formal report. The projects are assigned from the textbook (Small, 2003). Sixty-five percent of the grade is based on quizzes and tests with the remaining portion based on the other activities.

Traditional College Algebra

Traditional college algebra is a skill-based course. The topics covered are those found in most traditional textbooks with about 10% of the semester devoted to mathematical applications. Course sections are limited to 35 students. Students meet 150 minutes a week with an instructor who follows a lecture format. They spend 50 minutes a week in the computer lab where they get help on problems and work on homework assignments. Students are allowed access to graphing calculators but the amount of emphasis is up to the discretion of individual instructors. A software package bundled with the textbook (Wright, 2004) is required to complete all homework assignments. The assignments correspond to textbook sections and require students to correctly answer 80% of the problems provided. They do not receive partial credit for mistakes and must start over with a new set of problems if they do not successfully complete the problems for a section. While students are required to do the homework, it does not figure into the final grade which is based on four 50 minute exams and one comprehensive final exam given the weight of two regular exams.

Attitude Assessment

Six sections (121 students) of modeling-based college algebra and five sections (88 students) of traditional college algebra participated in the assessment. The students in these sections were administered the Fennema-Sherman Mathematics Attitude scales the first week of class and at the end of the semester. Data from students who participated in both administrations of the scales was evaluated. The mean scores are presented and discussed below. The Fennema-Sherman Mathematics Attitudes scales (1976) were designed to determine the level of students' attitudes toward mathematics with respect to several factors. The scales used in this study were: Confidence in Learning Mathematics, Usefulness of Mathematics, Mathematics Anxiety, and Effectance Motivation toward Mathematics (i.e., amount of involvement students have when doing mathematics). In each case, student responses to six positively worded statements and six negatively worded statements were gathered through a Leikert scale with five alternatives (strongly agree, agree, undecided, disagree, strongly disagree). Students were given a score between 12 and 60 for each scale with a higher score representing a more positive student attitude. The Split-half reliability for each scale is 0.87 or larger (Fennema & Sherman, 1976).

A paired t-test was used to determine whether the students' attitudes at the end of the course were significantly different than their attitudes at the beginning of the course. The analysis revealed statistically significant results in favor of students who completed the modeling-based course for two scales: Confidence in Learning Mathematics and Mathematics Anxiety. For Confidence in Learning Mathematics, the mean post-course attitude score of 42.83 was larger than the mean pre-course attitude score of 40.69 at the 1% level of significance. With respect to anxiety, a larger value reflects lower mathematics anxiety. At the same level of significance, results for Mathematics Anxiety (pre-course mean = 37.40, post-course mean = 39.12) reveal that students in modeling-based sections had a lower level of mathematics anxiety alter the course when compared with the level at which they began the course. As the numbers reflect, there was not a dramatic increase in confidence or dramatic decrease in mathematics anxiety. However, the statistical test revealed that the results were statistically significant. The data from students in the traditional course did not yield similar results. In fact, while the values were not statistically significant, the mean attitude scores reflected that students in the traditional sections had slightly higher confidence and lower mathematics anxiety before the course began compared to their post-course responses.

For Mathematics Usefulness, the modeling-based results reflect a pre-course to post-course increase in mean score while the results of the traditional sections reflect a decrease. However, the results for both groups were not statistically significant. There were no significant differences in the pre-course/post-course responses of students in both types of courses for the Effectance Motivation toward Mathematics scale. Lastly, an ANOVA was conducted with the data from each scale to determine if there were significant differences among class sections for the different approaches to college algebra. The ANOVA revealed that the responses of students in modeling-based college algebra sections were statistically similar and so were the responses of students in the traditional sections. Therefore, the results are representative of all college algebra sections that participated. The attitude assessment shows that participation in a modeling-based approach to college algebra can result in increases in students' attitudes toward mathematics. The modeling approach resulted in students having slightly more confidence in their ability to learn mathematics and less anxiety about mathematics. In the traditional sections, the students' confidence in learning mathematics and mathematics anxiety were unchanged after completing the course.

In Fall 2004, 284 students were enrolled in modeling-based sections while 988 students were enrolled in traditional sections. With respect to the traditional course, 20.34% of students did not complete the course while the corresponding percentage for the modeling-based course was 5.63%. Clearly, a significantly larger percentage of students completed the modeling-based course. While not a specific reflection of student attitude, the difference in withdrawal rates does reflect a greater willingness in students to stick with the modeling-based course.

Modeling-based Focus Groups

After 10 weeks of instruction, focus groups were conducted with students in the modeling-based course. Groups of 8 students in 50 minute sessions met with unbiased interviewers skilled in conducting focus groups. The discussions revolved around the course and the difference between the modeling-based approach and the students' prior algebra experiences. The questions to generate discussion were designed by the interviewers with input from the instructors. The students were not asked directly about their attitudes toward mathematics. Time was spent discussing a variety of topics including the textbook used, comparison of this course with prior mathematical experiences, suggestions for course improvements, and the ability of instructors to reach the ultimate goal of the course to help students think and reason mathematically. The student comments were shared with modeling-based instructors through a transcript of the discussions and all references to specific students were deleted.

When asked if they liked the course, two-thirds of the locus group participants responded in the affirmative. They recognized and appreciated the clear differences between the modeling-based course with emphasis on discussion and group work and their prior experiences in lecture-based courses. Seventy percent of the responses reflected that the students felt their learning experiences were enhanced by group collaboration. Three locus group participants were taking college algebra for the second time and had previously taken a large lecture course with computer-based homework assignments like the ones for the traditional course outlined above. These students expressed greater satisfaction with the modeling-based course when compared with their previous experience.

One-third of the responses reflected that students were dissatisfied with some aspect of the course. The general negative comments reveal that not all students can be reached with one particular teaching method. Most specific comments focused on the group projects. VCU is an urban institution with a large commuter population and students expressed dissatisfaction with having to meet outside of class to do group work. Others voiced concern with relying on other students work for a grade on a group assignment. In general, 75% of the focus group participants felt that instructors met the overall goal of the course and had helped them learn to think and reason mathematically. A similar number of comments noted that this approach is more applicable to real world situations when compared to the skill-based approach of their previous mathematical experiences or the experiences of other students.

Conclusion

The assessment outlined above reveals that the modeling-based approach to college algebra does have an impact on student attitudes toward mathematics. The responses to the Fennema-Sherman Mathematics Attitude scales were slightly better for students who completed the modeling-based course when compared with the responses of their traditional counterparts. Students completed the course with more confidence in their ability to do mathematics and less mathematics anxiety. More students completed the modeling-based course as compared to students who completed the traditional college algebra course. Focus group discussions yielded many positive comments from students who were taking the modeling-based course. Most of the negative comments were related to course specifics like group projects and use of the computer lab. Three-fourths of the participants expressed that the overall goal of the course to help them become mathematical thinkers--was met.

Based on the assessment outlined above, it appears that moving from a focus on skills in a lecture-based college algebra course to a course that revolves around learning to apply mathematical models to real-world problems does have a positive effect on students' attitudes toward mathematics. Changes to introductory mathematics courses are advocated by various mathematical organizations including the National Council of Teachers of Mathematics and the Mathematical Association of America (CUPM, 2004). The analysis of attitudes and withdrawal rates of students in two college algebra courses at VCU provides support for change as well.

References

Committee on the Undergraduate Program in Mathematics (2004). Undergraduate Programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide 2004. Washington, DC: Mathematical Association of America.

Fennema, E., & Sherman, J. (1976). Mathematics attitude scales: Instruments designed to measure attitudes toward the learning of mathematics by males and females. Journal for Research in Mathematics Education, 7, 324-326.

Fox, W., & West, R. (2001). College algebra drills or applications. PRIMUS, 11, 89-96.

Lacampagne, C., Blair, W., & Kaput, J. (Eds.) (1995). The Algebra Initiative Colloquium, Volume 1. Washington, DC: U.S. Department of Education. (ERIC Document Reproduction Service No. ED385436)

Oty, K, Elliott, B., McArthur, J., & Clark, B. (2000). An interdisciplinary algebra/science course. PRIMUS, 10, 29-41.

Small, D. 2003. Contemporary College Algebra: Data, Functions, and Modeling (5th ed.) Boston, MA: McGraw Hill College Custom Series.

Small, D. 2002. An Urgent Call to Improve Traditional College Algebra Programs. Focus: The Newsletter of the Mathematical Association of America, 22, 9.

Wright, D. 2004. Intermediate Algebra (5th ed.) Hawkes Publishing.

Aimee J. Ellington, Virginia Commonwealth University

Ellington is assistant professor of mathematics education. Her research interests include technology in the mathematics classroom and curriculum reform.