The textbook Calculus: A Modern Approach represents years of research into what a modern calculus course should be, but it does so without sacrificing topics from the traditional calculus curriculum that many feel are necessary to a complete calculus course. A quick perusal of the table of contents will show that topics like trigonometric substitutions, partial fractions, Taylor’s theorem, and the Mean Value Theorem are well-represented in Calculus: A Modern Approach. Our goal is not to “reform by omission” but instead, to “reform by modernization.”
That is, we present calculus with the most modern definitions of concepts, the most current uses of technology, and the most modern applications. For example, although we use the terms dependent variable and independent variable where appropriate, we often use the terms input variable and output variable when discussing functions. The discussion of the definition of the limit is enhanced by using the topological concept of a neighborhood of a point, and simple function approximation is used to provide a more modern presentation of the concept of the definite integral. Throughout the textbook we base our definitions and discussions on our current understanding of what calculus is and what it is used for.
Similarly, the use of technology is not only invited but is in some sections required. There are sections such as 1-4 on the definition of the limit, 4-1 on simple function approximation, and 7-6 on slope fields and equilibria that are based on the use of a graphing calculator. There are also sections such as 3-7 on least squares and 4-2 on approximating definite integrals that depend on the construction of tables of numerical data. In fact, nearly every section includes examples, exercises, and applications that utilize technology.
Specifically, exercises requiring the graph of a function are denoted by the word grapher and exercises requiring tables or the analysis of data are denoted by the word numerical. Exercises denoted by the phrase Computer Algebra System are infrequent, but they require the use of a computer algebra system. In addition, there are Write to Learn exercises that ask students to present their results in essay form, and there are Try it Out! exercises that provide opportunities for hands on activities involving calculus.
Each chapter concludes with a self-test that presents ideas in the chapter in a variety of question formats and with a Next Step essay that demonstrates how an idea or ideas from the chapter can be extended to new contexts. Each Next Step essay includes write to learn exercises and at least one activity designed for group learning. Finally, there is an Advanced Contexts section after each next step that can be used to challenge better students, as well as some additional precalculus review after some of the chapters.
The overall effect is only a modest change in the existing curriculum, but it is one that makes the calculus curriculum more suggestive of the motivation, concepts, and practices of modern mathematics, science, and engineering. The applications and the uses of technology also reflect our goal of more closely aligning calculus with its current applications in other fields, and similarly, we have also made some subtle changes in pedagogy that reflect some important results from research in mathematics education.
For example, research in mathematics education suggests that it is best to present new ideas in limited contexts. For this reason, our introduction to calculus in section 1-1 is an exploration of tangent lines to polynomials. This allows familiar ideas from algebra and analytic geometry to be used to develop a meaningful understanding of the methods and intentions of differential calculus. Similarly, our introduction to the integral is presented in the familiar context of a bar graph, which is also known as a simple function approximation of a given function.
Introduction within simple contexts leads to the use of recurring themes to revisit concepts time and time again with each visitation reinforcing the concept in a unique fashion. For example, section 1-7 uses the simple context in section 1-6 to introduce the concept of a rate of change, and then rates of change are fully developed in section 2-5. Rates of change are subsequently revisited in section 2-8, several times in chapter 3, and throughout chapter 7 on differential equations. As another example, consider that the chain rule is introduced in section 2-3, is reinforced with implicit differentiation in section 2-4, and is restated again in sections 2-5, 2-6, 2-7, and 2-8 in the context of transcendental functions. Additionally, the chain rule is restated again in sections 3-1, 4-4, 4-7, 6-1, 6-2, and on numerous occasions in later chapters. Similarly, monotonicity and concavity are revisited in three different chapters, and limits are reviewed and revisited time and time again.
The advantages to this approach are many and varied, but here we will mention only two. First, the early introduction of fundamental concepts means that students using this textbook spend far more time with the main ideas in calculus than they would have otherwise. For example, L’Hôpital’s rule occurs in the chapter 3, “Applications of the Derivative,” which means that students will have worked with these concepts several times by the point at which they would have encountered them for the first and perhaps only time in other textbooks.
The second advantage of our approach is that it allows calculus itself to be used as a context for introducing new ideas in calculus. In traditional settings, this practice is exemplified by the use of the tangent line concept in motivating Newton’s method, and it is this tangent concept reinforcement role that many refer to when discussing the importance of Newton’s method in the calculus curriculum.
In Calculus: A Modern Approach, calculus themes often “recur” as a context for new calculus concepts, much like in Newton’s method. The result is that concepts used to introduce new ideas are reinforced even as new concepts are introduced. For example, section 1-7 uses the concept of linearization in section 1-6 to introduce the concept of a rate of change. Likewise, the derivative form of the fundamental theorem, which is presented in section 4-4, is used to motivate the discussion of antiderivatives and the rules for antidifferentiation presented in section 4-5.
Moreover, by the middle of the textbook, calculus is often presented as a coherent context rather than as a collection of computational techniques. The first instance of this occurs in section 7-4, “Mathematical Modeling,” which explores how scientists use empirical data in combination with differential equations. In addition, calculus as a context for exploring ideas is used in section 7-7 for the study of equilibria , in section 8-2 for the study of discrete dynamical systems, in section 9-4 for the study of counting problems in combinatorics , and in a host of other sections including the multivariable chapters found at http://math.etsu.edu/multicalc/ .
Pragmatically, the use of recurring themes means that a course based on Calculus: A Modern Approach is both flexible and forgiving. Light coverage of a given section does not penalize students, because any concepts essential to the calculus curriculum will be revisited and explored in a similar context in a later section. In addition, recurring themes means that each section contains numerous examples that relate directly to the exercises, which is in direct contrast to many of the reform texts of the past.
In fact, we have designed the textbook so that the flexibility of recurring themes can be readily utilized. Each section in each chapter is comprised of 4 subsections and an exercise set. The first 3 subsections contain material that is important to later work and thus must be covered. However, with few exceptions, the fourth subsection is not essential to later work and can either be covered briefly or even omitted. In most sections, the concluding fourth subsection contains items such as additional graphical and numerical techniques, proofs of theorems, additional insights into previous material, and alternative techniques and identities.
There is also a great deal of flexibility in the use of technology. The text was prepared with the assumption that students would have a graphing calculator with some computer algebra abilities (e.g., with a TI-89). However, the course could be taught to students who have nothing more than a scientific calculator, primarily because the omission of graphing calculator exercises does not eliminate topics from the text. Alternatively, the textbook lends itself quite well to the use of more sophisticated technologies such as Maple and Mathematica , and we are already preparing supplements to indicate how such tools could greatly enhance and complement our approach.
Thus, it is conceivable that an instructor could progress through the course at breakneck speed by simply covering the first 3 subsections of each section and by only using the most modest amounts of technology. Or more desirably, an instructor could choose what topics to emphasize, how much coverage to provide to each topic, and how much technology to employ in that coverage. In either circumstance, the instructor can choose exercises and applications that best suit the needs of the students, whether they are mathematics majors, aspiring scientists, engineering students, or future businessmen.
Finally, let us briefly describe how our approach was developed and what impact it has had on our students in the 4 years that it has been used in the classroom. We began by developing a comprehensive plan for writing a calculus textbook, one that was based on exhaustive research on the following topics:
Development of our comprehensive plan also included extensive discussions with students, detailed examinations of existing calculus textbooks, and a model of mathematical learning incorporating much of what is currently known about concept acquisition and development ( Knisley , 2002).
The textbook is a direct result of that comprehensive plan. For example, it is well documented that the limit concept presents major difficulties for even our best students, and consequently, students have very little success in understanding the limit concept in an introductory calculus course (e.g., Davis and Vinner , 1986; Szydlik , 2000; Williams, 1991). However, introducing limits, derivatives, and tangent lines in the familiar context of polynomials allows students both to develop meaningful intuition about limits and to be exposed to the tangent concept independent of the limit context in which it will be rigorously defined in a later section.
Similarly, the presentation of the definite integral in chapter 4 was developed both with the modern concept of the integral and the interests of the student in mind. The goal was a definition of the integral that resembles definitions used in higher mathematics, engineering, and physics courses. The definition used in the text is the result of feedback and suggestions from a group of first semester calculus students who examined several different statements of the Riemann sum definition of the integral.
The result is a textbook that several of us have used successfully for the past 4 years. Departmental final exam scores for students in sections using Calculus: A Modern Approach are significantly higher than other sections. We have also documented superior performance on standardized test problems, such as from past AP and actuarial exams. Moreover, several papers and presentations, both faculty and undergraduate, can be directly attributed to exercises and Next Step material found in this text (e.g., Kerley and Knisley , 2001; Knisley , 1997).
However, the most profound evidence of our text’s success has been our opportunity to experience anew with our students the power and elegance of calculus. We have had students ask their chemistry professors for data to use in the mathematical modeling section. We have had groups of students ask us for more substantial and challenging problems in areas such as discrete dynamical systems, special functions, and combinatorics . Each year we receive gifts and cards expressing our student’s appreciation of their calculus experience.
Thus, we are convinced that our approach has allowed this textbook to advance in at least some small increment beyond what other books have done to capture the excitement and enjoyment that lured each of us into the study of higher mathematics. Indeed, we believe that our textbook excels at presenting calculus as growing and thriving, relevant and strong.
Thank you for exploring the textbook. We hope that once you have examined it, you will be as excited and enthusiastic as we are about presenting calculus in both as mathematically modern and as pedagogically sound a manner as is currently possible.
Jeff Knisley and Kevin Shirley
Davis, Robert and Vinner , Shlomo . “The notion of limit: Some seemingly unavoidable misconception stages.” The Journal of Mathematical Behavior, 5 (1986), 281-303.
Kerley, Lyndell and Knisley , Jeff. “Using Data to Motivate the Models Used in Introductory Mathematics Courses.” Primus, XI( 2), June 2001, 111-123.
Knisley, Jeff. Calculus: A Modern Perspective, The MAA Monthly, 104:8 (October, 1997) 724-727 .
Knisley , Jeff. “A 4-Stage Model of Mathematical Learning.” The Mathematics Educator, (12) 1, 2002, 11-16.
Szydlik , Jennifer E. “Mathematical Beliefs and Conceptual Understanding of the Limit of a Function.” Journal for Research in Mathematics Education 31(3) (2000): 258-276.
Williams, Steven. “Models of limit held by college calculus students.” Journal for Research in Mathematics Education, 22 (1991), 219-236.