Has the calculus reform project improved students' understanding of mathematics

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Has the calculus reform project improved students' understanding of mathematics?

Viewpoint: Yes, reform-based calculus provides students with a better grasp of the real-world applications and context of mathematical principles, and it also increases the participation of student populations that have been underserved by traditional teaching methods.

Viewpoint: No, the calculus reform project purges calculus of its mathematical rigor, resulting in a watered-down version that poorly prepares students for advanced mathematical and scientific training.

Calculus is quite literally the language of science and engineering. While the concepts and formalisms of calculus are more than 300 years old, they have never been more centrally important than they are today. Educators face the challenge of preparing students for careers that increasingly depend on science and technology. Traditionally, at colleges and universities calculus has served as a prerequisite for the study of any kind of science or engineering. In practice, the high failure rate in introductory calculus courses served to filter out many would-be science and engineering majors. During the 1980s, as more and more scientific and technological fields offered promising career opportunities, college administrators became critical of mathematics departments for the high failure and attrition rates typical of introductory calculus courses. Some educators began to consider new ways to teach calculus, hoping to improve performance and to make calculus a "pump" for prospective science students, rather than a filter. But to others these reform efforts seem merely to weaken the calculus curriculum, substituting faddish pedagogy for rigor and hard work.

At the core of the calculus reform debate is a long-lived problem common to most mathematical education. Should mathematics be taught primarily as a toolbox for its many applications, or should it be taught for its own sake as a challenging and important intellectual achievement? Both positions have their risks as well as rewards, which have been quite evident in the struggles surrounding the calculus curriculum. Critics of calculus reform suggest that students fail to grasp the general concepts and rigorous proofs that are the essence of calculus, while those who defend an application-based approach argue that students who focus solely on mathematical theorems and proofs often fail to understand how, why, or when to apply their knowledge. A newer, related twist to debates about teaching calculus is determining the best way to use computers in the curriculum. Computers can be used to perform tedious calculations and algebraic manipulations, and also to illustrate complex concepts such as graphing functions. But the value of computers to teaching calculus is diminished by the risk that students will learn more about manipulating a particular computer program than about calculus in general. Furthermore reliance on computer-generated solutions to problems can obscure important ideas, such as the difference between approximations and valid proofs.

The calculus reform debate has taken place in a wider context of concern about disparities in educational performance and opportunities for minority students. In the case of mathematics, and the scientific and technical courses that rely upon it, the traditional student population in the United States tended to be white (or Asian) and male. Proponents of calculus reform suggest that conventional teaching methods contributed to the dearth of successful female, black, or Hispanic students in calculus courses. They argue that reform strategies that emphasize problem-solving, group cooperation, and verbal accounts of mathematical problems would be more inclusive and lead to a more representative population of calculus students. If this is indeed successful, this could have broad impact on the representation of women, blacks, and Hispanics in many scientific and technical fields.

There is a great deal at risk with calculus reform. Because calculus is so essential to the successful study of science and engineering, poor mathematical preparation can cripple even the most talented student in those fields. The traditional curriculum, with its emphasis on mathematical rigor, theorems, proofs, and calculations, served the best students well. Those with adequate mathematical preparation and a capacity for hard work could emerge from a calculus sequence with a satisfactory grade and a good knowledge of the subject's fundamental concepts and techniques. Those with poor preparation or a lack of discipline, on the other hand, would often give up or fail. One of the primary goals of calculus reform has been to make it easier for all students to do well in calculus. While this could be good news for those who might otherwise have failed the subject entirely, better students—the traditional pool of future scientists and engineers—may pass through the calculus sequence knowing far less than they would have otherwise. Critics charge that while reformers claim they are democratizing the curriculum, they are really just watering it down.

The impetus to reform the calculus curriculum in the 1980s and 1990s came from concern over the large number of students who failed to progress through calculus and therefore be eligible for further study and careers in fields such as physics, engineering, and computer science. The growing economic importance of these fields ensured that college administrators would support educational reform efforts aimed at increasing enrollment and success rates in that pesky prerequisite, calculus. Reform-minded educators transformed the calculus curriculum. Innovations included computer-based learning, group study, reliance on learning by concrete examples, and verbal analysis of mathematical problems. But careful study of the traditional methods of problem solving and the rigorous demonstration of theorems and proofs were largely put aside. Reformers cheered a larger, more diverse group of students achieving passing grades in calculus, while critics moaned that the new curriculum doomed all calculus students to a lowest common denominator of merely superficial knowledge. Judging the success of calculus reform requires consideration not only of the respective curriculums, but also of the goals of calculus teaching and its position in the university curriculum. Is it for the talented few or the mediocre many? The debate over calculus reform has inadvertently posed this dichotomy, but surely successful reform should achieve both goals—rigorous training in important concepts and techniques, accessible to any committed student.

—LOREN BUTLER FEFFER

Viewpoint: Yes, reform-based calculus provides students with a better grasp of the real-world applications and context of mathematical principles, and it increases the participation of student populations that have been underserved by traditional teaching methods.

Calculus is an elegant intellectual achievement that can reduce complicated mathematical problems to simple but precise rules and procedures. Unfortunately, calculus has failed to become an important part of most college students' education for the same reason.

Taught for decades as merely rules and procedures, calculus, as it is traditionally taught, has had little or no meaning for most students. In addition, while teachers in other scientific classes and disciplines were using new technologies like computers to enhance their students' skills and prepare them for a high-tech marketplace, the calculator's first appearance in the 1970s was the only major technological advance in calculus education. Not surprisingly, at about this same time-period, the number of math majors at universities declined by 60%. Over the following decades, nearly 50% of the students who did take traditional calculus would fail or receive low grades.

Calculus is the study of "continuously" or "smooth" changing quantities (think of planets changing position or the movement of a car). Its applications range from physical concepts, such as velocity and acceleration, to geometric ideas, such as graphing curves. The ability to interpret and calculate change (for example, population or economic growth) can be an important skill for college graduates as they pursue a variety of careers, including in math and physics, biology, the health sciences, and social and economic sciences. As a result, more and more students should be learning calculus. But traditional calculus, with its emphasis on memorizing theorems and formulas regardless of understanding the concepts behind them, has slammed the door shut for most students.

In 1986, leading math educators from around the country attended a conference at Tulane University to look at the way calculus was taught, with an eye toward making it a more valuable and practical skill that would appeal to students. More and more educators were coming to the conclusion that most students were not grasping important concepts and skills, and the math problems and techniques being taught were too limited.

Two years later, the National Science Foundation (NSF) initiated a program funding efforts to design a new approach to calculus education. Out of this a radical idea emerged. Why not teach students to understand calculus in relation to real-world problems? By emphasizing active learning, mathematics instruction could become what it really should be for most students: training ground for becoming qualified scientists and other professionals in a variety of endeavors. This emphasis on "active" learning is the basis of the calculus reform movement.

What's Wrong with the Old Way?

The traditional approach to teaching calculus could be described as the "plug-and-chug" approach. The student is given a neatly packaged textbook problem and then trained to plug in the answers based on well-defined procedures. The ultimate goal for most students is not to understand the problem, but to get an answer that agrees with the one in the back of the book.

The results are two-fold. First of all, students who are required to enroll in a calculus course but have little prior preparation or aptitude for the field are failing calculus at an alarming rate. Secondly, even when students do chug along and learn how to do the calculations and succeed in the coursework, few can actually recognize a calculus-based problem outside of the classroom or even begin to think of applying calculus in real-world situations. This became more and more apparent as teachers of higher-level physics and engineering courses complained that the students entering their classes were not "prepared" to handle the calculus needed in their courses.

The essence of calculus is not to be able to do computations. Rather it is "proof," that is, a formal explanation of an observed pattern, like Sir Isaac Newton's proof explaining that the planets' orbits are subject to gravity. Newton certainly used calculations as the means of developing the proof; but they are not an end in and of themselves. Newton is not remembered as a great calculator but as a developer of the theory of gravity.

In the traditional calculus approach, the emphasis is placed on memorization of theoretical definitions. There is usually one right answer and a "preferred approach" to getting a solution, which is not reflective of real-world situations, whether it is in physics, economics, or medicine. As a result, students learn to do computations but not to understand. Repetition does instill learning, but with its emphasis on formulas and rigid rules, traditional calculus teaching falls dismally short of educating students in the realm of applying what they learn. In the end, many of those students who do well in the traditional calculus course achieved their success by memorizing rules and procedures to plug into homework and answers to exam questions.

Algebraic manipulation is important, but the thinking process is paramount. Asking students to memorize theoretical definitions with no real understanding leads to passivity. Nevertheless, most traditional calculus teachers expect their students to rise to a higher level without addressing students on a level that most of them can understand. In other words, students are expected to take giant steps in intellectual understanding without first having established a solid foundation.

Traditional calculus textbooks also have few problems in them that relate to real-world situations. This leads to disinterest in students who are not pursuing a career in mathematics or physics. In the end, traditional calculus education has undergone reform because it has committed a cardinal sin of education—failure to make calculus interesting or exciting to most of its students.

How Is the Reform Approach Better?

Reform-based calculus uses a radical new approach to teaching calculus based on real-world problems, computer technology, and more active participation by students (both individually and as a group). The goal is to make calculus more accessible to a wider group of students and to improve students' learning and comprehension of the field, moving away from the "chalk-and-talk" approach in which teachers lecture and students listen. Instead of the traditional approach in which students often work in isolation, more emphasis is also placed on interactions, both among students and between students and teachers.

For many students taking a traditional undergraduate college calculus course, a nagging question usually filters through their minds: "What on earth does this have to do with anything?" Calculus reform's emphasis on real-world problems helps to pique students' interest, motivate them, and increase their conceptual understanding. For example, real-world scenarios can be generated from numerous fields, including biology, economics, and even military science. They are often based on important issues that could impact everyone, like the growth of populations or the spread of epidemics. It is the investigation of practical problems that leads to formal definitions and procedures and not vice versa.

Integral to the new approach to teaching calculus is the "Rule of Three," that is, making sure students understand the interaction among numerical, symbolic, and graphic representations as opposed to the emphasis on algebraic solutions alone, as has been the custom in traditional calculus courses. (Some calculus reform proponents refer to the "Rule of Five," adding written and oral communication to the three basic rules.) The Rule of Three fosters the concept of looking at things from more than one perspective, which is valuable in more than just mathematics. It encourages understanding but gives students with weak manipulative skills the opportunity to grasp the basic concepts of calculus while strengthening their backgrounds.

One example of a problem that allows students to use multiple strategies to solve real-world problems would be population growth comparisons, or exponential growth. For example students may be asked to make a table showing the changing population of two cities and to determine in which year one city's population first exceeded the other city's population based on each city's populations in a specific year (say 1980) and growth percentages for all the following years (for example, 1.5% for city A as opposed to 4.5% for city B). The problem can be solved several ways, including using arithmetic to create one table one line at a time, numerically using tables together, graphically, and traditionally using logarithms. By using these separate approaches, students would know "automatically" that something in their solution of the problem was wrong if the answers used in different approaches did not match.

The additional emphasis on students being able to talk and write coherently about the "problem" and its "solution" also increases their interpretive and communications skills. It enables students to explain in a language he or she understands rather than using a foreign language, which is what mathematics is for many students. For example, students may be presented with the problem of determining how to choose the best telephone savings package and how much money would be saved by choosing a specific plan. In addition to the use of step functions to solve the problem, they then may be asked to write a letter to convince their telephone company to give them better rates and to include a written explanation of the mathematical justification for their request.

Technology

Traditional calculus education has lagged behind most other disciplines in integrating technology into the classroom. Most of the algebra needed to perform calculations can now be done by a computer, eliminating the need for the laborious and, for the most part, boring rote, paper-and-pencil calculus of traditional calculus learning.

Computer technology also improves learning through its use of graphics. Educators recognize that students comprehend things more readily when they can visualize it. With computers, students access well-conceived interactive computer graphics and can create their own graphics of almost any function as well. With software such as Mathematica and Matlab, students can quickly produce graphs of most functions, fostering a geometric approach to calculus, better understanding of concepts, and stronger connections between graphical and formulaic representations.

Several evaluations of students using computer algebra systems to do computations compared with students using traditional approaches showed that both sets of students had about the same manipulative skills for completing problems. However, students using the computer performed slightly and sometimes significantly better on conceptual problem solving.

Admittedly, technology in and of itself is not necessarily better. But the emphasis of reform is on understanding and not computer technology. For example, the old square root algorithm is no longer taught in grade school because it can be done with a calculator. But that does not mean that students can't be taught or grasp what the square root means. It is clear that computers help students to break free of the meaningless plug-and-chug approach to solving problems by fostering students' understanding of the interaction among numerical, symbolic, and graphic representations. The addition of the computer "laboratory" in many calculus reform programs also fosters communication as students work more in groups to solve problems.

Overlooked Benefits

An often overlooked but significant aspect of calculus reform is that it fosters an ongoing dialogue about the best ways to teach calculus. It is important to have the right mix of computer technology and hands-on approach, and many questions have been raised regarding this balance. How much technology should students use? How much of the mechanical skills should be discarded? How does a teacher develop a proper balance between theory and application? By probing and discussing these issues, calculus education can't help but improve.

Although the "demographics" of who studies and does well in science and math are changing, they traditionally have been white males. The reason why females and other minority students have not done as well may include social influences and access to quality education. Calculus reform can help these students. With less emphasis on formula memorization and traditional testing, even students with weak algebra skills can succeed in calculus and build confidence. The symbolic computational skills that are necessary are then more easily learned. Although more formal data is needed on outcomes due to reform calculus, many reform calculus teachers are reporting improved retention and passing rates.

The Future: A Synthesis

Throwing out the baby with the bath wash is not the intent of calculus reform. It is also not about technology. Calculus reform is about making calculus more accessible to more students and, as a result, improving the learning process. Although the content in many reform calculus courses has been greatly revised, these courses still focus on the traditional nature of calculus: applying it to problems of change and motion and developing precise definitions combined with rigorous statements of results. Reform calculus is also not a dilution of calculus education. Students discover that the courses are more challenging and demanding, but in an interesting way.

According to the authors of Assessing Calculus Reform Efforts , over 95% of the colleges and universities that incorporated calculus reform into their curriculums continued to use reform texts the next year, and very few faculty members who have tried the reform approach have gone on to abandon it. By emphasizing interactive, numerical, and geometric reasoning combined with communication skills, reform calculus has incorporated new approaches and materials to improve students' conceptual understanding and overall ability to learn calculus. Calculus reform represents a shift from an approach that seemed intent on "weeding out" students, focusing only on those who appear suitable for the rigors of calculus and scientific careers. In contrast, reform calculus "equalizes" the learning environment, enabling more and more students to appreciate the fundament nature of calculus, while opening up new vistas of understanding and career opportunities.

— DAVID PETECHUK

Viewpoint: No, the calculus reform project purges calculus of its mathematical rigor, resulting in a watered-down version that poorly prepares students for advanced mathematical and scientific training.

The essential difference between the reform and traditional approaches to teaching calculus is the emphasis placed on solving essential mathematical problems using a rigorous and sophisticated approach. Calculus reform requires only a superficial use of mathematical skills. It emphasizes real-life problems but does not believe that students must first be able to understand concepts and actually perform the algebraic equations necessary to do calculus. In contrast, traditional calculus focuses on teaching students mastery of basic calculus skills, including understanding, developing, and using theory and proofs.

In the late 1980s, with 40% of undergraduates failing introductory calculus, educators began to question their approach to teaching calculus. Not only was the poor student performance raising college administrators' eyebrows, but calculus teachers' own colleagues in science and engineering academic departments were complaining that students, after taking basic college calculus courses, were still unprepared for the calculus needed in their courses. Their solution, in the form of calculus reform, was to make the courses "easier" and to condemn computational skills and mathematical proofs as "boring" and "unnecessary." Increased emphasis was placed on computer learning and more conceptualization, including students writing about the problems and how they came up with solutions.

Trying to usurp basic skills with conceptual understanding is putting the horse before the cart. For example, while an emphasis on students writing about their calculus problems seems to be a good idea, the very essence of calculus is that it's a symbolic representation that presents a different, more scientific approach than everyday human language to describing and learning about the world.

Traditional calculus does not ignore conceptual understanding. However, it does emphasize problem-solving skills as the way to conceptual knowledge. Basic computational skills and understanding are inextricably linked in the world of mathematics. For example, on the basic arithmetic level, adding, subtracting, multiplying, and dividing reveal patterns and relationships that are absolutely vital to understanding algebra. Although modern, streamlined, computerized society may eschew the one-step-at-a-time approach, it is the ability to perform basic tasks that leads to progress, and this is especially true in mathematics. Once the "basic" computational skills are learned, students are then "free" to pursue conceptual understanding in a meaningful way.

High-Tech Math

Most adherents of calculus reform claim that the increasing use of computers in the classroom allows students to experiment and be creative in dealing with more open-ended questions, most of which have more than one single right answer. While the computer may give students more to talk about, it does little to enhance the computational skills needed for real understanding. For example, many reform calculus computer programs help students to avoid using l'Hôpital's rule for calculating limits or even to do calculations by integration by parts or partial fractions, all of which are basic to a true understanding of calculus.

Overall, the use of computers has not provided students with enough practice in standard algebraic manipulations. In addition, computers, coupled with poorly thought out calculus reform textbooks, often underexpose students or gives them no exposure at all to such basics of algebra and calculus as:

The definition of limits—a theoretical foundation of calculus that enables students to know what "e" to the "x" really means
The definition of continuity—a fundamental idea in mathematics that is often under-treated by calculus reform texts and computer programs
The intermediate value theorem—a theorem that is important in both theory and the practical use of finding roots of functions
Partial functions—a basic calculus topic that is necessary for science and engineering majors in later courses

Both computers and standard calculus reform texts also emphasize open-ended problems, which only furthers students' misunderstanding concerning the idea of proofs. In general, calculus reform proposes that if a mathematical proposition is true for a finite number of cases, then it is true in general. This leads to confusion between when something has been shown to be true and when it is likely to be true. By the overuse of general conclusions, calculus reform software programs and texts are giving students a false picture of mathematics. Telling students that something is always true because of three or four examples is a "lazy" arbitrary approach to mathematics that actually hinders students' critical thinking and understanding.

In addition to relying on programs to perform basic tasks, computers rob students of time that could be used to learn calculus instead of learning how to use the computer software. For most students, their short time learning calculus is the only time they would ever use such a program. Computers also take away from the time spent on human dynamics of teaching, such as adjusting the material to the audience, setting a pace for students to learn, and instilling critical thinking and reasoning. Not less but more teacher-student interaction is needed to help students become engaged in the process of learning calculus and to set standards for what understanding calculus really means. Finally, establishing expensive computer "labs" for calculus courses also drains much needed monetary resources that could be better used for faculty training and seed money for overall department improvements.

Race, Ethnicity, and Gender

The calculus reform movement also draws legitimacy from its proposal that the reform movement helps "level the playing" field for people of both sexes, regardless of race or culture. By de-emphasizing mathematical rigor, say the reformists, students from lower socioeconomic groups who did not have access to quality education or women who did not learn calculus because of social biases will now be able to "compete" with white and Asian males who dominate the math and engineering fields in the United States.

Culture undoubtedly plays an important role in academic achievement and in the opportunities and personal inclinations to pursue college and higher education. Nevertheless, racial and gender theories of learning must be viewed with caution. Is it "logical" to make courses "easier" so that more students do well rather than maintaining a high standard of integrity in a discipline that is admittedly difficult for many students? The reform movement seems to have gone astray by focusing on how to make calculus education easier. What is needed is a long, hard look at the discipline of mathematics education and how to improve teaching to obtain the same end results in terms of knowledge, mathematical ability, and understanding of calculus.

A famous example of how traditional educational approaches can be effective in helping minorities, women, and other groups thrive in mathematics is Jaime Escalante, whose success story in high school mathematics was made into the movie Stand and Deliver. Escalante did not seek to "dumb down" his teaching approach. The high school where he taught had a large minority population of Hispanics and working-class students and an extremely poor performance rate by its students in testing. In fact, the school was in peril of losing it accreditation in mathematics. Escalante turned the tables by insisting on the basics but making it exciting for his students. He went on to create a calculus class, with a significant number of his students going on to pass an advanced placement calculus exam. Even more important, more of his students went on to universities equipped with a strong understanding of basic algebra and calculus. The fact is that calculus reform movement came about, in part, because high schools failed to teach minorities, women, and others the basic algebra needed for calculus. Part of the answer to teaching calculus in college is to improve high school mathematics education and to set higher standards for teachers. No one is served by being required to learn less, which, in the case of minorities and women, only perpetuates stereotypes.

Calculus reform's failure in addressing specific student populations is further compounded by the educational reform notion of all for one. Universities using calculus reform are tending to use reform courses for all students, including those looking to pursue careers in math, engineering, and the physical sciences. Requiring science majors to take a three-semester calculus course that provides comprehensive coverage of theories and proofs is absolutely fundamental to their future performances, both in academia and in their chosen fields.

Real-World Problems

A good idea associated with calculus reform is the emphasis on real-world problems. Few would argue that students show more interest in learning something if they can apply it in a practical way. This is true not only for non-science majors in college, but also for students pursuing engineering and other science-based fields. However, using real-world problems still requires mathematical rigor. Just like the real-life example of Jaime Escalante in Stand and Deliver, the search should focus on how to better motivate students so that they will learn algebra and calculus as opposed to decreasing expectations by not requiring mathematical skills. Calculus is an analytical tool that represents the world in symbolic forms, and analysis is rooted in the practical.

Many of the newer calculus textbooks are beginning to recognize this fact and, as a result, are shying away from the most "radical" ideas of calculus reform. Instead, they are committed to the "roots" of calculus education in understanding and using mathematical models to describe and predict physical phenomena through the ability to use convergence, limits, derivatives, integrals, and all the other fundamentals of algebra and calculus. Real-world problems should be used to focus students' minds on practical problems, but in a way that motivates them to learn and use algebra and mathematics in calculus, not discard them.

Where's the Proof?

Calculus reform is not reform at all but a fad, that is, something followed with exaggerated zeal. Too many universities and calculus faculty have accepted calculus reform as being universally superior to traditional calculus. But there is little evidence to support this belief. Why are colleges and universities in such a rush to discard traditional methods of teaching calculus when the efficacy of calculus reform classes has not been established? For example, no one is sure if calculus reform is really increasing students' critical thinking and their understanding and mastery of mathematics. And there is little evidence that calculus reform programs are really making students excited about calculus and mathematical science. Many universities, such as the University of California, Los Angeles (UCLA), and the University of Iowa, have either flat out rejected calculus reform after developing and using reform courses, are in the process of moving away from reform courses, or are instilling a stronger traditional approach to them.

While the reform movement takes away from mathematical content in calculus courses, its emphases on group learning, computers, activities and projects, and writing has led to the evaluation of students and grading systems that are based more and more on the "processes" of learning as opposed to understanding content. In addition, grading approaches vary greatly from one class to another and from one university to another so that "standards" are practically non-existent. The result is that calculus reform results in large variations in the quality of teaching calculus and tends to confuse teachers on how to differentiate between success and failure in the course.

The trend to water-down calculus by purging it of its mathematical rigor in order to reduce failure rates and increase overall grade point averages is a dire mistake. It is not possible to provide students with an understanding of calculus by minimizing the component skills. The way to improve students' understanding and achievement in college calculus is to maintain standards based on the fact of life that hard work leads to success and results. Calculus should not be expected to apologize for being difficult for many students. Instead, calculus education should adhere to the principals of traditional calculus to provide students with skills in computation, conceptualization, and theory—skills that will help students succeed.

—DAVID PETECHUK

KEY TERMS

FUNCTION:

A mathematical relation in which each element of a set is assigned to exactly one element of (the same or) another set.

RULE OF THREE:

A central part of the calculus reform approach that states that every topic should be approached graphically, numerically, and algebraically, rather than the algebraically dominated traditional approach.

THEOREM:

A mathematical proposition that is provable on the basis of explicit assumptions or deductive logic. Some famous mathematical theorems include the Pythagorean Theorem and Fermat's Last Theorem.

Science in Dispute