Calculus for the 21st Century

David Bannard, Carolyn Eck, Marilyn Keir,
Kimberley G. McDowell, Bonnie Peterson,
H. W. Straley, Kathryn Trier, and Mary Sue Wyss

Philosophy of the Course

Traditionally, calculus has been presented from an analytical point of view often devoid of meaningful applications. Calculators and computers, with their powerful numeric, graphic, and symbolic tools, provide new opportunities for taking a multiple representation approach to the study of calculus. In particular, greater use of visualization, approximation, and prediction can be made in calculus instruction.

A modern calculus course should foster in students an appreciation and skill that allows them to apply their mathematical knowledge in a variety of practical situations. Ideally, successful completion of the course would provide students with the ability to pick up a newspaper and recognize the calculus that surrounds them. We hope students will apply their knowledge to situations that they face in their everyday lives.

Calculus students should look back on their learning experience with favor and in such a way that they desire to continue their pursuit of mathematics. Through technology, students may now take an active role in their learning. We are now able to create an environment which is rich with technology, nurtures curiosity, and promotes action. Mathematics is an experimental science and should be treated as such. Therefore, employing lesson plans that include laboratory activities, discovery exercises, individual projects, applied problems, writing exercises, and open-ended questions should be an integral part of the course.

A Sampling of the Course

It is not our purpose to detail a first year calculus course. However, the following illustrates an approach to calculus that utilizes hands-on experience and technology. This approach makes learning functions, limits, continuity, derivatives, integrals, approximation, and their applications a more enriching experience for the students and teachers.

LIMITS AND CONTINUITY

Limits are critical to the study of calculus. While the development of a rigorous definition is necessary, formal proofs may be de-emphasized in a first course. Limits should be approached numerically, graphically, and analytically. Graphing calculators are wonderful tools to help develop a clear-sighted concept of limits. An intuitive understanding of the e and d definition can be explored using the idea of local linearity.

Looking at problems like

(1 + )^x

both numerically and graphically greatly enhances understanding. Introducing L'Hopital's Rule early in the course is desirable. Students need to examine, graphically and analytically, the relationship between left and right hand limits, continuity, and local linearity.

DERIVATIVES AND THEIR APPLICATIONS

Because calculus is the study of change, the derivative and anti-derivative continue to be the focal point of this study. The definition of derivative and the relationship between differentiation and continuity must be emphasized as well as important theorems like The Mean Value Theorem. The derivatives of polynomial, rational, trigonometric, exponential, logarithmic and piece-wise functions must be studied. In addition, students need to have a working knowledge of implicit differentiation, logarithmic differentiation, the chain rule, product rule and quotient rule. Numerical estimates of the derivative should also be emphasized.

Applications of the derivative should include related rates, maximum/minimum, and motion problems. Questions in these areas should be realistic and focus on applications. The derivatives and their relationship to slopes, concavity, and the linearization of a curve continue to be important components of calculus. Newton's Method and similar iterative techniques should be part of the curriculum.

INTEGRALS AND THEIR APPLICATIONS

Relating motion and the anti-derivative to area and the Fundamental Theorem of Integral Calculus is a primary goal. Given a rate of change, a student should be able to construct the function. The integral as the infinite sum should be explored by several methods including rectangles, mid-point, trapezoid, and Simpson's Rule. With technology, these methods can be explored without tedious computations.

DIFFERENTIAL EQUATIONS

Differential equations are a common theme throughout a first year calculus course. They provide a wonderful opportunity for students to model real life situations. Numerical methods to solve differential equations, along with associated error analysis, help the student to understand the real world of applied mathematics. It is not necessary to solve differential equations solely by analytical methods when other approaches are just as enriching.

SEQUENCES AND SERIES

Students should have a thorough understanding of geometric series and the concept of estimating functions with infinite series. Graphical relationship, the ratio test, the comparison test, intervals of convergence, and error analysis should be addressed, especially with new technology.

Course Books and Resources

Several major calculus reform projects are currently in progress throughout the country. In selecting a calculus textbook, much consideration should be given to both the use of technology and the curriculum content. Major projects have been undertaken at Duke, Harvard, NCSSM, Ohio State, Oregon State, Smith, and St. Olaf's College among others. Associated with many of these are training institutes which are funded by the NSF. Other institutes and inservice opportunities exist. These programs are exciting opportunities for teachers to learn how to incorporate technology into the teaching of calculus and select curriculum materials based on current thinking in the field. Also, close attention should be paid to periodicals, newsletters and journals as a means for staying current with new ideas, technology, and trends in calculus reform.

Focus on the Future

The power of current and future technology can no longer be ignored in classroom instruction. We are faced with technology that is changing at an exponential pace. Teachers must look at how and what they teach in this environment of change. The availability of technology has caused changes in the curriculum. Some content will receive less emphasis, some content will receive more emphasis, and solutions to problems that were previously inaccessible are now possible. Open-ended problems and mathematical modeling need to be an integral part of the calculus curriculum. Writing and group activities are important to constructing and applying knowledge. Teachers must assume the role of a life-long learner and must convey this role to their students. Modes of assessment, including the Advanced Placement examination will, of necessity, change to reflect the use of technology in instruction.

Bibliography

Dick, Thomas P. and Charles M. Patton. Technology in Calculus. PWS-Kent Publishing Co., Boston, 1992.

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