Beyond the Limit, II

Last month, in "Beyond the Limit, I," I discussed some of the difficulties and misconceptions surrounding student understanding of limits in first-year calculus. I also raised the question of how serious these misconceptions actually are and introduced the work of Michael Oehrtman (2009, also see the preprint). This month I wish to explain what Oehrtman found as he analyzed student responses to the eleven problems that he posed and that are reproduced at the end of this column.

Rather than focusing on student mistakes and misconceptions, Oehrtman was interested in how students can use what they know or think they know about limits to reason through and explain some of the central ideas of calculus. He chose to focus on student metaphors, nontechnical ways of thinking and talking about limits that reflect individual experiences and that enable students to relate unfamiliar mathematical ideas to familiar concepts. Mathematics and science are full of useful metaphors. Oehrtman cites Max Black’s 1962 Models and Metaphors as the first serious study of how scientists use metaphors. Black paid particular attention to James Clerk Maxwell’s use of the "heuristic fiction" of thinking of electrical fields as incompressible fluids. In 2000 Lakoff and Núñez explained one route to an understanding of limit via a metaphorical map that takes the experience of an iterative process that terminates and maps it onto an iterative process that does not terminate, using the final stage of the terminating iteration as a metaphor for the limit of the infinite iterative process.

Oehrtman investigated the metaphors or nontechnical experiences on which students rely as they grapple with the notion of limit. His study involved 120 students who he followed throughout a yearlong single variable calculus sequence, observing the class throughout the year and gathering information from the students via pre- and post-course surveys, quizzes, and writing assignments, followed up with clinical interviews with 20 of the students. As he suspected and as his study confirmed, students employed a wide variety of metaphors for limit, many of which were reflected in significant idiosyncrasies in their understanding of limits.

He found eight distinct metaphors or ways of relating limits to previous experience, three of which he classified as weak because they were not used very often. Five were classified as strong because they were commonly employed. He then looked at how helpful these metaphors were in reasoning through to an explanation. An important point in mathematics is that, as with Maxwell, metaphors can be useful and even insightful. Thus, thinking of the derivative as a ratio of very small changes in the dependent and independent variables is a metaphor that is certainly not correct but that is very powerful in understanding many of the results of differential calculus.

It is also important to keep in mind that the metaphors for limits are fluid and often context specific. Oehrtman found that students used different metaphors in different situations and that if one metaphor was failing, many students would begin to transition to another that seemed more promising. Nevertheless, he did find that, across the board, some metaphors were more useful than others. In describing the metaphors that Oehrtman identified, I am going to expand beyond his work and incorporate into his framework some of the insights into student thinking that have been identified by other researchers.

The Weak Metaphors

Limit as Motion. One surprising finding was that the metaphor of limit as motion was very weak. This was initially unexpected because students consistently spoke of "approaching a limit," and the previous research literature had emphasized a dynamic interpretation of limit as the most common student misconception. Nevertheless, as he explored student use of the idea of "approaching," Oehrtman discovered that students were thinking of a sequence of discrete steps toward the limiting value rather than a continuous motion. The only question for which continuous motion did play a role was #10, explaining what is meant by continuity of a function of two variables. Here several students did speak of moving over the surface defined by this function without falling into holes or encountering cliffs, explaining continuity in terms of the physical topography of the surface.

Limit as Zooming. In the early days of graphing calculators, one frequently touted advantage was the ability to zoom in on the graph of a differentiable function to reveal its local linearity. This approach was emphasized by the instructor of the course whose students Oehrtman observed, but the students did not invoke this image on their own and, in fact, frequently misinterpreted its significance.

Limit as Informal Version of Correct Definition. While many students used the phrases "arbitrarily precise" and "sufficiently close" that they had heard in class, when pressed they defined these as meaning "very" or "very, very" or "very, very, very" close. None of the students used these phrases in the sense implied by the informal version of the correct definition: The limit is that value L to which the function is forced to be arbitrarily close by taking x sufficiently close but not equal to c.

The Strong Metaphors, in rough order of increasing productivity

Limit as Physical Limitation. The research literature is replete with evidence that a small but significant number of students never move beyond the colloquial definition of limit as a boundary that cannot be surpassed. While this view of limit is commonly held, none of the students in the study employed it directly in trying to answer the questions, yet it did surface in an interesting way. What Oehrtman found was a number of students who attempted to explain #6 and #9 by invoking a physical limit on how small a quantity could be. Thus, Torricelli’s trumpet has a finite volume because eventually the tail is too small to accommodate any matter, and the limit of the jagged line may look like a straight line, but it is really not. There is a smallest positive distance, perhaps the width of an atom, below which it cannot move. Thus, while the limit may look like a straight line, it is really microscopically jagged and so still has length \( \sqrt{2} \). This was, by far, the most counterproductive of all student metaphors for limit.

Limit with Infinity as Number. A number of students spoke of the last term of an infinite sequence or an infinitesimal as the smallest distance between a converging sequence and its limit. Sierpińska (1987) and Cornu (1991) refer to these students as "unconscious infinitists" who say "infinite" but think "very big." Infinitesimals can be useful metaphors as witnessed in the work of Leibniz, the Bernoullis, and Euler. Even Cauchy, while introducing the formal language of epsilons and deltas, explained continuity in the language of infinitesimals:

The function f(x) is continuous within given limits if between these limits an infinitely small increment i in the variable x produces always an infinitely small increment, f(x+i) – f(x), in the function itself. (as translated in Boyer, 1949, p. 277)

This metaphor was particularly strong in the students’ attempts to explain #5 through #8: L’Hôpital’s rule, Torricelli’s trumpet, the limit comparison test, and Taylor series. There are significant drawbacks. An obvious problem is the creation of a clear distinction between the number represented by \( 0.\overline{9} \) and 1. Many of these students, asked to explain why they are equal, argued instead that they are not. A less obvious but more insidious difficulty is that understanding infinity as very large number encourages belief in the generic limit property as described by Tall (1992), the assumption that any property held by all terms of the sequence must also be held by the limit. Question #9 is extremely problematic for students who attempt to employ this metaphor.

Limit as Proximity. The next two metaphors elaborate on a dynamic interpretation of limit in the sense actually employed by students, a description of closeness to the limit value. In exploring how students use this metaphor to explain the limit of a function or the meaning of continuity, we see a phenomenon—recorded by a number of other researchers—of the tendency to focus on the spatial proximity, one manifestation of which is to identify the limit as a point in the Cartesian plane located on or touched by the graph. In explaining #8, the Taylor series, students employing this metaphor talked of the graphs of the Taylor polynomials and emphasized their closeness to the graph of the sine. As Oehrtman explains, they essentially reinvented the L1 norm as a measure of closeness.

An additional problem with this metaphor was evident in explanations of the derivative as a limit. These students focused on the distance between the secant lines and the position of the tangent line, sometimes measuring the distance between two lines as the difference of their slopes, but sometimes referring to their physical separation. Students who relied on this metaphor had difficulty making the transition from spatial proximity to quantitative difference.

Limit as Collapse. This was the most interesting metaphor that Oehrtman encountered. While incorrect, it could be productive and insightful. It was a common student response to the inherent contradiction of an infinite sequence as an unending process of coming ever closer to the limit against the assertion that in some sense this limit value is equated with the sequence. The image is of a sequence that comes closer and closer until at some point it "collapses" onto the limit value.

Oehrtman’s choice of the term "collapse" arises from student use of this metaphor for question #11, why the derivative of the formula for volume yields the formula for surface area. Students spoke of small changes in the volume represented by thin outer shells that became thinner and thinner until they became the surface, collapsing down from three to two dimensions.

This phenomena had been observed earlier by Thompson (1994) as he explored student understanding of the fundamental theorem of calculus. In explaining the antiderivative part (finding the derivative of a function defined in terms of a definite integral), students may begin with the limit definition,

\[ \lim_{\Delta x \to 0} \frac{ \int_a^{x+\Delta x} f(t)\,dt-\int_a^x f(t)\,dt}{\Delta x}, \]

but they then ignore the denominator and view the difference as a thin rectangle of width \(\Delta x\) and height f(x) that collapses down to the one-dimensional height in the limit. This is very close to the way in which Newton and Leibniz first explained this result. In the other direction, they see an area as built up from one-dimensional lines, a metaphor that Finney, Demana, Waits, Kennedy, and I use in preparing students for the fundamental theorem of calculus (Exploration 2 on page 293 of Finney et al. 2012).

The collapse metaphor was also important for understanding of the derivative as a limit. Some selected passages from the transcript of a student working through the application of this metaphor to answer #2 are instructive.

You take your values and you squish them really small until … you can go no more, and magically that’s the limit. … As this gets smaller and this gets smaller [points at the vertical and horizontal changes], … you’re getting really, really close to the rise over the run of THIS [points at (3, f(3))]. And when you reach your limit, that’s what the rise over the run of this is [points at (3,f(3))], so I guess that’s the tangent, which is the derivative. Yeah. That does make sense. Because that’s what happens on a limit. (Oehrtman 2009, p. 411)

It is worth mentioning that this student made four cycles of attempts to answer question #2 before grasping at the collapse metaphor and finding in it an explanation that she found satisfying. Once she discovered this metaphor, Oehrtman reports that she applied it repeatedly and consistently across multiple representations and contexts. Furthermore, this was the most popular metaphor used by students in answering questions #2 and #4, the two that deal with the definition of the derivative.

We see in the collapse metaphor an attempt by students to connect their understanding of limit as part of an unending iterative process with the recognition that this process must be equated with a single value. One of the potential idiosyncrasies of this metaphor is the phenomenon, mentioned in last month’s column, of accepting both \(0.\overline{9} \) and 1 as limits of the sequence \( (0.9, 0.99, 0.999, \ldots) \) while still arguing that they are distinct.

Limit as Approximation. The last of the strong metaphors was both the most productive and the one that comes closest to the mathematically correct definition of the limit. This metaphor differs from mere proximity in its focus on numerical difference, rather than geometric distance, and the recognition of the need for a bound, either explicitly or implicitly described, on the difference between the limit and the terms that are approaching it. It is usually accompanied by recognition that this error can be made as small as one wishes. One student wrote,

In fact the power series for sin x will approximate a value infinitely close to the value of sin x and even a remainder can be calculated … The power series of sin x continues forever depending on how close you want your value to come to the value of sin x … The remainder is designed to show how much a power series deviates from the value of a function at a particular point … the power series or polynomial for sin x is an approximation of its value that can be as close of value as you want it to be. (Oehrtman 2009, p. 415)

Strictly speaking, approximation is not a metaphor for limit. It is an essential component of what a mathematician means by a limit. What is interesting is that although classroom instruction on limits had not focused on the notion of approximation as a way of understanding limits, many students instinctively drew on it as something from their experience that helped them to understand this concept.

Most students used the language of approximation to answer questions #3 and #8, and it was also popular, though not as common as the collapse metaphor, in explaining the meaning of the derivative, #2 and #4.

Conclusion. The point of this exploration of student metaphors for limit has not been to illustrate student errors and misconceptions, but rather to illuminate legitimate student attempts to build an understanding of the limit concepts that undergird calculus and to help instructors recognize the source of many of the idiosyncrasies they might encounter in student responses. The question before us as teachers is how to channel these attempts so that our students can build robust and productive ways of thinking about the fundamental ideas of calculus. In my next and final column on "Beyond the Limit," I will look at some of the approaches to teaching that have arisen from Oehrtman’s work.

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The following are abbreviated statements of the problems posed by Michael Oehrtman (2009) to 120 students in first-year calculus via pre-course and post-course surveys, quizzes, and other writing assignments as well as two hour-long clinical interviews with twenty of the students.

1. Explain the meaning of \( \displaystyle \lim_{x\to 1} \frac{x^3-1}{x-1} = 3. \) 
2. Let \( f(x) = x^2 + 1.\) Explain the meaning of \( \displaystyle \lim_{h\to 0} \frac{f(3+h)-f(3)}{h}.\) 
3. Explain why \( 0.\overline{9} = 1.\) 
4. Explain why the derivative \( \displaystyle f’(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}\) gives the instantaneous rate of change of f at x. 
5. Explain why L’Hôpital’s rule works. 
6. Explain how the solid obtained by revolving the graph of y = 1/x around the x-axis can have finite volume but infinite surface area. 
7. Explain why the limit comparison test works. 
8. Explain in what sense \( \displaystyle \sin x = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \frac{1}{7!}x^7 + \cdots . \) 
9. Explain how the length of each jagged line shown below can be \( \sqrt{2} \) while the limit has length 1. 
   
10. Explain what it means for a function of two variables to be continuous.
11. Explain why the derivative of the formula for the volume of a sphere, \( V = (4/3)\pi r^3 \), is the surface area of the sphere, \( dV/dr = 4\pi r^2 = A. \) 

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Black, M. (1962). Models and Metaphors: Studies in Language and Philosophy. Cornell, NY: Cornell University Press.

Boyer, C.B. (1949). The History of the Calculus and Its Conceptual Development. Reprinted 1959. New York, NY: Dover Publications.

Cornu, B. (1991). Limits. In D. Tall (Ed.) Advanced Mathematical Thinking. (pp. 153–166). Dordrecht, The Netherlands: KluwerAcademic.

Finney, R.L., Demana, F.D., Waits, B.K., Kennedy, D. (2012). Calculus: Graphical, Numerical, Algebraic, 4th ed. Boston, MA: Pearson.

Lakoff, G. & Núñez, R. (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic Books.

Oehrtman M. (2009). Collapsing Dimensions, Physical Limitation, and Other Student Metaphors for Limit Concepts. Journal For Research In Mathematics Education, 40(4), 396–426.

Oehrtman, M. (Preprint). Students’ Metaphors for Limit Concepts in Introductory Calculus. To appear in Lessons Learned from Research: Volume 2 Useful Research on Teaching Important Mathematics to All Students. NCTM

Sierpińska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18(4), 371-397.

Tall, D. (1992). The transition to advanced mathematical thinking: functions, limits, infinity, and proof. Chapter 20 in Grouws D.A. (ed.) Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 495-511.

Thompson, P.W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics. 26(2), 229–274.