# Would most first-year college students be better off taking a statistics course rather than calculus

# Would most first-year college students be better off taking a statistics course rather than calculus?

**Viewpoint:** Yes, most first-year college students would be better off taking a statistics course rather than calculus because statistics offers a greater variety of practical applications for a non-scientific career path.

**Viewpoint:** No, first-year college students would not be better off taking a statistics course rather than calculus because calculus is wider in application and teaches an intellectual rigor that students will find useful throughout their lives.

Statistics is a branch of mathematics devoted to the collection and analysis of numerical data. Calculus, on the other hand, is a mathematical discipline that deals with rates of change and motion. The two, it would seem, have little in common, yet they are linked inasmuch as they are at the center of an educational debate, here considered from two perspectives, as to which course of study is more appropriate for first-year college students.

The two disciplines are also historically linked, both being products of the early Enlightenment. They emerged in an intellectually fruitful period from the middle of the seventeenth to the beginning of the eighteenth centuries, and collectively they are the achievement of four of the greatest minds the world has ever produced. Each discipline benefited from the contributions of two major thinkers, but in the case of statistics and probability theory, French mathematicians Pierre de Fermat (1601-1665) and Blaise Pascal (1623-1662) worked together. By contrast, English physicist Sir Isaac Newton (1642-1727) and German philosopher Gottfried Wilhelm von Leibniz (1646-1716) presented two competing versions of calculus, and in later years, their respective adherents engaged in a bitter rivalry.

While statistics is the branch of mathematics concerned with the collection and analysis of numerical data relating to particular topics, probability theory is devoted to predicting the likelihood that a particular event will occur. The two are closely related, because a proper statistical database is necessary before probabilities can be accurately calculated. Furthermore, they emerged as areas of study together, and only later became differentiated.

Though statistics and probability theory can be, and are, applied in a number of serious pursuits, they also play a part in gambling and games of chance. It was a fascination with such games—albeit a professional rather than personal fascination—that led Fermat and Pascal to undertake the first important work in probability theory and statistics. The two disciplines remained closely linked until the end of the eighteenth century, by which time statistics emerged as an independent discipline.

By the beginning of the twentieth century, several factors contributed to growing respect for statistics and probability theory. In physics and chemistry, quantum mechanics—which maintains that atoms have specific energy levels governed by the distance between the electron and the nucleus—became the prevailing theory of matter. Since quantum mechanics also holds that it is impossible to predict the location of an electron with certainty, probability calculations became necessary to the work of physicists and chemists.

The early twentieth century also saw the rise of the social sciences, which rely heavily on statistical research. In order to earn a graduate degree today, a prospective psychologist or sociologist must undergo an intensive course of study in statistics, which aids in the interpretation of patterns involving complex variables—for instance, the relationship between schoolchildren's diet and their test scores, or between the age at which a person marries and his or her income.

Another factor behind the growing importance of probability and statistics was the emergence of the insurance industry. Insurance is an essential element of economically advanced societies, which literally could not exist if individuals and businesses had no way of protecting themselves financially against potential losses. In the world of insurance, statistics and probability theory are so important that an entire field of study, known as actuarial science, is devoted to analyzing the likelihood that certain events will or will not happen. Without actuarial analysis, which makes it possible to calculate risks, an insurance company could not make a profit and still charge reasonable prices for coverage.

Though the mathematics involved in calculating statistics go far beyond the education of the average person, the discipline itself is rooted in phenomena easily understandable to all. By contrast, calculus is likely to be more intellectually intimidating. Actually, there are two kinds of calculus: differential calculus, which is concerned with finding the instantaneous rate at which one quantity, called a *derivative*, changes with respect to another, and integral calculus, which deals with the inverse or opposite of a derivative. On the surface, this sounds rather abstract, though in fact calculus has numerous practical applications in areas ranging from rocket science to business.

The roots of calculus go back to ancient Greece, and the discipline owes much to the work of the medieval French mathematician Nicole d'Oresme (1323-1382). He advanced the study of motion, or kinematics, with his work on uniform acceleration. Galileo Galilei (1564-1642), building on Oresme's findings, proved that falling bodies experience uniform acceleration, and this in turn paved the way for Newton's own studies of gravitation and motion. In order to quantify those studies, Newton required a new form of mathematics, and therefore devised calculus.

One of the great controversies in the history of mathematics is the question of whether Newton or Leibniz deserves credit for the creation of calculus. In fact, both developed it independently, and though it appears that Newton finished his work first, Leibniz was the first to publish his. Over the years that followed, mathematicians and scientists would take sides along national lines: British scholars tended to support Newton's claims, and those of Germany and central Europe backed Leibniz. Even today, European mathematicians emphasize the form of calculus developed by Leibniz. (As noted below, Leibniz's calculus emphasizes integrals, and Newton's derivatives.)

During the eighteenth century, even as mathematicians debated the relative contributions of Newton and Leibniz—and some even questioned the validity of calculus as a mathematical discipline—calculus gained wide application in physics, astronomy, and engineering. This fact is illustrated by the work of the Bernoulli family. Influenced by Leibniz, a personal friend, Jakob Bernoulli (1645-1705) applied calculus to probability and statistics, and ultimately to physics, while his brother Johann (1667-1748) appears to have supplied much of the material in the first calculus textbook, by French mathematician Guillaume de L'Hôpital (1661-1704). Johann's sons all made contributions to calculus. Most notable among them was Daniel (1700-1782), who applied it in developing the principles of fluid dynamics, an area of physics that later made possible the invention of the airplane. In addition, the Bernoulli brothers' distinguished pupil, Leonhard Euler (1707-1783), used calculus in a number of ways that would gain application in physical studies of energy, wave motion, and other properties and phenomena.

The first-year college student attempting to choose between the study of statistics and calculus, then, has a tough choice. Each has merits to recommend it, as the essays that follow show. Statistics may seem more rooted in everyday life, while calculus may appear to draw the student into a lofty conversation with great mathematical minds; yet there is plenty of practical application for calculus, and statistics certainly brings the student into contact with great thinkers and great ideas.

—JUDSON KNIGHT

## Viewpoint: Yes, most first-year college students would be better off taking a statistics course rather than calculus because statistics offers a greater variety of practical applications for a non-scientific career path.

In today's highly competitive world, it can easily be said that a college education should prepare a student for the world of work. Certainly most students enter college today with the expectation of getting a job after graduation. They often express the desire to be wellrounded and marketable. Therefore, they need to be practical when deciding what courses to take. With the exception of math majors, most first-year college students would be far better off taking a statistics course rather than a calculus course. Why? The answer is simple: statistics are part of everyday life. A student who understands statistics has an advantage over those who do not. Understanding statistics can be useful in college, employment, and in life.

### The Scholastic Advantage

Taking a statistics course encourages a student to flex his or her intellectual muscles, but at the same time, is more practical than a calculus course. This is especially true for a first-year college student who may be uncertain about his or her career path. If someone plans to be a math major, then taking calculus makes perfect sense, but for someone pursuing another career, taking calculus in the first year of college could be a waste of time and money. Not all professionals will need a calculus background, but most professionals will need at least some knowledge of statistics. In fact, studying statistics helps first-year college students hone their critical and analytical thinking skills—skills they can draw upon throughout their college career regardless of what major they choose.

In the *Columbia Electronic Encyclopedia*, statistics is defined as "the science of collecting and classifying a group of facts according to their relative number and determining certain values that represent characteristics of the group." With that definition in mind, it's easy to see how understanding statistics would help a first-year college student. The main aim of most college-level introductory statistics courses is to get the student comfortable with such principles as statistical modeling, confidence intervals, and hypothesis testing, concepts that will help students become better consumers of information.

Most college students are required to write at least one paper in their first year, which often involves going to the library and combing through stacks of research material. Understanding statistical measures, such as mean and standard deviation, can certainly help them make sense of what they find. Students who truly understand what the numbers represent will not just be regurgitating facts and figures reported by others; instead, they will be able to come to their own conclusions based on the data presented. In essence, they will have the edge, because they will be better able to identify the strengths and weaknesses of another's research with the skill set that knowledge in statistics provides. An added bonus is that students who are able to think clinically and delve deeper into a particular subject are often rewarded with better grades.

### The Employment Advantage

Even before graduation many first-year college students find themselves looking for work to help pay for college expenses. A statistics course, coupled with one in business math, is going to be far more helpful in most job situations than calculus. Many college students get a job in a retail setting, for example, where calculus has little value, but being able to understand and interpret a survey could prove helpful to them not only as employees, but as consumers, too.

An article by Margaret Menzin and Robert Goldman for the Association for Women in Mathematics states that the United States Department of Labor forecasts a faster rate of job growth in the area of statistics than in computer science. Statisticians, they remind us, are utilized in a variety of fields from archeology to zoology. For example, statisticians are employed by many finance and insurance companies throughout the United States. Having a knowledge of statistics, whether a person pursues a career in statistics or not, will enhance a person's marketability and will most likely help him or her perform better on the job.

### The Life Advantage

Indeed, understanding statistics leads to a better understanding of the world around us. Certainly, if one wants to be an informed consumer, the study of statistics is valuable. Here is how a working knowledge of statistics can help. Say, for example, that someone hears on a television advertisement that four out of five doctors polled preferred XYZ medicine for tooth pain. The statement sounds impressive, but it does not mean that most doctors prefer that particular brand of pain reliever or that someone should choose it over another brand. How many doctors were actually polled? The statement does not mean very much if only five doctors were polled. Even if 1,000 doctors were polled, what types of doctors were they? Their specialty could influence their response. Knowing the particulars about the sample size and type are also important. Were they dentists, orthodontists, anesthesiologists, or a combination of these and other specialists? Asking questions about sampling is just one way to analyze a statistic. There are many other ways as well. A consumer-savvy person armed with some knowledge about statistics is less likely to be misled by clever advertising. After all, the marketing campaign might influence someone who does not know statistics to buy the product, but someone who knows statistics will know to ask the right questions.

Also, understanding statistical procedures can help students understand why certain tasks are done a certain way, which in some cases will help reduce stress. Understanding probability, for example, helps someone determine the likelihood that a certain event will occur. In fact, there are several everyday situations that will seem much less stressful and much more logical if someone understands statistical principles such as the queuing theory, for instance. Menzin and Goldman define the queuing theory as a theory that predicts what happens as people or things get in line. Queuing theory can be used to predict how traffic bunches up in toll booths or how long your report will take to print out of a shared printer. Say, for example, that a student is attempting to register for classes. When he or she gets to the registration hall and sees one long line that feeds into several windows, his or her first reaction might be to get angry and say, "Why don't they form several lines?" But someone who understands the queuing theory will realize that one line that feeds to several windows is actually more efficient and will reduce the wait time for everyone.

### Conclusion

Clearly, a first-year college student can benefit greatly from taking courses that have practical applications. Statistics is certainly one of those courses and can be beneficial to all students, not just math majors. Knowledge of statistics allows a student to intelligently question information he or she receives in the classroom, on the job, and in life.

—LEE A. PARADISE

## Viewpoint: No, first-year college students would not be better off taking a statistics course rather than calculus because calculus is wider in application and teaches an intellectual rigor that students will find useful throughout their lives.

In a November 28, 2001, *New York Times* piece, Richard Rothstein, a research associate of the Economic Policy Institute, made a compelling case for a renewed emphasis on statistics—as opposed to calculus—in the classroom. Though he was discussing mathematics teaching in high school rather than college, Rothstein's argument could as easily be applied to college education. Among the reasons he offered for teaching statistics rather than calculus was that few jobs make use of the latter, whereas statistics will aid students in understanding much of the data that bombards them from the news media.

While Rothstein's points are arguable, he makes an undeniably compelling case; however, his very emphasis on practicality undermines the assertion that statistics education is more important than calculus. On one level at least, it is precisely because calculus is not obviously a part of practical, everyday knowledge that it should be taught. Whereas one is assaulted with statistics at every turn—in newspapers and magazines, on television, and in public debates over issues such as health care or social security—calculus seems a more esoteric pursuit reserved for scientists and mathematicians. And this, ironically, makes it all the more appropriate for a young college student preparing to embark on a four-year education.

To some extent, teaching a first-year college student statistics is like teaching him or her to listen to popular music; the subject hardly needs to be emphasized, because the ambient media culture already does the job so well. Admittedly, there is considerable merit in studying statistics, not least because such study allows the individual to see through the sometimes deceptive claims made under the veil of supposedly unassailable statistical data. This deceptiveness is encapsulated in President Harry S. Truman's famous declaration that there are three kinds of untruth: "lies, damn lies, and statistics."

But while statistics will have practical application in the student's later life, calculus offers a degree of intellectual rigor that will equip the student to think about all sorts of issues in much greater depth. Teaching a first-year college student calculus is like teaching him or her the Greek classics, or logic, or the appreciation of Mozart and Beethoven—riches of the mind that the student might not seek out on his or her own, but which he or she will treasure for life.

### Calculus in Brief

Of all the subjects a student is likely to encounter, mathematical or otherwise, calculus is among the most intimidating.
Only the most scholastically advanced high-school students ever take calculus, and then only as seniors, after four years of study that include algebra, geometry, algebra II, and trigonometry. It might be comforting to learn that calculus is really very easy to understand—it *might* be, that is, if such were the case. In fact calculus is extremely challenging, much more so than statistics, and the student who emerges from a course in it is like an athlete who has undergone an extraordinarily rigorous form of physical training.

The branch of mathematics that deals with rates of change and motion, calculus is divided into two parts, differential calculus and integral calculus. Differential calculus is concerned with finding the instantaneous rate at which one quantity, called a derivative, changes with respect to another. Integral calculus deals with integrals, the inverse of derivatives.

What does all this mean? Suppose a car is going around a curve. Differential calculus would be used to determine its rate of change (that is, its speed) at any given point, but if its speed were already known, then integral calculus could be used to show its position at any given time. Imagine any object following a curved path: this is the territory of calculus, which measures very small changes.

### Applications of Calculus

Despite its apparently esoteric quality, the applications of calculus are myriad. Physicists working in the area of rocket science use it to predict the trajectory of a projectile when fired, while nuclear scientists apply calculus in designing particle accelerators and in studying the behavior of electrons. Calculus has also been used to test scientific theories regarding the origins of the universe and the formation of tornadoes and hurricanes.

Indeed, calculus is the language of science and engineering, a means whereby laws of the natural world can be expressed in mathematical form. It makes possible the analysis and evaluation of those physical laws, predictions regarding the behavior of systems subject to those principles, and even the discovery of new laws. Yet the importance of calculus extends far beyond the realm of science, engineering, and mathematics.

Though electricians apply scientific knowledge, their work is obviously much more practical and everyday in its emphasis than that of scientists, yet as Martha Groves reported in the *Los Angeles Times*, electrical workers' unions require apprentices to study calculus. Businesses also use calculus widely, to increase efficiency by maximizing production while minimizing costs, and for other purposes.

### Calculus, Change, and Infinitesimal Matter

A journal of the paint manufacturing industry might seem an odd place in which to make a case for the study of calculus, but that is precisely
what Joseph B. Scheller of Silberline Manufacturing Company of Tamaqua, Pennsylvania, did in a 1993 editorial for *American Paint & Coatings Journal*. Students, he maintained, should learn about the concept of change, an essential aspect of life, and particularly modern life. "What do I think we should teach young children?" he wrote near the conclusion of his editorial. "I think they should learn the concepts of physics and calculus … because they tell us about ourselves and the world in which we live."

The entire universe is characterized by constant movement—change in position over time—at every conceivable level. Galaxies are moving away from one another at a breathtaking rate, while our own planet spins on its axis, revolves around the Sun, and participates in the larger movements of our Solar System and galaxy. At the opposite end of the spectrum in terms of size, all matter is moving at the molecular, atomic, and subatomic levels.

Molecules and atoms move with respect to one another, while from the frame of reference of the atom's interior, electrons are moving at incredible rates of speed around the nucleus. Literally *everything* is moving and thereby generating heat; if it were not, it would be completely frozen. Hence the temperature of absolute zero (-459.67°F or-273.15°C) is defined as the point at which all molecular and atomic motion ceases—yet precisely because the cessation of that motion is impossible, absolute zero is impossible to reach.

What does this have to do with calculus? Two important things. First of all, calculus is all about movement and change—not merely movement through space, but movement through time, as measured by rates of change. Secondly, calculus addresses movement at almost inconceivably small intervals, and by opening up scientists' minds to the idea that space can be divided into extremely small parts, it paved the way for understanding the atomic structure of matter. At the same time, it is perhaps no accident that calculus—with its awareness of things too small to be seen with the naked eye—came into being around the same time that the microscope opened biologists to the world of microbes.

### Joining a Great Conversation

To study calculus is to join a great conversation involving some of the world's finest minds—a conversation that goes back to the fifth century b.c. and continues today. In his famous paradoxes, the Greek philosopher Zeno of Elea (c. 495-c. 430 b.c.) was perhaps the first to envision the idea of subdividing space into ever smaller increments. A century later, Eudoxus of Cnidus (c. 390-c. 342 b.c.) developed the so-called method of exhaustion as a means, for instance, of squaring the circle—that is, finding a way of creating a circle with the same area as a given square. Archimedes (287-212 b.c.) applied the method of exhaustion to squaring the circle by inscribing in a circle a number of boxes that approached the infinite.

The Greeks, however, were extremely uncomfortable with the ideas of infinite numbers and spaces on the one hand, or of infinitesimal ones on the other. Thus calculus did not make its appearance until the late seventeenth century, when Gottfried Wilhelm von Leibniz (1646-1716) and Sir Isaac Newton (1642-1727) developed it independently. There followed a bitter debate as to which man should properly be called the father of calculus, though in fact each staked out somewhat separate territories.

In general, Newton's work emphasized differentials and Leibniz's integrals, differences that resulted from each man's purposes and interests. As a physicist, Newton required calculus to help him explain the rates of change experienced by an accelerating object. Leibniz, on the other hand, was a philosopher, and therefore his motivations are much harder to explain scientifically. To put it in greatly simplified terms, Leibniz believed that the idea of space between objects is actually an illusion, and therefore he set out to show that physical space is actually filled with an infinite set of infinitely small objects.

### Challenging the Mind

Mathematicians did not immediately accept the field of calculus. Critics such as the Irish philosopher George Berkeley (1685-1753) claimed that the discipline was based on fundamental errors. Interestingly, critiques by Berkeley and others, which forced mathematicians to develop a firmer foundation for calculus, actually strengthened it as a field of study.

As calculus gained in strength over the following centuries, it would prove to be of immeasurable significance in a vast array of discoveries concerning the nature of matter, energy, motion, and the universe. In this sense, too, the study of calculus invites the student into a great conversation involving the most brilliant minds the world has produced. Admission to that conversation has nothing to do with gender, race, religion, or national origin; all that is required is a willingness to surrender old ways of thinking for newer, more subtle ones.

Calculus is a challenging subject, but it is an equal-opportunity challenger. For this reason, as reported in *Black Issues in Higher Education*, the College Board (which administers the SAT, or Scholastic Achievement Test) announced in 2002 that it would revise the test to include greater emphasis on calculus and algebra II. These subjects would be emphasized in place of vocabulary and analogy questions, which some critics have maintained are racially biased.

At the same time, as noted by Sid Kolpas in a review of Susan L. Ganter's *Calculus Renewal* for *Mathematics Teacher*, calculus teaching itself is entering a new and exciting phase. "Spurred by the mathematical needs of other subject areas, dramatic advances in cognitive science and brain research, and rapidly changing technology and workplace demands," Kolpas wrote, "calculus renewal seeks a transformation of the intimidating, traditionally taught calculus course. It seeks a brave new calculus with more real-world applications, hands-on activities, cooperative learning, discovery, writing, appropriate use of technological tools, and higher expectations for concept mastery." In these words, one glimpses a hint of the great intellectual challenge that awaits the student of calculus. Certainly there is nothing wrong with a student undergoing a course in statistics, but it is hard to imagine how that course could awaken the mind to the degree that calculus does.

—JUDSON KNIGHT

## Further Reading

Berlinski, David. *A Tour of Calculus*. New York:Pantheon Books, 1995.

"College Board Proposes SAT Overhaul." *Black Issues in Higher Education* 19, no. 4: 17.

*The Columbia Electronic Encyclopedia*. "Statistics"[cited July 22, 2002]. <http://www.inforplease.com/ce6/sci/A0846567.html>.

Ganter, Susan L., ed. *Calculus Renewal: Issues for Undergraduate Mathematics Education in the Next Decade*. New York: Kluwer Academic/Plenum Publishers, 2000.

Groves, Martha. "More Rigorous High School Study Urged: Panel Finds Graduating Seniors Are Often Ill-Prepared for College." *Los Angeles Times* (October 5, 2001): A-27.

Kolpas, Sid. "Calculus Renewal: Issues for Undergraduate Mathematics Education in the Next Decade." *Mathematics Teacher* 95, no. 1: 72.

Menzin, Margaret, and Robert Goldman."Careers in Mathematics" [cited July 22, 2002]. <http://math.usask.ca/document/netinfo/careers.html>.

Rothstein, Richard. "Statistics, a Tool for Life, Is Getting Short Shrift." *New York Times* (November 28, 2001): D-10.

Scheller, Joseph B. "A World of Change:There's Nothing Static About Modern Life, and Education Should Deliver That Message." *American Paint & Coatings Journal* 78, no. 21 (October 29, 1993): 20-21.

## KEY TERMS

### CALCULUS:

A branch of mathematics that deals with rates of change and motion.

### DIFFERENTIAL CALCULUS:

A branch of calculus concerned with finding the instantaneous rate at which one quantity, called a derivative, changes with respect to another.

### INTEGRAL CALCULUS:

A branch of calculus that deals with integrals, which identify the position of a changing object when its rate of change is known. An integral is the inverse of a derivative.

### MEAN:

An average value for a group of numerical observations.

### STANDARD DEVIATION:

A measure of how much the individual observations differ from the mean.

### STATISTICS:

A branch of mathematics concerned with the collection and analysis of numerical data.

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