Nemirovsky & Rubin
Ricardo Nemirovsky and Andee Rubin (1992) were interested in the critical elements and causes of changes in students thinking about derivatives, specifically students realizations that a function and its derivative may not resemble one another, contrary to most students preconceived notions. They write:
The current analysis of the teaching interviews focuses on what we call learning episodes; that is, episodes during which the student changed her view or adopted a different way of thinking. Our analysis strives to investigate the critical elements in that transition and why it took place. (p. 3)They based their work on three assumptions: 1) Every normal human being, from early stages in life, has some intuitive knowledge about the relationship between function and derivative. . . . [W]e construct complex bodies of knowledge that enable us to make sense of situations involving change. 2) The relationship between function and derivative is one of those notions that always remains open to further elaboration, with new and unresolved issues involving the fundamental nature of space, time, and number. 3) Students performance in solving problems involving the function/derivative relationship is strongly affected by contextual parameters. (pages 3-4)
We examine a particular tendency that appeared repeatedly in all three contexts: students tendency to assume resemblances between a function and its derivative. (p. 4)
[H]igh school students tend to solve problems of prediction between a function and its derivative by assuming partial resemblances between them. We observed this tendency in several physical contexts . . . These assumptions of resemblances lead to a particular approach to problems of prediction between a function and its derivative, characterized by forcing a match of global features of the two graphs (e.g. increasing/decreasing, sign) and by focusing on one of them (function or derivative) rather than their relationship.Nemirovsky and Rubin found that students use of a resemblance approach is not, as some have suggested, due to an inability to distinguish between position and velocity or between flow rate and volume.1 Rather, students have reasons for expecting the function and its derivative to have features in common. MBL environments, such as those used in this study, can help students to replace their resemblance approach with a more effective variational approach to the relationship between a function and its derivative. Examining one (either function or derivative), predicting the behavior of the other, and then testing the predictions in a physical context can help the student to build a better understanding of the relationship between the two. This understanding is built gradually. Students may revert to the resemblance approach when confronted with difficult questions or unfamiliar phenomena.
Use of resemblances is not the result of the students inability to distinguish between the function and the derivative. Several perceptual and cognitive aspects of situations of change support the plausibility of such resemblances. We described three such aspects: semantic cues, syntactic cues, and linguistic cues. . . .
[Replacing a faulty resemblance approach to derivative with a correct variational approach] is not a monolithic insight. Even at the end of the learning episode, the student used a new resemblance to account for some troublesome aspects of his experience. . . . This case study also supports the importance of our technique of using physical contexts to provide students with tools to explore mathematical ideas from a variety of directions, and gives us insight into how these tools help frame the interviewer/student discourse through which learning occurs. (p. 32 - 33)
We must certainly take every opportunity to lay foundations of ideas of rate of change throughout a pupils school life and, as with limits, not leave the study of this important idea until it is required in order to make sense of differentiation. . . . [T]he writer believes that no opportunity should be lost by teachers to develop these ideas, and that it is wrong to attempt to introduce calculus without a long and persistent study of graphs and rate of change. The same applies to ideas of limit. (page 243)Orton makes it clear, in the above quote and throughout the paper, that he believes certain concepts must be learned in a specific order. In particular, calculus should be preceded by extensive work with graphs, tangents, ratio, proportion, and rates of change. Difficulties are minimized, not by ignoring this order, but by taking it into account so thoroughly that all elementary and secondary school mathematics courses include work with these key concepts, laying a firm foundation for later study of calculus.
The purpose of this empirical research is to test the hypothesis that interactive computer programs, encouraging teacher demonstration and student investigation of a wide variety of examples and non-examples, may be used to help students develop a richer concept image capable of responding more appropriately to new situations. (p. 69)Three sections of sixteen-year-old high school students were taught with the experimental method. Five other sections of the same course were taught with the traditional method. Neither method is described in any detail. The experimental groups explored the meaning of tangent, including discussion of specific examples, such as y = |sin x|, which have no tangent at some point of interest. All Tall says about the instruction in the control groups is that the five control classes followed a more traditional strategy assuming an intuitive knowledge of the meaning of tangent.
The research emphasizes the difficulties embodied in the tangent concept, but suggests that the experiences of the experimental group helped them to develop a more coherent concept image, with an enhanced ability to transfer this knowledge to a new context. . . . However, potential conflicts remained, with a significant number of students retaining the notion of a generic tangent which touches the graph at a single point, giving difficulties when the tangent coincides with part of the graph.Given the difficulty of determining, from what Tall writes, exactly what it was about the experimental instruction that produced the learning documented through the two tests, we are left simply with the observation that students often have misconceptions about the tangent, and that some instruction remedies this problem more effectively than other instruction.
At a general level the research lends support to the theory that the computer may be used to focus on essential properties of a new concept by providing software that enables the user to manipulate examples and non-examples of the concept in a moderately complex context. This allows curriculum development to be more appropriate cognitively by giving students general ideas of concepts at an early age, to encourage discussion and the active construction of a shared meaning. (p. 75)
The purpose of the present study was to investigate the performance on some calculus application problems of a group of students who had previously experienced a traditional introductory calculus course, and thereby, to infer the role of their conceptual knowledge (or lack of it) in solving application problems. (p. 80)The first stage of the solution process, in which the application problem is translated from the original statement or description into mathematical symbols and relationships, is the most likely place for conceptual knowledge to be used. Thus, the study focused on this stage.
Using mathematical symbols may be an abstract operation if the symbols have no concrete reference: they are apart. The only context for the symbols is the symbols themselves. . . . An example of abstract-apart is knowing how to manipulate algebraic symbols without having any sense of what the letters stand for. On the other hand, abstract-general indicates that the mathematical objects involved are seen as generalizations of a variety of situations and so can be used appropriately in different looking situations. (pages 574-575)White and Mitchelmore concluded that the students weak concept of variable was responsible for most of their errors. They write:
Responses to the four research items strongly suggest that a major source of students difficulties in applying calculus lies in an underdeveloped concept of variable. In particular, students frequently treat variables as symbols to be manipulated, rather than as quantities to be related. Three examples of such a manipulation focus have been identified: failure to distinguish a general relationship from a specific value; searching for symbols to which to apply known procedures regardless of what the symbols refer to; and remembering procedures solely in terms of the symbols used when they were first learned. (p. 91)The sentence In the present study, the only detectable result of 24 hours of instruction intended to make the concept of rate of change more meaningful was an increase of manipulation-focus errors in symbolizing a derivative. would be amusing if it werent so tragic. Despite the researchers best efforts to help the students understand the derivative, the only discernible effect was an increase in errors. This underscores the magnitude of the problem. For those students who lack the necessary concept of variable, there are no quick fixes or easy outs. The concept is a difficult one, learned slowly and through much practice, and it is absolutely essential to further progress. This is a good example of what Tall termed a fundamental obstacle.
In the present study, the only detectable result of 24 hours of instruction intended to make the concept of rate of change more meaningful was an increase of manipulation-focus errors in symbolizing a derivative. Most students appeared to have an abstract-apart concept of variable that was blocking meaningful learning of calculus. . . . The inevitable conclusion is that a prerequisite to successful study of calculus is an abstract-general concept of a variable, at or near the point of reification. Even a concept-oriented calculus course is likely to be unsuccessful without this foundation. . . . Either entrance requirements for calculus courses should be made more stringent in terms of variable understanding, or an appropriate precalculus course should be offered at the university level. (p. 93)
Over half of the beginning calculus students in Pences study failed the course. This is as common as it is disastrous, and appears to be due, at least in part, to poor preparation. Pence found that many students understandings of the basic operations of multiplication and exponentiation are not sufficiently well developed to allow the students to succeed in calculus. White and Mitchelmore found that many students concepts of variable are similarly weak, and are substantially impeding the students efforts to learn calculus. Orton found that ratio, proportion, graphs, and tangents, basic concepts on which understanding of rate of change is constructed, are poorly understood by many students.
Fortunately, this research goes further than simply documenting the problem of poor preparation and its subsequent ill effects on students performance in calculus. Nemirovsky and Rubin found that students tend to expect a function and its derivative to resemble one another in a variety of ways, but they also found that students could make substantial progress in replacing this resemblance approach to the derivative with a more appropriate variational approach after only 150 minutes of instruction. Similarly, Tall found that, although students had a concept of a generic tangent that was inconsistent with the concept of tangent needed for the development of the derivative as the slope of a tangent, with appropriate instruction the students were able to develop a concept of tangency that would support study of the derivative. In addition to these two studies showing how instruction may help students overcome obstacles to learning calculus, we have the studies by Hauger that document how students use the knowledge they do have to reason effectively about rate of change.
Together, these studies show us where some of the problems are and what some of the solutions might be. As Orton states so clearly, we need to start addressing the problems of calculus instruction in grade school, giving the students more opportunities to work with graphs and rates of change. From the work of Pence and White and Mitchelmore, we see that concepts of variable and operations are also vital but weak, which suggests that these topics also should receive more attention in earlier courses. Once students begin a calculus course, Tall and Nemirovsky and Rubin have shown how instruction can be designed to help students develop the necessary concepts of tangent and the relationship between a function and its derivative. Hauger helps us to begin to see how students reason about rates of change, which moves us closer to developing more effective instruction. These studies give us enough information to improve instruction today, and suggest new directions for future research.
Clement, J., Mokros, J. R., & Schultz, K. (1985). Adolescents Graphing Skills: A Descriptive Analysis. (Technical report number TERC-TR-85-1). Cambridge, MA: Educational Technology Center.
Hauger, Garnet Smith (1995). Rate of change knowledge in high school and college students. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco (April).
Hauger, Garnet Smith (1997). Growth of knowledge of rate in four precalculus students. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago (March 24-28).
Monaghan, John (1991). Problems with the language of limits. For the Learning of Mathematics, 11, 3, 20-24.
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Murphy, Lisa (1998). Students Conceptions of Limit: A Review of Research Literature, with Attention to Methodology. Available on the web at: http://www.mste.uiuc.edu/murphy/Papers/LimitConceptsPaper.html
Nemirovsky, Ricardo, & Rubin, Andee (1992). Students tendency to assume resemblances between a function and its derivative. Cambridge, MA: TERC.
Orton, A. (1983a). Students understanding of integration. Educational Studies in Mathematics, 14, 1-18.
Orton, A. (1983b). Students understanding of differentiation. Educational Studies in Mathematics, 14, 235-250.
Pence, Barbara J. (1995). Relatinships between understandings of operations and success in beginning calculus. Paper presented at the Annual Meeting of the North American chater of the International Group for the Psychology of Mathematics Education, (17th, Columbus, Ohio, October 21-24).
Tall, David (1987). Constructing the Concept Image of a Tangent. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.) Proceedings of the Eleventh International Conference for the Psychology of Mathematics Education, (Vol. 3, pp. 69-75). Montréal, Canada: Université de Montréal.
Tall, David (1989). Different cognitive obstacles in a technical paradigm or A reaction to: Cognitive Obstacles Encountered in the Learning of Algebra. In Sigrid Wagner & Carolyn Kieran (Eds.), Research Issues in the Learning and Teaching of Algebra, Volume 4, 87-92. Reston, VA: National Council of Teachers of Mathematics.
White, Paul, & Mitchelmore, Michael (1992). Abstract thinking in rates of change and derivative. In B. Southwell, B. Perry, & K. Owens (Eds.) Proceedings of the Fifteenth Annual Conference of the Mathematics Education Research Group of Australasia, (pp. 574-581). Richmond, New South Wales: Mathematics Education Research Group of Australasia.
White, Paul, & Mitchelmore, Michael (1996). Conceptual knowledge in introductory calculus. Journal for Research in Mathematics Education, 27, 79-95.
1 Nemirovsky and Rubin mention the connection between the assumption that a function and its derivative resemble one another and the so-called slope/height confusion, discussed in the literature, wherein students appear to use the height of a graph to determine their answers when the slope is required, and vice versa. Nemirovsky and Rubin dont comment on the conclusions of other researchers on slope/height confusion, but they do note that their assumptions lead them to dismiss simplistic explanations for students performance, such as the student does not distinguish position and velocity. (page 4) For more on slope/height confusion, see my review of graph interpretation literature (Murphy, 1997).Back to the body of the paper.
2 Of course, this means that 40 of the 76 subjects, or 53% of the entire sample, failed the course. This is appalling, but from what I have read it is also quite common.Back to the body of the paper.
3 Jerry Uhl, creator of Calculus&Mathematica, has said that this confusion is reason enough to avoid presenting the derivative as the slope of the tangent. I have never agreed with him on this, thinking instead that it would be better simply to educate the students on what is meant by tangent. It is encouraging to find that Tall was able to do this, even if we dont really know how he did it.Back to the body of the paper.
4 Thats all the information we get on the interviews. We dont know how long they were, how they were structured, whether they were video or audio taped, or even whether the students were interviewed as a group or individually.Back to the body of the paper.
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This page last revised Januray 19, 2000