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:

- What are you approximating?
- What are the approximations?
- What are the errors?
- What are the bounds on the size of the errors?
- 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.

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.

- Explain the meaning of \( \displaystyle \lim_{x\to 1} \frac{x^3-1}{x-1} = 3. \)
- Let \( f(x) = x^2 + 1.\) Explain the meaning of \( \displaystyle \lim_{h\to 0} \frac{f(3+h)-f(3)}{h}.\)
- Explain why \( 0.\overline{9} = 1.\)
- 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*.
- Explain why L’Hôpital’s rule works.
- 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.
- Explain why the limit comparison test works.
- Explain in what sense \( \displaystyle \sin x = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \frac{1}{7!}x^7 + \cdots . \)
- Explain how the length of each jagged line shown below can be \( \sqrt{2} \) while the limit has length 1.
- Explain what it means for a function of two variables to be continuous.
- 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, \( dV/dr = 4\pi r^2 = A. \)

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.

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.