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Don Small
U.S. Military Academy
Introduction
This paper presents a case for transforming traditional college algebra from a failed program attempting to prepare students for calculus to one that enables students to address the needs of society, the workplace, and the quantitative aspects of disciplines. Characteristics of improved college algebra programs are described as well as the symbiotic relationship between a transformed college algebra and quantitative literacy.
Traditional college algebra courses are not working [3]. That was the strong consensus of the participants in a recent Conference to Improve College Algebra, held at the U. S. Military Academy, February 7-10, 2002. This conclusion was based on the courses’ outdated content, high FWD rates, and on the negative impact these courses have on student perceptions of mathematics. The large number of students enrolled in College Algebra creates an urgency to transform these courses into ones that do work.
Number of Students
College algebra has the largest enrollment (approximately 400,000 in fall 2000) of any college credit-bearing mathematics course [2]. (Another approximately 100,000 students are enrolled in combined college algebra /trigonometry courses.) This enrollment is approximately equal to the combined enrollment in all mainstream calculus courses, having increased from 73% in 1980. Precalculus has the second largest enrollment, which is about half that of college algebra. Almost all students are required to pass one of these courses, or a higher level mathematics course, as part of college distribution or major requirements. Some states, such as Texas and Louisiana, have legislated that students must pass a mathematics course at or above the level of college algebra before they are permitted to enter their third year of college. Thus college algebra and precalculus form the gateway to college mathematics for the large majority of students.
The combined enrollments in college algebra and precalculus increased 59% since 1980, while the enrollment in calculus I has remained relatively stable. This calls into question the traditional role of college algebra as a preparatory course for calculus. Commenting on this situation, Mercedes McGowen in her paper, “Redefining the First College-level Mathematics Course,” which she presented at the Conference to Improve College Algebra, said:
“A comparison of enrollments for all precalculus courses – particularly college algebra – with calculus I over the past twenty years indicates that the increasing enrollment in these courses has had little, if any, impact on the calculus I enrollments. There is a broad sense that the traditional college algebra course is
not the appropriate course – particularly for those not going on in the hard sciences. In fact, it appears that large numbers of students do not have learning experiences
in the college algebra courses which generate enthusiasm or enrollments in
calculus I and a subsequent program of studies in mathematics or other math-intensive fields.”
What then should be the role or roles of college algebra? If it is to be a stepping stone into calculus, then the traditional course must change as presently 10% or less of its students successfully complete calculus I. If college algebra is to be a service course in terms of satisfying distribution requirements, then it must change to emphasize applications across the disciplines. If it is to be a mathematics appreciation course, then it must change to emphasize student involvement in problem solving and discovery.
Content
The traditional content – factoring linear and quadratic polynomials, radicals, partial fractions, absolute values, inequalities, systems of equations, and so on---has its origins in the 1950s era when college algebra came into the curriculum as a capstone course for high school mathematics. This was long before the advent of graphing calculators or computer algebra systems and thus development of (hand) manipulation skills of algorithmic procedures was important for students going on to calculus. This thinking along with that era’s emphasis on drill and symbol manipulation continues to shape traditional courses. Today, however, these courses are not successful in launching students into a standard calculus track. The largest cohort (approximately one-third) of students passing college algebra go into schools of business. However their need for elementary data analysis, modeling real-world problems, using technology, gaining small-group experience, and developing communication skills are not addressed in traditional college algebra courses. In spite of containing numerous exercises involving exponential functions, the traditional courses do not prepare social or life science majors to model growth situations or economic majors to model the multiplier effects of increased spending. Nor does the traditional content address citizenship or workplace needs such as the ability to interpret data, construct a budget, design a schedule, plan a multifaceted event, understand round off, or optimize a procedure. In the view of Arnold Packer, Chair of the SCANS 2000 Center at Johns Hopkins University’s Institute for Policy Studies, “Interpreting data is more important than manipulation of algebraic skills that can be computerized.”
FWD rates
The percentage of students who receive a grade of F or D or who withdraw from the course---are unacceptably high. Several studies place the FWD rate in the 40-60% range, although there are several schools in which this rate is considerably higher. In particular, one large urban community college system has a withdrawal rate of 50%. There are many contributing factors to the FWD rate---high school preparation, placement, content, attitude, pace of the course, pedagogy, out of school commitments, etc [1]. However, the fact remains that several alternative college algebra programs, which focus on real-world problem solving, have lowered the FWD rate by 15-25 percentage points. Students in these programs are drawn from the same pool and were subject to the same placement procedures as students in the traditional programs.
Attitudes
Negative attitudes, generated by high FWD rates, abstract content, and low expectations are pervasive among both faculty and students in traditional college algebra courses. These attitudes create additional barriers to the majority of students struggling to complete a course in which they see little relevance. Chris Arney, Dean of Science and Mathematics at St. Rose College, said:
“Traditional College Algebra is a boring, archaic, torturous course that does
not help students solve problems or become better citizens. It turns off students
and discourages them from seeking more mathematics learning.”
Possibly more devastating is the fact that these attitudes influence life-long views of mathematics as College Algebra is the terminal mathematics course for the majority of students,
The urgency for creating alternative college algebra courses is heightened by the tremendous loss in student potential resulting from the traditional programs. Because of its gateway position in undergraduate programs, traditional college algebra courses block the academic opportunities and plans of approximately 200,000 students per semester. As educators, we can not accept this cost.
Vision for improved college algebra programs
The vision espoused at the Conference to Improve College Algebra was to create programs that empower all students to become competent and confident problem solvers. These real-world problem-based programs address the quantitative needs of other disciplines as well as those for citizenship and the workplace. The problem solving is to be understood in the sense of modeling as illustrated in the following diagram.
Figure 1. Problem Solving/Modeling Process
The symbol
manipulation type exercise that characterizes traditional courses
only involves the right-hand side of the Modeling Process diagram. An
example is to ask the student to factor the polynomial,
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.
The polynomial is the Mathematical Model and the factors,
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and
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,
are the Mathematical Result. This example is typical of the
traditional program in that the model is provided and only the
results are sought. In contrast, problem solving in an improved
college algebra program begins and ends within a real-world setting.
Students are expected to first create a mathematical model. This
often involves a communication skill of transforming a written or
verbal description into a mathematical description. The results are
then obtained, often by means of technology. The final stage is to
interpret the results in light of the real-world setting. This may
result in modifying the model to obtain a more realistic result
and/or “what-iffing” in order to gain a deeper
understanding of the situation. The interpretation stage encourages
conceptual abstraction that facilitates transferability to other
(mathematically) similar situations. The soda can problem provides a
nice illustration.
Soda can problem
Determine the dimensions of a 12 fluid ounce (355 ml) aluminum soda can in the shape of a closed cylinder that minimizes the amount of aluminum in the can. Use the following mathematical model:
Assume that the thickness of the aluminum is uniform, say one unit.
Let r be the radius of the can measured in centimeters.
Let h be the height of the can measured in centimeters.
Conversion factor:
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Objective function: minimize amount of aluminum,
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subject to the
constraint equation:
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.
Expressing h
in terms of r in the constraint equation and then substituting
for h in the objective function yields an aluminum
function in terms of r:
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Figure 2.
We see from Figure 2, that the aluminum function achieves a minimum value when r ~ 3.84
cm and thus h ~ 7.66 cm. Interpretation of these results suggests the height of the can should be approximately equal to its width. Although this makes sense mathematically, it does not reflect the general shape of a commercial soda can. This calls for a rethinking of the model, starting with the assumption. An examination of a soda can suggest that the top and bottom are thicker than the sides. (Students are encouraged to go to a physics lab to measure the thicknesses of a commercial soda can.) Reworking the model with the assumption that the ends are twice as thick as the side yields a minimum for r = 3.05 cm and h = 12.1 cm. This is a reasonable approximation to the dimensions of a commercial soda can. Reflecting on the problem, students are asked to identify similar situations requiring the optimization of an objective function subject to constraints.
The exercise problem characteristic of traditional college algebra courses is contrasted with the modeling problem characteristic of improved college algebra courses as follows:
1. The exercise problem emphasizes algebraic manipulation while the modeling
problem emphasizes conceptual understanding and realistic applications.
2. The exercise problem emphasizes solving in isolation while the modeling
problem emphasizes solving in context.
3. The exercise problem lends itself to drill work while the modeling problem
lends itself to inquiry.
4. The exercise problem remains in isolation while the modeling problem leads to
conceptual abstraction and transferability.
Improved college algebra courses incorporate strong communication components---reading, writing, presenting, and listening. For instance, an objective of these courses is for students to be able to draw informed opinions from a news article containing data. The use of technology to enhance conceptual understanding as well as for computing is another strong component of these courses. For example in the Soda Can Problem, the value of the radius that minimizes the amount of aluminum was determined from the plot of the aluminum function. The pedagogy associated with improved college algebra courses recognizes that student experiences in constructing their own understanding are more important than coverage of topics. Thus lecturing is restricted to a minimum in order to maximize opportunities to engage students in activities and small group projects.
The major characteristics of an improved college algebra program include:
Real-world problem-based: Introduce a topic is introduced through a real-world problem and then develop the mathematics necessary to solve the problem. Example problem: Schedule a multi-faceted process.
Modeling: Transform a real-world problem into a mathematical construct by using power and exponential functions, systems of equations, graphs, and difference equations. Primary emphasis is placed on creation of a model and on interpretation of the results. Example: Model the stopping time versus speed data presented in a driver’s manual by plotting the data and fitting a curve to the plot. The curve defines a stopping time function. Interpret how well this stopping time function models reality at small speeds. Revise the model, if necessary, to account for zero stopping time at zero speed. Use the resulting (revised) function to predict stopping times for speeds not given by the data. Revise the model to account for different road surfaces.
Elementary data analysis: Display data, extract information from data, and extract knowledge from the information. Example: Nutrition labels on soda cans provide an example for extracting information from data. The label on a 12-ounce Adirondack Ginger Ale can lists 55 mg of sodium representing 2% of the recommended daily value based on a 2,000 calorie per day diet. In comparison the label on a 5.5-ounce Welch’s Orange Juice can lists 15 mg of sodium representing 1% of the recommended daily value. What information concerning the recommended daily value of sodium can be extracted from this data?
Communication: Emphasize communication skills as needed in society and the workplace as well as in academia---reading, writing, presenting, and listening. Example: Students learn how to read, understand, and critique news articles that include quantitative information and to make informed decisions based on the articles.
Small group projects involving inquiry and inference: Provide experiences empowering students to become exploratory learners. Example: Analyze the soda preference of students by conducting a survey and comparing the results with data from the school’s dining hall or a local fast food restaurant.
Appropriate use of technology: Use technology to enhance conceptual understanding, visualization, and inquiry, as well as for computation. Example: Explore a model for paying off a credit card debt by changing the monthly payment, interest rate, size of debt, and so on. Plot the results to visually compare the different scenarios.
Student-centered rather than instructor-centered pedagogy: Place the focus on student learning rather than on covering content. For example, maximize hands-on activities and minimize lecturing.
What you test is what you get---WYTIWYG or rather---what you test is what students focus on---WYTIWSFO. What does this say about assessment in real-world problem based courses designed to help students become competent and confident problem solvers? How can the seven aspects listed as characteristic of improved college algebra courses be assessed? These are very difficult questions to answer and answering them may be the biggest barrier to making curricular change. We can assign students to write an essay, but then how do we grade it? Do we grade grammar, spelling, number of words, and so on or grade just on meaning? Do we feel competent to do this? Traditionally mathematics courses have addressed problem solving by focusing on well-defined, well-structured problems. However, many problems that arise in society or the workplace are ill defined and ill structured. For example, what are the three best predictors of success in college?
Traditional testing is not an adequate means of assessing student performance in an improved college algebra course. Traditional methods are teacher-centered in the sense of minimizing grading time, maximizing coverage, and focusing on well-defined, well-constructed skill type questions. In contrast, student-centered assessment needs to focus on process as well as results. It involves more subjectivity, more creativity, and more grading time than do the present traditional methods.
Collaboration with faculty in other disciplines and with representatives from the workplace is important to the improvement of college algebra and in on-going assessments of the programs. This collaboration is particularly important in the development or transformation of a course in order to ensure that content will align with student interests and needs. In addition, the collaboration establishes bridges to other disciplines that enhance opportunities for strengthening quantitative literacy throughout the academic program.
The past five years have seen the rise of a national movement to improve college algebra.
This is evidenced by the large increase in the number of sessions at professional meetings devoted to improving college algebra and the number of college algebra workshops and conferences such as the Conference to Improve College Algebra held at West Point in 2002. The growth in the number of elementary modeling courses being offered as alternatives to college algebra as well as texts being written for improved college algebra courses are other indications of this national movement. Four samples of such texts are listed in the References [5].
Improved college algebra: A base for quantitative literacy programs
Improved college algebra courses provide the focus, content, and interdisciplinary aspects on which to establish college-wide quantitative literacy programs. Lynn Steen describes “quantitative literacy,” sometimes called “numeracy,” as the quantitative reasoning capabilities required of citizens in today’s information age [4]. Extended experience in problem solving in the modeling sense provides these quantitative reasoning capabilities. Thus a quantitative literacy program involves a two-step process.
Students understand the problem-solving process of transforming a
problem situation into a mathematical description (model), solving, and
then interpreting the results; and students develop confidence in their ability
to apply this modeling process.
2. Students gain extensive practice in problem solving in a variety of situations.
With the emphasis on modeling, communication, and appropriate use of technology, improved college algebra courses provide the ideal opportunity for students to address the first step. Development of student self-confidence is facilitated by applying the problem-solving process in a variety of settings under the mentoring environment of a first year course. The second step, practice, needs to be addressed by all faculty so that students continue to hone their problem solving skills and strengthen their self-confidence as they progress through their academic careers. The interdisciplinary collaboration associated with improved college algebra courses facilitates this joint approach to developing students’ quantitative capabilities.
An emphasis on quantitative literacy throughout the curriculum adds meaning and purpose to college algebra courses. In particular, attention to problem solving in the modeling sense throughout the curriculum serves as a laboratory for college algebra and thus extends these courses into programs. Thus the symbiotic relationship between college algebra and quantitative literacy is to the betterment of both programs.
Summary
As the primary college gateway course for thousands of students, college algebra has a great potential as a service course to address the quantitative needs of society, the workplace, and other disciplines. However traditional college algebra, based on a 1950’s curriculum as part of the preparation sequence for calculus, does not fulfill this potential. To do so, college algebra needs to be transformed by refocusing both the content and pedagogy in order to develop competent and confident problem-solvers. The content needs to be real-world problem-based, to emphasize problem solving in the modeling sense, and to include elementary data analysis. Student-centered pedagogy involving development of communication skills, appropriate use of technology, and small group activities and projects should be designed to create student confidence and positive experiences. Student-centered rather than instructor-centered assessment procedures needs to be developed.
The gateway function of college algebra means that transformed or improved courses provide a basis on which to develop a college-wide quantitative literacy program. The interdisciplinary collaboration that is important to the development and ongoing assessment of an improved college algebra course provides opportunities to link problem solving to the quantitative needs of other disciplines. In this sense, these disciplines provide laboratory experiences for college algebra students. Interdisciplinary collaboration is essential to realize the potential of the symbiotic relationship between college algebra and quantitative literacy programs.
Improved college algebra courses better serve the approximately 90% of students who do not enter into math-intensive programs as well as providing a more effective preparation for those going on to calculus I than do the traditional algorithmic type courses. Several improved college algebra courses have shown that FWD rates can be significantly lowered and positive student attitudes obtained. More importantly these courses have demonstrated, in a variety of schools, ways in which college algebra can be transformed from extracting an unacceptable cost to providing a valuable asset in educational programs.
References
[1] Herriott, Scott R., “Changes in College Algebra,” paper in this volume.
[2] McGowen, Mercedes A., “Redefining the First College-level Mathematics Course,” paper presented at the U.S. Military Academy Conference to Improve College Algebra.
[3] Small, Don, “An Urgent Call to Improve Traditional College Algebra Programs,” MAA Focus, May/June 2002 issue. (Summary of the Conference to Improve College Algebra held at the U.S. Military Academy, February 7-10, 2002.)
[4] Steen, Lynn A., Mathematics and Democracy: The Case for Quantitative Literacy, the Woodrow Wilson National Fellowship Foundation, Washington, D.C., 2001
[5] Samples of texts written for improved College Algebra programs:
Crauder, Bruce, Evans and Alan Noell, Functions and Change: A Modeling Alternative to College Algebra, Houghton-Mifflin, Boston, 1999
Herriott, Scott R., College Algebra through Functions and Models, Brooks-Cole, Pacific Grove, CA., 2002.
Kime, Linda A. and Clark, Judy, Explorations in College Algebra, 2nd ed., Wiley, New York 2001
Small, Don, Contemporary College Algebra: Data, Functions, Modeling, 5th ed., McGraw-Hill, New York, 2004