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  

   

Phone:  (423)  439-­‐7065  

Email:  Knisleyj@etsu.edu  

   

   

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.        

   

   

   

   

   

   

   

   

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.  

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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.