As a final assignment in the course for which I wrote this paper, I was asked two questions calling for a summary of what I learned in the process of writing the paper. The two questions were What have you learned about your area of interest? and What have you learned about the plurality of research methods? My answers to these two questions are followed by the paper itself:

Limit: Cornerstone of the Derivative

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**What have you learned about your area of interest?**

Students often cling tenaciously to their intuitive ideas of limit, which are usually dynamic, despite efforts to help them develop more sophisticated, formal concepts. For example, students often continue to insist that a function cannot reach its limit, even after they have worked with many examples of functions that do reach their limits. They often view such disconfirming examples as exceptions to valid general principles, and thus continue to hold their original ideas even after being presented with the counter-examples.

Since the formal definition is in disagreement with students' dynamic, intuitive ideas of limit, this persistence of their informal ideas means that students often are unable to make sense of the formal definition, and may be unwilling to try. They may view the formal definition as a useless bit of arcane trivia that their teachers insist on covering, rather than seeing it as an expression of the essence of the idea of limit. In this case, students may respond based on their informal dynamic ideas when asked about a particular example, and give a completely different response when asked for the definition of limit. Such a student makes no effort to reconcile the two ideas, and so often does not notice when the two responses are in direct conflict with one another.

Teachers and textbooks may unintentionally reinforce students' preference for intuitive ideas and disregard of formal definitions. Students tend to prefer strategies that they see as practical, enabling them to complete their homework and attain acceptable scores on exams. When homework and exams consist of repetitive problems that students can solve without recourse to a formal concept or definition of limit, students are encouraged to regard formal definitions and concepts as irrelevant. Since many homework problems can be solved by the application of memorized procedures, even the informal concepts may be devalued in the minds of students intent on completing the required problems with a minimum of effort.

Disagreement between formal definitions and informal concepts is only one example of a situation in which a student may hold two mutually contradictory ideas and not notice a conflict. Even when the formal definition is not involved, students working with their informal ideas may respond to essentially the same problem in different and contradictory ways when it is presented in different forms. For example, although the sum of the infinite series 0.9 + 0.09 + 0.009 + 0.0009 + ... , is the same as the limit of the sequence of partial sums of the series, 0.9, 0.99, 0.999, 0.9999, ... , and also the same as the repeating decimal 0.9999..., students may say that one of these representations is equal to 1 while another is not. Apparently, the students hold several concept images, and select one according to the form of the question.

The wording of the question plays a role in the selection of the particular concept image that the student brings into play when presented with a problem. Words and phrases that mathematicians consider to be synonymous may have very different connotations for students. To design instruction that will help students develop the desired concept image, one must first understand their intuitive ideas and the language in which these ideas are expressed.

It is tempting to think of the students' informal ideas as stumbling blocks along the path toward formal concepts and definitions, and indeed there is some truth in this. However, these stumbling blocks are also the building blocks from which the more sophisticated concepts will be constructed. No complicated concept is ever acquired in an instant. The formal concept of limit must be built up through a process of forming, accepting, rejecting, modifying, and connecting more primitive and intuitive concepts.

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**What have you learned about the plurality of research methods?**

The nature of the research question suggests the method. If the researcher wants to know in detail what a student's concepts of limit are, and has only a vague idea of what those concepts might be, then qualitative methods are most appropriate. Questions can be open-ended, so that the interviewer is not required to anticipate the answers, and very rich data can be gathered. However, conducting interviews and interpreting the data is very time-consuming, so often it is not possible to have a large number of subjects. This prevents the researcher from generalizing to a population.

If the researcher wants to know the prevalence of certain concepts within a population or populations, then a larger number of subjects is required. This makes it impractical to spend a lot of time interviewing each subject and analyzing data from the interviews. Thus, for larger numbers of subjects, more quantitative methods are desirable. The easiest way to gather and process data from a large sample is to administer a survey consisting of multiple-choice questions. This method requires the researcher to anticipate the students' concepts in some detail, since subjects cannot select an option that does not appear on the survey. Time and attention to survey design will be rewarded in the final result.

Sometimes a researcher wants to use a large enough sample to permit conclusions to be drawn about a population, but does not have quite as clear an idea of the subjects' concepts as would be required to design an effective multiple-choice instrument, or needs to avoid influencing students' responses by suggesting answers. In either case, open-ended, short-answer questions may be appropriate. Responses will take more effort to analyze than is the case for multiple-choice items, but it may be worth this investment to get a better picture of the students' concepts. Similarly, short interviews conducted according to a precisely specified protocol may serve as a compromise between the desire for the rich picture of qualitative methods and the need for the large sample size of quantitative methods, and may allow the researcher to avoid influencing the subjects' responses by suggesting answers.

For example, one study I read (but did not put into my review) involved interviews with 110 subjects. Despite the qualitative connotations of using interviews, this was essentially a quantitative study. Interviews were highly structured, involving carefully planned tasks. Data from the interviews were coded according to an elaborate, detailed system, and results were reported as proportions of the sample giving different types of responses. This study had an advantage over those using multiple-choice instruments, in that the researcher could get all of the responses subjects had to the questions, not just those included in some preconceived list, and students' responses were not guided by proffered choices. However, this advantage was obtained at the cost of 220 hours of interviews and presumably a great deal of time spent coding the data.

Not only does the nature of the question determine the method, also the method determines the type of answer that can be found. Once committed to long interviews with a few students, for example, the researcher may be able to construct a detailed portrait of those students' ideas, but cannot find characteristics of a population. Similarly, once a multiple-choice instrument has been designed, the researcher is able to find some characteristics of a large population, but is limited to only those results that can be found from students' selections among the proffered choices.

One paper that I read but did not include in my review used multiple-choice questions to determine students' concepts of the role of definition in mathematics. Several statements were presented, and for each one the student was asked to specify whether it was an axiom, a postulate, a fact, a definition, or a theorem. The researcher presented the proportions of the sample giving each response for each statement, and then ran into trouble. It wasn't at all clear what any of this data meant about students' concepts of definitions. As the researcher noted, a statement may be a definition in one development of a topic, and a consequence of the definition in another development. He was counting on the students not to be sophisticated enough to be aware of this, but still it meant that there were no right answers to any of his questions. He developed a rationale for considering some answers "correct," but it was not convincing. In the end he was stuck, unable to draw any meaningful conclusions or to convince the reader of his claims, because he was limited to the results of a badly designed questionnaire. [He got published anyway. I wonder why.]

By contrast, some of the researchers using qualitative methods may have had equally vague ideas of what their subjects' concepts were, but once they had several hours of tape of each subject explaining his or her ideas, the researchers were able to get some useful information from their data.

These different methods, giving different kinds of answers to different kinds of questions, can be used in complementary ways. Interviews can be used to develop and improve a survey instrument. Surveys can be used to identify students to interview, or to discover the proportion of a population sharing the views of a particular interview subject. Any researcher, attempting to answer various aspects of a broad question, may use methodology from all along the qualitative-quantitative spectrum at different points in the same research project.

Since I am interested in the response of a population of students to various methods of instruction, I will use quantitative methods. However, this review has made me even more aware of the difficulty of constructing a multiple-choice instrument that will measure what I want it to measure. Qualitative methods, including interviews with students, could be very helpful to me in my efforts to construct an instrument that will meet the needs of my study. Compromise methods, such as open-ended questions on surveys, will probably be useful throughout the development of my study.

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My dissertation research involves helping students to understand the concept of the derivative. For this literature review, I concentrated on the concept of *limit.* *Limits *are used to make more precise the idea of an instantaneous rate of change and to develop the formal definition of the derivative. Thus, the concept of limit is fundamental to understanding the derivative. In another paper (Murphy, 1999), I discuss rate of change, which is also central to students' understanding of derivative.

My choice of topics for these reviews is based on the assumption that mathematical concepts are built one upon another, with students' understanding of each concept resting on certain previous concepts, and in turn helping to lay the foundation for later concepts. In particular, it assumes that the specific concepts of limit and rate of change are necessary for the successful study of derivative. This idea of mathematics as being in some sense *linear*, with concepts built up in a necessary order, is not universally accepted.

Some researchers(Footnote 1) claim that this ordering of concepts is not due to the nature of mathematics, but is instead a quirk of our educational system. They maintain that this artificially induced ordering of concepts holds some students back, keeping them out of mathematics. A student fails to grasp some concept that we consider to be an important prerequisite for calculus, for example, and is thus kept out of calculus. According to this theory, such a student might be very successful in calculus if only we could give up our insistence that concepts be learned in a specific order. By this reasoning, the traditional ordering--or, indeed, any fixed ordering--of topics in the curriculum is inherently inequitable, because it is thought to contribute to preventing some students from learning mathematics. Racial and ethnic minority students are believed to be particularly likely to be injured by an insistence on prerequisites.

In deciding to approach students' learning and understanding of the derivative by studying students' conceptions of rate of change and limit, I am assuming that this theory--that there is no natural ordering of mathematical concepts--is false. However, this should not be taken as a wholesale endorsement of the status quo. The natural ordering, to the extent that such exists, may not always match the traditional(Footnote 2) ordering. Indeed, the two cannot be entirely the same. In mathematical terms, a syllabus is a "total ordering"--for any two concepts in a particular syllabus, it is possible to say which is encountered first in that syllabus. By contrast, the natural ordering of mathematical concepts is a "partial ordering"--some pairs of concepts have a natural order, but other pairs may be learned equally well in either order.

There are problems inherent even in the natural ordering. David Tall, who coauthored one of the papers reviewed here (Tall & Vinner, 1981), states this explicitly (Tall, 1989):

The implication of Piagetian stage theory is that there are certain fundamental obstacles that occur for us all. . . . I postulate that the reason for the belief in fundamental obstacles arises from the fact that certain concepts have a degree of complexity that makes it necessary to become acquainted with them in a certain order. For example, fractions are, of necessity, more complicated than whole numbers, and experience with operations on whole numbers leads to the implicit generalization that "multiplication makes bigger," which leads to a cognitive obstacle when the individual meets the multiplication of fractions less than one.However, some topics that are traditionally taught in a certain order may not have the a priori property that one concept is essentially more complex than the other. For instance, fractions are usually met in traditional syllabuses before negative numbers, but there is no reason why, given an appropriate context, the two topics should not be taught in the reverse order. (page 88, italics added) |

Tall goes on to suggest that traditional orderings should be reconsidered in an effort to prevent as many obstacles as possible, while recognizing that certain orderings are necessary and give rise to fundamental obstacles that cannot be prevented. As Tall indicates, assuming that there is a natural partial ordering does not end the discussion. One must determine what specific concepts naturally precede other concepts. In some cases, there will not be sufficient difference in complexity or dependence of concepts to create a natural ordering, but there still may be advantages to one ordering over another for certain students or in certain situations. When an ordering has been determined, one must discover the fundamental obstacles inherent in this ordering, and consider how they may be surmounted.

The researchers whose work is reviewed here appear to have assumed that there exists some natural or inherent ordering of limit and certain underlying concepts. Although not all papers address this assumption explicitly, this research investigates the relationships among these concepts, thus helping the reader to distinguish natural orderings from orderings imposed by custom or whim. In addition, this research aims to identify obstacles created by prior knowledge or concepts, many of which are partially correct but come with connotations that impede the learning of more formal concepts of limit. Some of these papers also address ways to help students surmount these obstacles.

At this point I hope it is clear that none of these researchers are advocating inequity. Rather, they believe that the most effective methods for teaching mathematics to all students must take into account the inherent ordering of mathematical concepts. Under this assumption, eliminating relevant prerequisites can be expected to lead to failure for under-prepared students, not to increased access to mathematics. Thus, equity is advanced, not by rejection of order, but by improving our understanding of natural orderings and their consequences, and basing instruction on this understanding.

In addition to this assumption of a natural ordering of mathematical concepts, the studies described here have other features in common. All were intended to discover how students think about limit. Some were interested in studying how students change their concepts, others simply took snapshots of students' thinking at some moment in time, but in all cases the object of study was the student's conceptions. These are studies of *properties,* since they concentrate on identifying and describing concepts, thought processes, and ways that concepts are changed.

All of these researchers treated the individual student as the unit of analysis. There exist studies (not related to limits, as far as I can find) in which researchers study the classroom culture, and consider what ideas are taken as shared and how these evolve through negotiation and other social processes. The studies discussed here do not have that perspective. Nearly all interviews are with one student at a time, rather than groups of students.(Footnote 3) To the extent that the classroom culture influences the ways that students learn and think about the derivative, this focus on the individual to the exclusion of the group limits the researcher's ability to fully understand the process of concept development. However, it has the advantage of narrowing the field of study to a manageable size. It would be interesting to compare and contrast these studies with studies on the development of the same mathematical concepts in groups of students working together, if such studies could be found. I suspect that examining concept formation in calculus from both the intrapersonal and the interpersonal perspectives would provide additional insight, which could be used to improve instruction.

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**Tall and Vinner**

The paper by David Tall and Shlomo Vinner (1981) is primarily conceptual. In support of their ideas, they include reports of some empirical studies not reported elsewhere.(Footnote 4) Since this is intended as a review of empirical studies, I will examine the empirical work discussed in their paper. Other than one brief mention of an interview with a group of four students, the work reported here is primarily quantitative. Surveys were given to large groups of undergraduates, 36 in one study and 70 in another. Short answers to free-response questions were categorized, and the numbers of each type were reported and discussed.

These studies all revolve around concept images (the student's image of the concept), concept definitions (words used, either by the student or the teacher, to specify the concept), and concept definition images (the student's image of the meaning of the concept definition). Tall and Vinner write:

Many concepts we meet in mathematics have been encountered in some form or other before they are formally defined and a complex cognitive structure exists in the mind of every individual, yielding a variety of personal mental images when a concept is evoked. In this paper we formulate a number of general ideas intended to be helpful in analysing these phenomena and apply them to the specific concepts of continuity and limits. (page 151) |

In a questionnaire for mathematics students arriving at an English university, Tall asked students to find the limit lim n->° (1 + 9/10 + 9/100 + 9/1000 + ... + 9/10n), give a definition of limit, and say whether 0.999... (repeating) = 1 or not, giving a reason for their answer. Although it seems clear that the requested limit must be 1.999..., which could be 2 only if 0.999... = 1, fourteen of the 36 students (39%) correctly gave the limit as 2, but incorrectly said that 0.999... is less than 1. Tall and Vinner remark "Clearly the two questions evoked different parts of the concept image of the limiting process." (page 159)

In a subsequent test, the same students were asked to write several decimals as fractions, including the repeating decimals 0.333... and 0.999... . Thirteen of the fourteen students who had previously said 0.999... < 1 now said 0.999... = 1, often indicating in their responses an awareness of cognitive conflict; they crossed out answers and wrote things such as "0.999... = 3 x 0.333... = 3 x 1/3 = rubbish." (page 159) Tall and Vinner use this to illustrate their point about the significance of the difference between *potential cognitive conflict,* in which a student holds two concept images that are inherently contradictory, and *actual cognitive conflict,* in which the student calls on both concept images at the same time and becomes aware of the contradiction. These fourteen students displayed potential cognitive conflict in their results to the first questionnaire, but did not experience actual cognitive conflict until presented with the questions on the second test.

While actual cognitive conflict may lead to refinement of the concept image and/or the concept definition image, this is not always the case. Students who experience actual cognitive conflict when presented with an anomalous example may preserve their sense of equilibrium by discounting the example. Four students who had indicated that they believed that no term of a series can be equal to the limit of the series were shown the example 0, 1/4, 0, 1/8, 0, 1/12, ..., which can also be written sn = {0 for odd n, 1/2n for even n}. In this series, many of the terms are equal to the limit of 0. The students all concluded that this was not a real series, but was two. The series 1/4, 1/8, 1/12, ... *tended to* zero, while the terms 0, 0, 0, ... were *equal *to zero. Although the given series fits the formal definition of a series, it failed to fit the students' concept image, so they discarded it, insisting that it was not really a series. Tall and Vinner write: "This is a typical phenomenon occurring with a strong concept image and a weak concept definition image that permeates the whole university study of analysis, especially when there are potential conflict factors between the two." (page 160)

Finally, Tall and Vinner gave a questionnaire to seventy first-year university students, all of whom had previously earned grades of A or B in A-level mathematics in English high schools. First, the students were asked to explain what is meant by lim *x*->1 (*x*3 - 1) / (*x* - 1) = 3. On the next page, they were asked to write down a definition of lim *x*->a *f*(*x*) = *c* if they knew one. Responses were categorized according to correctness and type, either formal or dynamic.(Footnote 5) The 31 students who gave a dynamic definition all used a dynamic approach to the example, as did most of the 21 students who did not give a definition. Of the four students who gave a correct formal definition, three used a formal approach to the example, and one used a dynamic approach.

All of this is as might be expected--students generally used the same approach to the example as to the definition. However, there was one exception. Only four of the fourteen students who gave incorrect formal definitions used their formal definitions in their approach to the example. The other ten used a dynamic approach to explain the example. For these ten students, the request for a definition apparently evokes a different concept image than is evoked by a specific example. Since they gave incorrect definitions, it appears that their concept definition image may have been weak or faulty. Their failure to use their concept definitions on the example suggests that their concept definition images may be in conflict with, or disconnected from, their concept images of limit.

Tall and Vinner go on to describe similar work with students' concept images and concept definition images related to continuity. As with limits, the students may have a concept image of continuity that works well for simple problems, but that is at best weakly connected to a formal definition. Many have little or no concept definition image. The students often cling to these informal ideas, creating an obstacle to their learning of the more formal concept and preventing them from understanding how continuity relates to the unusual problems and anomalous examples Tall and Vinner used in their study.

Tall and Vinner conclude by writing "The difficulty of forming an appropriate concept image, and the coercive effects of an inappropriate one having potential conflicts, can seriously hinder the development of the formal theory in the mind of the individual student." (page 169)

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**Monaghan**

John Monaghan (1991) studied the effects of the language used in teaching and learning limits. Words taken from natural language, having familiar everyday meanings, are given subtly different meanings in mathematics. Students may confuse the two. Monaghan writes:

The phenomena reported on here were part of a wider study which investigated mathematically able adolescents' conceptions of the basic notions behind the calculus. This paper reports only on those aspects of the study that examined students' understanding of the language used by teachers to communicate calculus concepts. . . . This article deals solely with ambiguities inherent in the four phrases tends to, approaches, converges, and limit. (page 20) |

Monaghan administered a questionnaire to 54 English high school students who had passed O-level exams in mathematics at the age of 16 and were in their first year of A-level studies. Twenty-seven were studying A-level mathematics and 27 were not. A month after the questionnaire, the subjects were interviewed. The following year, a revised form of the questionnaire was administered to 190 students, 114 studying A-level mathematics, from schools following the same O-level and A-level schedule as in the first sample. The subjects in the second sample were not interviewed.

Several items with Likert-type responses (yes / think so / unsure / think not / no) were common to both questionnaires. Each item asked students the same question in four different ways, using the four phrases *tends to*, *approaches*, *converges*, and *limit*. One item asked about the sequence of numbers 0.9, 0.99, 0.999, 0.9999, . . . with suggested limits of 1 and 0.999 repeating. Six other items related to continuous curves given by sketches. All questions were asked about convergence to 0, although two of the curves actually converged to 1. The other four did converge to zero. In addition to these items, students in the first sample were asked to write four sentences, each illustrating the use of one of the four phrases. Students in the second sample were asked to write sentences using the word *limit*, but not the other three phrases.

Although a mathematician might say that the sentences: "The curve tends to 0," "The curve approaches 0," "The curve converges to 0," and "The curve has 0 as a limit" all mean the same thing, the students often agreed with one but disagreed with another in reference to the same curve or sequence. For example, 66% of the second sample thought the given sequence of numbers tended to 1, but only 22% thought it converged to 1.(Footnote 6)

Quantitative analysis of the Likert-type items indicated that the students understood these four phrases to have very different meanings. Monaghan examined the responses to the interviews and the sentence-writing task to determine the nature of these different concepts. *Limit* was often thought of as a bound. Some examples involved legal bounds, such as speed limits. It is possible to exceed the speed limit, although it is forbidden. Other conceptions of limit involved a bound on physical or mental abilities, such as a limit to the height one could jump or a limit to one's patience.

*Approaches* was seen in a more dynamic way. It involved things moving toward other things, sometimes with the idea that the object being approached would eventually be reached ("The train approached the station.") and sometimes with the suggestion that the moving object might neither reach the object being approached nor even get particularly close. In the Likert-type items, the graph of the function *f*(*x*) = 1 + 1/*x* was viewed by about a third of the first sample and half of the second as approaching 0, since the curve decreases as *x* increases, even though it always remains above its asymptote at *y* = 1.

In responses to the sentence-writing task, nearly all examples of *tends to* involved personal inclination, as in "he tends to wear jeans" or "she tends to drink a lot." In the Likert-type items providing mathematical examples, students generally treated *tends to* as having the same meaning as *approaches*. Both are seen as dynamic, evoking movement, and generally referring to motion of an object that does not reach the point approached.

*Converge* was also seen as dynamic, but usually involved two continuous objects, as in "the roads converge" or "the light rays converge." Other references involved discrete objects making contact, such as "the footballers converged on the ball." (Footnote 7) One student said that two sequences could converge to one another, as light rays might converge, but that it didn't make sense to talk about one sequence converging. Also, *converging* more often involved reaching or touching, rather than getting closer without touching, as was implied by *tends to* and *approaches*.

Monaghan quotes Tall and Vinner (1981), who wrote: "We shall call the portion of the concept image which is activated as a particular time the evoked concept image. At differing times, seemingly different conflicting images may be evoked." (page 24 in Monaghan, page 152 in Tall and Vinner) Monaghan suggests that mathematically synonymous terms bring different connotations from their natural language meanings, and thus play a role in evoking different concept images for the same mathematical concept. In conclusion, Monaghan recommends: "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)

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**Williams**

Steven Williams (1991) examined students' conceptions of limit and their responses to the challenge to those conceptions posed by the presentation of anomalous examples. He writes:

In an effort (a) to explore in more detail what Cornu called students' spontaneous models of limit, and (b) to study the means by which these models can be altered and made more rigorous, the study investigated the views of 10 second-semester calculus students as to what constituted a limit and then presented them with descriptions and examples of limits that conflicted with their views. (p. 219)This study was conducted under the assumption that most calculus students form rudimentary limit models . . . and has sought specifically to (a) document and describe the existence of such models, (b) attempt to use exposing and discrepant events to encourage a change in students' understandings of limit so that they more nearly resemble the formal definition, and (c) document the factors affecting such change. The subset of data reported here focuses on two models of limit found to be most common among students in the study. (pp. 220-221) |

Williams gave a brief questionnaire on limits to 341 students enrolled in second semester calculus as a university. Students were asked to mark each of six statements about limits as true or false, select which one of the statements they thought best described a limit, and write a few sentences explaining what they thought a limit was. Williams presents a summary of the results of the questionnaire, giving the proportion of students who selected each of the suggested views of limit. From the results of this questionnaire, he selected ten subjects whose questionnaires each clearly and unambiguously presented one of the four common informal views of limit. He selected four students holding each of the two most common concepts of limit, and one holding each of the other two common concepts.

Each subject was interviewed five times over a period of seven weeks. The first session, which was an hour long, concentrated on articulating the students' present concept of limit. The next three sessions, half an hour each, began with students explaining their views of limit in relation to statements presented by the interviewer, and proceeded to the subjects working limit problems that brought to the subjects' attention difficulties with their current conceptions and provided incentive for them to alter their viewpoints. In the final session, which was an hour long, Williams presented the subjects with each of the three views of limit they had given in the three previous sessions, and asked them to explain why their views of limit had or had not changed.

From these interviews, Williams made some generalizations about how the students viewed limits. He writes:

[S]tudents often considered the ease and practicality of a model of limit more important than mathematical formality. This is particularly true in the sense that models of limit that allow them to deal with the realities of limits in the classroom, the kind they see on tests, tend to be seen as sufficient for the purposes of most students. It was noted by several students that neither formal nor dynamic models of limit figure heavily in the procedures students use to work problems from their calculus class; their formal knowledge (e.g., substituting values into continuous functions, factoring and canceling, using conjugates, employing l'H™pital's rule) is largely separate from their conceptual knowledge. (p. 233) The data of this study confirm students' procedural, dynamic view of limit, that is, as an idealization of evaluating the function at points successively closer to a point of interest. The data also suggest that there are numerous idiosyncratic variations on this theme, some of them extremely difficult to dislodge. Given the complex nature of cognitive change, it is not surprising that the students in the study failed to adopt a more formal view of limit after only five sessions. . . . [T]he data suggest that the attitudes toward practicality and mathematical truth displayed by the subjects did interfere with conceptual change. Specifically, students' views of mathematical truth, the value they place on practicality and simplicity in models of limit, the everyday demands of calculus class, their previous experience graphing functions, and their faith in the a priori existence of graphs combine to make it difficult for them to appreciate the need for a more formal definition of limit. . . . The end result for the students in this study is a lack of appreciation for formal thinking, which effectively removed any motivation to learn what is, after all, a very formal definition of limit. . . . All this suggests that improving students' understanding of limits from a formal viewpoint requires careful and explicit instruction, which accounts for the rich variety of limit models students bring to the classroom as well as the sorts of knowledge they value. In some sense, their prior knowledge of graphs and functions must be deconstructed, to expose the underlying assumptions that formal definitions attempt to address. (p. 235) |

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**Davis and Vinner**

Robert Davis and Shlomo Vinner (1986) worked with teachers at University High School in Urbana to develop a mathematics sequence from beginning arithmetic through two years of calculus, in which concepts were taught first and skills were built on that understanding. Their paper "The Notion of Limit: Some Seemingly Unavoidable Misconception Stages" examines the calculus sequence, focusing on the students' development of the concept of limit. Rather than finding, as they had hoped, that their method produced greater student understanding, Davis and Vinner found that changes in instruction did not create as great a change in learning as they had expected. They write:

One goal, then, of this present study was to determine how successful this "start with understanding" approach has been, in the case of the 2-year calculus course. As we shall see, getting evidence that "early understanding is possible" is difficult, and the present study can claim little or none. We shall consider why obtaining evidence is so difficult. (page 283, italics in the original) |

During the first year of the calculus sequence, the students demonstrated mastery of the limit concept by proving theorems, stating definitions, and producing example sequences to illustrate weaknesses in incorrect definitions. On the first day of the following school year, the fifteen students were asked to informally describe the intuitive concept of the limit of a sequence and give a precise formal definition. The understanding that they had demonstrated the year before was not always evident in their responses.

Davis and Vinner suggest that many of the students' errors are *retrieval errors*, similar in nature to reaching for a phone book and picking up last year's edition instead of this year's. The students had learned the new concepts, but retained their naive concepts. When asked about limits, they retrieved the older, naive concepts, and not the newer, school-taught ones. For instance, many students said that a sequence can never reach its limit, despite having worked competently the previous year with sequences that did reach their limits.

These were unusually bright students who had been carefully taught in a conceptually oriented curriculum intended to prevent them from making the mistakes common to average students in traditional curricula, but these errors appeared nonetheless. By examining the students' free responses and considering their ideas in their own words, this qualitative study was able to find what and how these students think about limits. It found that they were not much different from less gifted students in traditional curricula. The same naive concepts were present and affected the students' responses.

Davis and Vinner suggest five explanations for their results: (1) the influence of language; (2) assembling mathematical representations from pre-mathematical fragments; (3) building concepts within mathematics; (4) the influence of specific examples; and (5) misinterpreting one's own experience. The first two have to do with familiar words and phrases, such as "limit," "approaching," and "going to," and other extra-mathematical ideas that students bring to their work and use to develop and express their mathematical ideas. These words and ideas may bring with them misleading connotations. In an effort to prevent confusion, Davis and Vinner avoided all use of the word "limit" until after the concept was thoroughly developed, referring instead to the "associated number" of a sequence, but they were not certain that this was sufficient to prevent students from mixing the mathematical concept of limit with their ideas of the everyday meaning of the word limit, as in "speed limit" and so on.

Their third explanation also relates to preexisting knowledge, but, unlike the first two, refers specifically to mathematical knowledge. They write:

Even if words and ideas from outside of mathematics could be excluded from a student's concept image, and if one could work entirely within mathematics, it would still be necessary for each student to build mental representations gradually. One cannot put anything as complex as limit into a single idea that can appear instantaneously in complete and mature form. Some parts of the idea will get adequate representation before other parts will. It is probably inevitable that these parts will not be perfectly representative of the whole. Thus there will be stages in the student's development of mental representations where parts of the representation will be reasonably adequate and mature, but other parts will not be. (page 300, italics in the original) |

Davis and Vinner designed their instruction to prevent students from making the usual errors and misconceptions, even going so far as to avoid the use of the word "limit," but found the same "errors" cropping up despite their efforts. In the passage quoted above, they suggest that these misconceptions are actually partial conceptions, steps along the road to a full understanding of limit. The title of their paper characterizes these partial conceptions as "seemingly unavoidable," suggesting that they are inevitable, and perhaps even necessary, as students construct an understanding of limit.

This relates to Tall's idea about fundamental obstacles, inherent in the natural ordering of concepts. One has to build representations gradually, and certain steps in that construction may come in a necessary order, but that does not mean that the construction will proceed effortlessly. One cannot come to the study of limits without some preexisting ideas of the meanings of language and of some mathematical concepts. Indeed, it does not seem desirable that one should, even if it were possible. Yet, those preexisting ideas, useful though they may be, also contain obstacles. The needed stepping-stone is at one and the same time a stumbling block.

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**Lauten, Graham, & Ferrini-Mundy**

A. Darien Lauten, Karen Graham, & Joan Ferrini-Mundy conducted two one-hour clinical interviews with each of five calculus students, two taking an advanced placement calculus course in a high school and three taking first-semester calculus at a state university. They write:

The purposes of this study were: (a) to examine students' understandings and concept images of functions and limits; (b) to explore the ways in which their understandings of these conceptual areas may relate to one another; (c) to attend to the students' propensity to use of the graphics calculator; and (d) to identify, in a preliminary fashion, instances where there is evidence that the students' understanding and or processes for solving problems seem to have been influenced by the available technology. (page 228) |

This paper reports on interviews with Amy, one of the five subjects, but does not indicate whether she was a high school student or a college student. The writers mention that Amy had had access to the calculator for only two weeks before the interviews began, and later note "Amy's unprompted use of the calculator was minimal in the course of these interviews." (page 235) Since Lauten, Graham, and Ferrini-Mundy were interested in students' spontaneous use of the calculator, Amy's lack of unprompted calculator use limited their ability to accomplish parts (c) and (d) of their purpose.(Footnote 8)

The interviewer presented Amy with six statements about limits, (Footnote 9) and asked her to say whether she agreed or disagreed with each one and to explain why. Amy agreed with all six, but preferred the first one, "A limit describes how a function moves as x moves toward a certain point." When presented with example functions graphed on the calculator, Amy used the trace feature, which allows the user to move the cursor along the curve of the function and read off the x and y values of each point. Both of these responses are in keeping with the general observation that Amy appeared to have very dynamic concepts of function and limit. To Amy, a function is a particle moving along a path, and a limit means the particle is getting close to some fixed point, but not quite reaching it. Lauten, Graham, and Ferrini-Mundy put it this way:

In considering Amy's responses to this discussion, she seemed to use "it," "the x value," "the y value," "the curve," and "the function" interchangeably. In using each of these terms, she seemed to indicate that they were all on the graph of the function and moving along the curve. (page 233)
In the case of Amy, the most striking feature of her thinking relative to functions was her image of the |

Lauten, Graham, and Ferrini-Mundy note that Amy seemed to employ different concepts of function and limit in different contexts, depending on whether the function was represented as a list of points, a graph, or an algebraic formula. As was the case for ten of the students in Tall and Vinner's survey of 70 university students, Amy's concept image of limit appears to have little or no connection to the formal concept definition. It is not clear that she had a concept definition image at all. Lauten, Graham, and Ferrini-Mundy write:

In the limit interview, Amy seemed comfortable with her view that points moved along a curve and never quite reached the limit point. However, it did not bother her at all to plug in an x value to get a limit when that was possible in an equation. The connections between the algebraic manipulation, formal definition, and graphical interpretation seemed unclear to her. Her expression was "It's all vague anyway." In fact, the formal definition seemed to have no meaning for her. Her working definitions, although appearing inconsistent, seemed to satisfy her and serve her well. (page 234)There was some evidence that Amy handled equivalent problems quite differently depending on whether the context was a graphical or an analytical one. (page 236) |

Lauten, Graham, and Ferrini-Mundy present a detailed picture of Amy's conceptions, and provide several tasks and questions that proved useful in eliciting from Amy responses that revealed those conceptions. From this, we know a lot about Amy and something about revealing interview items, but next to nothing about high school or college students as a population. However, we do have a starting point for further study. As Lauten, Graham, and Ferrini-Mundy note:

Case studies of this type are especially useful in generating hypotheses and issues to be explored in subsequent research. . . . For example, given what we learned in the case of Amy, we have a number of questions about how widespread this tendency to "trace along the curve" might be in students who have had experience with graphing calculators. (page 235) |

Since other research has found that most students hold a dynamic view of limit, even when not using graphing calculators, and since Amy had used the calculator for only two weeks prior to the interviews, I suspect that Amy's very dynamic concept of limit had little to do with the curve-tracing capabilities of the calculator. However, there is no way to know this from what has been reported to date. As Lauten, Graham, and Ferrini-Mundy suggest, their work sets the stage for further research into that question. The questions and tasks that proved useful in the study of Amy's conceptions could be adapted to create survey items, which could then be used to study how widespread Amy's very dynamic concept of limit is, and to what extent it is related to the use of various technological aids.

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John Monaghan decided to study the ways that language about limits is interpreted by English high school students. To draw conclusions about a population, he needed a reasonably large sample, so he chose a quantitative methodology. To use this approach effectively, Monaghan had to have a very precise idea of what he wanted to study. His general questions were the same as in the other studies in this review: "How do students think about limits?" and "What factors affect the ways students think about limits?" However, to design his survey instrument, he needed a great deal more focus. First, he narrowed his questions to concentrate on the language used to describe limits, and the meanings that these words and phrases have for the students. Next, he selected four words or phrases commonly used with reference to limits, and acquired information (Footnote 10) about what types of misconceptions might be associated with the limit concept or with one or more of those phrases. Finally, he developed items that would allow him to detect some sign of these misconceptions from students' responses on a five-point, Likert-type scale. To this, he added a few items asking students to write sentences using each of the four phrases. After administering this survey once, he interviewed the subjects to try to get a clearer idea of what their ideas about limit were and how these ideas related to the students' responses to his survey items. Based on this, he revised his survey to prepare to use it with a larger sample.

Since Monaghan interviewed only the subjects in the first sample, and not those in the second one, it appears that a major purpose of the interviews was to improve the survey for later use. He also makes use of the insight from the interviews, together with the responses to the sentence-writing task, in explaining the data from the survey. Thus, the qualitative portion of this study was in a supporting role, undergirding the quantitative work, which was the main focus.

To use a multiple-choice format effectively, Monaghan had to plan his items carefully and focus them precisely on the desired concepts. For instance, to determine whether students associated some of the phrases with curves that get closer to some bound as x increases, but do not have that bound as a limit, Monaghan used the graph of *f*(*x*) = 1 + 1/*x* and asked whether it had a limit at 0 (or approached, or tended to, or converged to 0). In this example, 0 is a lower bound, and the decreasing function does get progressively closer to 0 as *x* increases, but it never gets closer than 1 unit from 0, so 0 is not a limit of the function. Using this example let Monaghan find the proportion of his subjects that associated this incorrect concept image with each of the phrases in his study. In particular, it allowed him to discover that a substantial proportion of his subjects associated this image with the word "approaching," and that far fewer associated it with any of the other phrases. From his sample he can generalize to the population and conclude that approximately a third to a half of 16-year-old English high school students who have passed O-levels in mathematics (Footnote 11) may have this incorrect concept image associated with the word "approaching" as it is used with reference to limits. This information could lead to the development of instructional approaches to clear up this particular misconception.

The multiple-choice format has the advantage of allowing Monaghan to study a large sample and thus get results that can be generalized to a population, but the disadvantage of restricting him to finding only those things for which his survey was designed to look. The subjects may very well have held other misconceptions about limits that were not revealed due to the lack of a suitable item. In contrast, qualitative studies, involving interviews with selected students, can reveal student conceptions that were previously unsuspected by the interviewer. However, the impracticality of interviewing large numbers of students can leave the researcher with little or no idea of how widespread those concepts may be.

The free-response question provides a middle ground. Compared to multiple choice questions, free-response questions have the qualitative-study advantage of allowing the student to provide responses which the researcher did not anticipate, but also the qualitative-study disadvantage of increasing the time and effort required to process the results. However, free-response questions are not as labor-intensive to administer and interpret as are interviews, so they retain some of the quantitative-study advantage of permitting one to efficiently process data from a large sample. Monaghan used four free-response items in the first administration of his survey, with 54 subjects. He used this information in interpreting the results of the multiple-choice questions as well as in improving the instrument, so it appears to have been worth the effort. However, in the second administration of his survey, with 190 subjects, he used only one free-response question. He doesn't say why he cut the other three, but the thought of trying to read and make sense of 760 responses to open-ended questions gives one some idea of why he would choose not to do this.

Comparing Monaghan's study to the two quantitative studies by Tall and Vinner reveals a difference in where the effort of the study is concentrated. A multiple-choice format leaves no room for new information once the instrument is complete. All that is left is to find the proportion of students selecting each response. Thus, Monaghan's questions had to be as finely tuned as he could make them. Once this work was done, scoring the responses may have taken only minutes. Tall and Vinner, by contrast, did not have to focus their questions quite so precisely on expected student concepts. By asking free-response questions, they left open the possibility of getting responses that they had not anticipated. This may reduce the effort involved in preparing the instrument, but increases the effort required to process the results. Tall and Vinner could not simply count the number of students selecting each option. They had to read the students' words, identify categories, group the responses, refine the categories, regroup the responses, and so on. This may be part of the reason why Tall and Vinner worked with smaller samples than Monaghan used.

Like Tall and Vinner, Davis and Vinner used free-response questions, asking students to give an informal explanation and a formal definition of limit. Designing these questions appears to have been very straightforward--you want to know what students think about limit, so you ask them what a limit is. It didn't take the same meticulous planning that is required for a good multiple-choice instrument, because Davis and Vinner didn't have to know what concepts the students might have, or what responses they might give. However, substantial effort was required to interpret the results.

The studies by Tall and Vinner, Davis and Vinner, and Monaghan provide snapshots of students conceptions at the moments when the data were collected. They do not show learning, wrestling with cognitive conflict, or any kind of change over time. One might see some change over time with repeated snapshots, but that still is not a moving picture. Williams wanted to know how students' concepts might be changed by consideration of anomalous examples, so he needed to work with the same subjects over an extended period of time. To get a detailed moving picture, he decided to conduct teaching interviews with each of this subjects individually for a total of three and a half hours per subject, in five sessions spread over seven weeks. This allowed him to customize both the instruction and the assessment according to the concepts the student expressed, and to get much richer information about the students' concepts and their responses to disconfirming examples. However, it limited him to many fewer subjects than Tall and Vinner, Davis and Vinner, or Monaghan used.

Despite this limitation, Williams was able to get some idea of how common each of these concepts is in the population of interest, because he started with a quantitative, multiple-choice survey of 341 students. This survey was designed to support the main study, by telling him how common each conception was and by guiding his selection of interview subjects. Williams did not select subjects at random, as would be appropriate for a quantitative study, because he did not need for them to be representative in the sense that subjects in a quantitative study are expected to be. Instead, he deliberately selected students who, in their responses to his survey, were able to clearly and unambiguously articulate one of the four most common views of limit. Rather than being truly representative, the subjects were chosen more as archetypes of each of the four common conceptions.

Williams's study runs in a spiral of sorts; he started with some idea of what the common conceptions are, designed a survey to reveal them, used the survey to find the frequency of each of these conceptions in the population of interest, used this distribution to determine how many subjects with each conception to interview, selected and interviewed ten subjects exhibiting the desired conceptions, and learned from those interviews more detail about what the common conceptions are and how firmly they are held. From this point, one could imagine a new and improved survey, based on what was learned in the interviews. The qualitative and quantitative aspects of this study complement one another and serve as preparation for one another, as was the case in Monaghan's study. In contrast to Monaghan's study, Williams's work places the qualitative work at center stage, with the quantitative work in a supporting role.

Lauten, Graham, & Ferrini-Mundy have created a detailed picture of the concept of limit held by a student they call Amy. Amy's ideas appear to be consistent with those found in many of the larger studies. Like Tall and Vinner's subjects, Amy seemed to call on different portions of her concept image in response to different forms of questions. Like the majority of subjects in the studies by Williams and by Tall and Vinner, Amy held a dynamic view of limit, which was independent of the formal definition of limit, and appeared to have a strong, although somewhat erroneous, concept image of limit, with little or no concept definition image. Williams notes that his subjects appeared to have little interest in the formal definition. They considered it to be irrelevant to the practical work of answering questions on homework, quizzes, and exams. Lauten, Graham, & Ferrini-Mundy found the same to be true of Amy; she had no use for the formal definition.

Amy's concept image contained what Tall and Vinner called potential cognitive conflict, in that she sometimes said functions can't reach their limits and other times used the function's formula to calculate a limit by substituting in the corresponding value of the independent variable. The same potential cognitive conflict existed in many of Davis and Vinner's students, who said that functions cannot reach their limits, despite having had worked competently the previous semester with functions that reached their limits. In both cases, the students fail to recognize the disconfirming evidence and cling to the image of a function getting close to a limit without ever reaching it. From Lauten, Graham, & Ferrini-Mundy's report, it appears that this potential cognitive conflict never produced actual cognitive conflict for Amy during this study.

By interviewing the same student for a total of two hours, Lauten, Graham, & Ferrini-Mundy were able to get much more detail about that student's conceptions than could be obtained from a survey. In the end, they knew a great deal about Amy. However, we have to turn to the other studies, with larger samples, to discover whether Amy's ideas are similar to, or different from, those of her peers. As noted above, other studies have shown some of Amy's ideas to be common among high school and college students. Other of Amy's ideas have not yet been the subject of larger studies, but could be. Lauten, Graham, & Ferrini-Mundy suggest as much when they write: "Case studies of this type are especially useful in generating hypotheses and issues to be explored in subsequent research." (page 235)

It appears that they see this study of Amy as possibly laying the groundwork for another study, with a larger number of subjects, to determine how common this tendency is. While they do not suggest any particular method, studies using a number of subjects large enough to determine frequencies in a population tend to be more quantitative. As has been noted above, quantitative studies, particularly those using multiple-choice items, require the researcher to know with some precision what is likely to be found (although not in what proportions). With their study of Amy, Lauten, Graham, & Ferrini-Mundy have provided some guidance for the construction of an appropriate instrument.

In summary, the quantitative and qualitative methods used in these studies complement one another and serve to prepare for one another. Monaghan used interviews with his first sample to prepare the survey instrument for his second sample. Williams used a survey to prepare for interviews. The interviews with Amy give rich detail about one student's conceptions, and leave one wondering whether these conceptions are common. A survey could help answer that question. In the study of students' conceptions of limits, the relationship between quantitative and qualitative methods is one of cooperation and complementarity.

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Davis, Robert B., & Vinner, Shlomo (1986). The notion of limit: Some seemingly unavoidable misconception stages. *Journal of Mathematical Behavior, 5*, 281-303.

Lauten, A. Darien, Graham, Karen, & Ferrini-Mundy, Joan (1994). Student understanding of basic calculus concepts: Interaction with the graphics calculator. *Journal of Mathematical Behavior, 13*, 225-237.

Monaghan, John (1991). Problems with the language of limits. *For the Learning of Mathematics, 11*, 3, 20-24.

Murphy, Lisa D. (1999). *Students Conceptions of Rate of Change*. (In progress.)

Sierpivska, Anna (1987). Humanities students and epistemological obstacles related to limits. *Educational Studies in Mathematics, 18*, 371-397.

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.

Tall, David, & Vinner, Shlomo (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. *Educational Studies in Mathematics, 12*, 151-169.

Williams, Steven R. (1991). Models of limit held by college calculus students. *Journal for Research in Mathematics Education, 22*, 219-236.

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Footnote 1

This theory comes out of a course I took two years ago, C&I 407, "Research Issues in Mathematics Education: Historical and Contemporary Issues in Equity," with Prof. Rochelle Gutierrez. I have emailed her for references, but with no success. She does remember presenting the idea to our class, but doesn't remember the specific papers from which she got the information. She gave me the names of a couple of researchers whom she remembers as working in that area, but an ERIC search on their names turned up nothing relevant. If either of us comes up with anything, I'll revise this paper accordingly.

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Footnote 2

I am reluctant to use the word traditional. Its operational definition often appears to be "whatever the particular teacher or school was doing before the researcher intervened." However, in this instance, that definition will suffice.

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Footnote 3

I did find one study, by Anna Sierpivska (1987), describing interactions within a group of six students who discussed problems related to limits and to infinity. Even though she writes that there are two dimensions of the didactical situation, cognitive and social, it still appears that the cognitive processes of the individuals, rather than the social processes of the group, are her primary objects of study. After describing the general situation, she devotes several pages to the stories of two of the students, treating each one separately and talking about that student's personal transformations in thinking about the problem at hand. She applies various terminology and coding to attitudes and beliefs expressed by individual students, but does not talk about the group as a whole. I started out to include her study in this review, but it was long and cumbersome (my summary ran to four pages), and did not appear to add much insight to this discussion, so I cut it. I'm trying to avoid making this paper twice as long as assigned.

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Footnote 4

Or reported in forms inaccessible to me, such as presentations at conferences whose proceedings are not on ERIC microfiche.

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Footnote 5

Formal would be along the lines of "For every *e* > 0 there is some *d* > 0 so that if |*x* - *a*| < *d* and *x* is not *a* then |*f*(*x*) - *c*| < *e*." Dynamic would be something like "As *x* gets closer to *a*, *f*(*x*) gets closer to *c*."

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Footnote 6

Percentages of students agreeing with a particular phrasing were given for some but not all items. I would have preferred a table giving all results

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Footnote 7

Monaghan considers the example of the footballers to represent a different category of thought from the roads and light rays, but I'm not sure it does. If one thinks of the roughly linear paths traveled by the footballers as they run toward the ball, rather than thinking of the footballers as discrete objects, then this example is very similar to the converging roads and light rays.

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Footnote 8

This looks to me like a major flaw in the study. If they want to know how the available technology influences Amy's thinking and problem-solving processes, it seems to me that they would be better off to wait until she had gotten used to the calculator, learned what it would do, and formed some habits involving calculator use. To put her in the unfamiliar setting of the interview and expect her to spontaneously use an unfamiliar tool strikes me as unrealistic.

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Footnote 9

These six statements came from Williams (1991), which is discussed elsewhere in this review.

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Footnote 10

Probably starting with a literature review, although he doesn't say.

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Footnote 11

He probably also could break it down according to whether they studied A-levels in mathematics or not, but this information was not in his paper.

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