Beyond the Limit, III

In my last two columns, Beyond the Limit, I and Beyond the Limit, II, I looked at common student difficulties with the concept of limit and explained Michael Oehrtman’s investigations into the metaphors that students use when they try to apply the concept of limit to problems of first-year calculus. The point of this exploration is to identify the most productive and useful ways of thinking about limits so that we can channel calculus instruction toward these understandings. In this month’s column, I will describe Oehrtman’s suggestions for how to accomplish this.

In the MAA Notes volume Making the Connection (Carlson and Rasmussen 2008), Oehrtman focuses on the last of the strong metaphors described in his 2009 paper, that of limit as approximation. The point of building instruction around this approach is that it arises spontaneously from the students themselves, providing what Tall refers to as a cognitive root:

Rather than deal initially with formal definitions that contain elements unfamiliar to the learner, it is preferable to attempt to find an approach that builds on concepts that have the dual role of being familiar to the students and providing the basis for later mathematical development. Such a concept I call a cognitive root. (Tall 1992, p. 497)

While limit as approximation was not always the most commonly employed way of understanding limits—that honor frequently went to limit as collapse—it has several major advantages. First, because it does arise spontaneously from many of the students, we know that it is easily accessible to many of them. Second, it is the metaphor that comes closest to the mathematically correct definition of limit. This is important. Because it comes so close to the formal understanding of limits, it provides a means for students to reason consistently throughout the course, providing coherence and making it easier to transfer this understanding to novel situations. Finally, explorations of how to approximate quantities such as instantaneous velocity or the force on a dam provide direct connections between the concepts of calculus and the modeling situations students will encounter in engineering or the sciences.

In student responses to the eleven problems that Oehrtman posed to the 120 students in his study (reproduced at the end of this column), Oehrtman found that the approximation metaphor was employed by 11% in answering questions #1 and #2, 26% for #6, 35% for #4, 70% for #3, and 74% for #8. This last asked to explain in what sense \( \displaystyle \sin x = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \frac{1}{7!}x^7 + \cdots . \) Oehrtman quotes at length one of these explanations that I wish to reproduce here because it amply demonstrates how, without any explicit instruction in ideas of approximation, it was the metaphor instinctively seized by a student trying to verbalize what she knew about Taylor series.

When calculating a Taylor polynomial, the accuracy of the approximation becomes greater with each successive term. This can be illustrated by graphing a function such as sin(x) and its various polynomial approximations. If one such polynomial with a finite number of terms is centered around some origin, the difference in y-values between the points along the polynomial and the points along the original curve (sin x) will be greater the further the x-values are from the origin. If more terms are added to the polynomial, it will hug the curves of the sin function more closely, and this error will decrease. As one continues to add more and more terms, the polynomial becomes a very good approximation of the curve. Locally, at the origin, it will be very difficult to tell the difference between sin(x) and its polynomial approximation. If you were to travel out away from the origin however, you would find that the polynomial becomes more and more loosely fitted around the curve, until at some point it goes off in its own direction and you would have to deal once again with a substantial error the further you went in that direction. Adding more terms to the polynomial in his case increases the distance that you have to travel before it veers away from the approximated function, and decreases the error at any one x-value. Eventually, if an infinite number of terms could be calculated, the error would decrease to zero, the distance you would have to travel to see the polynomial veer away would become infinite, and the two functions would become equal. This is a very important and useful characteristic, as it allows for the approximation of complicated functions. By using polynomials with an appropriate number of terms, one can find approximations with reasonable accuracy. (Oehrtman 2008, pp. 72–3)

I am certain that this explanation echoes much of what this student heard and saw in the classroom. I read in it much of what I say when I explain Taylor series. Yet this observation is useful because it demonstrates which images and explanations have resonated with this student.

For Oehrtman, this explanation is classified as approximation not just because that word appears frequently in the student’s explanation, but because the student combines it with a sense of the numerical size of the error. This is very different from the metaphor of limit as proximity. For this student, what is important is not just that the graphs of the polynomials are spatially close but that she can control the size of the error.

Given this observation, Oehrtman has built a series of activities designed to encourage and strengthen student reliance on the metaphor of approximation, several of which are described in his 2008 article, many more of which are coming available on his new website, CLEAR Calculus (contact Mike for access to the posted materials by writing to michael.oehrtman@gmail.com). Thus, instead of introducing the slope of the tangent as the limit of the slopes of secant lines, he chooses a particular point on a particular curve, in this case x = –1 on y = 2^x, and introduces the secants as lines whose slopes approximate the slope of the tangent line. He gets students to identify those secant lines that provide an upper bound on the slope of the tangent and those that provide a lower bound and then has them explore secant lines that tighten these bounds until they can approximate the slope of the tangent to within an error of 0.0001. The word limit need never arise.

In a similar vein, integrals are approached via Riemann sums, but not as the limit of these sums. Rather, these sums provide approximations to the desired quantity. One can identify those approximations that overshoot and those that undershoot the true value and adjust the partitioning of the interval to make the error as small as one wishes.

Each of Oehrtman’s activities is built around five questions:

1. What are you approximating?
2. What are the approximations?
3. What are the errors?
4. What are the bounds on the size of the errors?
5. How can the error be made smaller than any predetermined bound?

As Oehrtman explains, the last two are intentionally reciprocal: Given a choice of approximation, what are the bounds on the error? Given a bound on the error, what approximation will achieve it?

While this approach provides a route through calculus that does not require the use of the word “limit,” Oehrtman does not avoid it. For those students who will pursue mathematics, it is a term that will come up in other contexts. For the serious student of mathematics, it is absolutely essential. What Oehrtman does recognize is that performing a series of exercises in which one finds limits or explains why they do not exist has little or no bearing on the development of a robust personal understanding of derivatives and integrals.

Not surprisingly, students whose understanding of limits is deeply rooted in the concept of approximations, including the reciprocal processes of determining the bound from the approximation and finding an approximation that will satisfy a particular bound, find it much easier to grasp the formal epsilon-delta definition of limit. In fact, Oehrtman, Swinyard, and Martin (2014) have documented the relative ease with which students schooled in this approach are able to rediscover the mathematically correct definition of limit for themselves.

This is not a new insight. In Emil Artin’s A Freshman Honors Course in Calculus and Analytic Geometry Taught at Princeton University (published by MAA in 1958), he talks about approximations to the slope of the tangent line before introducing limit as “the number approached by the approximations to the slope” (page 23). The Five Colleges Calculus Project, Calculus in Context, also begins with approximations, as do Calculus with Applications by Peter Lax and Maria Terrell and The Sensible Calculus Program by Martin Flashman.

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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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Artin, E. (1958). A Freshman Honors Course in Calculus and Analytic Geometry Taught at Princeton University. Buffalo, NY: Committee on the Undergraduate Program of the Mathematical Association of America

Callahan, J., Hoffman, K., Cox, D., O’Shea, D., Pollatsek, H., Senechal, L. (2008). Calculus in Context: The Five College Calculus Project. Accessed August 11, 2014. www.math.smith.edu/Local/cicintro/book.pdf

Carlson, M.P. & Rasmussen, C. (Eds.). (2008). Making the Connection: Research and Teaching in Undergraduate Mathematics Education. MAA Notes #73. Washington, DC: Mathematical Association of America.

Flashman, M. (2014). The Sensible Calculus Program. Accessed August 11, 2014. users.humboldt.edu/flashman/senscalc.Core.html

Lax, P. & Terrell, M.S. (2014). Calculus with Applications, Second Edition. New York, NY: Springer.

Oehrtman, M. (2008). Layers of abstraction: theory and design for the instruction of limit concepts. Pages 65–80 in Carlson & Rasmussen (Eds.), Making the Connection: Research and Teaching in Undergraduate Mathematics Education. MAA Notes #73. Washington, DC: Mathematical Association of America.

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., Swinyard, C., Martin, J. (2014). Problems and solutions in students’ reinvention of a definition for sequence convergence. The Journal of Mathematical Behavior, 33, 131–148.

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.