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.

Saturday, March 1, 2014

Collective Action by STEM Disciplinary Societies

At the end of January, it was my great pleasure to be part of the leadership for a meeting at the MAA Carriage House of representatives of a collection of STEM disciplinary societies [1] and concerned educational associations [2] to consider ways that these societies can coordinate efforts to increase their collective impact on undergraduate education. Across academia, but especially at research universities, most faculty identify first with their discipline and department and only secondarily with their university. Disciplinary societies therefore have the potential to impact how faculty think about their teaching and how willing they are to reach outside their own department in seeking ideas and support for improving undergraduate education.

Many disciplinary societies are actively promoting effective methods for engaging students to improve both what they learn and their desire to persist. The American Physical Society and the American Association of Physics Teachers have been particularly effective in this regard. See, for example, the Physics Education Research User’s Guide,, described in my column “Learning from the Physicists,” July, 2012. Over the past several years, the life sciences community, scattered over some 147 disciplinary societies, has come together to produce a joint report, Vision and Change in Undergraduate Biology Education: A Call to Action [3]. Recognizing that it is not sufficient to issue a report, Vision and Change continues to seek ways to implement the changes it champions. One outgrowth has been PULSE, the Partnership for Undergraduate Life Sciences Education, which is building communities that share experiences of department-level implementation of the Vision and Change recommendations. Inspired by the example of PULSE, the mathematics community began last summer to build a comparable effort, INGenIOuS, Investing in the Next Generation through Innovative and Outstanding Strategies.

We have much to learn from each other. Beyond just sharing information, an ability to offer comparable statements of vision and comparable programs to promulgate effective practices would increase their collective impact. This would be especially true if the disciplinary societies were to establish and promote linkages that enable individuals to connect with others at their university who are working toward the same ends but within other departments.

With these goals in mind, 28 representatives of disciplinary societies and educational associations met at the MAA Carriage House in Washington, DC on January 30–31 for an NSF-sponsored workshop [4] entitled ISSUES, Integration of Strategies that Support Undergraduate Education in STEM, to look for opportunities to work collectively. As preparation, most of the societies provided a summary of their current activities directed toward faculty development and the improvement of undergraduate education. These Profiles can be found within the ISSUES website at A summary of the workshop is available at

The workshop identified five concrete areas in which disciplinary societies could increase their effectiveness by sharing and coordinating their efforts:
  1. Supporting Early Career Faculty. Within the disciplinary societies, the task is to develop workshops for and build communities of early career faculty, as well as partnering with the Discipline-Based Educational Research community to assess the long-term effectiveness of this work. On individual campuses, the task is to work with deans and chairs to build cross-disciplinary networks of faculty who have been through these experiences, supported by networks of mentors both from the individual’s profession and from within the individual’s home institution.
  2. Strengthening Departments. There is a need to increase the value placed on the department chair and to provide support for the chair by supplying tools for departmental self-assessment of teaching effectiveness together with practical suggestions that chairs and departmental leaders can implement to improve teaching effectiveness.
  3. Communicating Career Pathways. We need to increase the diversity of students within our disciplines by increasing student awareness of the variety of pathways that are available to them, actively recruiting students to these pathways, preparing them for a variety of careers, and introducing them to a network of potential employers.
  4. Shifting Cultural Norms. Disciplinary societies should strive to move their members toward embracing teaching practices that align with what educational research has shown to be most effective and toward a mindset of continual efforts to improve undergraduate teaching and learning. This can be accomplished through policy statements, rubrics for assessing effective educational processes, and active promotion of these practices. Part of our collective goal should be the adoption of consistent language that reinforces this message across disciplinary boundaries.
  5. Measuring the Impact of Our Own Programs for Improving Undergraduate Education. The disciplinary societies can benefit from developing common rubrics for assessing the effectiveness of their own programs and using these to help frame discussion and dialog across the societies.

On point 1, we are already working with the Association of American Universities (AAU) to put together a pilot project on AAU campuses that will build local networks of faculty from multiple disciplines who have each been through an early career professional development program run by their disciplinary society. On point 5, we are beginning the task of gathering information from the disciplinary societies about their experiences with assessment of their own programs. Within the next months, we hope to see progress on all of these agendas.


Footnotes and References:

[1] The disciplinary societies that were represented were the American Association of Physics Teachers, American Chemical Society, American Geophysical Union, American Institute of Biological Sciences, American Institute of Physics, American Mathematical Society, American Physical Society, American Psychological Association, American Society for Engineering Education, American Society for Microbiology, American Statistical Association, Mathematical Association of America, National Association of Biology Teachers, National Association of Geoscience Teachers, and the Society for Industrial and Applied Mathematics.

[2] The educational associations that were represented included the American Association for the Advancement of Science, Association of American Universities, Association of Public Land-Grant Universities, Howard Hughes Medical Institute, National Academy of Sciences, National Science Foundation, and Project Kaleidoscope of the Association of American Colleges and Universities.

[3] Brewer, C.A., and Smith, D. (eds.). 2011. Vision and Change in Undergraduate Biology Education: A Call to Action. Washington, DC: American Association for the Advancement of Science. Available at

[4] The workshop was made possible by a grant from the National Science Foundation, #1344418. The opinions expressed here do not necessarily reflect those of NSF.

Saturday, February 1, 2014

Mathematics for the Biological Sciences

MAA has just published a Notes volume, Undergraduate Mathematics for the Life Sciences: Models, Processes, and Directions [1] that provides examples and advice for mathematics departments that want to reach out to the growing population of biological science majors.

Biological science majors have replaced prospective engineers as the largest group of students taking regular Calculus I. From the MAA’s Calculus Survey [2], just over 28% of all students in mainstream Calculus I intend to pursue a major in the biological sciences, the largest single group of majors in this course. It is larger than engineers (just under 28%) or the combined physical science (7%), computer science (7%), and mathematical science majors (1%). For women in mainstream Calculus I, 42% intend a biological science major. For Black or Hispanic students, 34% are going into biological sciences. This dominance is certain to only increase. As the graph in Figure 1 illustrates, the growth in science, engineering, and mathematical sciences majors is occurring almost exclusively in the biological sciences.

Figure 1: Number of full-time entering freshmen who identified a STEM field as their most likely major. Data from The American Freshman surveys. [2]
Mathematics has done well by encouraging students who have to study mathematics to continue its study. Mathematics departments actually graduate more majors than the number of students who enter with the intention of pursuing a math major. In 2012, 18,842 students graduated with a Bachelor’s degree in mathematics.[3] Four years earlier, in 2008, only 11,583 entered a full-time program with the intention of majoring in mathematics.[4] Even after subtracting the roughly 5,000 students per year who are heading into K-12 mathematics teaching and who get a degree in mathematics but identify education as their intended field when they enter, we see that mathematics—uniquely among the major STEM disciplines—still has a net gain in majors. If we are to maintain this happy state of affairs, then we need to convince our audience that mathematics is relevant to its interests.

Mathematics departments are recognizing this fact. The 2010 CBMS report revealed that 41% of those at research universities had added interdisciplinary courses in mathematics and biology within the past five years. As the new MAA Notes volume illustrates, there is a tremendous amount of experimentation under way.

This volume begins with descriptions of thirteen programs that range from calculus for biology majors, to programs that draw calculus and statistics together into a year-long course, to bioinformatics, to research programs for biology majors that incorporate significant quantitative analysis. The institutions include large universities: Illinois at Urbana-Champaign, Ohio State, and the Universities of Minnesota, Nebraska-Lincoln, and Utah. There also are smaller places: Benedictine University, Macalester College, University of Richmond, Chicago State, Sweet Briar College, University of Wisconsin-Stout, and East Tennessee State.

The volume continues with a collection of essays on “Processes.” These are nine accounts of the trials and tribulations of getting such a program started and keeping it going. This is particularly useful because these essays describe both programs that have survived, moving beyond the small group of individuals who initiated them, and programs that have failed or are failing, the ones that have not managed to establish themselves as a permanent feature of the local curriculum. The final four essays, labeled “Directions,” speak to opportunities and needs.

Unfortunately, this book is only available as a pdf file ($25) or as Print-on-Demand ($43), so you probably will not see a display copy at MAA meetings. But it is well worth checking out.


[1] Ledder, G., J.P. Carpenter, and T.D. Comar (Eds.) (2013). Undergraduate Mathematics for the Life Sciences: Models, Processes, and Directions. MAA Notes #81. Washington, DC: Mathematical Association of America.

[2] Higher Education Research Institute. (Multiple years). The American Freshman.

[3] NCES. (2013). Digest of Education Statistics. Table 322.10.

[4] Higher Education Research Institute. (2008). The American Freshman: National norms for fall 2008.

Wednesday, January 1, 2014

MAA Calculus Study: Seven Characteristics of Successful Calculus Programs

By David Bressoud and Chris Rasmussen

In these days of tight budgets and pressure to improve retention rates for science and engineering majors, many mathematics departments want to know what works, what are the most effective means of improving the effectiveness of calculus instruction. This was the impetus behind the study of Characteristics of Successful Programs in College Calculus undertaken by the MAA. The study consisted of a national survey in fall 2010, followed by case study visits to 17 institutions that were identified as “successful” because of their success in retention and the maintenance of “productive disposition,” defined in [NRC 2001] as “habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.”

Our survey revealed that Calculus I, as taught in our colleges and universities, is extremely efficient at lowering student confidence, enjoyment of mathematics, and desire to continue in a field that requires further mathematics. The institutions we selected bucked this trend. This report draws on our experiences at all 17 colleges and universities, but focuses on the insights drawn from those universities that offer a Ph.D. in mathematics, the universities that both produce the largest numbers of science and engineering majors and that often struggle with how to balance the maintenance of high quality research with attention to undergraduate education.

Case studies were conducted in the fall of 2012 at five of these universities: two large public research universities, one large private research university, one public technical institute, and one private technical institute. We shall refer to these as

  • ETI: Eastern Technological Institute. Private university. Data from nine sections of calculus with an average enrollment of 33.
  • MTI: Midwestern Technological Institute. Public university. Data from seven sections with an average enrollment of 38 and one with an enrollment of 110.
  • MPU: Large Midwestern Public University. Data from 41 sections with an average enrollment of 27.
  • WPU: Large Western Public University. Data from four sections with an average enrollment of 200.
  • WPR: Large Western Private University. Data from three sections with an average enrollment of 196 and one section with 32 students.
In addition to productive disposition and improved retention rates, the five also had noticeably higher grades (see Figure 1), cutting the DFW rate from 25% across all doctoral universities to only 15% at the case study sites. The difference was in B’s and C’s. The five case study universities actually gave out a slightly lower percentage of A’s than the overall average.

We identified seven characteristics of the calculus programs at these five universities, characteristics that, as applicable, were also found at the other twelve:

  1. Regular use of local data to guide curricular and structural modifications. In his description of the MAA study of Models that Work [Tucker, 1995], Alan Tucker wrote, “No matter how successful their current programs are, faculty members in the visited departments are not yet satisfied with the programs. Experimentation is continuous.” [Tucker, 1996] We found that not only was this true of the successful programs we studied, these universities used the annual gathering and sharing of data on retention and grade distributions to guide this continuous experimentation. A bad semester was not dismissed as an anomaly, but was viewed as an opportunity to understand what went wrong and what could be done to avoid a similar occurrence.
  2. Attention to the effectiveness of placement procedures. Though this could be considered part of the first characteristic of successful programs, it received so much attention from all of the universities that we have elevated it to the level of a separate point. These universities evaluate and adjust their placement procedures on an annual basis. We also found a great deal of attention paid to those students near the cut-off, paying particular attention to programs in support of those allowed into Calculus I but most at risk and working with those who did not quite make the cut so that they were placed in programs that addressed their actual needs.
  3. Coordination of instruction, including the building of communities of practice. As Tucker reported in 1996, “There is a great diversity of instructional and curricular approaches, varying from one visited department to another, and even varying within a single department.” We found this, but we also found that those teaching Calculus were in regular communication with the other instructors of this class. Of course, where classes were taught by graduate teaching assistants, there was much tighter coordination of instruction. In all cases, we found that common exams were used. The simple act of creation of such exams fostered communication among those teaching the course. In some cases, communication about teaching was much more intentional, sharing innovative pedagogies, assignments, and approaches to particular aspects of the curriculum. In all cases there was also a course coordinator, a position that was not rotating but a more or less permanent position with commensurate reduction in teaching load.
  4. Construction of challenging and engaging courses. This is reflected in an observation that Tucker made in 1996: “Faculty members communicate explicitly and implicitly that the material studied by their students is important and that they expect their students to be successful in mathematical studies.” It also is the first example of effective educational practice in Student Success in College [Kuh et al, 2010, p 11]: “Challenging intellectual and creative work is central to student learning and collegiate quality.” None of the successful programs we studied believed that one could improve retention by making the course easier. Instructors used textbooks and selected problems that required students to delve into concepts and to work on modeling-type problems, or even problems involving proofs. Interviews with students—most of whom had taken calculus in high school—revealed that they felt academically challenged in ways that went far beyond their high school courses. 
  5. Use of student-centered pedagogies and active-learning strategies. This is the second example of effective educational practice in [Kuh et al, 2010]: “Students learn more when they are intensely involved in their education and have opportunities to think about and apply what they are learning in different settings.” As the first author learned twenty years ago when he surveyed Calculus I students at Penn State [Bressoud, 1994], few students know how to study or what it means to engage the mathematics, and most take a very passive role when attending a lecture. Active-learning strategies force students to engage the mathematical ideas and confront their own misconceptions. The successful programs we studied made much greater use of group projects and student presentations. 
  6. Effective training of graduate teaching assistants. Graduate students play an important role in calculus instruction at all universities with doctoral programs, whether as teaching assistants in the breakout sections for large lectures or as the instructors of their own classes. The most successful universities have developed extensive programs for training, monitoring, and supporting these instructors. Running a successful training program is not a task that can be handed off to a single person. While there is always one coordinator, their effectiveness requires a core of faculty who are willing to participate in the graduate students’ training that takes place before the start of the fall term and to assist in visiting classes and providing feedback.
  7. Proactive student support services, including the fostering of student academic and social integration. This is a broad category that ranges from the building of a student-faculty community within the mathematics department to the specifics of support mechanisms for at-risk students. These are addressed in three of the effective practices identified in [Kuh et al, 2010]: “Student Interactions with Faculty Members,” “Enriching Educational Experiences,” and “Supportive Campus Environment.” The first is mentioned in [Tucker, 1996]: “Extensive student-faculty interaction characterizes both the teaching and learning of mathematics, both inside and outside of the classroom.” The universities we visited had rich programs of extra-curricular activities within the Mathematics Department. They also had a variety of responses to supporting at-risk students. These included stretching Calculus I over two terms to allow for supplemental instruction in precalculus topics, providing “fallback” courses for students who discovered after the first exam that they were in trouble in Calculus, and working with student support services to ensure that students who were struggling got the help they needed. There also were heavily utilized learning centers that attracted all students as places to gather, work on assignments, and get help as needed. Often, these were centers dedicated solely to helping students in Calculus. What was common among all of the successful calculus programs was attention to the support of all students and a willingness to monitor and adjust the programs designed to help them.
There were some dramatic differences between instruction at the doctoral universities that were selected for the case study visits and instruction at all doctoral universities (see Table 1). Where the section size facilitated this—at ETI, MTI, MPU, and one section of WPR—instructors made much less use of lecture and much more use of students working together, holding discussions, and making presentations. Three of the five have almost universal use of online homework, and a fourth uses it for half of the sections. Graphing calculators were allowed on exams in two of the five universities, though use was not consistent across sections. The most striking difference between these five universities and the overall survey was the number of instructors who ask students to explain their thinking.

Instructors at the case study sites still consider themselves to be fairly traditional (see Figure 2), though slightly less so than the national average. They also tend to agree with the statement, “Calculus students learn best from lectures, provided they are clear and well organized” (see Figure 3). Interestingly, not a single instructor at any of the case study sites strongly agreed with this statement. On the other hand, the instructors at the case study sites were slightly less likely to disagree with it. They clumped heavily toward mild agreement, suggesting an attitude of keeping an open mind and a willingness to try an approach that might be more productive.


Bressoud, D. 1994. Student attitudes in first semester calculus. MAA Focus, vol 14, pages 6–7.

Kuh, G.D.,  J. Kinzie, J.H. Schuh, E.J. Whitt. 2010. Student Success in College: Creating Conditions that Matter. Jossey-Bass.

National Research Council. 2001. Adding It Up: Helping Children Learn Mathematics. Kilpatrick, Swafford, and Findell (Eds.). National Academy Press.

Tucker, A. 1995. Models that Work: Case Studies in Effective Undergraduate Mathematics Programs. MAA Notes #38. Mathematical Association of America.

Tucker, A. 1996. Models that Work: Case Studies in Effective Undergraduate Mathematics Programs. Notices of the AMS, vol 43, pages 1356–1358.

Characteristics of Successful Programs in College Calculus is supported by NSF #0910240. The opinions expressed in this article do not necessarily reflect those of the National Science Foundation.