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.

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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.

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.