This is Chapter VI of the MAA Report on the “Algebra, Gateway to a Technological Future Conference,” edited by Victor J. Katz. The conference was held at MAA Headquarters’ in Washington, D.C in 2007
College Algebra
William Haver, Donald Small, Aimee Ellington, Barbara Edwards,
Vernon M. Kays, John Haddock, Rob Kimball
Approximately 700,000 students annually enroll in College Algebra courses, most of which focus on algebraic manipulation. Students are told to learn the procedures for factoring polynomials, “simplifying” radicals, “solving” equations with absolute values, and “solving” inequalities. Students are expected to learn to follow the same procedure demonstrated to them by the instructor. In these courses, students are not expected to use the solutions in any context outside of mathematics. The course typically serves as a terminal course for students in many majors as well as a prerequisite to courses such as pre-calculus, statistics, business calculus, finite mathematics, and mathematics for
elementary education majors. While the goals of the course are based on what some think are the necessary prerequisites for calculus, fewer than 10% of the students ultimately enroll in calculus. When businesses and other employees discuss the quantitative skills of their non-technical workforce they are, to a large degree, discussing individuals who have enrolled in College Algebra. The course fails to emphasize those
abilities that faculty from other disciplines and representatives from industry expect from these students. Furthermore, by the most conservative estimates, fewer than 50% of the students who enroll in the course receive a grade of A, B, or C. Fortunately, with support from the National Science Foundation, a great deal has been learned about
College Algebra and about materials and approaches that can be used to create a course that is more valuable to students. This refocused College Algebra course can result in a greater retention and in a higher success rate. The national mathematics organizations have endorsed the need for such a change. A great deal of
inertia exists, and it is recognized that offering a renewed course must take into account the fact that most sections of this course are taught by part time instructors and graduate teaching assistants. However, the opportunity exists to make a profound change for this large body of students. Based upon what is known concerning College Algebra, the working group proposes a long-term program that would produce a dramatic change in College Algebra nationwide. Although seemingly expensive in total, such a program
would cost very little for each enrolled student.
What is known about College Algebra
With support from the NSF for studies, workshops and pilot projects, and by professional organizations such as the Mathematical Association of America (MAA) and American Mathematical Association of Two Year Colleges (AMATYC), a great deal has been learned about the College Algebra experience in the United States:
• Annually 650,000 to 750,000 college students enroll in College Algebra. This number is firm, subject only to varying names of the course at different institutions. The definitive Conference Board of the Mathematical Sciences study provides detailed information on these enrollment numbers (Lutzer et al, 2002).
• Fewer than 10% of the students who enroll in College Algebra intend to prepare for technical careers and a much smaller percentage end up entering the workforce in technical fields. Again, there is definitive evidence to support this situation that occurs at all types of institutions. Typical findings are that even at research universities, only 9% of College Algebra students ultimately register for calculus and about 1% enroll in third semester calculus (Dunbar, 2006). Comparable data exists for community colleges and other types of institutions (Herriott, 2006, Agras, 2005, McGowen, 2006).
• Nationwide more than 45% of students enrolled in College Algebra either withdraw or receive a grade of D or F. For example, a community college in Georgia has a DWF rate of 49% among the 2,300 students who enrolled in College Algebra (Herriott, 2006); at a two-year college in Florida the DFW rate was 45% (Agras, 2005); at a university in Virginia the DFW rate was 63% for students enrolled in large lecture sections and 51% for students enrolled in classes of 35 students (Ellington et al, 2006).
• When given an opportunity, faculty from other disciplines and representatives from business, industry, and commerce have consistently called for mathematics departments to make major changes in the content of College Algebra. Particularly noteworthy are the sentiments voiced by prospective employers (Steen et al, 2007) and the results of 11 weekend workshops of representatives of 17 different partner disciplines (Ganter et al, 2004). They uniformly recommend: that algebraic techniques not be the focus; that there be a strong emphasis on conceptual understanding; that communication skills should be stressed; and that the courses focus on mathematical modeling and realistic problem
solving. (See for example Gordon, S., 2006).
• The curriculum committees of national mathematics organizations have uniformly called for replacing the current College Algebra course with one in which students address problems presented as real world situations by creating and interpreting mathematical models. These recommendations have been consistent and persistent, coming from AMATYC (Wood et al, 2006) and MAA (Pollatsek, 2004 and Ganter et al, 2004). Most recently MAA’s Curriculum Renewal Across the First Two Years committee has approved CRAFTY’s College Algebra Guidelines, which describe the renewed College Algebra course that is the recommended offering for all colleges and universities. A
copy of the guidelines is attached to this report.
• There is widespread interest in mathematics departments concerning offering modeling-based College Algebra and also an expressed need for support for instructors, adjuncts and graduate assistants who teach most sections of courses at this level. When department liaisons were asked if they would like to participate in a proposed NSF-supported MAA research/pilot study project to offer modeling based and control sections of College Algebra, more than 210 departments responded positively within six days. Of the 11 institutions that were invited to participate in the project, 10 offered the modeling based sections in both the Spring and Fall semesters of 2006 and all 10 determined, based upon their experiences, that they would continue to offer modeling sections beyond the period of the grant. They also expressed the view that more professional development was needed for their instructional staffs and that an extended period of time would be needed to fully implement modeling based courses.
• With support from NSF a large number of exemplary materials have been developed and put in place, although on a very small scale. The materials address the areas stressed by faculty from other disciplines and representatives from industry and, in addition, the student success rate has increased. These materials are described and the successes documented on these small scale efforts in a number of publications: Hastings, 2006; Ellington, 2005a; Fox et al, 2001; Gordon, F., 2006; and Johnson, 2004. For example, at one community college (Agras, 2005) 71% of students (n = 159) who took a modeling course received a grade of A, B, or C, while only 55% of students in traditional
courses did so. At one university (Ellington, 2005b), 20.3 % of students (n = 989) withdrew from the traditional College Algebra course while only 5.6% of students (n = 284) withdrew from the modeling course. In a companion study (Ellington et al, 2006), 32% more students enrolled in a modeling course completed that course and a subsequent course than was the case for students enrolled in a traditional
course. In a four year institution (Oty et al, 2000), 84% of students (n = 73) reported that they found a modeling based course to be interesting while only 42% of students (n = 79) found their traditional course interesting. It is reported that 55% of students (n = 178) in a modeling course maintained or improved their grade in a subsequent course, whereas only 10% of students (n = 212) in the control group did so (Norwood, 1995).
Despite all of these findings, the overall majority of students nationwide are enrolled in College Algebra courses that focus on algebraic manipulation. A large scale effort by the mathematics community would produce a huge change with profound implications nationwide. Such an effort should consist of four components:
CA1. Sustained Support to Enable Large Numbers of Institutions to Refocus College Algebra
We recommend that support be given to large numbers of institutions to change their College Algebra program. Each participating institution would engage in a four year implementation period that would include participation in an initial workshop followed by on-going mentoring, site visits, faculty development, material and curriculum development, presentations, publications and research. It is essential that this ongoing
professional development fully involve those individuals who actually teach College Algebra at the institution; typically this includes graduate teaching assistants and adjunct instructors. The four year period allows for a phased implementation of the various components of refocused College Algebra: content changes including use of data and modeling; small-group in-class activities; and out-of-class projects. Based upon the evidence described above, the working group is confident that there would be widespread participation by mathematics departments. Participating institutions would provide a significant portion of the overall costs. Grants could be provided to two to four professional organizations or consortiums of institutions enabling 300 colleges and universities to develop and fully implement refocused College Algebra courses.
Assuming these efforts are successful, the materials, experience and research generated by this number of institutions would tip the balance and change common practice nationwide.
CA2. Research on Impact of Refocused College Algebra on Student Learning
We recommend two or three in-depth, multi-year, longitudinal research projects to study all aspects of the development and implementation of refocused College Algebra with an emphasis on determining the impact of well-designed and well-supported refocused College Algebra courses on student achievement and understanding as well as persistence in future mathematically-related coursework. Each project would
include a research team and a consortium of colleges and universities that are committed to refocusing College Algebra in a comprehensive manner by refining curriculum and professional development materials for use at their institution and providing extensive development and support of those teaching the
sections of the course: full-time faculty, adjunct instructors and/or graduate teaching assistants. In short, each institution would be attempting to offer a well-conceived refocused College Algebra program. While some support would be provided for the curriculum and faculty development components, the major focus would be on research assessing the effects of change on faculty and institutions and on student learning.
CA3. Electronic Library of Exemplary College Algebra Resources
We recommend the support of projects to provide departments and individual instructors with resources (electronic and video) to enable and equip them to teach refocused College Algebra. Resources would include classroom activities (examples and homework problems, data sets, spreadsheet files), extended projects (designed for collaboration and outside class work), samples of student work, and videos of lessons
that clearly show how an instructor might use these resources to create a student centered classroom in which students are actively engaged in learning that promotes critical thinking, communication skills, and higher order thinking skills.
Instructors often attend sessions and workshops on the teaching of College Algebra. There is much written on the subject. However, instructors, because of their own experiences regarding how they were taught (usually in a lecture format, large classes, and little engagement) are not able to internalize and easily implement the ideas they either hear about or read. Examples of exemplary lessons that demonstrate best
practices, classroom activities (with teacher notes), and long-term projects (with notes on implementation) could enable large numbers of departments and instructors to put the new ideas about a refocused College Algebra into practice. By utilizing technology, we can provide these to instructors who might use them to prepare for the next day’s lesson and/or get ideas on how they can improve their own lessons. The videos
would demonstrate the best practices using a lesson on content that is appropriate for College Algebra. Working with the projects that develop and deliver the workshops designed to change College Algebra, the creators of the videotaped lessons will first create models to demonstrate the important ingredients to changing teaching habits. As the resource expands, it will help to define content for College Algebra We recommend that five to ten small curriculum development projects be supported as well as two
major projects: one to collect, revise, catalogue and disseminate exemplary lessons, classroom activities, and long-term projects and a second to develop the videotaped models.
CA4. National Resource Database on College Algebra
We recommend a long-term project to prepare and maintain a national resource database that would include summary information on funded projects, textbooks, research articles, etc. An evaluation of College Algebra as related to retention and other student successes would be a central component of the database and all
projects would be required to provide this data as a part of their funding requirements. The project could be based on a TIMSS-like Model with several features:
• Curriculum Analysis (the Intended College Algebra Curriculum); the textbook content analysis, and the implemented curriculum.(Robitaille, 1993; Schmidt, McKnight et al, 1997; Schmidt, McKnight et al, 2001).
• Description of the nature of the faculty, the students, the courses, retention and success rates using qualitative and quantitative tools.
• Information that is known and that is being developed in research and evaluation of such research and implementation projects, e.g., NSF projects, Title III projects, Title V1 (traditionally Hispanic colleges), Tribal colleges programs, and related projects.
The knowledge of positive results and dissemination of these results is currently not widespread. There needs to be an ongoing identification of exemplary programs based on this meta-style analysis. After conclusion of the funding period, the work should be continued for another five years through the competitive grant process.
References
Agras, Norma (2005). Who Takes College Algebra? How Do They Do? Newsletter of the HBCU College Algebra Reform Consortium. 61, pp. 1.
Dunbar, Steven (2006). Enrollment Flow to and from Courses below Calculus. In Hastings, N. A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus: pp. 28–42.
Ellington, A. J. (2005a). A Modeling-based Approach to College Algebra. Academic Exchange Quarterly, 9 (3), pp. 131–135.
——— (2005b). A Modeling-based College Algebra Course and Its Effect on Student Achievement. Primus , 15 (3), pp. 193–214.
Ellington, A. J. & Haver, W. E. (2006). The Impact of Assessing Introductory Mathematics Courses. In B. Madison (Ed.), Assessment in Lower Level Collegiate Mathematics. Tallahassee , FL: Association for Institutional Research, pp. 76–96.
Fox, William & West, Richard (2001). College Algebra Drills or Applications? Primus, 11, pp. 89–96.
Ganter, Susan and William Barker, editors (2004). Curriculum Foundations Project: Voices of the Partner Disciplines. MAA Report, Mathematical Association of America, Washington, DC.
Gordon, Florence (2006). Assessing What Students Learn: Reform versus Traditional Precalculus and Follow-up
Calculus. In Hastings, N., A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus, pp. 181–192.
Gordon, Sheldon (2006). Where do We Go From Here? Creating a National Initiative to Refocus the Courses below Calculus. In Hastings, N., A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus, pp. 274–282.
Hastings, N (2006). A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus. Washington: Mathematical Association of America.
Herriott, Scott (2006). Changes in College Algebra. In Hastings, N., A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus, pp. 90–100.
Johnson, Laurie (2004). Making Mathematics Relevant: College Algebra Reform at Trinity College. Mathematics and Education Reform Newsletter, pp. 1–2.
Lutzer, E., Maxwell J. and Rodi, S. (2002). Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States: Fall 2000 CBMS Survey, Providence: American Mathematical Society.
McGowen, Mercedes (2006). Who Are the Students Who Take Pre-calculus? In Hastings, N., A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus), pp. 15–27.
Norwood, Karen (1995). The Effects of the Use of Problem Solving and Cooperative Learning on the Mathematics Achievement of Underprepared College Freshmen. Primus, 5, pp. 229–252.
Oty, Karla; Elliot, Brett; McArthur, John; & Clark, Byron (2000). An Interdisciplinary Algebra/Science Course. Primus, 10, pp 29–41.
Pollatsek, H., et al (2004). Undergraduate Programs and Courses in the Mathematical Sciences: CUPM Guideline 2004. Washington: Mathematical Association of America.
Robitaille, D. F., Ed. (1993). Curriculum Frameworks for Mathematics and Science. Vancouver: Pacific Educational Press.
Schmidt, W. H., C. C. McKnight, et al. (2001). Why Schools Matter: A Cross-national Comparison of Curriculum and Learning. San Francisco: Jossey-Bass.
38 Algebra: Gateway to a Technological Future
——— (1997). Many Visions, Many Aims: A Cross-National Investigation of Curricular Intentions in School Mathematics. Boston: Kluwer Academic Publishers.
Small, Don (2002). A Grade Report on Contemporary College Algebra. Vision-Potential: Newsletter of the HBCU College Algebra Reform Consortium. 39, pp 1.
Steen, L. and Madison, B. (2003). Quantitative Literacy: Why Numeracy Matters for Schools and Colleges. Princeton: National Council on Education and the Disciplines.
Wood, S. et al (2006). Beyond Crossroads: Implementing Mathematics Standards in the First Two Years of College, American Mathematical Association of Two-Year Colleges.
Addendum: College Algebra Guidelines
These guidelines represent the recommendations of the MAA/CUPM subcommittee, Curriculum Renewal Across the First Two Years, concerning the nature of the College Algebra course that can serve as a terminal course as well as a pre-requisite to courses such as pre-calculus, statistics, business calculus, finite mathematics, and mathematics for elementary education majors.
Fundamental Experience
College Algebra provides students a college level academic experience that emphasizes the use of algebra and functions in problem solving and modeling, provides a foundation in quantitative literacy, supplies the algebra and other mathematics needed in partner disciplines, and helps meet quantitative needs in, and outside of, academia. Students address problems presented as real world situations by creating and interpreting mathematical models. Solutions to the problems are formulated, validated, and analyzed using mental, paper and pencil, algebraic, and technology-based techniques as appropriate.
Course Goals
• Involve students in a meaningful and positive, intellectually engaging, mathematical
experience;
• Provide students with opportunities to analyze, synthesize, and work collaboratively on
explorations and reports;
• Develop students’ logical reasoning skills needed by informed and productive citizens;
• Strengthen students’ algebraic and quantitative abilities useful in the study of other
disciplines;
• Develop students’ mastery of those algebraic techniques necessary for problem-
solving and mathematical modeling;
• Improve students’ ability to communicate mathematical ideas clearly in oral and written
form;
• Develop students’ competence and confidence in their problem-solving ability;
• Develop students’ ability to use technology for understanding and doing mathematics;
• Enable and encourage students to take additional coursework in the mathematical
Sciences
Competencies
1. Problem Solving
Goals for students include
VI College Algebra 39
• solving problems presented in the context of real world situations with emphasis on model creation and interpretation;
• developing a personal framework of problem solving techniques (e.g., read the problem at least twice; define variables; sketch and label a diagram; list what is given; restate the question asked; identify variables and parameters; use analytical, numerical and graphical solution methods as appropriate; determine plausibility of and interpret solutions);
• creating, interpreting, and revising models and solutions of problems.
2. Functions and Equations
Goals for the students include
• understanding the concepts of function and rate of change;
• effectively using multiple perspectives (symbolic, numeric, graphic, and verbal) to explore elementary functions;
• investigating linear, exponential, power, polynomial, logarithmic, and periodic functions, as appropriate;
• recognizing and using standard transformations such as translations and dilations with graphs of elementary functions;
• using systems of equations to model real world situations;
• solving systems of equations using a variety of methods;
• mastering algebraic techniques and manipulations necessary for problem-solving and modeling in this course.
3. Data Analysis
Goals for the students include
• collecting (in scientific discovery or activities, or from the Internet, textbooks, or periodicals), displaying, summarizing, and interpreting data in various forms;
• applying algebraic transformations to linearize data for analysis;
• fitting an appropriate curve to a scatter plot and using the resulting function for prediction and analysis;
• determining the appropriateness of a model via scientific reasoning.
Emphasis in Pedagogy
Goals for the instructor include
• facilitating the development of students’ competence and confidence in their problem-solving abilities;
• utilizing and developing algebraic techniques as needed in the context of solving problems;
• emphasizing the development of conceptual understanding of the mathematics by the students;
• facilitating the improvement of students’ written and oral mathematical communication skills;
• providing a classroom atmosphere that is conducive to exploratory learning, risk-taking, and perseverance;
40 Algebra: Gateway to a Technological Future
• providing student-centered, activity-based instruction, including small group activities and projects;
• using technology (computer, calculator, spreadsheet, computer algebra system) appropriately as a tool in problem-solving and exploration;
• conducting ongoing assessment activities designed to determine when mid-course adjustments are warranted.
Assessment
• Assessment tools will measure students’ attainment of course competencies, including:
o solving problems and interpreting results using algebraic tools;
o building and interpreting models and predicting results;
o communicating processes and solutions orally and in writing;
o making quantitative and algebraic arguments;
o reading and interpreting data presented in various forms.
• Assessment tools will include
o individual quizzes;
o individual examinations;
o additional activities or assignments, such as
individual or group homework, projects, and activities;
individual or group oral presentations;
portfolios that demonstrate student growth;
group quizzes and exams.
• The course will be assessed by analyzing its effectiveness in:
o facilitating student achievement of the course competencies;
o positively affecting student attitudes about mathematics;
o preparing students for subsequent courses in mathematics and mathematics-dependent disciplines; preparing students for subsequent endeavors in and outside