Jeffrey O. Bennett, University of Colorado at Boulder
William L. Briggs, University of Colorado at Denver
This article appeared in AMATYC Review, Fall 1999
“Philosophy is written in this grand book – I mean the universe – which stands continuous open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics.” — Galileo
“I think it’s not so much a question of what students study, as how we equip them to lead a life that involves regular use of certain skills [of quantitative reasoning].” — Keith Devlin, quoted in Forman (1997).
Introduction
When it comes to mathematics, we know what we must teach students majoring in science, engineering, or mathematics (“SEM” for short). These students need particular calculus-based skills that they will use regularly in their careers. But what about the rest of the collegiate population, the “non-SEM” students that easily comprise half of all students we teach in mathematics courses in this country? These students outnumber their SEM peers among all college students, and they represent a large majority among students at two-year colleges (National Science Board, 1998). Nevertheless, many institutions have given little serious thought to the development of appropriate mathematics curricula for non-SEM students. One result is that approaches to non-SEM mathematics courses are all over the board. Another, of which we are constantly reminded in conversations with mathematics teachers across the United States, is widespread dissatisfaction with these courses among both faculty and students.
We have spent much of the past twelve years working on the problem of mathematics for non-SEM students. In this paper, we will present some observations regarding the nature of the problem, along with our opinions regarding its solution.
Initial Goals: Ours and Theirs
The first step in developing any kind of curriculum must be to identify the goals to be achieved. As mathematicians, we resonate with the above quotation from Galileo and would like our students to feel the same sense of awe when they think of mathematics. Thus we might state an initial goal for mathematics courses for non-SEM students as follows: Show our students some of the beauty and utility of mathematics.
Unfortunately, we almost immediately encounter a problem in trying to achieve this goal: the vast majority of non-SEM students enter mathematics courses with very negative attitudes toward mathematics. In our courses at the University of Colorado, we have found that almost all of our students will openly declare themselves – with no hint of embarrassment – to be either “math-phobics” or “math-loathers.” The math-phobics are afraid of mathematics and have done poorly in past (high school) mathematics classes. The math-loathers have done well in past mathematics courses and therefore are not afraid of mathematics; they just believe that it is worthless and irrelevant to their lives. Thus there is a conflict between “our goal” of demonstrating the beauty and utility of mathematics and “their goal” of getting away from mathematics as far and as fast as they can.
An Overarching Responsibility
This conflict between our initial goal and their initial goal requires that we find a way to bring the two goals into line. The key involves thinking about our obligations as teachers. As mathematicians, we possess skills and knowledge that are of critical importance to our students. It is our responsibility to transmit this information to our students.
If we succeed, we will find that, in the process, we have also shown students some of the beauty and utility of mathematics. Moreover, the students will find that what they fear or loathe is not real mathematics, but rather a caricature of mathematics (such as the common student belief that mathematics is nothing more than a set of skills) that has been set up for them by bad experiences in past courses. Once they see the relevance of mathematics to their own lives, their goals and ours will be one and the same.
Refined Goals: Mathematics for College, Careers, and Life
We are now ready to transform our initial goal of showing students some of the beauty and utility of mathematics into a set of goals that also meets our overarching responsibility of transmitting to students the mathematics that they need. The only question is, what type of mathematics do students need to know? We believe the answer is three-fold:
- Students need mathematics for college. Most students take other collegiate courses, such as core courses in natural and social sciences, in which they will use mathematical skills.
- Students need mathematics for their careers. Nearly all careers today (and the average student will have several careers in a lifetime) require an ability to reason with quantitative information and to discuss quantitative issues clearly and cogently.
- Students need mathematics to understand the issues that confront them in daily life. In part, this means being able to deal with issues of personal finance. But it also means dealing with citizenship issues – such as the economy, public health, and the environment – on which they will be called upon to vote intelligently.
Note that these goals – mathematics for college, careers, and life – are goals that we and our students can share. Moreover, we have found that enunciating these goals to students almost immediately begins to break down the negative attitudes that they bring to mathematics. For the first time in most of their lives, students see teachers as being on “their side,” working with them to help them learn crucial and relevant skills.
It is important to recognize that the latter goal, mathematics for life, is more pervasive and more subtle than is commonly realized. In our courses, we frequently use current events to illustrate how thoroughly mathematics permeates everyday life. Virtually every major issue of our day is at least partly quantitative in nature; consider, for example, the issues surrounding the 2000 census, the federal budget, “saving” social security, finding an appropriate government settlement with tobacco companies, the statistics of crime rates, the science and economics of global warming, and even the recent impeachment of the President. The mathematics in these subjects may be fairly explicit or quite subtle, but in all cases it is necessary for a full understanding of the issues.
What Course Meets These Goals?
Now that we have a clear set of goals, it is possible to begin designing a course for non-SEM students. A brief description of how we arrived at our own course is instructive. In 1987, one of us (Bennett) was a member of an interdisciplinary faculty committee charged with developing a new mathematics requirement for non-SEM students at the University of Colorado at Boulder. The committee consisted of a dozen members; about half of whom were drawn from the Mathematics Department, while the rest represented disciplines such as astronomy, biology, psychology, geography, and economics.
Early in the discussions, several committee members suggested that all students should be expected to complete a course in calculus. While this may be a noble objective, it was quickly discarded as impractical: most non-SEM students are unprepared for calculus upon entry to college and therefore could not meet this objective within the constraints of a one- or two-semester general education mathematics requirement.
With calculus ruled out, the committee next considered the only other existing mathematics courses taught at the University of Colorado at the time: pre-calculus courses such as College Algebra or Trigonometry. But after only brief discussion, the faculty committee also rejected these courses for the new requirement, largely for the following three reasons:
- While the importance of algebra and other pre-calculus courses for SEM students is clear, it is much more difficult to make the case for non-SEM students. Aside from a bit of algebra, most non-SEM students will not use the skills learned in these courses in their other college courses, their careers, or their daily lives.
- At the University of Colorado, all entering students have already taken at least one year’s worth of high school algebra. The committee members therefore felt that it would be largely redundant to require students to take College Algebra – although this is commonly done. To paraphrase a colleague (Darrell Abney), such an approach means teaching the students essentially the same mathematics they were taught in high school, only this time “teaching it to them louder.”
- The committee members recognized that whatever mathematics course was required, it would be the last mathematics course that most of these students would ever take. The members therefore saw this requirement as a tremendous opportunity to teach students something new and important about mathematics.
Thus the committee decided that a new course would need to be created, and set about the task of identifying specific content.
Content Areas
Designing a new course means deciding upon content, so the University of Colorado committee focused most of its discussion on identifying appropriate content. In the end, the committee identified four major content areas. The original statement from the committee was a bit more terse, but we now identify the four areas as follows:
- Logic, Critical Thinking, and Problem Solving: Students should learn skills that will enable them to construct a logical argument based on rules of inference and to develop strategies for solving quantitative problems.
- Number Sense and Estimation: Students should become “numerate,” or able to make sense of the numbers that confront them in the modern world. For example, students should be able to give meaning to a billion dollars, and distinguish it from a million dollars or a trillion dollars. Part of developing such number sense involves making simple calculations or estimates to put numbers in perspective. As a simple example, a student should be able to quickly figure out that a star athlete earning $10 million per year earns about 400 times more than the average American.
- Statistical Interpretation and Basic Probability: Reports about statistical research (for example, concerning diet or disease) are ubiquitous in the news. Students must have the stools needed to interpret this research. Note the emphasis on interpretation. While it is certainly useful to show students how to calculate a mean, a standard deviation, or a margin of error, our non-SEM students will rarely perform such calculations once they leave our course. But they will encounter such statistics in the news, and we must equip them to interpret these statistics critically. Because statistical interpretation involves inference from samples to populations, it also requires a basic understanding of probability. This study of probability can then be easily extended to relevant topics including lotteries, casino gambling, risk assessment, and disease and drug testing.
- Interpreting Graphs and Models: Graphical displays of numbers abound in modern media, so learning how to create and interpret graphs is clearly important. Though it is a bit less obvious, an understanding of modeling is equally important, because many major issues today (such as economic or environmental issues) are studied through mathematical models. While we do not expect non-SEM students to create sophisticated mathematical models, we must teach them how to interpret what they read or hear about models. For example, they should know enough to question the assumptions of a model before accepting its predictions, and they should understand the difference between linear and exponential growth. We group graphing and modeling together because we have found that one of the easiest ways to teach students about modeling is by presenting graphs as simple mathematical models.
These areas and their associated skills form the core of what is often called quantitative reasoning. Interestingly, we have found that almost every group that identifies appropriate content areas for non-SEM students reaches similar conclusions. For example, the AMATYC standards (AMATYC, 1995) list the following standards (identifiers in parentheses are from the AMATYC report): (I-1) problem solving; (I-2) modeling; (I-3) reasoning; (I-4) connecting with other disciplines; (I-5) communicating; (I-7) developing mathematical power; and (C-1) number sense. Although this list is organized differently from our list above, we maintain that both say essentially the same thing. A similar set of goals was enunciated in the MAA report on quantitative reasoning (MAA, 1995; p. 10).
It’s Not Remedial
Before we continue, it’s important to point out that our four quantitative reasoning content areas are certainly not remedial, even though they do not necessarily include much in the way of formal algebra. As a first example, consider the following quotation taken from a front-page article in the New York Times (4/20/97):
Teen-age smoking rates are still lower than in the 1970’s. But the percentage of 12th graders who smoked daily last year jumped 20 percent since 1991, to 22 percent. The rate among 10th graders jumped 45 percent, to 18.3 percent, and the rate for 8th graders is up 44 percent, to 10.4 percent.
Most non-SEM students (and many SEM students as well) have a very difficult time interpreting the use of percentages in this statement. But simply teaching percentages in a remedial sense – that is, divide two numbers and multiply by 100% – will not solve the problem. Instead, students need well-developed critical thinking skills (Content Area 1) and number sense (Content Area 2) to interpret this statement, and these aptitudes are often not emphasized in standard high school mathematics curricula. Thus learning to interpret this statement is not remedial, even though it deals only with percentages.
As a second example, consider the federal budget (as it stands in March 1999). The politicians are very proud that, after decades of deficits, the government ran a $69 billion surplus in 1998. But looking deeper, you’ll find that some people claim there was no surplus, but rather a $30 billion deficit. Moreover, while you might expect a surplus to reduce the national debt, the debt actually rose in 1998 by $113 billion!
From the standpoint of mathematical manipulation, reconciling these budget numbers requires nothing more than addition and subtraction. But that does not make it easy or remedial. In fact, understanding federal budget numbers requires skills from all four quantitative reasoning content areas: logic to follow the convoluted path by which the numbers are derived; number sense to understand the meaning of the numbers; statistical interpretation to understand how economic data are measured; and modeling to understand how the government forecasts future surpluses or deficits.
The Key to Success: A Context-Driven Approach
There are many possible ways to integrate the four content areas into a course syllabus, but we believe most of them can be categorized either as “content-driven” or “context-driven.” A couple of examples should clarify the difference and illustrate why we believe the latter approach is superior.
Consider the topic of logic and critical thinking. A “content-driven” approach looks at logic as a mathematical content area that students should study for its own sake. This approach therefore begins by establishing the important mathematical ideas behind logic, such as sets, truth tables, and Venn diagrams, and then shows students how these ideas are useful. This approach is common in textbooks for non-SEM students. However, this content-driven approach immediately sets up the conflict discussed earlier between instructors’ initial goals and students’ initial goals: the instructor is trying to teach some mathematics, which the students would rather avoid.
In contrast, a “context-driven” approach begins by establishing a context that helps students understand why they should care about logic. In our course, for example, we begin by discussing common logical fallacies and critical thinking problems that appear in everyday situations (such as how to choose between the fare alternatives when buying an airplane ticket), thereby showing students the immediate relevance of logic to their lives. Only then do we introduce the essential mathematical ideas. In the end, both approaches to logic cover the same mathematical content. But the context-driven approach is far more successful because it sets up a shared goal between instructor and students – discussing a topic that all can agree is important to college, careers, and life.
As a second example, consider payments on loans, such as student loans, credit cards, or mortgages. In a content-driven approach, loan payments are taught as an application of exponential growth; after all, the formula for loan payments can be derived from compound interest considerations. Thus a content-driven approach starts with the mathematics of exponential growth, and eventually shows students that this has relevance to loan payments. Again, because students are shown the mathematics before being shown its relevance, this approach fails to bridge the gap between the initial goals of students and instructors.
A context-driven approach recognizes that most students have loans of some type (usually credit cards or student loans, and sometimes mortgages) and the topic of loan payments therefore can engage students in the more general topic of exponential growth. Thus, in our course, we teach students about loan payments before we cover exponential growth more generally. Moreover, whereas the content-driven approach is usually finished once it reaches the loan payment formula (which students can rightly argue is something they will not use, since banks or real estate agents usually do the calculations for them), the formula is only the beginning in the context-driven approach. We continue discussions of loan payments to show students how they can avoid getting into credit card trouble, how they can make decisions between adjustable-rate and fixed-rate mortgages, and how closing costs and fees can affect the cost of a loan. These are mathematical topics that involve logic, problem solving, number sense, and modeling. (Loan payments are only one of many contexts in which exponential growth can be introduced.)
In fact, when it comes to actual course construction, the four content areas identified earlier can be covered through an almost endless variety of specific applications. For example, if you want to cover the mathematics of voting, do it in the context of actual elections rather than starting from mathematical theory. If you want to cover geometry, do it in the context of art or architecture that students will appreciate, rather than as a set of abstract ideas. If you want to cover exponential modeling, begin by observing a population with a fixed doubling time. If you want to solve algebraic equations, do it in the context of linear models or inverse percentage problems. Whatever the topic, a context-driven approach will allow you to convey the beauty and utility of mathematics because your students will perceive that they are working with you toward common goals.
Practical Teaching Considerations
We now address several of the most frequently asked questions about teaching courses of the type described in this paper.
- What prerequisite mathematical background is required? Ideally, students have taken a year or more of high school algebra, but this is not absolutely necessary. The context-driven approach works for all types of students, and as the examples in this paper describe, you can make a non-remedial course without integrating much algebra. Basically, an algebra prerequisite will only affect whether you are able to cover certain specific applications.
- Our school requires a course that is equivalent in level to college algebra. Does this course fit the bill? Our course is so different from college algebra that “equivalence” is difficult to interpret. If it means that students learn just as much new (to them) mathematics, the answer is an emphatic yes. In fact, we believe that students learn far more mathematics in our course than they would in a college algebra course – regardless of the extent to which we cover topics (such as linear and exponential modeling) with a formal algebra content.
- Is this necessarily a terminal mathematics course? Nearly all of our students enter our course with the intention of it being their last mathematics course. However, typically 5-10% of our students decide to take more mathematics after completing our course. In addition, many of our students go on to take discipline-based courses that involve mathematics; for example, social science students usually take some type of statistics course within their major, and business students often take accounting or other finance courses.
- Should the course format be lecture or discussion? We prefer a discussion format. Cooperative and group learning strategies are also successful. However, at many schools large lectures are unavoidable. In such cases, it is very helpful to have a recitation in addition to the lectures.
- How do you evaluate students? Ideally, students do a lot of homework that forms the bulk of their grade, along with exams that do not involve memorization or heavy time pressure. In practice, most instructors do not have adequate resources for all the grading entailed in this ideal strategy. One approach we like in this case is to grade only selected homework problems (letting students check the rest themselves from a solution set). You can overcome students objections to turning-in problems that you do not grade by basing exams almost directly on the homework (including problems you did not grade); students who work hard on the homework thereby see a tangible reward on the exam.
- What technology is required? The only required technology is a scientific calculator. You can easily incorporate additional technology, such as using spreadsheets for statistics or financial calculations, but any time you spend teaching a technological tool means less time for covering application areas. Given the enormous number of applications that we’d like to cover with our students, we’ve tended to stay away from additional technology with one exception: we now expect all our students to make use of the Web.
- Is it true that this type of course is a challenge for instructors? Yes. Although the mathematical level of this course is only general education, it is a challenging course to teach, at least for first-time instructors. A high premium must be placed on making each class stimulating and motivating. Examples and applications must be current, relevant, and carefully selected. However, we have also found that the rewards of teaching this course – especially in seeing the changing attitudes of your students toward mathematics – more than make up for the extra effort.
Summary
In conclusion, we believe that all non-SEM students – including liberal arts students, business students, and pre-service elementary teachers – should take a mathematics course (one or two semesters) with the content and approach described in this paper. All of these students need to be competent in the four content areas described, and a course using a context-driven approach is the best way to build that competence. Such a course is rich and rigorous, and can serve as the cornerstone in the collegiate mathematical training of our students. By touching on topics of interdisciplinary interest, the course provides lasting benefits for students in their future courses, careers, and lifetimes.
References
AMATYC (American Mathematical Association for Two-Year Colleges), 1995, Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus
Bennett, Jeffrey O. and Briggs, William L., 1999, Using and Understanding Mathematics – A Quantitative Reasoning Approach, Addison Wesley
Forman, Susan L., 1997, “Afterword: Through Mathematicians’ Eyes,” in Why Numbers Count: Quantitative Literacy for Tomorrow’s America (ed., Lynn Arthur Steen), The College Board
MAA (Mathematical Association of America), 1995, Quantitative Reasoning for College Graduates: A Complement to the Standards
National Science Board, Science and Engineering Indicators - 1998. Arlington, VA: National Science Foundation, 1998 (NSB 98-1).
Paulos, John A., 1995, A Mathematician Reads the Newspaper, Basic Books