Mel S. Sabella
E. F. Redish

Student Understanding of Topics in Calculus

Introduction

For most students in mathematics, science, and engineering, calculus is the entry-point to undergraduate mathematics. Because of its importance in such a wide range of disciplines, and its size of enrollment, there have been many research studies in the student understanding of calculus. The studies indicate that students enrolled in the traditional university calculus class have a very superficial and incomplete understanding of many of the basic concepts in calculus.

There has been much concern in the failure to develop a conceptual understanding of calculus topics because of the rote, manipulative learning that takes place in an introductory course (Cipra; Steen; White). The traditional course in university calculus "is driven by content features, including organization of content into broad categories, segmenting of the categories, sequencing of topics, and specific lessons by topic (Ferrini-Mundy 628)." Romberg and Tufte argue that students view mathematics as a static collection of concepts and skills to be mastered one by one. Students are required to solve, sketch, find, graph, evaluate, determine, and calculate in a straightforward fashion (Ferrini-Mundy 627)." They are rarely engaged in "higher level" problems. This failure of the traditional calculus curriculum has led to the calculus reform effort. Because of the importance of calculus in many disciplines there has been a great deal of research into student learning of calculus. The existing research provides a lot of implications for curriculum development and reform.

This paper provides an outline of the current research in student understanding of topics in calculus. The intent of the paper is to provide an overview of specific difficulties based on education research. The topics presented are:

• Functions and Variables
• Limits and Continuity
• Derivatives
• Integrals

References to various articles are provided for a more detailed discussion.

Functions and Variables

The idea of a function underlies many central concepts in calculus such as limit, derivative, and integral. The importance of the idea of a function has led to much research in student understanding of this topic. There are a number of papers, specifically those by Dreyfus, Eisenberg, Vinner, and Monk, which deal exclusively with student understanding of the function. The papers by Dreyfus and Vinner, and Monk also contain questions that may be used to assess student performance.

Researchers have shown that many students have a primitive concept of a function. These students enter a calculus course unable to provide a general definition of a function and when prompted are only able to provide examples of functions. Even students who have a Dirichelet-Bourbaki definition, describing a correspondence between two non-empty sets, revert to a more primitive definition when asked certain types of questions.

Dreyfus and Eisenberg found in interviews that some students state that a relation is a function only when it can be represented by a single formula (Ferrini-Mundy 630). Vinner and Dreyfus show that even when students have a Dirichelet-Bourbaki definition of a function, their behavior might be based on the formula conception when working on identification or construction tasks. Their study was performed on several groups of first-year college students in two Israeli institutions and some junior high school mathematics teachers who had not majored in mathematics. Their work showed that many who gave the Dirichelet-Bourbaki definition for the function concept later used a formula definition stating that a particular graph could not be a function because "two parts of [the] graph . . . have different characters [and therefore] cannot be described by a single formula (Vinner 363)."

The study done by Dreyfus and Vinner also showed some students rejected certain graphs of functions because of a correspondence being discontinuous at one point or a correspondence having a point of exception. The study also probes student understanding in construction tasks and shows that "students usually pay less attention to the conceptual aspects of a given notion and more attention to its computational or operational aspects (Vinner 364).

The interviews by Dreyfus and Eisenberg also show that students view algebraic data and graphical data as separate; a graphical representation with no formula has no meaning for most entering calculus students. Heid points out that

traditional course development of graphs requires only that students master the methods of using zeros of derivatives to sketch graphs of quadratic and cubic functions and that they use the general notion of checking functional behavior near undefined function values for a small, well-defined set of functions whose graphs have linear asymptotes. Typically, instructors confine graphing assignments to requiring a sketch for a graph like that of or or They seldom ask students to draw conclusions on the basis of their graphs or to comment on the relationship between two different graphs (Heid 7-8).

In her experimental class taught to college students, Heid stresses instruction which encourages students to reason deeply from and about graphs using computer technology. Most traditional calculus courses offer little opportunity for students to develop a deep conceptual understanding of the graph and do not promote an understanding of the connection between an algebraic representation and a graphical representation.

Students also feel that when they are given a function they are expected to do something with it, such as substitute a value (Graham 1989). They view the function as a static quantity, thinking of only one point at a time according to Monk. Monk's study shows a distinction between a "point-wise understanding" and an "across-time understanding." A point-wise view considers the function as a correspondence between two sets or between two variables whereas an across-time view deals with how a change in one variable leads to a change in others. An across-time understanding requires the student to view the function at infinitely many points (or with a continuously changing variable), instead of one point at a time. The study indicates "that, at least in simple situations, students have a confident and secure Point-wise understanding of functions, but even at the end of the first or second quarter of calculus . . . they are still struggling to use functions in an Across-Time manner (Monk 2)." This may be partially due to the point-wise view being presented in calculus and pre-calculus texts in their introductions of the concept of a function, even though an across-time view is more applicable to topics in calculus. Monk's study also indicates that the pre-requisite for an across-time understanding is a point-wise understanding, although "it does not seem to be the case that an Across-Time understanding comes easily and automatically after a Point-wise understanding has been developed (Monk 5)." This seems to be the case in three of the four questions analyzed by Monk.

White and Mitchelmore show that students have a primitive understanding in the concept of a variable. The study involved four questions and each question had four versions. Version A required the students to do more translation from English to mathematical symbolism while version D required them to do very little translation. These questions were used in their research performed on first year university students, all of whom had studied calculus in secondary school. They found that students treated variables as symbols to be manipulated rather than as quantities to be related. In the problems that were given, the students were unable to distinguish between a general relation and a statement of a specific variable. This underdeveloped concept of a variable made it difficult to identify and symbolize an appropriate variable by translating one or more quantities and therefore define a usable function.

Rozier and Viennot also see students treating variables in a primitive manner. Their study showed how students reduce the number of variables, or take all the variables into account, but in a simplified way, when dealing with thermodynamic problems. One way the students reduce the number of variables is in their "tendency in coping with multivariable problems . . . to forget some relevant variables (Rozier 160)." Because of a "preferential association" between two variables, we see students relating only these two variables and ignoring the others. Rozier and Viennot also see students reducing the number of variables by combining two variables and treating them as one. Linear causal reasoning is another way students are able to deal with only two variables at a time even when the "changing physical quantities are all supposed to change simultaneously (Rozier 165)." This linear reasoning involving successive steps allows the student to relate two variables while keeping the others constant during each step. Thinking about more than two variables at a time seems to be a very difficult task for students and this difficulty surfaces in various topics such as thermodynamics.

Limits and Continuity

Confrey argues that students enter a calculus course in one of three ways: with a discrete understanding of continuous ideas; with an independent transition to the idea of continuity; or with an algorithmic approach (Ferrini-Mundy 630). Tall states that part of the difficulty in the conceptual understanding of a limit is in the colloquial meaning of the terms used when referring to limits. To many students the statement "we can make s_n as close to s as we want" means we can make s_n close but not coincident. Other research, done by Ferrini-Mundy and Graham in 1989, shows that when students are asked to evaluate limits of the form lim f(x) as x a they are quite successful, but when asked for a geometric interpretation, students showed very little understanding. This shows the independence of the algebraic and graphical representations. One interview revealed that the notion of "approaching" was not part of the student's understanding of the limit. The student stated that limit problems were simply functions to be evaluated and that "the graph can't help me find an answer (Ferrini-Mundy 630)." Heid shows similar student difficulty in the understanding of limits in two student populations at Penn State. One population was taught using graphical and symbol manipulation computer programs while the other population was taught by more traditional methods. Both populations seemed to "identify the notion of a limit with a process rather than a number, focusing on the "getting close to" rather than on the number being approached. This confusion about a limit permeated their explanations of derivatives as they "described derivatives as approximations for slopes of tangent lines rather than as being equal to the slope (Heid 17)."

Derivatives

The articles by Heid and Orton provided the most useful information about student understanding of the derivative. Heid conducts a study performed on two groups of college calculus students. One group used computer software extensively in the course while the other group was taught by more traditional methods. Orton's article focuses exclusively on student understanding of differentiation in college and pre-college students. His analysis is based on tasks presented to students in interviews.

Orton shows, based on conceptual tasks given to 110 students majoring in mathematics, that students have little intuitive understanding as well as some fundamental misconceptions about the derivative. The routine aspects of differentiation could be performed quite well by almost all the students interviewed. However, when the students were presented with a function they had not seen before, the frequency of errors increased indicating strong reliance on algorithmic steps without a conceptual understanding. Other areas of student difficulty are related to the tangent as the limit of a set of secants and to the ideas of a rate of change at a point versus the average rate of change over an interval (Orton₂ 237-8). Heid also notes that "the notion of derivative as slope or rate of change, or of second derivative as a measure of concavity, fades quickly with disuse because students learn to rely on memorized procedures for a small number of exercise types (Heid 9)."

Based on the interview tasks assigned to college students training to be teachers of mathematics, Orton outlined the following items and how well the students performed on the items. Responses were graded on a five point scale in order to carry out statistical analysis. Mean scores for each topic range from 0 to 4 and error classification is based on the scheme described by Donaldson (Orton₁ 236). Structural errors are those "which arose from some failure to appreciate the relationships involved in the problem or to grasp some principle essential to solution." Executive errors were those which involved failure to carry out manipulations, though the principles involved may have been understood. Arbitrary errors were said to be those in which the subject behaved arbitrarily and failed to take into account the constraints laid down in what was given (Orton₁ 4).

Table 1. Student understanding of differentiation.

Orton suggests that the basic concepts be revisited throughout the students' mathematical education. He thinks the initial approach to calculus should be informal and should involve both numerical and graphical exploration, first using real life data followed by a more algebraic development of the fundamental ideas (Orton₂ 243-4).

Integral

Orton's paper entitled, "Students' Understanding of Integration," was the only paper that dealt exclusively with the topic of integration. Both his differentiation paper and his integration paper use the same students in an interview format. This paper was the precursor to his paper on differentiation and contains more background information on the students and the error classification.

Orton's study on integration shows that students are "able to apply, with some facility, the basic techniques of integration . . . [but] further probing indicates that they posses fundamental misunderstanding about the underlying concepts (Ferrini-Mundy 631)." Orton's results, summarized below, indicate that the procedure of breaking up an area or volume, making use of a limit process, and providing the reasons why such a method works were not part of the students understanding of the integral.

Table 2. Student understanding of integration.

The limiting process, essential in calculus, is generally overlooked before it is suddenly required. This leads to a great deal of difficulty in this topic. When Orton asked whether it was possible to obtain an exact answer for the area under the curve y=x² by taking more and more rectangles under the curve only 10 students stated that a limiting process was required. 69 students indicated that by taking more and more rectangles under the curve that they could obtain better and better approximations but that such a procedure would never produce the correct answer. To address the difficulty in understanding the limiting process Orton suggests that activities in which the students can explore the idea of a limit in an intuitive way have to be developed. In these activities the nature of approximations should be emphasized.

Tables 1 and 2 show what we can expect in terms of student understanding of differentiation and integration. Expectations of student performance in various topics should be taken in to account when developing computer curriculum. Orton suggests the use of computers and calculators to facilitate a conceptual understanding of the derivative and integral. In particular he suggests the use of technology to aid in the explorations of the approximation process stating that "the calculator does provide us with the opportunity of numerical approaches to calculus, and better understanding of the arithmetic may lead to better understanding of the algebraically equivalent procedures (Orton₁ 11)."

Use of Technology

As indicated by the research described above, students show little intuition about the concepts and processes of calculus. By mimicking examples and doing homework problems similar to examples solved in the text students develop misconceptions based on trying to adapt prior knowledge to new situations (Ferrini-Mundy 631). Students are never given the opportunity to develop a conceptual knowledge of topics in a traditional calculus class. Heid shows in her comparison study that students in the traditional class showed no evidence of attempts to reason from basic principles and lacked detail in defining concepts such as the derivative. Without a clear conceptual understanding, students base decisions about which procedure to apply on the given symbols and ignore the meaning behind the symbols (White 88).

To remedy this, Thompson suggests that the curriculum should

be problem based, promote reflective abstraction; contain . . . questions that focus on relationships; have as its objective a cognitive structure that allows one to think with the structure of the subject matter; and allow students to generate feedback from which they can judge the efficacy of their methods of thinking (Thompson 200).

One area being discussed in mathematics reform is the use of technology in the calculus course. Many topics in mathematics have characteristics which indicate that a computer aided learning environment may be effective in promoting student understanding.

[Topics in mathematics which lend themselves to computer implementation] have visual aspects which can be well represented on a computer screen; they have transformational aspects which necessitate a dynamic implementation; they have technical computational aspects which are not very relevant to the essence of the topic and are thus better being taken care of by the computer; and they are intimately connected to the relationship between two different representations of the same concept; these two representations can be dealt with in parallel by the computer program (Dreyfus and Halevi 44).

Using the computer as a tool for performing the procedures of calculus and algebra can free students to explore applications (Hsaio, 1984/85: Tall, 1986). The course can then de-emphasize skills and concentrate on the underlying concepts. Students in the study done by Heid stated that they enjoyed the computer work because it freed them from the manipulative work and gave them confidence in results which were based on their reasoning. It also allowed them to focus more attention on the global aspects of problem solving. Heid's study showed that students using the computers understood the concepts as well as, and in most cases better than the students in the comparison class. After only three weeks of work on traditional skills the experimental class performed almost as well on the final examination (a skills test) as the comparison class which met 200 minutes a week for 15 weeks.

Research in education must be considered when designing computer software. Kaput and Thompson state that to use "technology . . . one must continually rethink pedagogical and curricular motives and contexts. To exploit the real power of the technology is to transgress most of the boundaries of the school mathematics practice
. . . Normally a powerful technology quickly outruns the activity-boundaries of its initial design (Kaput 681)." Often such technology-based tools are "designed by people steeped in the technology but without deep insight into the problems of mathematics education (Kaput 682)." People developing software and hardware should therefore work to discover something about the learning process as it occurs with the support of that software or hardware (Ferrini-Mundy 632). Damarin and White offer the following characteristics for high quality educational software:

1. Appropriate: The program should preserve the integrity of the subject matter and respect the integrity of the learner. The instructional goals of the program should be appropriate to the intended user and the format of the presentation should be designed to incorporate appropriate learning theory.
2. Friendly: The user should be able to interact easily and naturally with the software with a minimum of confusion.
3. Simple: The structure of the program should be as clear-cut and direct as possible. Rules for using the software should not be complicated.
4. Flexible: The software should lend itself to use in a variety of related learning situations. The software should be adaptable to the varying needs of teachers and learners.
5. Robust: The software should be designed to accept unusual responses and be able to process them in a manner meaningful to the user.
6. Constructible: The topic selected for development must be such that a meaningful instructional program can be designed within the limitations of the available hardware and software tools.
7. Verifiable: The software, embodying the concept to be taught, must correctly model the computer experience planned by the designers.
8. Parsimonious: The software should make effective use of the computer capabilities available (Damarin 39).

Computer software must also be aware of topics which have proved to be difficult for students in a calculus course. Based on the research in student understanding, in the list below, we suggest topics and difficulties that must be addressed.

In the topic of functions and variables students were found to:

• view a function as a single formula
• have trouble viewing functions in an across-time manner
• not see the connection between algebraic and graphical representations of a function
• treat variables simply as symbols to be manipulated

In the topic of limits and continuity the research showed that:

• often students' conceptual understanding is based on colloquial meaning of terms involved
• students often focus on the getting close to rather than on the number being approached
• students often identify the notion of a limit with a process rather than a number

When dealing with derivatives students have difficulty with the following items:

• rate of change from straight line graph
• average rate of change from curve
• use of the -symbolism
• rate, average rate, and instantaneous rate
• differentiation as a limit

And when learning integration the following items proved difficult:

• limit of sequence equals area under graph
• volume of revolution
• integral of sum equals sum of integrals

References

• Cipra, B. A., (1988). Calculus: Crisis Looms in Mathematic's Future, Science, 239 1491-1492
• Confrey, J., (1980). Conceptual Change, Number Concepts and the Introduction to Calculus, Unpublished doctoral dissertation.
• Damarin, Suzanne K., and Carol M. White, (1986). Examining a Model for Courseware Development, Journal of Computers in Mathematics and Science Teaching, 6 (1) 38-43.
• Dreyfus, T., and Tirza Halevi, (1990/91). QuadFun-A case Study of Pupil Computer Interaction, Journal of Computers in Math and Science Teaching, 10 (2) 43-48.
• Dreyfus, T., and T. Eisenberg, (1983). The Function Concept in College Students: Linearity, Smoothness, and Periodicity, Unpublished manuscript.
• Ferrini-Mundy, Joan and Karen G. Graham, (1991). An Overview of the Calculus Curriculum Reform Effort: Issues for Learning, Teaching, and Curriculum Development, American Mathematical Monthly, 98 (7) 627-635.
• Graham, K. G., and J. Ferrini-Mundy, (1989). An Exploration of Student Understanding of Central Concepts in Calculus, paper presented at the Annual Meeting of the American Educational Research Association.
• Heid, Kathleen M., (1988). Resequencing Skills and Concepts in Applied Calculus Using the Computer as a Tool, Journal for Research in Mathematics Education, 19 (1) 3-25.
• Hsaio, F. S., (1984/85). A New CAI Approach to Teaching Calculus. Computers in Mathematics and Science Teaching, 4 (2) 29-36.
• Kaput, J. J., and Patrick W. Thompson, (1994). Technology in Mathematics Education Research: The First 25 Years in the JRME, Journal for Research in Mathematics Education, 25 (6) 676-684.
• Monk, G. S., (1987). Students' Understanding of Functions in Calculus Courses, Unpublished paper.
• Orton₁, A., (1983). Students' Understanding of Integration, Educational Studies in Mathematics, 14 1-18.
• Orton₂, A., (1983). Students' Understanding of Differentiation, Educational Studies in Mathematics, 14 235-250.
• Raymond, Anne M., (1994). Assessment in Mathematics Education: What are some of the Alternatives in Alternative Assessment, Contemporary Education, 66 (1) 13-17.
• Romberg, T. A., and Fredric W. Tufte, (1987). Mathematics Curriculum Engineering: Some Suggestions from Cognitive Science, The Monitoring of School Mathematics: Background Papers, (2)
• Rozier, S., and L. Viennot, (1991). Students' reasoning in thermodynamics, International Journal of Science Education, 13 159-170.
• Steen, L., (Ed.) (1988). Calculus for a New Century. Washington, DC: Mathematical Association of America.
• Tall, D. O., and R. L. Schwarzenberger (1978). Conflicts in the Learning of Real Numbers and Limits, Mathematics Teaching, 83 (44-49)
• Thompson , W. P., (1985). Experience, Problem Solving, and Learning Mathematics: Considerations in Developing Mathematics Curricula, Teaching and Learning Mathematical Problem Solving, 189-236
• Vinner, S., and Tommy Dreyfus, (1989). Images and Definitions for the Concept of a Function, Journal for Research in Mathematics Education, 20 (4) 356-366.
• White, Paul and Michael Mitchelmore, (1996). Conceptual Knowledge in Introductory Calculus, Journal for Research in Mathematics Education, 27 (1) 79-95.

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Please address questions and comments to Mel Sabella.
email: msabella@delphi.umd.edu