Presented at the
Conference to Improve College Algebra
U.S. Military Academy
West Point, NY 10996
February 7-10, 2002
By Arnold Packer
College Algebra can stay on its current path of a technique-driven curriculum. In this case, techniques are emphasized and applications or problems are chosen that are susceptible to the specific technique. Data shows this approach is unsuccessful in attracting students who have other choices or encouraging students to proceed on to calculus.
Or, College Algebra can become problem-based quantitative mathematics. In the latter case, the curriculum-design task is choosing generic problems and the mathematical techniques needed to solve them. In this alternative approach, College Algebra is divided into two roughly equal parts. The first half of this new course is "Mathematics for Planning." Students learn how to efficiently allocate four kinds of resources: money, as in budgets; time, as in schedules; space, as in architecture or space planning; and staff, as in staff requirements. The second unit is called "Modeling Systems" and teaches students to understand, monitor, and design systems. Students learn ? at a deep level ? what exactly graphic and symbolic representations of reality imply. A pilot study, using the second approach in Algebra II led to student success.
As another paper to be presented at this conference makes clear, College Algebra is the last mathematics course many students take. A majority may enter the classroom having already decided that it will be their final mathematics course. Recent data, contained in the preliminary report of an MAA Task Force, indicate that only one in ten College Algebra (CA) students go on to take a full length calculus sequence. Not coincidentally, one in ten of these students are enrolled in a mathematics intensive field i.
Many would skip College Algebra if they did not have to pass it to get the degree they need to enter their chosen career field. Enrollment in CA tends to fall dramatically when colleges make quantitative reasoning or intermediate algebra the requirement. Finally, a few years after finishing the course, getting their degree, and starting their professional life, they cannot recall anything they learned. Or, equivalently, they have never used anything they learned in College Algebra.
All of this is unfortunate and related. Mathematics courses that seem hard, boring, and irrelevant prior to College Algebra establish the expectation that College Algebra will be more of the same. Moreover, the course ? as conventionally taught ? does nothing but confirm the foreboding.
Look at a typical description:
This course is a modern introduction to the nature of mathematics as a logical system. The structure of the number system is developed axiomatically and extended by logical reasoning to cover essential algebraic topics: algebraic expression, functions, and theory of equations.
Who decided that "algebraic expression, functions, and theory of equations" is essential, and if so, essential to who or what? The course covers the following topics: Radicals, Complex Numbers, Quadratic Equations, Absolute Value and Polynomial Functions, Equations, Synthetic Division, the Remainder, Factor, and Rational and Conjugate Root Theorems, Linear-Quadratic and Quadratic-Quadratic Systems, Determinants and Cramer's Rule, and Systems of Linear Inequalities.
That is a long list of topics; yet, it is only half the topics listed. How much can students learn in two days and what will they remember? How close is this official curricula to the one actually taught? Is this the topic list the mathematics department would present if it were an elective course for those not majoring in mathematics or engineering or intending to go on to graduate or professional school? For too many students it looks like ? and is -- a painful experience that they would prefer to skip.
Presumably, these kinds of questions led to this conference at West Point. What should CA accomplish? One way to answer is to consider how to evaluate a changed CA course. What empirical evidence would this audience want to see before they called a new CA course successful? More students taking subsequent mathematics courses would please some. More students leaving college who could be deemed quantitatively literate would please others.
College algebra can stay on its current path of a technique-driven curriculum. In this case, techniques are emphasized and applications or problems can be chosen that are susceptible to the specific technique. Or, CA can become problem-based quantitative mathematics. In the latter case, the decision is what generic problems should be included and what mathematical techniques are needed to solve them.
Quantitative Literacy and SCANS
The term Quantitatively Literacy is defined extensively in Lynn Steen's book Mathematics and Democracy, the Case for Quantitatively Literacy. Let me be specific. To me, quantitative literacy implies, at a minimum, the mathematical competency to solve problems in the SCANS domains.
SCANS is the acronym for the Secretary's Commission on Achieving Necessary Skills. Ten years ago, this commission of 31 senior Human Resource executives and educators and union officials issued two reports, entitled: What Work Requires of Schools and Learning a Living. The so-called SCANS skills include the ability to use basic and advanced skills, such as mathematics and problem-solving to solve problems in five domains. The two math-intensive SCANS problem domains are planning and systems.
Let me be specific. I would divide CA into two parts and spend one-half the semester on each. The first half of this new course would be called something like "Mathematics for Planning" or "Mathematics for Resource Allocation."
Planning, according to SCANS, means the process of allocating four kinds of resources. The four are: money, as in budgets, time, as in schedules, space, as in architecture or space planning, and staff, as in staff requirements. What mathematics skills are needed to solve problems in these domains? In what follows I'll be mentioning some examples from applications we are running in Baltimore high schools' Algebra I and II and community college algebra (and other) courses.
Preparing or evaluating budgets require the ability to work with matrices expressed in spreadsheets as well as the algebraic equations imbedded in the program. These tasks do not require inverting a matrix manually or many of the things taught in a matrix algebra class. Mathematics faculty at a community college told me that teaching spreadsheets was not "mathematics" and they do not do it in their classes. The algebra requirements do not include irrational or complex numbers; but, instead, require knowing how to express complex relationships algebraically. For example, we ask our Algebra I students to figure costs of printing brochures where there is a step function between 1000 and more copies.
This application takes place in the context of developing a marketing plan for a tourism company. The ultimate mathematical question is choosing between a four-page and eight-page brochure; the former is less costly but also less effective.
Schedules require the ability to handle different numbering systems ? decimal, clock-time based on 60, and calendar-time based on 28, 30, 31, and (in leap years) 29 days in the month. Students should know how to figure out when a heat-treatment will be completed if it has to be treated for 108 hours. Students should be able to use Gantt and Pert charts and Harvard Project Planner to plan more complex undertaking. In our Baltimore work, students have to schedule their presentations and all the tasks that need to be completed on the way.
Space planners need to understand geometry but not necessarily proving something about the interior angles of parallelograms. In another application, dealing with a business plan for a retail store, students need to design their store. This includes looking at sight lines for discouraging shoplifting and dealing with the conical shape of light emanating from overhead fixtures to lay out a lighting plan.
Students should know that most real world allocation situations require trade-offs to live within constraints. They should also know something about determining objective functions: What does the decision maker want to maximize and minimize? In business, it may be profits or costs. In a health application, it might be some weighted average of efficacy and side-effects. In one of our community college applications, students use decision theory to locate a factory. They balance objectives for environmental problems and economic cost. By the way, the model includes some Gaussian equations for particle dispersal. Students know it is there, but need not really understand it.
To an economist, resource allocation problems bring to mind optimization under constraints. This, in turn, brings to mind differential calculus and linear programming. Most realistic problems turn out to much too complex for calculus but it is worthwhile for students to understand the concepts of maxima, minima, and inflexion points. They should know the difference between the first and second derivatives and rates of change and acceleration. Actually solving calculus problems is, for almost any job except teaching calculus, unnecessary.
Linear programming is more useful. In another application, students (community college this time) are asked to put together a plan for introducing a new product (an electric car). Inequalities and very simplified linear programming are used to find the optimum allocation of the advertising budget among alternative media. The larger problem, which includes allocating funds to training and R&D, is too complicated, however. Successful students use spreadsheet simulation to find a satisfactory, if not optimum, solution.
The second half of my recommended College Algebra course would cover systems, the second of the math-intensive SCANS problem domains. The unit would be called "Modeling Systems." The SCANS commission recommended that students be able to understand, monitor, and design systems.
Understanding mathematical models of systems requires that students grasp ? at a deep level ? what exactly graphic and symbolic representations of reality imply. Clearly, this lesson should be repeated in physical and social science courses, but it is crucial in applied mathematics. Students completing two years of college should comprehend positive and negative feedback loops. All of this requires some coordination with the social and physical science courses. Can students understand an epidemiological model and the positive feedback loops that let the epidemic spread? Do they grasp the negative feedback that finally brings the epidemic to an end? Can they understand a model of an inventory control system and its stability-enhancing negative feedback? (The latter is part of what is needed in our application on the business plan.) For the vast majority of students, being able to answer these and similar questions is more important than Cramer's rule.
One of our other community college applications deals with statistical process control. Students obtain randomly generated data from a production process, make hypotheses regarding the cause of the quality problems, and test their hypotheses. Clearly, statistics is involved.
To understand or build models of systems, students should know something about linear models with and without uncertainty. They need to know some functions (but not necessarily trigonometric functions) such as the normal, binomial, and exponential. Do they know the 80/20 rule? Most importantly, can they use and build mathematical models to predict system performance?
In neither part of this new CA course would I use "x" or "y" as variables. Let students visualize the reality by using meaningful symbols: p for population and t for time in the epidemiological models, r for revenue and c for cost, and so on. Many think this a horrible idea. The very power of mathematics is its ability to generalize ? to use the same technique in a variety of fields.
But the abstract "general" approach to mathematics is not necessarily a big favor for those who "love" mathematics and science. "No scientist thinks in equations," said Einstein, who employed visual images and muscular feelings. The mathematician S.M. Ulam said that he uses "mental images and tactile sensations to perform calculations, replacing numerical values with the weights and sizes of imagined objects." Joshua Lederberg becomes "an actor in a biological process, to know how [to] behave as if I were a chromosome" ii
Lynn Steen, in his introduction to Why Numbers Count refers to scientific mathematics in which mathematical variables always stand for physical quantities ? "a measurement with a unit and implicit degree of accuracy"iii. Jim Rutherford says "? citizens need to possess certain basic mathematical capabilities understood in association with relevant scientific and technological knowledge." (Italics in the original.)iv
Mathematics educators, properly, want their students to understand the power of mathematics to solve general problems, ones that are not rooted in an existing situation. That point can be made near the end of the mathematics course and demonstrated to students. Teach that the equation for velocity can be used in many equations relating to change. True generality can be saved for those mathematics students who will still be taking mathematics courses beyond CA.
There may be more such students if mathematics was less abstract in the earlier years of school. Indeed, the empirical evidence is that mathematics educators are not achieving their goals by their current practice. By the year 2000, U.S. students were to be first in science and math. By any measure they are not. NAEP scores in mathematics (for 17 year olds) in 1996 were only three percent better than they were in 1982. Their average NAEP score was 307, meaning that the average 17-year old can compute with decimals, fractions and percents, recognize geometric figures, solve simple equations, and use moderately complex reasoning. The average among blacks (286) and Hispanics (292) were below 300, meaning the ability to do the four arithmetic operations with whole numbers and solve one-step problemsv. Over half of the students entering the California College system need to take a "developmental" course. Over one in four college freshmen feels that they will need tutoring or remedial work in math. This compares to one in ten for English, science, and foreign language. vi
What happens when students get to college ? which they are doing in greater and greater numbers. In the paper she prepared for this conference Mercedes McGowan asks "why we are attracting fewer and fewer students into our mathematics-intensive programs?" She points out that, in four-year colleges, enrollment in mainstream calculus is declining in absolute and relative terms. About 405,000 such students were 24% of mathematics enrollment in 1980; by 2000 the enrollment had fallen by 53,000 and the share to 20 %. In light of these trends I would hope that all recognize that the practice of mathematical education must be improved ? and quickly.
Why Is It Important and for Whom?
Many of you may know about Bob Moses and his Algebra Project. Moses, Harlem-born in 1935, attended Hamilton College graduate school at Harvard University. During the 1960's, Moses worked with the Student Nonviolent Coordinating Committee (SNCC) to increase voter registration in Mississippi (for which he was later awarded a Macarthur "genius" award). In the 1980's he decided that "the absence of mathematics literacy in urban and rural communities throughout this country is an issue as urgent as the lack of registered Black voters in Mississippi was in 1961." [p.5]
Moses makes the point that symbolic representations of reality are the keys to using the new technology and algebra is the place where youngsters "learn this symbolism." [p 13] His effort is the Algebra Project that is focused, with good results, on middle schools with the hope that it will lead to minorities and others going on to college and being able to take college-level, rather than remedial, courses. He, Moses, mentions one college where 90% of the entering minority students take the non-credit course. Other studies indicate that taking non-credit remedial courses is a predictor of dropping out of college ? especially if student fails the course.
I came across some numbers last week, by Gerald Bracey that I found startling. Remember, we ? the United States ? was supposed to be first in mathematics and science by the year 2000. The study of 28 industrial countries, by the Organization for Economic Cooperation and Development (OECD) found instead we were only average in mathematical literacy ? the ability to apply mathematics to "real life" questions. If, however, we isolate the data for white students the U.S comes in 7th. Black and Hispanic students come in 27th in the list. Something must change at all levels of mathematics education, and college algebra ? typically the first and last college mathematics course ? has to change too.
Why is mathematics a required course in most colleges? A cynic might read the following, which I quote. "There are many factors that will make change difficult. Courses below calculus, like college algebra, are often departmental "cash cows." They may have large enrollments and can be taught relatively cheaply by part-timers or TA's." The same document, more idealistically, says "?students will need to develop mathematical thinking to support lifelong learning?to read and assess quantitative arguments they will encounter?for diverse workforce needs? [and] prepare them for potentially multiple job changes." [Emphasis added.]
I want to concentrate on the idealism and emphasize the word encounter in answering the question of: Why require mathematics?
The students believe ? correctly ? that they will not encounter complex numbers or Rational and Conjugate Root Theorems. They may well, however, encounter budgeting and scheduling ? in their roles as citizen, worker, or consumer. They are unlikely to encounter Linear-Quadratic and Quadratic-Quadratic Systems; but ? as citizen, worker, or consumer ? they will have to interpret statistical data. Determinants and Cramer's rule will not come up; building and interpreting results from a mathematical model may.
The goals of College Algebra should be to get students to internalize mathematics and come to understand certain ideas conceptually. Bob Moses and most others who have been successful with all students know that these goals can be obtained only if the mathematics is connected to everyday life. What does that mean for College Algebra?
In my judgment it means starting with the applications ? not the techniques. What problems should all college graduates ? those from two-year and four-year colleges who do not major in a math-intensive subject ? be able to solve? It is not how to factor a polynomial or complete a square; it is how to put together a budget and schedule for a proposal or building project or sales campaign. It is not how to solve a quadratic equation; it is how to use a statistical process control or model a production or health system.
I mentioned before that CA is a cash cow that many mathematics departments do not want to tamper with, especially if it means smaller classes and higher costs. If this attitude is maintained too long both the cash and the cow may disappear as students opt for useful and interesting quantitative literacy that teaches them how to solve problems they will encounter and be paid to solve.
- Mercedes McGowan
- Robert S. and Michele Root-Bernstein Learning to Think With Emotion, The Chronicle of Higher Education, January 14, 2000, p A64
- Lynn Steen, Preface: the New Literacy, in Why Numbers Count.
- F. James Rutherford, Thinking Quantitatively about Science in Why Numbers Count.
- Do You Know The Good News About American Education? Center on Education Policy, Washington D.C., 2000.p13
- This Year?s Freshmen: A Statistical Profile, The Chronicle of Higher Education, 1/28/00 p A50.