This is the second half of a paper on college students' concepts of rate of change and tangent. It may make more sense if you start with the first half.

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)

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)
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)

Believing that every person, even before formally studying calculus, has some idea of the relationship between a function and its derivative and is open to refining that concept, they chose to work with high school students who had not yet taken a calculus course. To consider ways in which context influences students’ ways of thinking about problems, they created three different microcomputer-based laboratory (MBL) environments in which students could explore the relationship between a function and its derivative. One involved a motion sensor, which produced graphs of the position and velocity of a car over a period of time. Another involved an air flow sensor, which produced graphs of the volume of air in a balloon and rate of air flow into the balloon. The third involved numerical integration, with a sequence of numbers displayed, accompanied by its sequence of partial sums. In each environment, students were asked questions about the behavior of the apparatus and encouraged to experiment to find the answers. For example, in the air flow environment, students explored the volume as an accumulation of the air flow. They were asked to control the flow rate in such a way as to produce a certain response in the volume of air in the balloon. Often, the results were not what they had anticipated. They revised their concepts to take into account this new information, and tried other approaches.

In each of the three environments, high school students participated in 75-minute individual teaching interviews. Each subject participated in two interviews in the same environment, for a total of 150 minutes per subject. Interviews were videotaped with two cameras, one focused on the computer screen and one on the interaction between the subject and the interviewer. During these two interviews, subjects were presented with a total of 15 different problems. Analogous problems were used in each of the three environments. Six subjects are included in the present study, two in each environment. The bulk of this paper consists of description and analysis of interviews with one student, who worked in the air flow environment. His case is used as an example to illustrate the findings of all of the interviews.

Over the course of the 150 minutes of the teaching interviews, the student showed clear signs of improving his understanding of the relationship between a function and its derivative. He started with a resemblance approach, expecting the derivative to resemble the function in attributes such as sign, direction of change, and shape. Gradually he began to adopt a variational approach, expecting the derivative to be positive when the function was increasing, rather than when the function was positive, and so on.

From their analysis of the interviews, Nemirovsky and Rubin conclude:
[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.

Use of resemblances is not the result of the student’s 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)
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.

In their stating their assumptions, Nemirovsky and Rubin indicate that they see concept formation in mathematics as a process that begins early in life and continually builds new understandings from old. While they don’t talk about any particular order to this construction, they do describe a progression from simple to complex. They note that the assumption of resemblances between a function and its derivative is quite common, perhaps even universal. This suggests that the assumption of resemblances may be one of the “fundamental obstacles” of which Tall (1989) wrote. Thus, Nemirovsky and Rubin’s work could be interpreted as answering Tall’s call for ways to help students surmount fundamental obstacles.

In 1983, A. Orton published two papers in Educational Studies in Mathematics, both taken from the same study. The second paper, “Students’ Understanding of Differentiation” (Orton, 1983b), is of primary interest here, although I have made some use of the first, “Students’ Understanding of Integration” (Orton, 1983a), which gave information about the study as a whole. According to the earlier paper, “A major purpose of the research study on which this paper is based was to investigate students’ understanding of integration and differentiation.” (page 1)

One hundred and ten students, 55 males and 55 females, were interviewed twice, for an hour each time. Subjects ranged in age from 16 to 22. All were studying mathematics, some in high schools and some in college. The college students intended to become teachers of mathematics. Although all data were collected through interviews, Orton’s work is definitely quantitative. For purposes of analysis, he organized his complex interview tasks into 38 items, each involving a single skill or concept. In some cases, information from several tasks was combined to form an item. Responses to items were coded on a five point scale. Subjects’ ages and scores on a test of general intelligence were also included in the analysis. The 21 items relating to differentiation and related concepts formed the basis of the second paper.

After describing the types of errors that were observed in response to each item, Orton concludes that several earlier concepts form the foundation on which understanding of calculus is built, and that these concepts are weak or lacking in many students. The idea of rate of change involves ratio and proportion, which are poorly understood by many students. Graphs, and particularly tangents to curves, are important in developing ideas of rate of change, but students’ understanding of graphs is often weak. These two factors lead to a poor understanding of rate of change. Orton suggests that greater attention should be paid to ratio, proportion, graphs, and tangents throughout a student’s education. He writes:
We must certainly take every opportunity to lay foundations of ideas of rate of change throughout a pupil’s 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.

Barbara Pence conducted a study (Pence, 1995) “investigating the relationships between understandings of operations and understandings of concepts studied during first semester calculus.” (page 1) She administered a brief questionnaire to 76 students in first semester calculus. In addition, she collected information on her subjects’ midterms, final course grades, and second semester calculus course grades (where applicable). Her questionnaire contained six demographic and attitudinal items, and two achievement items.

This paper is based on one of those achievement items. The students were given a number line showing the locations of 0, 1, and x. The point x was located at about 1.4, as shown on the sketch below:


Subjects were asked to label points corresponding to 2, 2 x, x2, and 2 x. Responses were grouped into the following six categories: (1) 5 students could not successfully locate any of the four points, (2) 34 students could locate 2, but none of the other three points, (3) 19 students could locate 2 and 2 x, but not x2 or 2 x (4) 2 students could locate all points except 2 x, (5) 2 students could locate all points except x2 (6) 14 students could locate all four points. All responses fit into one of the above categories.

To me, the most striking thing about these results is the fact that 39 students fell into the first two categories. That is, over half of the students entering calculus did not know that 2 x is twice as far from 0 as x is. Presumably, all or nearly all of these students could calculate a value for 2 x if given a value for x, but without a numerical value they are lost. Pence suggests that the students understand operations such as multiplication at the process level, meaning that they view operations as instructions for processes to be carried out, but do not understand them at the object level. That is, students see 2 x as an instruction to multiply x by 2, which they can do if given a value for x, but they do not understand 2 x as an object, representing a quantity twice as large as x. The students have not encapsulated the multiplication process in a way that allows them to think about 2 x as an object. If they cannot think of 2 x as an object, then they will have trouble dealing with other processes operating on 2 x.

As one might expect, Pence found that success in calculus was strongly related to performance on this item. Of the 60 students in categories 1 through 4, 35, or 58%, failed the first semester calculus course. Of the 16 students in categories 5 and 6, only 5, or 31%, failed.2 That is, the failure rate for students in categories 1 through 4 was approximately double that of students in categories 5 and 6. On a scale of A = 4.0, the average grade earned was 1.1 for students in categories 1 through 4 and 2.6 for students in categories 5 and 6.

Pence regarded categories 4 and 5 as parallel, but they showed a great difference in course grade. This may be an artifact of the small numbers of students in these categories, since each category contained only two students. Both of the students in category 4 failed, while both of the students in category 5 passed, with an average grade of 3.5. Omitting these four students leaves categories 1 through 3 with a failure rate of 57% and an average grade of 1.2, while category 6 has a failure rate of 36% and an average grade of 2.5.

Pence takes these results to indicate that encapsulating operations and working with them at the object level is important for the study of calculus. Students who are not able to do this are much more likely to fail calculus than students who understand operations at an object level. According to Pence, operations are an important cognitive root in calculus. That is, they are (assumed to be) familiar and well understood, so they can form a basis for later mathematical development. Students who lack this base are much less likely to succeed in calculus.

David Tall examined students’ response to instruction using interactive computer programs (Tall, 1987). He writes:
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.”

Both groups were tested once after instruction on the gradient, and again after more detailed instruction on the tangent. A group of university students took the same tests, without receiving any instruction. Both tests used the same six graphs of functions. For the gradient test, the students were asked, for each of the six functions, whether the gradient was defined at the point where x = 0, if so what it was, and if not why not. For the later function test, the students were asked the same questions about the existence of the tangent at the point x = 0.

Results were compared for each question. Generally the experimental sections significantly outperformed the control sections, and occasionally outperformed the university students as well. This indicates that the experimental instruction was superior to the traditional instruction for teaching the concepts of gradient and tangent. It does not give the reader any idea of how important the computer work may have been to this process. From what little Tall says about the instruction, it appears that the experimental groups devoted significantly more time and attention to the concept of tangent. This alone would be expected to result in the experimental group having a greater understanding of tangent, regardless of technological differences.

Tall concludes:
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.

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)
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.

The concept Tall calls the “generic tangent,” a line which touches the curve at exactly one point, is one I have encountered elsewhere. Students are reluctant to call a line “tangent” to the curve y = f(x) at the point x = a if the line also touches or crosses the curve at some other point x = b.3 Since tangency is a local property, the relationship between the line and the curve at some other point is irrelevant. However, perhaps because of high school geometry courses dealing with tangents to circles, students often insist that the tangent is not allowed to touch the curve at any other point.

In this regard, Tall’s study appears to be in agreement with Monaghan (1991), who found that the terms used to describe a concept already have meanings for the student, meanings which may conflict with the new concept being presented. Monaghan suggested that “students should be led to explore and discuss their own conceptions and to realise how everyday meanings of mathematical phrases can direct them into fallacious interpretations.” (page 24) Apparently, Tall did lead his students to explore the meaning of “tangent” in some detail, and thereby reduced the frequency of fallacious interpretations.

Tall does not indicate whether he considers the misunderstanding of the meaning of tangent to be a fundamental obstacle. It does appear to be inherent in the usual way in which examples of tangency are introduced. Students first see lines tangent to circles, conclude (or are told) that the tangent touches the circle in only one place, and then incorrectly transfer this definition of tangent to their later study of tangents to functions. However, it appears possible that this might be avoided by introducing other kinds of tangency with or before tangents to circles, or by altering the way in which tangents to circles are presented.

White and Mitchelmore
In their study of the role of conceptual knowledge (White & Mitchelmore, 1996), Paul White and Michael Mitchelmore examined application problems because they believe that conceptual knowledge plays a larger role in application problems than in more routine symbolic manipulation problems. They write:
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.

Four application problems were written, with four versions of each problem. All four versions of a single problem involved precisely the same manipulations, and had the same solution. The differences among the versions of a single problem consisted in different amounts of translation into mathematical symbols and relationships required before the student could perform the manipulations needed to obtain the solution.

White taught 40 students for six weeks in an introductory calculus course at a university. All subjects had previously taken a high school calculus course. On the basis of their performance in an algebra course completed the previous semester, the students were placed into four parallel groups of ten students each. All students received the same instruction. The subjects were tested four times: before, during, immediately after, and six weeks after the instruction. At each test administration, each group of ten subjects worked one version of each of the four application problems. The versions were distributed so that in each test administration, each version was completed by one group. Thus, data from all versions of each question were available at each testing point.

The numbers of correct responses by problem, data collection period, and version were examined to determine the effect of the instruction and the effect of being required to perform the varying amounts of conceptual work involved in translating the application problems into mathematical symbols and relationships. For each version of each problem, White and Mitchelmore also tabulated the numbers of partially correct responses in which the student successfully translated the application problem into mathematical symbols and relationships and then made a manipulation error. There were very few such responses on any version of any question.

Four students per group, selected before the start of instruction, were interviewed within three days of each data collection. White and Mitchelmore write: “These interviews served to clarify and expand on written responses so that student reasoning could be better identified.”4 In addition, the interviews confirmed that the students were not aware that the questions were essentially the same on each of the four tests. White and Mitchelmore discuss the interviews and the responses to the open-ended questions, noting which types of errors and misconceptions are most common in subjects’ responses. This lends a qualitative component to what appears to be primarily a quantitative study.

In examining their data, White and Mitchelmore noted that students’ concepts of variable were often lacking. In an earlier paper on the same study (White & Mitchelmore, 1992), they define their terms “abstract-apart” and “abstract-general,” which they use to describe the concepts of variable held by their subjects:
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)

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)
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 weren’t 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.”

White and Mitchelmore have identified this obstacle, described it, and demonstrated the importance for students of surmounting it. The next step along this line is to develop and test methods of helping students to develop an abstract-general concept of variable. If the ways in which this concept is formed were better understood, it might be possible to incorporate appropriate instruction into the pre-calculus curriculum, so that fewer students would arrive in calculus courses still lacking this vital concept.

Lessons Learned

Over half of the beginning calculus students in Pence’s 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.

Murphy, Lisa (1997). Graphing Misinterpretations and Microcomputer-Based Laboratory Instruction, with Emphasis on Kinematics. Available on the web at:

Murphy, Lisa (1998). Students’ Conceptions of Limit: A Review of Research Literature, with Attention to Methodology. Available on the web at:

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 don’t 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).

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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.

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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 don’t really know how he did it.

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4 That’s all the information we get on the interviews. We don’t 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.

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This page last revised Januray 19, 2000