# Developing Guidelines for Reform

### 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 p, 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.