## Literature Review: Student Understanding in Calculus

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Mel S. Sabella
E. F. Redish

Student Understanding of Topics in Calculus

Introduction

#### 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

#### 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

Derivatives

#### 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.

 Description Mean Scores (0-4) Error Classification gradient of tangent to curve by differentiation 3.76 Structural and Executive substitution and increases from equations 3.68 Structural and Executive significance of rates of change from differentiation 3.62 Structural and Executive carrying out differentiation 3.50 Executive limits of geometric sequences 2.78 Structural infinite geometric sequences 2.56 Structural stationary points on a graph 2.54 Structural and Arbitrary rate of change from straight line graph 2.02 Structural average rate of change from curve 1.92 Structural and Executive use of the -symbolism 1.40 Structural rate, average rate, and instantaneous rate 1.18 Structural differentiation as a limit 1.14 Structural

Integral

#### 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.

 Description Mean Scores (0-4) Error Classification simplification of sum of areas of rectangles 3.52 Executive heights of rectangles under graphs 3.42 Structural and Executive carrying out integration 3.40 Structural, Arbitrary and Executive sequence of approximations to area under graph 3.22 Executive limits of sequences of numbers 3.06 Structural and Executive limits from general terms 2.90 Structural and Executive complications in area calculations 2.78 Structural, Arbitrary and Executive limit from sequence of fractions + from general term 2.48 Structural limit of sequence equals area under graph 1.00 Structural volume of revolution 0.88 Structural integral of sum equals sum of integrals 0.60 Structural

#### 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

#### 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).

#### 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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Comments and questions may be directed to E. F. Redish