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
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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,
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.
Figure 2: Chess without Allegories
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