From the Authors…
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:
1. How calculus is used in modern science, mathematics, and engineering 2. What research in mathematics education tells us about teaching calculus 3. What issues and debates occurred in the past 150 years of calculus
instruction
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.
Sincerely,
Jeff Knisley and Kevin Shirley
References
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.
What motivated us to write yet another
Calculus Textbook? Calculus occupies a pivotal position in math and science education. Typically, it is the first exposure our students have to higher mathematics, it is the first encounter with modern concepts of rigor and proof, it is the foundation for much of the mathematics used engineering courses, and it is the mathematical language that will be used by scientists to express many of the most important ideas in science.
We are of the opinion that calculus textbooks are not presenting calculus as the foundation of modern mathematics, engineering, science, and technology. The needs of modern scientists seem to have little influence on the calculus course, the "rigor" in calculus is uneven and largely unmotivated, and many of the applications seem out of date and out of touch.
We wrote this book as a first step in addressing the foundational role of calculus. However, it soon became apparent that "modernizing" the calculus course would also require an examination of pedagogical issues as well. In fact, we decided that to be truly effective, a calculus textbook would have to address 3 issues in particular:
• How students learn mathematics and in particular, calculus • How calculus is used in modern mathematical, engineering, and
scientific applications • How best to use technology, reform, and traditional techniques to
address the first two issues
Over the next few years, we researched these issues until we had addressed them to our satisfaction. We also worked with students to ascertain their preferences between different topics, definitions, and applications. These efforts resulted in a detailed plan for writing the textbook, and the implementation of that plan has now culminated in the textbook itself.
Why do we use the term "Modern Approach?" The teaching of calculus has changed a great deal over the past 300 years. Originally, calculus was introduced with differentials and was rigorously based on Taylor's theorem, with integration considered the inverse of differentiation. Cauchy changed all that by showing that calculus followed from the Mean Value Theorem and that integration is the limit of a sum. Weierstrass and Lesbesgue changed it again by making limits and integrals set-‐theoretic and by replacing the Mean Value theorem with results that followed from absolute continuity and uniform convergence. In this century, differential forms, operator theory, numerical analysis, and dynamical systems have continued the ongoing transformation of what calculus is and what it is used for.
How then to present calculus with a consistent interpretation while maintaining at least some semblance of rigor? After struggling for quite some time, we identified two important ideas which would allow us to do just that. First, we recognized that two themes have been central to calculus since the days of Newton to the present and will continue to be central for years to come. These two themes are differential equations and integration. That is, nearly all the theory and application of calculus is reflected in the study of differential equations and the theory of integration.
Second, we realized that the most modern realizations of calculus are also the best. That means using algebra to study the derivative and using simple functions to define integrals. It also means including applications which involve data, developing the idea of a mathematical model, and using sequences to study discrete dynamical systems. Thus, our "modern approach" is one that presents calculus as the foundation of modern mathematics, science, and engineering.
How is our book different from other Calculus textbooks?
Although we present topics numerically, graphically, analytically, and literally, we are not just another reformed textbook. Our goal is to present calculus as a coherent body of knowledge and to do so with as much rigor as is possible for students in a first course. However, great care has been taken to present the calculus content in a way that incorporates what is known about how students best learn mathematics.
Calculus: A Modern Approach begins with the differentiation of polynomials, because the derivative of a polynomial can be defined algebraically. We then introduce the limit as a means of extending the theory of the derivative to a broader class of functions. The Mean Value Theorem is introduced and used to complete the elementary theory of the derivative, although the Mean Value Theorem is not proven at this time.
Once the theory of the derivative is completed, the exponential, logarithmic and trigonometric functions are defined and studied. Much of this study is motivated and developed using the fact that the elementary functions are either the solutions or inverses of the solutions to linear differential equations.
Chapter 4 introduces integration with a modern definition of the Riemann integral. Antiderivatives are intimately connected to the Fundamental theorem. Applications of the integral, differential equations and modeling, Taylor's series, and Fourier series then follow.
What’s Wrong with Calculus
Jeff Knisley (with Kevin Shirley)
Introduction
The fundamental theorem of calculus has not only made calculus one of the
most powerful intellectual tools known to man, but it has also created a dichotomy
that makes calculus very difficult to teach. Should calculus be presented as the
taproot of geometry? Or should it be presented as the tip of the analysis iceberg? Is
calculus the first step toward an understanding of the topology of the real line? Or is
calculus the first step in the exploration of manifolds and the geometry of
mechanics?
Of course, the answer is “both,” and therein lies the crisis. This idea was
explored in detail by Halmos in a 1974 essay, and likewise, many generations of
mathematicians have declared in their own words that “all calculus books are bad.”
In fact, it was once accepted that traditional approaches are flawed, as evidenced by
so many of us saying we did not know calculus until graduate school. At one time
nearly everyone talking about “Calculus: a Pump, not a Filter” and the need for a
“Lean and Lively Calculus.” Reform textbooks are also widely viewed as flawed
products, so much so that they are driving calculus instructors back to the
traditional approaches they once condemned. After centuries of discussing how
calculus should be taught, we still today find that most mathematicians do not learn
calculus until they are in graduate school.
So is there anything wrong with calculus? If so, what is it? What should
calculus be about? How can calculus be presented to the general population in a
meaningful way? These were the questions we began to address when we decided
to write our own calculus book. This paper presents the answers we formulated in
the course of writing that calculus book. Hopefully, even those who do not accept
our calculus book as part of the solution will find this discussion helpful in
identifying the problem and how it might be approached.
Evidence of a Crisis in Calculus
It is unlikely that there will ever be a means of teaching calculus that allows every theorem to be proven
rigorously, every concept to be developed completely, and every meaningful application to be explored.
This is true in many introductory math courses and is not evidence of any crisis in calculus, in our opinion.
However, there is a great deal of evidence of a crisis in calculus that has nothing to do with its being an
introductory course. We focus on 3 categories of such evidence. We admit up front that the analysis below
is solely our opinion and is in all likelihood biased by our desire to have compelling reasons for writing our
own calculus textbook.
How Mathematicians Discuss Calculus Among Themselves
The crisis is evident in how we as mathematicians discuss calculus with each
other. In fact, nearly all discussions of calculus I have been involved in are stilted
and disjoint. It is as if our knowledge of calculus is rote rather than logical, or as if
we are using a different part of our brain when we begin discussing ideas from a
first year calculus course. Seemingly, even the most proficient mathematicians
struggle with calculus and fail in much the same way that our students do.
For example, I read a test question written by an accomplished researcher
that asked for the “tangent line to a function at a given point.” Although mixing
function terminology with geometric terminology is admittedly a minor
misstatement at worst, it is only the tip of the iceberg. I have heard calculus
instructors make statements like “locally a tangent line intersects a curve at only
one point” and then almost immediately make the contradictory statement “the
tangent line to a line is the line itself.” And this is still rather tame compared to
statements like “Riemann sums converge to the function, so the integral converges
to the area under the function,” and the following mind twister I once overheard
from a hallway outside of a classroom: “if a sequence converges conditionally, then
so does its series—or not at all, unless its sequence converges to 0.”
Also, Calculus books are full of errors even though they are written and
reviewed by mathematicians. There are exercise instructions imploring the student
to “Let F(x) be the antiderivative of f(x) in which C=0.” (Calculus, Stewart, 4th
edition, page 522) (to see why this does not make sense, consider that
F(x)=sin2(x)+C and G(x)=–cos2(x)+C are both antiderivatives of p(x)=sin(2x) for any
value of C). There are also nonsensical definitions of limits of powers (Calculus,
Thomas/Finney, 9th edition, page 61), and flawed chain rule proofs (Calculus,
Larson/Hostetler/Edwards, 6th edition). And I am picking on these three because
they are arguably among the best calculus textbooks available. Space does not
permit the number of errors in calculus books we actually uncovered, including
large numbers of errors in reformed textbooks.
In contrast, mathematicians and textbooks rarely make nonsensical
statements when discussing trigonometry, or linear algebra, or even measure
theory. My suspicion is that many of us could not understand calculus we were
being taught at the time, so we relied primarily on memorization in our first calculus
course. The result is that when we try to fall back on our calculus background, it
comes out more like a memorized poem than a well-‐understood collection of
concepts (more on this idea later).
How Calculus Students use Calculus
If our own stilted conversations about calculus are not enough, then consider
the evidence all around us that even our best students do not learn calculus in a
calculus course. Check any computer algebra system to see how well our students
picked up on the necessity of the “+C” when computing antiderivatives.
Consider also that we make a great many seemingly absurd statements in a
calculus course, yet even the most inquisitive students display no intellectual
curiosity of any kind. Have you ever had a student ask how an average can be equal
to the ratio of two differences? Do any of them ever snicker when they first hear the
oxymoronic statement “C is an arbitrary constant?” Research has established that
even our best students reduce limits to a set of rules to be memorized. Not
surprisingly, many instructors and many students view Calculus as a course which
reinforces algebra and trigonometry and does little else.
We need not belabor this point, because the evidence over the past decade
has been overwhelming in showing that student's are not learning much calculus in
our calculus courses, including studies on retention of the material, ability to adapt
their calculus experience to new settings, and so on. Suffice it to say that student
performance is sufficiently low to support a decade of annual calls to new reform
ideas.
How Relevant Calculus Courses are to the Other Sciences
In spite of a calculus course that is saturated with “applications,” laden with
references to physics and chemistry, and packed with numerical techniques, most of
our colleagues in other disciplines see very little relationship between their fields
and the calculus course. In fact, except for possibly in colleges of engineering, our
colleagues tend to think of our calculus sequence as quaint and curious, important
but irrelevant. And most engineers will tell you that they have to "undo" much of
what the calculus course does to their students.
Of course, this is due in part to the fact that much of the calculus taught in
those courses is irrelevant. Simpson’s rule is no longer used for numerical
integration (except by mathematicians). The “applications of the integral” do not
even resemble the ways in which those concepts are examined in their respective
fields. That is not to say that applications of the integral are not important, but
rather that that mathematicians do not know how calculus is used outside of a
calculus textbook and thus skew all applications toward mathematical contexts and
away from their natural settings.
Moreover, the calculus that our colleagues receive makes them struggle mightily when calculus does occur
in their area. They teach their own statistics courses (usually quite poorly) as if calculus and statistics were
not inextricably intertwined. ( Can we actually expect to give students a meaningful introduction to the
Central Limit Theorem without any concept of a limit?) And even when instructors in other disciplines do
discuss small quantities and local approximation, they tend to hand-wave through any real use of calculus
and ignore all but the basic concepts of tangents and areas.
What is surprising is that many of these instructors will say that they rarely use calculus, if ever, or that all
they need are a few simple derivatives. What is not surprising, however, is that the most “mysterious”
topics in science are often those that rely heavily on calculus. The study of fields in physics is essentially
an exploration of the definition of the integral and the fundamental theorem of calculus, and yet it is the
rare student who has any grasp of Maxwell’s equations, how they are derived, and what they imply.
Clearly, our calculus course does not prepare scientists in other fields to
recognize, understand, and utilize the calculus that many of their fields are based
upon. Thus, when it comes to calculus, we don’t get it the first time around, our
colleagues don’t get it, and our students are still not getting it. It’s no wonder that
one of the most common occurrences in higher education is that of a non-‐
mathematics faculty member discovering that something they were doing is
calculus. And at the very least, we feel justified in asserting that there still is a crisis
in calculus instruction.
How to address that crisis it the topic in the next paper, "Facing the Crisis."
Facing the Crisis In Calculus
When it comes to calculus, we don’t get it the first time around, our colleagues don’t get it, and our
students are still not getting it. It’s no wonder that one of the most common occurrences in higher
education is that of a non-mathematics faculty member discovering that something they were doing is
calculus.
Something is wrong with calculus instruction, and the problem may be with
the calculus curriculum itself. Admittedly, there are ideas in calculus that will never
be accessible in a first course, and this will never be corrected. We certainly not
saying that any textbook we write will cure all that is wrong with calculus.
However, there are many problems that can and should be corrected, as we point
out below.
Circular Associations
Research indicates that learning mathematics depends heavily on the ability
to make connections between similar concepts. Indeed, a particularly strong way of
presenting a theorem is by placing it in the form “the following are equivalent.” For
example, in trigonometry, the association between right triangles and trigonometric
functions is fundamental.
However, in the calculus curriculum, many of the associations are
circular. All too often a given concept is associated with a concept that is defined in
terms of the original concept. Such connections increase the complexity of a concept
without shedding any insight on the concept itself. Not surprisingly, concepts
motivated with circular associations are the ones most often memorized with little
or no comprehension.
Consider, for example, tangent lines. The standard approach is to use secant
lines to motivate the difference quotient, after which the derivative is defined to be a
limit of difference quotients. The implication is that the derivative is the slope of the
tangent line, except that the tangent line itself is never defined. What then is a
tangent line, according to the standard treatment? It is, of course, the line through
the point whose slope is the derivative.
The result is that students do not develop any intuition about what a tangent line is, and conversely, their
understanding of the derivative is not aided by the consideration of tangent lines. Instead, tangent lines
become a metaphor for differentiation, important but without real meaning. In general, circular
associations often seem quite profound without actually revealing anything at all.
To further illustrate, let me list additional examples from calculus along with some of the confusion that
arises as a result. (This list is not exhaustive. Indeed, this is likely only a very small sample of the
confusion in calculus we ourselves create).
1. Review of Functions: A function f is defined to be a relationship
between two sets, and then y=f(x) is defined to be another way of writing
the function. However, y=f(x) as used in calculus is the equation of a
curve in analytic geometry. It is no wonder that so many of us mix
geometric notions of tangent lines with numerical notions of local linear
approximations of functions.
2. Limits: Intuitive approaches to the limit are abundant with circular
associations, but I want to pick on the formal definition of the limit. Most
undergraduate analysis courses begin with sequences because sequences
give us a means of actually associating “x is approaching a” with “f(x) is
approaching L.” Cauchy’s definition of the limit—i.e., the formal
definition—does not define the idea of “approaching.” However, calculus
texts routinely argue that “approaching” means that there is a δ>0 very,
very close to 0 that forces x to be very, very close to a, which in turn
forces f(x) to get very, very close to L, so close that it is within ε>0 of
L even ε>0 when is itself very, very close to 0. That is, "approaching" is
defined to mean "satisfies Cauchy's definition," and then Cauchy's
definition is said to imply approaching. To see why this association is
circular, consider that if f(x)=L is constant, then it is within any ε of L
regardless of the value of x—it need not be anywhere close to a given
value a, much less approaching it.
3. Derivatives: Derivatives are applied to differentiable functions, where a
function is differentiable at a point if its derivative exists at that point.
Differentiability as an independent concept is only briefly explored.
4. Definite Integral: In most calculus courses, antiderivatives are
introduced without motivation and then a few sections later, the
fundamental theorem implies an association between definite integrals
and antiderivatives—an association our students have assumed all along
(after all, both use the same symbol). Thus, the amazing connection
between differentiation and integration is anti-‐climatic, at best.
5. Applications of the Integral: The motivation behind “applications of
the integral” is to associate the definition of the integral with concepts
other than area. However, these contrived applications are outside of
most mathematicians’ training, which means mathematicians must use
the definite integral to define the ideas in the application itself. For
example, work becomes an integral of force, instead of the proper
interpretation that work can be associated with force via an integral.
8. Techniques of Integration: We learn certain techniques to evaluate
integrals because there are integrals that can be evaluated with those
techniques. There are other techniques and other integrals, but those
techniques are not considered because those integrals do not appear in
the text.
9. Sequences and Series; Convergence Tests: We learn convergence tests
for certains types of series because there are series that can be tested
with those convergence tests. There are other series and other
convergence tests, but those convergence tests are not considered
because those series are not introduced in the text.
10. Taylor’s Theorem: Taylor Series: Taylor Polynomials: Taylor, Taylor,
Taylor, Taylor! In almost any calculus text, the 2 or 3 sections on Taylor
series follow section after section of unmotivated convergence tests, and
in those few short sections the word Taylor is used so many times that it
is no wonder that students never seem to understand what all those
different Taylor things are all about.
Although calculus is but the tip of the analysis iceberg, many of the problems
mentioned above can be fixed with nothing more than a little reorganization, the
omission of a few extraneous ideas, and the expansion of a few underdeveloped
topics. For example, is there not enough material and sufficient conceptual
importance to warrant an entire, separate chapter on Taylor Polynomials, Taylor’s
theorem, and Taylor’s Series?
The Mean Value Theorem
Even when reformed textbooks include the Mean Value theorem (as well
they should), they seldom include a proof of the Mean Value theorem. Traditional
textbooks, on the other hand, place a great deal of emphasis on the route from
extreme value theorem to Rolle's theorem to the Mean Value theorem. The extreme
value theorem itself is never proven, since a proof requires the Heine-‐Borel theorem
or its equivalent.
Instead, we assume the extreme value theorem is obvious. We simply tell the
students that if f(x) is continuous on [a,b], then it must look like the picture shown in
figure 1, thus proving that there is a c in [a,b] such that f(c) maximizes f over [a,b].
Figure 1: "Proof" of the Extreme Value Theorem
As innocent as it may seem, the assumption that all continuous functions resemble
the curve in figure 1 is what prevented eighteenth century mathematicians from
seeing the lack of rigor in their study of calculus.
Graphs of continuous functions can differ radically from the curve in figure 1.
Indeed, a continuous function can have an infinite number of relative extrema over a
closed interval [a,b], For instance, self-‐similarity implies an infinite number of
relative maxima for the fractal interpolation function shown below.
Figure 2: A Fractal Interpolation Function
It is not at all obvious to our students that the continuous fractal function attains the
supremum of those maxima.
Thus, the extreme value theorem is far less obvious than the Mean Value
theorem itself, and indeed, the fact that a continuous function attains its maximum
over a closed interval is a remarkable result. Unfortunately, when our majors
encounter the far from trivial proof of the extreme value theorem in an analysis
course, they usually miss the point. And it is because their traditional calculus
course misleads them into thinking about continuity solely as in figure 1.
Proof by Picture
The practice of “proof by picture” is almost always flawed. For example, the
intermediate value theorem is justified with the same type of picture that is used to
justify the extreme value theorem. This further reinforces the notion that
“continuity means piecewise analytic with a cusp here or there.” And this leads to
“differentiability means piecewise analytic with possibly a cusp here or a vertical
tangent there.”
Moreover, a “proof by picture” often gives students a fuzzy, unsophisticated
view of rigor, and I would argue that Math Reasoning courses exist almost
exclusively as a tool to address the flawed concepts of rigor and proof inherent in
most calculus courses. As an example, consider that Newton's method is visualized
but never proven. As one of my students once said, “If you change the picture, you
get a whole different method.” That is, we rely on a picture as sole justification that
Newton’s method “usually” works.
There are many other occasions when claims of rigor are based on
illustrations (derivative of the sine function, multivariable second derivative test),
and in many cases, the diagrams which claim to be proofs are misleading or biased
toward special cases. Can continuity of the sine and cosine functions really be
inferred from the unit circle? Diagrams and illustrations are appropriate in calculus
and should be used. However, they should be as illustrations of concepts, not as
proofs of theorems.
Infinity is not a Number
At the risk of beating a dead horse, let me mention one more problem with
the use of pictures and diagrams. Too often, calculus textbooks use infinity as a
number, such as when they use pictures to justify writing
However, doing so immediately requires a student to exhibit a level of
sophistication that many professional mathematicians seldom reach.
In particular, infinity as a number requires the arithmetic implied by
indeterminate forms. Thus, it means that students must be able to relate a limit
such as
to the limit calculation below which results in an indeterminate form:
The task then becomes trying to distinguish the occasional use of infinity as a
number from other uses of infinity when it is not appropriate to use it as a number
(such as in the sum of the limits theorem). The individual limits above do not
exist—even though we have been using infinity as a number—so that the limit of a
sum theorem does not apply. That is, infinity as a number may be too confusing for
an introductory course.
Theorem Now, Proof Years Later
I have been told that calculus textbooks should be intuitive but with some
rigor. Certainly, all intuition and no rigor is a pseudo-‐intellectual exercise that
usually results in little more than sophisticated cave drawings. But to be completely
rigorous, a calculus course would have to begin with sequences, series, and the
topology of the real line, which may work against teaching the physics major to be
able to think of speed as the ratio of a small change in distance ds to a small change
in time dt.
However, a theorem should be motivated even when it is not proven, and yet
many theorems in calculus are stated without even the slightest suggestion as to
why they are true. Error bounds in numerical integration rarely have even the
slightest justification. Taylor's theorem usually descends from on high.
Indeed, textbooks often present calculus as theoretical overkill with theorems that will not be proven to our
students for years to come (if at all). To illustrate, suppose a student is asked to test the following series for
convergence:
The comparison test fails because n2 -‐ 0.5< n2. Today’s calculus courses either
ignore such problems or require the use of the limit comparison test, which by that
point in the course is little more than another pie-‐in-‐the-‐sky, memorize-‐or-‐die
convergence test.
But would it not make more sense to simply re-‐index the series,
and then use the comparison test? Do we really need to introduce another
unmotivated, unjustified convergence test? Indeed, references to high-‐powered
theorems and relatively inaccessible techniques are not nearly as necessary as
traditional books would have us think.
And even when a proof is included, it is not clear why that theorem deserved
proof when something else did not. We prove that the limit of a sum is the sum of
the limits, but almost never is it shown that the limit of a product is the product of
the limits. We prove the sandwich theorem, we prove that differentiable at a point
implies continuous at a point, we prove that derivative positive on an interval
implies function increasing on that interval.
But we don’t prove the intermediate value theorem, the extreme value
theorem, the chain rule, the convergence of Newton’s method, Taylor’s theorem, the
ratio test, the root test, the monotone convergence theorem for sequences, and on,
and on, and on.
Where on earth does the end correction formula for Simpson’s rule come
from? Or Simpson’s rule itself, for that matter? Is concavity defined in most
Calculus courses? Don’t we need uniform convergence in order to differentiate and
integrate series? If we are going to use conditionally convergent series, then
shouldn’t we say that certain rearrangements of conditionally convergent series
lead to different sums?
Calculus is not completely rigorous, nor should it be. However, calculus
courses would be better served if there was a strategy and a consistency in selecting
which theorems should have proofs, which definitions should be completely
rigorous, and which algorithms should be completely justified. The fact that there is
no rhyme or reason to when we do prove a theorem and when we don’t prove one is
a terrible way to introduce our students to higher mathematics. It is no wonder
they enter their math reasoning and modern algebra courses with absolutely no
concept of what it means to prove a theorem.
Some Perspective on Concepts
Calculus is and should be concept-‐based. However, according to Webster, a
concept is nothing more than a general notion or idea. That is, concepts are
essentially a first refinement of intuition, and math based on intuition is to be
avoided, as we learned the hard way 150 years ago. Instead, mathematicians long
ago realized that rigorous definitions must be used to place concepts in a
mathematical setting. Thus, mathematics, like all of the other sciences, is concept-‐
based, but only after the concepts have been made into definitions.
Unfortunately, in many calculus courses, concepts are often explored without
ever being rigorously defined. The results may be entertaining, but they are not
mathematical. There simply is too much exploration which does not lead to
definition. Indeed, no meaningful theorems can be built upon or even implied by
such a foundation.
For example, a common practice is to use “zooming” to explore limits.
However, we can zoom till we drop, and yet we will not have obtained more than
one or two loose conjectures about limits. A better practice would be to zoom a few
times to get a feel for what a definition of the limit should be, and then use the
zooming process as motivation for a rigorous definition of the limit. Having
captured the “zooming to estimate limits” concept in a definition, we can now begin
to prove theorems based on the definition, and those theorems will imply new
technologies, which will in turn lead to new concepts, new explorations, and
eventually, new definitions and theorems.
I think that we must be very careful when using the “rule of 3” or when
incorporating technology into the curriculum. Visualization is a powerful tool, but
visualization did not take mankind from projectile motion to general relativity, nor
could it. It was the definition—theorem—proof cycle which allowed us to move
from the obvious to the spectacular. Number crunching and numerical simulations
cannot be arranged into a comprehensive theory of statistics. Instead, the numerical
algorithms of statistics are implied by the theoretical results. Thus, if the “rule of 3”
is ever used to imply that the visual and the graphical are on equal terms with the
analytical, then it has failed, regardless of how the students fare in the course.
Concepts are why we study mathematics, but they are not what we study in
mathematics. Calculus courses—indeed, all mathematics courses—should
emphasize that doing mathematics means definitions, theorems, proofs, and
examples. Visualization and conceptualization are useful and commendable, but
they should never be the centerpieces of a mathematics course.
Summary of “Facing the Crisis”
Thus, there is much that is wrong with calculus and a great deal of evidence
that the crisis in calculus continues. Admittedly, no course will ever be able to
address all these difficulties, but that should not keep us from trying to correct as
much as is possible.
Moreover, the flaws with the calculus curriculum are further compounded by
the fact that calculus has become a high school course, a community college course,
and even an online course. Indeed, the calculus curriculum is poised to confuse and
befuddle on a grander scale than we have ever seen before.
Developing Guidelines for Reform
Jeff Knisley (with Kevin Shirley)
The Current State of Calculus
In spite of the many reform textbooks and innovative approaches developed
in the past few years, there is a rather strictly-‐defined formula for the 1st year of
calculus, which goes pretty much as follows (I encourage you to review it in detail):
1. Review of Functions (perhaps with rates of change)
2. Limits Intuitively and Rigorously
3. Asymptotes and Continuity
4. Tangent line; Instantaneous Rate of Change
5. Derivatives and Derivative Rules
6. Rates of Change; Related Rates
7. Mean Value Theorem
8. Optimization and Curve Sketching
9. Antiderivative: Substitution
10. Definite Integral; Numerical Integration
11. Applications of the Integral
12. Techniques of Integration; Improper Integrals
13. Sequences and Series; Convergence Tests
14. Taylor’s Theorem; Taylor Series
There are variations, of course. Traditional courses place exponentials and inverse
trigonometric functions between components 11 and 12, while reformed courses
cover them in component 1. Fourier series are sometimes covered, and perhaps
soon it will constitute an item 15, while Newton’s method and differentials seem to
float between components 5 through 8.
There are definitely two different methodologies for presenting these
components. Traditional textbooks introduce concepts via definitions and then
proceed by stating theorems, mixing in some technology, and providing examples.
Reform textbooks attempt to present each concept in three different ways—
numerically, graphically, and analytically, with theorems and techniques as
consequences.
In both approaches—traditional and reformed—the limit of a function at a
point is the only one studied in detail, although many different notions of limit are
used throughout the text. Studies have shown that the limit concept is not well
developed in even the best calculus students. Perhaps as a consequence, definite
integrals, sequences, and series are also not well understood by most calculus
students.
Why No Fourier Series
In the latter half of the nineteenth century, a crisis developed in mathematics
that affected every aspect of our understanding of the field. In calculus, the crisis
was revealed by the strange convergence properties of Fourier Series. There are
Fourier Series which converge only for irrational multiples of π, and a Fourier Series
was used by Weierstrass to define a function that is continuous at every point but
differentiable at no point. A Fourier series can even represent the Dirac Delta
function, which is not a function at all!
The crisis resonated to the very foundation of mathematics, and as a result,
mathematics was given a new foundation, a foundation based on sets, mappings and
transformations. And once this new foundation had set, twentieth century
mathematics and science was built upon it. As a result, Fourier Series are no longer
the ragged edge of mathematics and science, but instead have become the
centerpiece of a new scientific revolution which will continue well into the third
millenium.
However, Fourier series do remain the ragged edge of calculus instruction, as
do many other topics essential to 21st century science and mathematics. Although
the “mapping” definition is used to introduce functions, functions are used
throughout both traditional and reformed calculus in the sense of analytic geometry.
In fact, the working definition of the function concept for most of our students is an
equation of the form
y = “an expression in x”
Moreover, few calculus books even attempt to provide anything resembling a
modern definition of the integral, although such a definition and the concept of
measure theory it spawned are foundational to 20th century math and science. And
even when current textbooks do make a nod at Fourier Series, mappings, the
definition of the integral, curve-‐fitting, and mathematical models, they are still
treated as the ragged edge rather than as centerpieces of 21st century math and
science. That is, they are little more than “bonus sections” in most courses.
How Students Learn Mathematics
Our first step in designing a successful calculus course was to develop a
model of how students learn mathematics. To do so, we incorporated results from
research in mathematics education, as well as results from cognitive, educational,
and applied psychology. The result is a model that is the subject of a paper entitled
“A Research-‐Based Model of Mathematical Learning (submitted to the Mathematics
Teacher). It is also available at http://math.etsu.edu/knisleyj.
In this document, we present only a brief description of this model and refer the reader to the document
above for details. To begin with, each of us acquires a new concept by progressing through 4 stages of
understanding:
• Allegorization: A new concept is described figuratively in a familiar context
in terms of known concepts.
• Integration: Comparison, measurement, and exploration are used to
distinguish the new concept from known concepts.
• Analysis: The new concept becomes part of the existing knowledge base.
Explanations and connections are used to “flesh out” the new concept.
• Synthesis: The new concept acquires its own unique identity and thus
becomes a tool for strategy development and further allegorization.
A student’s individual learning style is a measure of how far she has progressed
through the 4 stages described above:
• Allegorizers: Cannot distinguish the new concept from known concepts.
• Integrators: Realize that the concept is new, but do not see how the new
concept relates to familiar, well-‐known concepts.
• Analyzers: See the relationship of the new concept to known concepts, but
lack the information that reveals the concept’s unique character.
• Synthesizers: Have mastered the new concept and can use it to solve
problems, develop strategies (i.e., new theory), and create allegories.
Moreover, a student can be an “analyzer” for one topic but only an “allegorizer” of
another, although in practice a student’s style tends to remain constant over a range
of similar concepts.
If this 4 stage process fails before a student has reached the analysis stage,
then that student will almost invariably switch to a “memorize and regurgitate”
form of learning—a learning style known as heuristic reasoning. Even the best
students resort to heuristic reasoning if they can’t “get it,” as is evidenced by several
studies on how students learn limits in an introductory calculus course. It is likely
that even the best mathematicians among us also resorted to heuristic reasoning in
their introductory calculus course, and as a result, even today most mathematicians
discuss calculus as if discussing a poem they once memorized.
Moreover, synthesis requires creativity, so the degree to which a student can
synthesize is a function of their talent level. As a result, the instructor becomes
instrumental in the synthesis stage, since she must provide much of the creative
activity used to finish the study of existing topics and must develop most of the
allegories used to introduce new topics.
Thus, we feel that the goal of any calculus course is to lead the student
through allegorization and integration to an analytical understanding of calculus. In
doing so, the undesirable practice of heuristic reasoning will be avoided, and the
instructor can act as an agent for synthesis for the majority of students who will
have limited ability to use mathematical concepts creatively.
The 4 step model described above is primarily a tool for an instructor to use to
implement effective teaching strategies. To incorporate it into a strategy for
textbook development, this model is reinterpreted as a spiraling approach to
learning mathematics. In particular, the 4 stage model of mathematical learning
implies 4 principles to guide our development of the text.
• Allegory: Concepts should be introduced in as simple a setting as possible
• Integration: New concepts should be defined as soon as possible, and then
those definitions should be explored graphically and numerically
• Analysis: Once a concept has been defined and explored, its unique
character should be revealed through computation, connections, recurring
themes, and theorems.
• Synthesis: Written assignments, projects, group learning, and advanced
contexts should be used to challenge students to use concepts creatively and
completely
In some sense, this gives structure to the “Rule of 3,” although any resemblances of
our model to the “Rule of 3” are more likely due to our choice of vocabulary than to
our adoption of its ideas.
Restructuring the Calculus Textbook
It follows that an effective calculus textbook should begin by presenting
concepts in as accessible a context as is possible. In our opinion, that means that
concepts such as limits, derivatives, and rates of change should be introduced in the
context of polynomials, since this is likely to be the setting most familiar to the
majority of our students. The spiraling approach then implies that once the
concepts have been introduced in the context of polynomials, they can be extended
to algebraic and transcendental functions. This “upward spiral” in the study of a
concept also allows the most important themes in calculus to be repeated over and
over again.
This was, in fact, the original motivation of the chapter in most traditional
textbooks which covers “applications of the integral.” Specifically, such chapters
were originally intended to reinforce the definition of the definite integral as a limit
of Riemann sums, although the concept of limit used in the definition was rather
vague and unfamiliar. Unfortunately, this is about all of the spiraling a traditional
calculus textbook has ever attempted, and now even that has disappeared for all
intents and purposes.
More to the point, spiraling leads us to begin a textbook with the calculus of
polynomials, since the tangent line and derivative concepts can be explored
algebraically in this setting. The second chapter then introduces the broader
concepts of differential calculus, such as continuity and differentiability. The third
chapter then spirals upward to define and study the calculus of exponentials,
logarithms, and trigonometric functions.
Likewise, integration also spirals, in that antiderivatives are first explored and
utilized in chapter 4. The definite integral and its relationship to the antiderivative
are then established in chapter 5, and then the integral is applied to new functions
and new settings in the 6th chapter.
This spiraling idea continues through sequences, series, and multivariable
calculus, although as we progress in the course, we are more and more justified in
assuming that students are learning calculus in the fashion that we are presenting it.
Thus, by the end of the textbook, mathematics hopefully can be presented in more of
a definition-‐theorem-‐example format which is so desirable to mathematicians.
Redesigning the Calculus Course
Once we had developed the concept of the spiraling approach, had examined
what is in calculus already, and had determined what should be in calculus that is
not, we realized that we were doing more than simply modifying the 14 part outline
and adding a few “bells and whistles.” In fact, in order to incorporate all the new
material, we would have to leave out a few familiar topics common in traditional
treatments.
Thus, our last task was to determine the big picture for calculus—what it is
about, what a calculus course should intend to do, and why we cover what we cover.
That is, rather than reform how calculus is taught, textbook development should
begin with a careful consideration of why calculus is taught in the first place. In
particular, we argue that an effective calculus book should concentrate on the
calculus necessary to modern science and mathematics, including the use of data
and the modeling of real world phenomena.
This leads to an initial goal of examining how calculus is used today by working mathematicians and
scientists, and then using that to develop criteria for determining which concepts should be included in a
calculus textbook. To do so, we recognized that differential equations and integration were central to
Calculus in its inception and have remained in the center ever since. These two themes can thus be used to
motivate both the theoretical development and the applications of calculus.
While these two themes do not encompass all that is desired to be known
about calculus, they serve as a useful criteria—i.e., a litmus test—for which concepts
are to be included in a calculus book and what aspects of an included concept should
be emphasized.
The Role of Technology
At this point, we have decided that the key to developing an effective
textbook is to begin as simply as possible, spiral upward through the major topics of
limits, derivatives, integrals, sequences, and series, and as we progress through each
major theme to develop criteria for including topics motivated by their importance
to the study of differential equations and integration. But how does the use of
technology relate to this development?
To begin with, technology should not be used as a context for allegorization
unless it is clear that every student is intimately familiar with the technology being
used. Instead, technology should be used as a tool for integration—i.e., for visually
and numerically comparing new concepts to known concepts. On a limited scale,
technology can also be used for analysis and synthesis, although talented students
should certainly be encouraged to explore and utilize technology in a creative
fashion.
Specifically, our model of learning and the types of technology now being used
lead us to the following guidelines for the use of technology in the single-‐variable
portions of the book:
1. Graphing Functions: Graphing is used for verification, for exploration, and
for problem solving. For example, graphing is used to develop and utilize the
formal definition of the limit.
2. Constructing Tables of Numerical Values: In the business world, one often
“runs the numbers to see what they say.” We likewise see great value
having students produce tables of numerical values when they are
introduced to a concept.
3. Symbolic Calculation: When computer algebra systems are part of the
problem-‐solving process, then they can reinforce both a concept and its
notation. For example, we suggest the use of computer algebra systems for
optimization problems in which the derivatives are very difficult to compute
by hand.
With the exception of symbolic calculation, these tasks can be performed with a graphing calculator.
However, in the multivariable sections there are concepts that require more extensive calculation, and as a
result, multivariable calculus should be enhanced with computer-based technology and powerful computer
algebra systems.
Conclusion
Finally, we have arrived at a set of guidelines for developing an effective
textbook. Motivated by the 4-‐stage model of mathematical learning, the book
should use a spiraling approach to repeatedly revisit key ideas at successively
greater levels of sophistication. The topics to be encountered in this spiraling
approach are to be motivated by the two themes of differential equations and
integration, and technology is to be used as a tool for comparison and exploration of
concepts once they have been introduced allegorically and defined rigorously.
Of course, many other issues are involved in writing a calculus textbook—
ongoing changes in client disciplines such as engineering, the desire to cover the
fundamental theorem of calculus in the first semester, limits on how much our
colleagues will allow the course to change, and many others. Thus, what begins as a
well-‐defined program for developing a textbook quickly becomes a juggling act in
which we attempt to preserve our original strategy while reflecting on issues such
as AP exam requirements and the like.
Thus, the final result is unlikely to be exactly the product we intended.
However, it is hoped that in the end we will have produced a textbook which
presents calculus as growing and thriving, relevant and strong.
A Four-Stage Model
of
Mathematical Learning
Jeff Knisley
Department of Mathematics
East Tennessee State University
Box 70663
Johnson City, TN 37614-‐0663
Introduction
Research in education and applied psychology has produced a number of
insights into how students think and learn, but all too often, the resulting impact on
actual classroom instruction is uneven and unpredictable. In response, many in
higher education are translating research in education into models of learning
specific to their own disciplines (Felder, et.al, 2000) (Buriak, McNurlen, and Harper,
1995). These models in turn are used to reform teaching methods, to reinvent
existing courses, and even to suggest new courses.
Research in mathematics education has been no less productive, but
implementation of that research often leads to difficult questions such as “how
much technology is appropriate,” and “in which situations is a given teaching
method most effective.” In response, this paper combines personal observations
and education research into a model of mathematical learning. The result is in the
spirit of the models mentioned above, in that it can be used to guide the
development of curricular and instructional reform.
Before presenting this model, however, let me offer this qualifier. Good
teaching begins with a genuine concern for students and an enthusiasm for the
subject. Any benefits derived from this model are in addition to that concern and
enthusiasm, for I believe that nothing can ever or should ever replace the invaluable
and mutually beneficial teacher-‐student relationship.
Some Results From Education Research
This section briefly reviews the research results in mathematics education
and applied psychology that most apply to this paper. This is far from exhaustive
and no effort is made to justify the conclusions in this section. Interested readers
are referred to the references for more information.
Decades of research in education suggest that students utilize individual
learning styles (Felder, 1996). Instruction should therefore be multifaceted to
accommodate the variety of learning styles. The literature in support of this
assertion is vast and includes textbooks, learning style inventories, and resources
for classroom implementation (e.g., Dunn and Dunn, 1993).
Moreover, decades of research in applied psychology suggest that problem
solving is best accomplished with a strategy-‐building approach. Indeed, studies of
individual differences in skill acquisition that suggest that the fastest learners are
those who develop strategies for concept formation (Eyring, Johnson, and Francis,
1993). Thus, any model of mathematical learning must include strategy building as
a learning style.
As a result, I believe that the learning model most applicable to mathematics
is Kolb’s learning model (see Evans, et al., 1998, for a discussion of Kolb’s model). In
the Kolb model, a student’s learning style is determined by two factors—whether
the student prefers the concrete to the abstract, and whether the student prefers
active experimentation to reflective observation. This results in 4 types of learners:
• Concrete, reflective: Those who build on previous experience.
• Concrete, active: Those who learn by trial and error.
• Abstract, reflective: Those who learn from detailed explanations.
• Abstract, active: Those who learn by developing individual strategies
Although other models also apply to mathematics, there is evidence that
differentiating into learning styles may be more important than the individual style
descriptions themselves (Felder, 1996).
Finally, let us label and describe the undesirable “memorize and regurgitate”
method of learning. Heuristic reasoning is a thought process in which a set of
patterns and their associated actions are memorized, so that when a new concept is
introduced, the closest pattern determines the action taken (Pearl, 1984).
Unfortunately, the criteria used to determine closeness are often inappropriate and
frequently lead to incorrect results.
For example, if a student incorrectly reduces the expression
to the expression x2+2x, then that student likely used visual criteria to determine
that the closest pattern was the root of a given power. That is, heuristic reasoning is
knowledge without understanding, a short circuit in learning that often prevents
critical thinking. Moreover, such an arbitrary and unreliable approach to problem
solving must surely be responsible for much of the “math anxiety” that so often
plagues students in introductory courses.
Kolb Learning in a Mathematical Context
The model in this paper is based on the idea that Kolb’s learning styles
translate directly into mathematical learning styles. For example, “concrete,
reflective” learners are those who use previous knowledge to construct allegories of
new ideas.1[1] In mathematics courses, these are the students who approach
problems by trying to mimic an example in the textbook. In similar fashion, the
other three Kolb learning styles also translate into mathematical learning styles:
• Allegorizers: These students prefer form over function, and thus, they often
ignore details. They address problems by seeking similar approaches in
previous examples.
• Integrators: These students rely heavily on comparisons of new ideas to
known ideas. They address problems by relying on their “common sense”
insights—i.e., by comparing the problem to problems they can solve.
• Analyzers: These students desire logical explanations and algorithms. They
solve problems with a logical, step-‐by-‐step progression that begins with the
initial assumptions and concludes with the solution.
• Synthesizers: These students see concepts as tools for constructing new
ideas and approaches. They solve problems by developing individual
strategies and new approaches.
1[1] An allegory is a figurative description of an unknown idea in a familiar context.
Moreover, several years of observation, experimentation, and student interaction
suggest to me that these are the only four learning styles, although certainly more
research into this assertion is warranted.
For example, in one experiment, I made sure that each student knew the Pythagorean theorem and had a ruler. I then asked them to find the length of the hypotenuse of a right triangle with sides of length 2¼” and 3”, respectively.
Figure 1: Right Triangle with Unknown Hypotenuse
Some students flipped through the textbook looking for a similar example, many measured the hypotenuse with their ruler, some used the Pythagorean theorem directly, and a handful realized that the triangle is a 3-4-5 triangle (in units of ¼).
However, there were no other styles utilized, and similarly, in other
experiments I have conducted, only a bare handful of students have ever utilized
styles other than the four mentioned above. In addition, I have observed that the
learning style of a given student varies from topic to topic, and unfortunately that
when a student’s learning style is not successful, that student will almost always
resort to heuristic reasoning.
Four Stages of Mathematical Learning
Thus, the question becomes, “What leads a student to choose a given style
when presented with a new concept?” I have concluded that variations in learning
style are often due to how successful a student has been in translating a new idea
into a well-‐understood concept. Indeed, it appears that each of us acquires a new
concept by progressing through 4 distinct stages of understanding:
• Allegorization: A new concept is described figuratively in a familiar context
in terms of known concepts.
• Integration: Comparison, measurement, and exploration are used to
distinguish the new concept from known concepts.
• Analysis: The new concept becomes part of the existing knowledge base.
Explanations and connections are used to “flesh out” the new concept.
• Synthesis: The new concept acquires its own unique identity and thus
becomes a tool for strategy development and further allegorization.
It then follows that the learning style of a student is a measure of how far she has
progressed through the 4 stages described above:
• Allegorizers: Cannot distinguish the new concept from known concepts.
• Integrators: Realize that the concept is new, but do not see how the new
concept relates to familiar, well-‐known concepts.
• Analyzers: See the relationship of the new concept to known concepts, but
lack the information that reveals the concept’s unique character.
• Synthesizers: Have mastered the new concept and can use it to solve
problems, develop strategies (i.e., new theory), and create allegories.
It also follows that a student’s learning style can vary, although in practice a
student’s style tends to remain constant over a range of similar concepts.
The Importance of Allegories
This model suggests that learning a new concept begins with allegory
development. That is, learning begins with a figurative description of a new concept
in a familiar context. Moreover, the failure to allegorize leads to a heuristic
approach. That is, if a student has no allegorical description of a concept, then he
will likely resort to a “memorize and associate” style of learning.
Consider, for example, teaching the game of chess without the use of
allegories. We would begin by presenting an 8 by 8 grid in which players 1 and 2
receive tokens labeled A, B, C, D, E, and F arranged as shown in figure 2.
We would then explain that valid moves for a token are determined by the token’s
type and that the goal of the game is to immobilize the other player’s “F” token. In
response, students would likely memorize valid moves for each token and would
use visual cues to motivate token movement—i.e., not much fun.
Clearly, learning requires allegory development. Indeed, people learn and
enjoy chess because the game pieces themselves are allegories within the context of
medieval military figures. For example, pawns are numerous but have limited
abilities, knights can “leap over objects,” and queens have unlimited power.
Capturing the king is the allegory for winning the game. In fact, a vast array of
video and board games owe their popularity to their allegories of real-‐life people,
places, and events.
In my own teaching, I have found that arithmetic is one of the most useful
and most enjoyable contexts for allegories in mathematics. For example, many of us
already use the multiplication of integers, such as in
to motivate the fact that abac=ab+c. In addition, visual and physical models also serve
as appropriate contexts for allegories as long as they are easily understood and
presented in a familiar fashion.
Components of Integration
Once a new concept has been introduced allegorically, it must be integrated
into the existing knowledge base. I believe that this process of integration begins
with a definition, since a definition assigns a label to a new concept and places it
within a mathematical setting. Once defined, the concept can be compared and
contrasted with known concepts.
Visualization, experimentation, and exploration play key roles in integration.
Indeed, visual comparisons are the most powerful, and explorations and
experiments are ways of comparing new phenomena to well-‐studied, well-‐
understood phenomena. As a result, the use of technology is often desirable at this
point as a visualization tool.
For example, once exponential growth has been allegorized and defined,
students may best be served by comparisons of the new phenomenon of exponential
growth to the known phenomenon of linear growth. Indeed, suppose that students
are told that there are two options for receiving a monetary prize—either $1000 a
month for 60 months or the total that results from an investment of $100 at 20%
interest each month for 60 months. The visual comparison of these options reveals
the differences and similarities between exponential and linear growth (see figure 3
below). In particular, exponential growth appears to be almost linear to begin with,
and thus for the first few months option 1 will have a greater value. However, as
time passes, the exponential overtakes and grows increasingly faster than the linear
option, so that after 60 months, option 1 is worth $60,000 while option 2 is worth
$4,695,626.
Figure 3: Visual Comparison of Linear and Exponential Growth
Analysis and Synthesis
In short, analysis means that the student is thinking critically about the new
concept. That is, the new concept takes on its own character, and the student’s
desire is to learn as much as possible about that character. Analyzers want to know
the history of the concept, the techniques for using it, and the explanations of its
different attributes. Moreover, the new concept also becomes one of many
characters, so that analyzers also want to know connections to existing concepts as
well as the sphere of influence of the new concept within their existing knowledge
base.
As a result, analyzers desire a great deal of information in a short period of
time, and thus, it is entirely appropriate to lecture to a group of analyzers.
Unfortunately, the current situation is one in which we assume that all of our
students are analyzers for every concept, which means that we deliver massive
amounts of information to students who have not even realized that they are
encountering a new idea. This, in fact, appears to be the case for the limit concept in
calculus. Studies have shown that almost no one completes a calculus course with
any meaningful understanding of limits (Szydlik, 2000). Instead, most students
resort to heuristics to survive the initial exposure to the limit process.
Finally, synthesis is essentially mastery of the topic, in that the new concept
becomes a tool the student can use to develop individual strategies for solving
problems. For example, even though games often depend heavily on allegories, the
fun part of a game is analyzing it and developing new strategies for winning.
Indeed, all of us would like to reach the point in any game where we are in control—
that is, the point where we are synthesizing our own strategies and then using those
strategies to develop our own allegories of new concepts.
The Role of the Teacher
As mentioned in the introduction, the value of this 4-‐stage model of
mathematical learning is that it can be used as a guide to implementing reform
methods and curriculum. For example, we can use this model to describe and
explore the role of the teacher in a reformed mathematics course.
To begin with, synthesis is a creative act, and thus, not all students will be
able to synthesize with a given concept. Moreover, appropriate allegories are based
on a student’s cultural background, and as a result, new allegories must be
developed continually. Finally, some concepts require more allegorization,
integration, and analysis than others. Simply put, this model does not allow us to
reduce mathematical learning to an automated process with 4 regimented steps.
As a result, there must be an intermediary—i.e., a teacher—who develops
allegories for the students, who determines how much allegorization, integration,
and analysis should be used in presenting a concept, and who insures that students
learn to think critically about each concept. And once students can think critically,
the teacher will need to synthesize for many of the students by presenting problem-‐
solving strategies and creating new allegories.
To be more specific, this model suggests the following roles for the teacher in
each of the 4 stages of concept acquisition:
• Allegorization: Teacher is a storyteller.
• Integration: Teacher is a guide
• Analysis: Teacher is an expert
• Synthesis: Teacher is a coach.
Space does not permit me to elaborate on each role, but let me point out one
that I feel should not be neglected. Students who have talent are too often bored or
even stifled in our educational system. If we accept that a coach is someone who
applies discipline and structure to creativity, then clearly these are students who
need to be coached. In particular, teachers need to insure that synthesizers realize
that there is creativity in mathematics, and they need to show that such creativity is
both enjoyable and rewarding.
The Role of Technology
Although reform ideas such as the use of technology, group learning, and the
rule of 4 are valuable and effective, their implementation often requires a great
expenditure of valuable class time. If not used wisely, reform ideas can easily lead
to courses which have depth but no breadth, which is entirely inappropriate for a
college-‐level curriculum.
However, the 4-‐stage model of learning allows us to develop a strategy for
implementing reform that leads to little, if any, sacrifice of course content. To
illustrate this assertion, I will limit my comments to the incorporation of technology
into the curriculum.
Suppose that we have a concept that lends itself to the use of technology. To
determine how best to utilize that technology, we need to first determine which of
the four stages best describes that technology, and then we need to restrict our use
of that technology to that stage of the presentation of that concept. Moreover, if we
determine that students generally need very little time in that stage, then we may
not want to use that technology at all.
For example, suppose we have an “applet” that demonstrates the
convergence of Riemann sums to the area under a curve. There is no comparison of
known ideas to unknown ideas, nor does this applet aid in distinguishing the
concept of the integral from any other concept (such as the concept of the
antiderivative). Thus, it is not appropriate (in my opinion) for integration, analysis,
or synthesis.
However, if the applet is simple to understand and easy to use, then it should
serve as an excellent visual context for introducing the concept of the integral. Thus,
I would use the applet as an allegory for the definite integral. I would introduce it as
an illustration of the next concept we want to consider, and then I would use it to
motivate the definitions of partition, Riemann Sum, and ultimately, the definite
integral.
In fact, I might decide that a couple of well-‐drawn pictures are just as
effective as the applet, and thus, I might feel justified in avoiding the time and effort
needed to present the applet and describe how it is used. Or I might decide that I
really want to use the applet, and consequently, I might design an assignment that
asks them to compare applet results to results produced with pencil and paper.
Moreover, my usage will vary from semester to semester. In a given
semester I may decide that individual and class needs dictate that I spend more time
allegorizing the definite integral concept than I did in another semester. Or I may
present the applet simply as an opportunity to challenge a group of synthesizers to
produce a better applet with the promise that I will use their in place of the one
presented.
Regardless, my usage or non-‐usage of the technology is guided by the model’s
identification of what role that technology can play in presenting a certain concept.
It is amazing to me how much initially impressive technology actually has very little
instructional value with respect to this model.
Conclusion
Finally, I want to re-‐iterate my belief that the model is effective only as a tool
in the hands of an enthusiastic teacher who wants to enhance the student-‐teacher
relationship. In fact, I suspect that many teachers use this model already, although
they have not formalized it. Many of us already measure a hypotenuse with a ruler
in order to corroborate the use of the Pythagorean theorem, and we do so because
we know that once the student sees that the measurements and the theorems
produce the same results, they will use the theorem independent of any
measurements.
Nonetheless, this model has become an invaluable tool in my teaching. It
allows me to diagnose student needs quickly and effectively, it helps me budget my
time and my use of technology, and it increases my students’ confidence in my
ability to lead them to success in the course. I hope it will be of equal value to my
fellow educators in the mathematics profession.
References
Bloom, B. S. Taxonomy of Educational Objectives. David McKay Company. New York.
1956.
Buriak, Philip, Brian McNurlen, and Joe Harper. “System Model for Learning.” In
Proceedings of the ASEE/IEEE Frontiers in Education Conference 2a (1995).
Dunn, R.S., and Dunn, K.J. Teaching Secondary Students Through Their Individual
Learning Styles: Practical Approaches for Grades 7-‐12. Allyn & Bacon, 1993.
Evans, Nancy J., Deanna S. Forney, and Florence Guido-‐DiBrito. Student Development
in College: Theory, Research, and Practice. Jossey-‐Bass. 1998.
Eyring, James D., Debra Steele Johnson, and David J. Francis. “A Cross-‐Level Units of
Analysis Approach to Individual Differences in Skill Acquisition.” Journal of Applied
Psychology 78(5) (May 1993): 805 – 814.
Felder, Richard M. “Matters of Style.” ASEE Prism 6(4) (December 1996):18-‐23.
Felder, Richard M., Donald R. Woods, James E. Stice, and Armando Rugarcia. “The
Future of Engineering Education: II. Teaching Methods that Work.” Chem. Engr.
Education. 34(1) (January, 2000): 26-‐39.
Lee, Frank J., John R. Anderson, and Michael P. Matessa. “Components of Dynamic
Skill Acquisition.” In Proceedings of the Seventeenth Annual Conference of the
Cognitive Science Society (1995): 506-‐511.
Pearl, J. Heuristics: Intelligent Search Strategies for Computer Problem-‐Solving.
Addison-‐Wesley, Reading, MA, 1984.
Szydlik, Jennifer Earles. “Mathematical Beliefs and Conceptual Understanding of the
Limit of a Function.” Journal for Research in Mathematics Education 31(3) (March
2000): 258-‐276.
Calculus: A Modern Approach
By
Jeff Knisley and Kevin Shirley2[1]
Introduction
Calculus occupies a pivotal position in the mathematics curriculum. It is the
mathematical foundation for much of the science, mathematics, and engineering
curriculum at a university. For the aspiring mathematics student, it is a first
exposure to rigorous mathematics. For the future engineer, it is an introduction to
the modeling and approximation techniques used throughout an engineering
2[1] COPYRIGHT © 1999 by Jeff Knisley, East Tennessee State University. All rights reserved.
curriculum. For the future scientist, it is the mathematical language that will be
used to express many of the most important scientific concepts.
Consequently, it is imperative that calculus be presented as it is used and
understood by today’s engineers, scientists, and mathematicians. Rigor in calculus
should prepare students for rigor in higher-‐level mathematics courses. Modeling
and approximation in calculus should resemble the techniques and methods
currently in use. Concepts, definitions, terminology, and interpretation in calculus
should be as current as possible whenever possible.
Although a completely modern calculus text is neither possible nor perhaps
desirable, our motivation for writing this textbook was to present calculus as the
foundation of modern science, engineering, and mathematics. To accomplish such a
goal, however, we realized that such a textbook would also need to address
pedagogical issues, as well as issues related to the use of technology, the rule of
three, and other similar issues.
In particular, we began this textbook by identifying 3 issues central to the
development of a more modern and truly effective calculus text:
1. How students learn mathematics, and in particular, calculus.
2. How calculus is used and conceptualized in modern science, engineering,
and mathematics
3. What combination of technology, reform methods, and traditional
techniques best address 1 and 2
In addition, we wanted to create a textbook that had a coherent structure that
allowed ideas to flow from one section to the next.
Once we had addressed these issues to our satisfaction, we developed a
comprehensive plan for producing the best possible textbook. The plan we
developed is almost a book itself, and parts of the plan have been published or
submitted for publication in scholarly journals. The original versions of the plan
documents can be found at http://faculty.etsu.edu/knisleyj/calculus . In addition to
the plan, we also redeveloped many calculus concepts to reflect modern thinking
about those concepts, and we also developed a list of all the topics that we felt
should be included in a calculus course that reflects the needs of today’s
mathematicians and scientists.
Finally, we used these materials to write the actual textbook. It combines
technology, reform, and tradition in a way that we feel best serves today’s students.
It is based on research into how students learn mathematics. Most importantly, it
uses relevant applications and reformulated definitions to present calculus as the
foundation of modern mathematics, science and engineering.
Incorporating Research into
Mathematical Learning
Soon after we began exploring how students learn mathematics and calculus, we realized that the first
few chapters would have to differ markedly from traditional and even reformed approaches. For example,
several studies have shown that even our best calculus students fail to grasp the limit concept (several such
studies have appeared over the past decade in the Journal for Research in Mathematics Education). Many
of the unique features in the first 2 were designed to address these shortcomings in learning limits.
Studies have also shown that although each person has their own unique learning style, there are some
aspects of learning mathematics that all of us have in common. Based on these commonalities, we
developed a model of how students learn mathematics. Details of this model can be found at at
http://faculty.etsu.edu/knisleyj/calculus. To summarize, our efforts lead to the following 4 principles:
•Concepts should be introduced in as simple a setting as possible
•Definitions should be developed and utilized as soon as possible
•Concepts should be reinforced with recurring themes, written assignments, and
technology
•Computation and rigor are important goals in the learning of mathematics
A framework for the textbook was then constructed and reconstructed until we felt that it best reflected
these 4 principles.
For example, the first principle—that concepts should be introduced in as
simple a setting as possible—led us to introduce limits, derivatives, tangent lines,
and rates of change in the simple setting of polynomials. This approach allows us to
establish fundamental ideas in calculus very early. Students are using derivative
rules by the second week of the course, and they have been introduced to the
recurring themes of the limits, the chain rule, and differential equations by the end
of the third week.
However, we do not cater to students, nor do we compromise the presentation
of calculus in any way. Once students have been exposed to differential calculus in
the context of polynomials, chapter two presents them with rigorous definitions and
proofs of basic theorems. By chapter 3, Applications of the Derivative, students have
encountered all of the material they would have encountered in a traditional
approach—and then some—but without much of the confusion and frustration they
might have otherwise developed.
Perhaps as importantly, this approach allows us to revisit the fundamental concepts of calculus over
and over again. For example, the chain rule is explicitly revisited at least twice in each of the first, second,
third, fourth, sixth, and ninth chapters. Monotonicity and concavity are revisited in three different chapters.
Limits are reviewed and revisited time and time again, and the definition of the integral occurs in a vast
array of settings.
We call such repetition the use of recurring themes, and we call the use of recurring themes a
spiraling approach to calculus. After having used this textbook for the past 4 years, we have found that the
average student tends to master the derivative rules, including the chain rule. They also tend to have a
rather sophisticated understanding of the many different roles of the derivative, and they have at least a
working knowledge of what limits are all about.
In addition, many of the components of the book are designed to reduce the
frustration and confusion expressed by so many students when trying to learn
mathematics. Many of the examples were developed in concert with the exercise
sets, and many of the sections were developed to not only introduce new ideas, but
also to reinforce ideas from earlier sections.
Finally, spiraling and the use of recurring themes make the book very flexible.
Some sections can be covered less rigorously than others, because many of the ideas
presented in a given section will occur again in a later section. The organization of
the text further enhances this flexibility. Each section is comprised of 4 subsections
and an exercise set. The first 3 subsections may be essential to later work, but the
fourth subsection is not essential to later work and can either be covered briefly or
even omitted. In general, these subsections are devoted to additional graphical and
numerical techniques, proofs of theorems, additional insights into previous
material, and alternative techniques and identities.
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. Or more
desirably, an instructor could choose what topics to emphasize and how much
coverage to provide to each topic. In either circumstance, the instructor can choose
exercises and applications that best suit the needs of the students, whether they be
mathematics majors, aspiring scientists, engineering students, or future
businessmen.
Calculus as the Foundation of Modern Science and Math
This textbook also presents a calculus course that best serves the needs of
science and mathematics as it enters the twenty-‐first century. To do so, we
recognized that differential equations and integration were central to Calculus in its
inception and have remained in the center ever since. In this textbook, these two
themes are often used to motivate both techniques and applications of calculus.
Much of the differential calculus is motivated by concepts related to differential
equations, and once the fundamental theorem is introduced, much of the material is
motivated by applications of the integral.
Moreover, many of the topics covered in the later sections of the book are
relatively new to calculus. There are sections on mathematical modeling, discrete
dynamical systems, Fourier series, and digital filtering. The multivariable chapters
include concepts like separation of variables to solve partial differential equations
and the fundamental form of a surface. In fact, the multivariable chapters constitute
an online multivariable calculus course that is located at
http://math.etsu.edu/MultiCalc/ .
We also realized that one of the major goals of any calculus course is that of
preparing students for further study in mathematics, science, and engineering. As a
result, we have for several years worked with students to develop definitions of
concepts that reflect modern treatments of those concepts while remaining
accessible to the average student. For example, open intervals are incorporated
into the definition of the limit, thus giving it a slightly more topological flavor. The
definitions of differentiability and integrability are independent of the definitions of
the derivative and the integral, which reflects more advanced treatments of
differentiation and integration. Indeed, the definite integral of a function is defined
to be a limit of simple function approximations, thus preparing students for future
work with modern definitions of the integral.
Finally, the textbook also contains many applications of calculus that are
currently relevant, including mathematical biology, mathematical modeling,
geometric probability, curve-‐fitting, quantum mechanics, and a host of others.
There is also a capstone chapter after the multivariable chapters that applies all the
calculus presented in the textbook to the analysis of the inverse square law and its
many applications.
Implementation of the Plan
Although the textbook is based on models of mathematical learning and the
desire for relevant content, the original plan had to be modified due to pragmatic
considerations. For example, modifications were made to address the needs of AP
calculus courses. In addition, the original organization of the book was altered so
that it more closely resembled the content organization of other calculus textbooks.
In addition, we made a conscious effort to use the “rule of 3” whenever possible,
which is to say that many concepts are presented numerically, graphically, and
analytically. In some sections, the use of technology is essential both to a
presentation of the material and in the exercises, and in other sections, the use of
technology is deprecated in favor of traditional pencil and paper skill development.
In addition, we feel that the ability to read and write mathematics is essential in
today’s world. We have worked closely with students in developing the writing
style for the text, and the result has been found to be very readable. Also,
throughout the text are various “Write to Learn” exercises that ask for students to
write short essays communicating their understanding of a given problem or
concept. There are also short essays called “Next Steps” which are themselves
followed by a collection of “Write to Learn” and group exercises.
There is extensive review material in the textbook. Placement of precalculus
review material corresponds roughly to its initial occurrence in the study of
calculus. We feel that this serves the students better than an all-‐at-‐once review at
the beginning that is forgotten by the time those topics appear later on.
Thus, this textbook augments our plan with what we feel to be the best from
traditional textbooks, the reform movement, and the use of technology. Moreover, it
was designed to address both pedagogical and pragmatic considerations. Finally,
this textbook attempts to fully develop student comprehension, thus leading the
student to appreciate that calculus remains a field of study that is growing and
thriving, relevant and strong.
Structure of the Textbook
The textbook is highly structured and the content of the course is rigorously organized. Let’s begin
our description of the structure of the textbook by examining the organization of individual sections. In
particular, each section is organized according to the following:
1. Each Section has 4 subsections: Each subsection introduces one or two new concepts followed
by examples of these new ideas follow.
2. The 4th Subsection typically presents ideas that are not necessary in later sections. The
fourth subsection often consists of proofs of theorems, additional applications, or additional
examples, thus giving an instructor some discretion in how best to teach the course.
3. After each of the 1st three subsections, there is a “Check your Reading” question. These
questions assess a student’s comprehension of the material just read and can be used to facilitate
either discussion either in class or online via threaded message forums.
4. Exercise Sets are Graded and Correspond Closely to the Examples: These problem sets drill
the techniques encountered in the section, whether they be graphical, numerical or analytical.
5. Applications problems include “Write to Learn” and Discussion Problems: In addition, we
periodically include problems that are more challenging than usual. These are marked by an
asterisk (*).
The sections are written in what we call a “tutorial style.” In particular, the sections
have been designed to be as readable as possible, and the examples are written to be
as self-‐explanatory as possible.
The coverage of integration in this textbook also differs from traditional
treatments, which in large part is due to our desire for this textbook to introduce
calculus as a foundation for modern mathematics and science. Although definite
integrals are defined to be limits of Riemann sums and the fundamental theorem is
proven rigorously, definite integral is defined to be a limit of simple function
approximations. While such a definition is no more difficult for students than the
typical Cauchy-‐style definition with Riemann sums, it better prepares future
mathematicians, statisticians, and engineers for the modern concepts of integration
and function approximation they will encounter later. It must be emphasized that
we have been using this textbook for several semesters, and while our treatment of
integration is different, students do not find it especially difficult.
Uses of Technology
Technology is utilized throughout the textbook. Indeed, the multivariable
portions of the textbook, chapters 9 – 13, are being taught as a web-‐based,
technology-‐intensive course, and will be available either in printed form or as on
online course once the textbook is published. In the single-‐variable portions of the
book, chapters 1-‐8, technology is used in at least 3 different ways:
1. Graphing Functions: Graphing is used for verification, for exploration,
and for problem solving. For example, graphing is used to develop and
utilize the formal definition of the limit.
2. Constructing Tables of Numerical Values: In the business world, one
often “runs the numbers to see what they say.” We likewise see great value
having students produce tables of numerical values when they are
introduced to a concept. For example, in the multivariable sections we use
tables of numerical values to explore limits in two variables.
3. Symbolic Calculation: When computer algebra systems are used to solve
problems other than rote calculations, they can reinforce both a concept and
its notation. For example, we suggest the use of computer algebra systems
for optimization problems in which the derivatives can be very difficult to
compute by hand.
However, although technology is utilized throughout the textbook, the single
variable portions can be completed with no more than a simple graphing calculator
(i.e., one without symbolic capabilities).
Conclusion
Our textbook is not traditional, nor was it written as an action or reaction to any movement. Instead,
this book is an implementation of a plan developed through years of research. We identified best practices
in traditional approaches, the reform movement, and the use of instructional technology. We collected
information and developed models of how students learn mathematics. We reviewed the use of calculus
concepts in modern mathematics, science, and engineering.
We also tried to write the textbook that would best meet the needs of the wide variety of instructors
who would be using it. Along these lines, we would like to acknowledge the many contributions of those
who explored and reviewed this project during its development. Our treatment of the limit concept was
greatly improved by insights and examples from A. Shadi Tahvildar-Zadeh. Many improvements in the
first two chapters are due to insights from Dr. Debra Knisley. The use of technology was much improved
by techniques and insights from Dr. Lyndell Kerley. Many others have also contributed (and their names
will be listed below once the book is published).
We thank all those who have contributed to this project, and we recommend it to anyone who wants to
present calculus in an accessible, coherent, and relevant fashion.