David M. Bressoud March, 2007
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Last month I finished my discussion of the recommendations in the CUPM Curriculum Guide 2004, but I want to use these Launchings to continue to explore issues of concern to CUPM, issues that span the range of our undergraduate courses and programs in the mathematical sciences. In February’s column, I discussed the disappearance of two standard courses at the advanced end of what we teach. This month, I wish to turn your attention to the elephant in the living room of most mathematics departments, the College Algebra course. I wrote about this course in October 2005. The reason for returning to it now is that CRAFTY, the CUPM subcommittee on Curricular Renewal Across the First Two Years, has just finished its College Algebra Guidelines, a document endorsed by CUPM and now available for download at www.maa.org/cupm/crafty.
There is a problem defining College Algebra. It usually—but not always—is a course that carries college credit. It usually—but not always—covers material that should have been mastered in high school. Despite the ambiguity of the name, most of us can identify the course or courses in our own departments that fit under this rubric. In some places, it is synonymous with or goes under the name of Precalculus; in others it is a prerequisite to Precalculus. The intention of such a course should be to prepare students for success in more advanced mathematics, especially Calculus. In actuality, few students who take College Algebra go on to successfully complete Calculus. All too often, this course becomes a place to park students who need to satisfy a quantitative distribution requirement and who have no intention of taking any further mathematics.
According to the latest CBMS (Conference Board of the Mathematical Sciences) data, this course constitutes a fifth of the mathematics taught in 2-year colleges and almost half of the mathematics taught at 4-year colleges. CBMS groups college math courses into Precollege (courses such as elementary algebra that often do not carry college credit), Precalculus or Introductory (this includes College Algebra and the entire constellation of courses that are roughly at the level of College Algebra including Trigonometry, Algebra & Trigonometry combined, Elementary Functions, Introduction to Modeling, and Precalculus), and Calculus or Calculus Level (for 2-year colleges this is mainstream Calculus I through Differential Equations, for 4-year colleges it includes these courses plus Discrete Mathematics and Linear Algebra). Totals for 2-year colleges include statistics and computer science. Statistics and computer science courses at 4-year colleges are counted separately. Tables with graphs are shown below. Data is taken from [1,2,3].
2-year college mathematics enrollments, fall term (thousands)
Year |
Precollege |
Precalculus |
Calculus |
Total |
1980 |
441 |
180 |
86 |
1048 |
1985 |
482 |
188 |
97 |
1034 |
1990 |
724 |
245 |
128 |
1393 |
1995 |
800 |
295 |
129 |
1498 |
2000 |
763 |
274 |
106 |
1386 |
2005 |
964 |
321 |
107 |
1696 |
4-year college mathematics enrollments, fall term
(thousands) [4]
Year |
Precollege |
Introductory |
Calculus Level |
Total |
1980 |
241 |
603 |
590 |
1525 |
1985 |
251 |
593 |
637 |
1619 |
1990 |
261 |
592 |
648 |
1621 |
1995 |
222 |
614 |
539 |
1471 |
2000 |
218 |
723 |
570 |
1614 |
2005 |
201 |
706 |
587 |
1607 |
Despite the huge number of students enrolled in this course, few colleges or universities are willing to devote significant resources to teaching it. The result is that it ill serves both those for whom this is the last opportunity to develop an appreciation for mathematics and those who would continue into STEM (Science, Technology, Engineering, and Mathematical Sciences) or other quantitatively rich majors. The refusal to face and deal with problems in this course is short-sighted. A poorly taught course can perpetuate the popular perception among our students and graduates that mathematics consists of arcane procedures in which one can be trained, but which have little real value and are beyond comprehension by ordinary mortals. It can increase the isolation of the mathematics department as other departments see us constricting the supply of potential majors to their programs.
Whether your College Algebra, by whatever name it may go, seems to be working or not, the first step in building and maintaining a healthy program is to know who your students are. What is their background? Why are they taking this class? What are their goals? How do other departments see this course, and what are their expectations for it? And then you need an honest assessment of your success in meeting these goals and expectations.
Even if all seems to be well, this time of assessment is also an opportunity to reassess your own departmental goals for this course. CRAFTY’s College Algebra Guidelines can be helpful here. They suggest nine goals for this course, goals that begin with strengthening algebraic abilities and developing student mastery of algebraic techniques, but that go on to encompass the development of logical reasoning skills, competence and confidence in problem solving, experience in analysis and synthesis of mathematical ideas, the ability to communicate mathematics in both oral and written formats, and an appreciation for and the ability to work with appropriate technologies. The remaining two goals are of a softer quality, but are no less important: to provide a “meaningful, positive, intellectually engaging mathematical experience” and to “enable and encourage [I would add ‘entice’] students to take additional coursework in the mathematical sciences.” The College Algebra Guidelines go on to flesh out these goals in greater detail, to discuss how to attain them, and to suggest means of assessing the success of our efforts.
In particular, CRAFTY has focused on the teaching of College Algebra within the context of modeling as a means of addressing these goals. To measure the effectiveness of this approach, CRAFTY has been conducting a comparison study at several universities. At each institution, some sections of College Algebra are taught with a modeling emphasis, others without modeling. Preliminary data show that students in the course with modeling emphasis receive fewer Ds, Fs, and Ws. The long term implications of this use of modeling—the effect on enrollment and performance in subsequent courses—are being followed, but data is not yet available. Watch the CRAFTY website for additional information on this experiment.
It may seem that CRAFTY has taken two modest goals that are already tough to achieve and only increased our difficulties by adding many others. The fact is that those other goals always have been, or should have been, components of what we mean when we say that we want to strengthen the algebraic abilities of our students. They should be part of the package that we desire for our students. To lose sight of them almost certainly means so narrowing our focus that failure becomes likely if not inevitable.
Broadening our vision of what we want to accomplish in these courses also makes it much harder to continue to deceive ourselves that these courses can be taught on the cheap. To do the job we owe to our students and to our colleagues in other departments, we must draw on the time, energy, and effort of our best faculty.
[1] Lutzer, David J., James W. Maxwell, and Stephen B.
Rodi, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences
in the United States, Fall 2000 CBMS Survey, American Mathematical Society,
Tables TYR.4, page 131 and A.1, pages 175–178.
[2] CBMS Survey, Fall 2005, preliminary table S.2 at www.math.wm.edu/~lutzer/cbms2005/Ch1Pdf/05Ch1TableS.2.pdf.
[3] CBMS Survey, Fall 2005, preliminary table TYE.4 at www.math.wm.edu/~lutzer/cbms2005/Ch6Pdf/05Ch6TableTYE.4.pdf.
[4] I cannot help but point out that while the total number of Bachelors degrees has increased by over 50% during the past quarter century (from 934,800 in 1980–81 to 1,439,262 in 2004–05), the number of students taking any course in mathematics at 4-year institutions has been flat. Note that this excludes statistics, computer science, and any courses taught outside a mathematics department.
Do you know of programs, projects, or ideas that should be included in the CUPM Illustrative Resources?
Submit resources at www.maa.org/cupm/cupm_ir_submit.cfm.
We would appreciate more examples that document experiences with the use of
technology as well as examples of interdisciplinary cooperation.
David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, he was one of the writers for the Curriculum Guide, and he currently serves as Chair of the CUPM. He wrote this column with help from his colleagues in CUPM, but it does not reflect an official position of the committee. You can reach him at bressoud@macalester.edu. |