Wednesday, October 1, 2014

The Pitfalls of Precalculus

Many of the students who aspire to study science or engineering never manage to get through calculus. For many if not most of them, the stumbling block is not calculus as such; it is an inadequate grasp of the mathematics of precalculus. This is why most colleges and universities administer a placement exam and offer a course called Precalculus. This is a high school course, but most colleges and universities offer it for students who want to take calculus but are believed to lack the mathematical foundation needed to succeed. Does precalculus in college actually work? Do the college students who take precalculus go on to succeed in calculus? Does a college course in precalculus even help student performance in calculus? Recent research by Sonnert and Sadler suggests that the benefits of college precalculus are marginal at best. At worst, it can be damaging.

Precalculus, as the name suggests, should be a course that students take to prepare for calculus. In fact, the University of Illinois at Urbana-Champaign has changed the name of this course to Preparation for Calculus, emphasizing the fact that this is not and should not be taken as a course to satisfy a mathematics requirement, but only as preparation for success in calculus. Unfortunately, precalculus often proves to be terminal, even for students who do well. A variety of local studies have shown that large numbers of students who are successful in precalculus choose not to continue on to calculus. A Texas Tech study [Jarrett 2000] found that a third of their students who earned a B or higher in their precalculus course failed to enroll in Calculus I. There are similar data from Arizona State University [Thompson et al 2007]: Among declared Engineering majors who earned a C or higher in Precalculus, 38% failed to enroll in Calculus I. It was worse for other majors: 55% of Physical Science majors, 56% of Mathematical Science majors, and 65% of Life Science majors who earned a C or higher in Precalculus failed to enroll in Calculus I. Herriott and Dunbar [2009] reported comparable data from the University of Nebraska-Lincoln and a collection of colleges in Illinois.
What about the students who do go on to Calculus I? Has a semester of college-level precalculus helped them? Gerhard Sonnert and Philip Sadler have just reported on the first large-scale study to address this question [Sonnert & Sadler 2014]. What they found is that for students whose high school mathematical preparation lies between the mean and one standard deviation below the mean (roughly the 15th to 50th percentile) of all Calculus I students, taking precalculus in college produced a small and not statistically significant improvement in their expected grade: around 1 point on a 100-point scale. More than one standard deviation below the mean, their study showed worse grades in Calculus I if they also took precalculus, but here the numbers are so small that the results are problematic. For students whose high school preparation was 0.3 standard deviations or more above the mean, taking precalculus in college was associated with a reduction in their Calculus I grades by 6 or more points, a result that was statistically significant at p < 0.05 for students at 0.3 standard deviations and numerous higher cut-offs.
The Sonnert-Sadler analysis relies on data collected in their FICS-Math study (Factors Influencing College Success in Mathematics, NSF #0813702) that collected both extensive background information and final grades of approximately 10,500 students enrolled in Calculus I in fall 2009. From their own analysis and that of others, once you control for variables such as gender and socio-economic status, high school preparation is the best predictor of success in college calculus. Sonnert and Sadler used a hierarchical logistic regression to identify six high school variables that were highly correlated with success and from which they could build a score to describe readiness for calculus. The variables they identified are
  1. SAT/ACT math scores
  2. Took precalculus in high school
  3. Grade in high school precalculus
  4. Took non-AP calculus in high school
  5. Took AP Calculus in high school
  6. Grade on AP Calculus exam
Because of the large sample with which they were working, they were able to find substantial numbers of students with comparable readiness scores, many of whom had taken precalculus in college, many of whom had not. At each level of preparation, from 1.8 standard deviations below the mean to 1.0 standard deviations above, in increments of 0.1 standard deviations, they were able to simulate a discontinuity regression, comparable to the one described in my column from January 2012, First, Do No Harm. This demonstrated the results reported above.

The exceedingly modest gains for students below the mean are not very surprising. Across all subjects, effective remediation is tough to pull off. There are some precalculus programs that seem to be helping. Identification and investigation of these will be part of MAA’s next study of the precalculus/calculus sequence, Progress through Calculus (NSF #1420389). Some of the more promising directions include courses that weave precalculus review into the introductory calculus course. The University of Illinois has also had success with its use of ALEKS in their Preparation for Calculus. Unfortunately, the vast majority of college precalculus is still taught as recapitulation of the course students took and failed to master in high school, just coming at them a lot faster.

The harm that appears to be done by putting good students into precalculus is a more intriguing result, perhaps a reflection of the damage done to their self-confidence.


Herriott, S.R. & Dunbar, S.R. 2009. Who takes college algebra? Primus. 19:1, 74–87.

Jarrett, E. 2000. Evaluating Persistence and Performance of ‘Successful’ Precalculus Students in Subsequent Mathematics Courses, M.S. Thesis, Texas Tech University.

Sonnert, G. & Sadler, P. 2014. The impact of taking a college pre-calculus course on students’ college calculus performance. International Journal of Mathematical Education in Science and Technology. DOI: 10.1080/0020739X.2014.920532

Thompson, P. et al. 2007. Failing the future: Problems of persistence and retention in Science, Technology, Engineering, and Mathematics (STEM) majors at Arizona State University. Report prepared & submitted by the Provost’s Freshman STEM Improvement Committee.

Monday, September 1, 2014

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 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 = 2x, 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.

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, \( dV/dr = 4\pi r^2 = A. \) 

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.

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.

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.

Friday, August 1, 2014

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.

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, \( dV/dr = 4\pi r^2 = A. \) 

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.

Monday, June 30, 2014

Beyond the Limit, I

In my May column, FDWK+B, I said that I would love to ignore limits until we get to infinite series. One of my readers called me out on this, asking how I would motivate the definition of the derivative. Beginning this month and continuing through September, I would like to use my postings to give a brief overview of some of the problems with limit as an organizing principle for first-year calculus and to describe research that supports a better approach.

To a mathematician, the limit of f(x) as x approaches c is informally defined as that value L to which the function is forced to be arbitrarily close by taking x sufficiently close (but not equal) to c. In most calculus texts, this provides the foundation for the definition of the derivative: The derivative of f at c is the limit as x approaches c of the average rate of change of f over the interval from x to c. Most calculus texts also invoke the concept of limit in defining the definite integral, though here its application is much more sophisticated.

There are many pedagogical problems with this approach. The very first is that any definition of limit that is mathematically correct makes little sense to most students. Starting with a highly abstract definition and then moving toward instances of its application is exactly the opposite of how we know people learn. This problem is compounded by the fact that first-year calculus does not really use the limit definitions of derivative or integral. Students develop many ways of understanding derivatives and integrals, but limits, especially as correctly defined, are almost never employed as a tool with which first-year calculus students tackle the problems they need to solve in either differential or integral calculus. The chapter on limits, with its attendant and rather idiosyncratic problems, is viewed as an isolated set of procedures to be mastered.

This student perception of the material on limits as purely procedural was illustrated in a Canadian study (Hardy 2009) of students who had just been through a lesson in which they were shown how to find limits of rational functions at a value of x at which both numerator and denominator were zero. Hardy ran individual observations of 28 students as they worked through a set of problems that were superficially similar to what they had seen in class, but in fact should have been simpler. Students were asked to find \(\lim_{x\to 2} (x+3)/(x^2-9)\). This was solved correctly by all but one of the students, although most them first performed the unnecessary step of factoring x+3 out of both numerator and denominator. When faced with \( \lim_{x\to 1} (x-1)/(x^2+x) \), the fraction of students who could solve this fell to 82%. Many were confused by the fact that x–1 is not a factor of the denominator. The problem \( \lim_{x \to 5} (x^2-4)/(x^2-25) \) evoked an even stronger expectation that x–5 must be a factor of both numerator and denominator. It was correctly solved by only 43% of the students.

The Canadian study hints at what forty years of investigations of student understandings and misunderstandings of limits have confirmed: Student understanding of limit is tied up with the process of finding limits. Even when students are able to transcend the mere mastery of a set of procedures, almost all get caught in the language of “approaching” a limit, what many researchers have referred to as a dynamic interpretation of limit, and are unable to get beyond the idea of a limit as something to which you simply come closer and closer.

Many studies have explored common misconceptions that arise from this dynamic interpretation. One is that each term of a convergent sequence must be closer to the limit than the previous term. Another is that no term of the convergent sequence can equal the limit. A third, and even more problematic interpretation, is to understand the word “limit” as a reference to the entire process of moving a point along the graph of a function or listing the terms of a sequence, a misconception that, unfortunately, may be reinforced by dynamic software. This plays out in one particularly interesting error that was observed by Tall and Vinner (1981): They encountered students who would agree that the sequence 0.9, 0.99, 0.999, … converges to \(0.\overline{9} \) and that this sequence also converges to 1, but they would still hold to the belief that these two limits are not equal. In drilling into student beliefs, it was discovered that \(0.\overline{9} \) is often understood not as a number, but as a process. As such it may be approaching 1, but it never equals 1. Tied up in this is student understanding of the term “converge” as describing some sort of equivalence.

Words that we assume have clear meanings are often interpreted in surprising ways by our students. As David Tall has repeatedly shown (for example, see Tall & Vinner, 1981), a student’s concept image or understanding of what a term means will always trump the concept definition, the actual definition of that term. Thus, Oehrtman (2009) has found that when faced with a mathematically correct definition of limit—that value L to which the function is forced to be arbitrarily close by taking x sufficiently close but not equal to c—most students read the definition through the lens of their understanding that limit means that as x gets closer to c, f(x) gets closer to L. “Sufficiently close” is understood to mean “very close” and “arbitrarily close” becomes “very, very close,” and the definition is transformed in the student’s mind to the statement that the function is very, very close to L when x is very close to c.

That raises an interesting and inadequately explored question: Is this so bad? When we use the terminology of limits to define derivatives and definite integrals, is it sufficient if students understand the derivative as that value to which the average rates are getting closer or the definite integral as that value to which Riemann sums get progressively closer? There can be some rough edges that may need to be dealt with individually such as the belief that the limit definition of the derivative does not apply to linear functions and Riemann sums cannot be used to define the integral of a constant function (since they give the exact value, not something that is getting closer), but it may well be that students with this understanding of limits do okay and get what they need from the course.

There has been one very thorough study that directly addresses this question, published by Michael Oehrtman in 2009. This involved 120 students in first-year calculus at “a major southwestern university,” over half of whom had also completed a course of calculus in high school. Oehrtman chose eleven questions, described below, that would force a student to draw on her or his understanding of limit. Through pre-course and post-course surveys, quizzes, and other writing assignments as well as clinical interviews with twenty of the students chosen because they had given interesting answers, he probed the metaphors they were using to think through and explain fundamental aspects of calculus.

The following are abbreviated statements of the problems he posed, all of which ask for explanations of ideas that I think most mathematicians would agree are central to understanding calculus:
  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, \( dV/dr = 4\pi r^2 = A. \) 

In next month’s column, I will summarize Oehrtman’s findings. I then will show how they have led to a fresh approach to the teaching of calculus that avoids many of the pitfalls surrounding limits.

Hardy, N. (2009). Students' Perceptions of Institutional Practices: The Case of Limits of Functions in College Level Calculus Courses. Educational Studies In Mathematics, 72(3), 341–358.

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

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.

Sunday, June 1, 2014

AP Calculus and the Common Core

In February, 2013, Trevor Packer, Senior Vice-President for the Advanced Placement Program and Instruction at The College Board, appeared before the American Association of School Administrators (AASA), the professional society for school superintendents, to provide information about the Advanced Placement Program. Following that session, he had a short video interview in which he was asked to comment on the relationship between the Common Core State Standards (CCSS) and the College Board’s Advanced Placement Program. What he said about CCSS and AP Calculus has, unfortunately, been misreported. With Trevor Packer’s encouragement, I would like to attempt a clarification: There is no conflict between the Common Core and AP Calculus. In fact, it is just the opposite. If faithfully implemented, the Common Core can improve the preparation of students for AP Calculus or any college-level calculus.

In the AASA summary of the conference proceedings, “College Board: Reconciling AP Exams With Common Core,” Packer’s comment on AP Calculus was reported as follows:
“Despite these measures, there are still difficulties in reconciling many AP courses with the Common Core. In particular, AP Calculus is in conflict with the Common Core, Packer said, and it lies outside the sequence of the Common Core because of the fear that it may unnecessarily rush students into advanced math classes for which they are not prepared. 
“The College Board suggests a solution to the problem of AP Calculus ‘If you’re worried about AP Calculus and fidelity to the Common Core, we recommend AP Statistics and AP Computer Science,’ he told conference attendees.”
This article, written by a high school junior serving as an intern at the meeting, is not in line with the video of Packer’s remarks, “College Board’s Trevor Packer on Common Core and AP Curriculum,” where he says that
“AP Calculus sits outside of the Common Core. The Calculus is not part of the Common Core sequence, and in fact the Common Core asks that educators slow down the progressions for math so that students learn college-ready math very, very well. So that can involve a sequence that does not culminate in AP Calculus. There may still be a track toward AP Calculus for students who are interested in majoring in Engineering or other STEM disciplines, but by and large, the Common Core math sequence is best suited to prepare students for AP Statistics or AP Computer Science, which have dependencies on the math requirements of the Common Core.”
The assertion of a conflict between the Common Core and AP Calculus was a misinterpretation on the part of the student. Nevertheless, this lack of clear articulation between Common Core and AP Calculus is easy to misinterpret.

Packer’s remarks arose from concerns that I and others have expressed about the headlong rush to calculus in high school (see, in particular, MAA/NCTM Joint Position on Calculus). As I pointed out in last month’s column (FDWK+B, May, 2014), almost 700,000 students begin the study of calculus while in high school each year. Not all of them are in AP programs. Not all in an AP program take or even intend to take an AP Calculus exam. But we are now closing in on 400,000 students who take either the AB or BC Calculus exam each year, a number that is still growing at roughly 6% per year with no sign that we have reached an inflection point. Over half the students in Calculus I in our colleges and universities have already completed a calculus course while in high school. At our leading universities, the fraction is over three-quarters. Unfortunately, merely studying calculus in high school does not mean that these students are ready for college-level calculus and the subsequent mathematics courses required for engineering or the mathematical or physical sciences.

The problem for many students who enter with the aspiration of a STEM degree is inadequate proficiency at the level of precalculus: facility with algebra; understanding of trigonometric, exponential, and logarithmic functions; and comprehension of the varied and interconnected ways of viewing functions. Packer speaks of slowing down the progressions through mathematics. This is in response to a shared concern that the rush to get to calculus while in high school can interfere with the development of a solid foundation on which to build mathematical proficiency. Much of the impetus for the Common Core State Standards in Mathematics comes from the recognition that there are clear benchmarks consisting of skills and understandings that must be mastered before students are ready to move on to the next level of abstraction and sophistication. Failure to achieve those benchmarks at the appropriate point in a student’s mathematical development risks seriously handicapping future mathematical achievement.

The Common Core was designed as a common core, a set of expectations we intend for all students. There is an intentional gap between where the Common Core in Mathematics ends and where mathematics at the level of calculus begins. This gap is partially filled with the additional topics marked with a “+” in the Common Core State Standards in Mathematics, topics that usually get the required level of attention in a course called Precalculus. As the name suggests, Precalculus is the course that prepares students for calculus. This is the articulation problem to which Packer alludes. Completing the Common Core does not mean one is ready for the study of AP Calculus or any other calculus. It means one is ready for a number of options that include AP Statistics, AP Computer Science, or a Precalculus class.

There is no conflict between AP Calculus and the Common Core. Rather, there is an expectation that if the Common Core is faithfully implemented, then students will be better prepared when they get to AP Calculus and the courses that follow it.

Thursday, May 1, 2014


I am very pleased to announce that I will be joining the team of authors for the AP Calculus text Calculus: Graphical, Numerical, Algebraic by Finney, Demana, Waits, and Kennedy (commonly known as FDWK). Ross Finney has not been an active member of the team for some years (he died in 2000), and Frank Demana and Bert Waits are easing out of their roles, but their names reflect the incredible pedigree of this text. It began with George Thomas in 1951 and has variously been known as Thomas; Thomas & Finney; Finney & Thomas; Finney, Thomas, Demana, Waits; and Finney, Demana, Waits, Kennedy.

I was fortunate to be able to get to know George Thomas after he retired from MIT and moved to State College, Pennsylvania. I knew him as an extremely modest and gentle person with a continuing fascination with mathematics. I have long admired Frank Demana and Bert Waits for their pioneering work in the Calculus Reform efforts. Dan Kennedy and I have known each other for many years through the AP Program, and it is a particular delight for me now to be collaborating with him.

I also am very happy to be joining an effort aimed at high school calculus. Roughly one million U.S. students begin the study of calculus each year, and close to 700,000 of them, at least two-thirds of the total, start this journey in high school. This is the place where one can have the greatest impact in shaping students’ understanding of calculus.

There are limitations that I, as an author of “niche textbooks” for which I can take whatever approach I wish, find constraining. First of all, the text has to be closely tied to the AP Calculus syllabus and exams, which, in their turn, are closely tied to the curricula as enacted at the major universities, the big consumers of AP Calculus results. The emphasis on limits is one of those limitations. I would love to ignore them until we get to infinite series, but that really is not an option.

Second, the books I write for my own pleasure can assume whatever level of sophistication on the part of the reader I choose to impose. I recognize that this text will be used by teachers and students for whom digressions and elaborations may be more confusing than helpful. That said, I do hope to push both teachers and students a little and to open more perspectives, especially historical perspectives, on this subject.

Third, I am now working for the behemoth that is Pearson. I’ve worked with Pearson people on several projects and have always found them to be intelligent, conscientious, and seriously concerned with producing quality products. Nevertheless, this is a mass-market endeavor that travels with its own peculiar baggage of demands and constraints. I am pleased that in the face of so much pressure to bulk up with every tidbit relevant to Calculus, FDWK has managed to maintain a lean profile of only 717 pages (16 fewer than the first edition of Thomas).

Also on the plus side is the large and talented staff that will be working with us to produce the next edition of this text. As I observed in my contribution to “Musing on MOOCS,” which appeared in the Notices of the AMS this past January, the real revolution in education created by the online world is not the disappearance of the live instructor but the richness of supporting resources that instructors can now draw upon. Robert Ghrist argued that the ease with which individuals can produce their own online materials will eliminate the need for big publishers. I argued that the situation is exactly the opposite: “The problem is that few of us will have the time to develop our own materials, and anyone who searches for such resources online is quickly inundated with options. In an era of overwhelming choices, it is the reputable bundlers who will dominate.” MAA is one reputable supplier, as evidenced by WeBWorK (see my column from April, 2009). Pearson is well aware of this need and is actively building these supports.

By an opportune coincidence, I also am working with Karen Marrongelle and Karen Graham on the calculus chapter for the next version of the NCTM Handbook of Research on Learning and Teaching Mathematics.  This means that I am currently steeped in the accumulated research on how students understand and misunderstand the key concepts of calculus. I expect to translate some of this knowledge into the shaping of future editions of FDWK, and I also hope to share some of what I’m learning in future Launchings columns.

Tuesday, April 1, 2014

Age Is Not the Problem

Edward Frenkel recently resurrected an old complaint in his Los Angeles Times op-ed, “How our 1,000-year–old math curriculum cheats America’s kids.” He observes that no one would exclude an appreciation for the beauty of art or music from the need to build technique. Why do we do that in mathematics? As I said, this is an old complaint. Possibly no one has voiced it more eloquently than Paul Lockhart in A Mathematician’s Lament, the theme of Keith Devlin’s 2008 MAA column, “Lockhart’s Lament.” Enough time has passed that it is worth my while to bring this lament back to the attention of the readers of MAA columns. I also want to respond to Frenkel’s post. I have two problems with what he writes.

The first is the suggestion that we spend too much time on “old” mathematics and not enough on what is “new.” I share Frenkel’s disappointment that too few have any appreciation of mathematics as a fresh, creative, and self-renewing field of study. Frenkel himself has made a significant contribution toward correcting this. In his recent book, Love and Math, he has opened a window for the educated layperson to glimpse the fascination of the Langland’s program. But I disagree with Frenkel’s solution of devoting “just 20% of class time [to] opening students’ eyes to the power and exquisite harmony of modern math.” There is power and exquisite harmony in everything from early Babylonian and Egyptian discoveries through Euclid’s Elements to the Arithmetica of Diophantus and the development of trigonometry in the astronomical centers of Alexandria and India, all of which were accomplished more than a millennium ago and are still capable of inspiring awe. 

In fact, I believe that one of the worst things we could do is to create a dichotomy in students’ minds between beautiful modern math and ugly old math. We must communicate the timeless beauty of all real mathematics. The challenge of the educator is to engage students in rediscovering this beauty for themselves, not outside of the standard curriculum, but embedded within it. The question of how to accomplish this leads to my second problem with Frenkel.

Frenkel makes the implicit assumption that what we need is a wake-up call, that it is time to recognize that mathematics education must do more than create procedural facility. In fact, the need to combine the development of technical ability with an appreciation for the ideas that motivate and justify the mathematics that we teach goes back at least a century to Felix Klein and his Elementary Mathematics from an Advanced Standpoint. It is front and center in the Practice Standards of the Common Core State Standards in Mathematics. It was a driving concern of Paul Sally at the University of Chicago, who we so recently and unfortunately lost. It continues to motivate Al Cuoco and his staff engaged in the development of the materials of the Mathematical Practice Institute. It lies at the root of Richard Rusczyk’s creation of the Art of Problem Solving. It permeates the efforts of literally thousands of us who are struggling to enable each of our students to encounter the thrill of mathematical exploration and discovery.

As we know, it takes more than good curricular materials and good intentions to accomplish this. It requires educators who understand mathematics both broadly and deeply and can bring this expertise to their teaching. Many are working to spread this knowledge among all who would teach mathematics to our children. This is the inspiration behind the reports of the Conference Board of the Mathematical Sciences on The Mathematical Education of Teachers. It is a goal of the Math Circles, in particular the Math Teachers’ Circles that reach those who too often are unaware of the exciting opportunities for exploration and discovery within the curricula they teach.

The mathematician’s lament is still all too relevant, but it is neither unheard nor unheeded. I am encouraged by the many talented and dedicated individuals and organizations working to meet its challenge.