Sunday, January 1, 2017

IJRUME: Approximation in Calculus

You can now follow me on Twitter @dbressoud.

In an earlier column, "Beyond the Limit, III," I talked about how Michael Oehrtman and colleagues have been able to use approximation as a unifying theme for single variable calculus that helps students avoid many of the confusing aspects of the language of limits. I also pointed out that this is hardly a new idea, having been used by many textbook authors including Emil Artin in A Freshman Honors Course in Calculus and Analytic Geometry and Peter Lax and Maria Terrell in Calculus with Applications. The IJRUME research paper I wish to highlight this month, “A study of calculus instructors’ perceptions of approximation as a unifying thread of the first-year calculus” by Sofronas et al., looks at how common this approach actually is.

The authors address four research questions:

  1. Do calculus instructors perceive approximation to be important to student understanding of first-year calculus? 
  2. Do calculus instructors report emphasizing approximation as a central concept and-or unifying thread in the first-year calculus? 
  3. Which approximation ideas do calculus instructors believe are “worthwhile” to address in first-year calculus?  
  4. Are there any differences between demographic groups with respect to the approximation ideas they teach in first-year calculus courses? 
They surveyed calculus instructors at 182 colleges and universities, collecting 279 responses.

To the first two questions, 89% agreed that approximation is important, but only 51% considered it a central concept, and only 40% found that it provides a unifying thread (see Figure 1). For those who did consider it central and-or unifying, the reasons that they gave included: (a) it illuminates reasons for studying calculus, (b) most functions are not elementary and approximation is helpful in dealing with such functions, (c) approximation facilitates the understanding of fundamental concepts including limit, derivative, integral, and series, (d) linear approximations lie at the foundation of differential calculus, and (e) an emphasis on approximation resonates with the instructors personal interests in applied mathematics or numerical analysis.
Figure 1: Graph depicting participants’ perceptions of approximation (N=214).
 Source: Sofronas et al. 2015.

For those who did not consider approximation to be central or unifying, many stated that it is not sufficiently universal, only important in a few contexts such as motivating the definition of the derivative at a point or the value of a definite integral. Many stated other unifying threads such as limit or the study of change. Some objected to an emphasis on approximation because of its inevitable ties to the use of technology. There were also a large number of obstacles to the use of approximation that instructors identified. These included: (a) an overcrowded syllabus that left no room for the instructor to develop a unifying thread, (b) required adherence to a curriculum emphasizing procedural facility, (c) students with weak preparation who are not prepared to understand the subtleties of approximation arguments, (d) lack of access to technology, (e) lack of familiarity with how to use approximation ideas in developing calculus. I personally find these obstacles to be very sad, in particular the assumption on the part of many instructors that the only way to get through the required syllabus or to enable students to pass the course is to focus exclusively on memorizing procedures.

Jumping ahead to the fourth question, the authors found that the single factor that most highly correlated with emphasizing approximation as a central concept and-or unifying thread was having served on either a local or national calculus committee. Not surprisingly, this factor was also highly correlated with number of years teaching calculus, rank, being the recipient of a teaching award, and having published or presented on a calculus topic.

To the third research question, the combined list of topics gleaned from all of the responses truly spans first-year calculus: numerical limits, definition of limit, definition of the derivative, derivative values, tangent line approximations, differentials, error estimation, function change, functions roots and Newton’s method, linearization, integration, Riemann sums, Taylor polynomials and Taylor series, Newton’s second law, Einstein’s equation for force, L’Hospital’s rule, Euler’s method, and the approximation of irrational numbers. One unexpected outcome of the survey is that several of the respondents commented that answering this survey about their use of approximation in first-year calculus opened their eyes to the opportunity to use it as a unifying theme. As one respondent wrote,
I agree that approximation is an important concept AND after taking this survey I can see teaching calculus using approximation as the main theme. The rate of change theme offers many opportunities for real-life applications but I can see how using approximations from the beginning would offer other opportunities. It is an interesting idea, and I would love to incorporate more of this theme into my lessons.
For those who are interested in following up on the use of approximation as a unifying thread, this article also supplies a wealth of background information that includes a discussion of the different ways in which approximation can be used and the research evidence for its effectiveness as a guiding theme in developing student understanding of limits, derivatives, integrals, and series.


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

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

Sofronas, K.S., DeFranco, T.C., Swaminathan, H., Gorgievski, N., Vinsonhaler, C., Wiseman, B., Escolas, S. (2015). A study of calculus instructors’ perceptions of approximation as a unifying thread of the first-year calculus. Int. J. Res. Undergrad. Math. Ed. 1:386–412 DOI 10.1007/s40753-015- 0019-5

Thursday, December 1, 2016

IJRUME: Peer-Assisted Reflection

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The second paper I want to discuss from the International Journal of Research in Undergraduate Mathematics Education is a description of part of the doctoral work done by Daniel Reinholz, who earned his PhD at Berkeley in 2014 under the direction of Alan Schoenfeld. It consists of an investigation of the use of Peer-Assisted Reflection (PAR) in calculus [1].

Macintosh HD:Users:macalester:Dropbox:Personal:Launchings:2016:2016-12:21565092511_2e41878c9c_b.jpg
Daniel Reinholz. Photo Credit: David Bressoud
PAR addresses an aspect of learning to do mathematics that Schoenfeld refers to as “self-reflection or monitoring and control” in his chapter on “Learning to Think Mathematically” [4]. As he observed in his problem-solving course at Berkeley, most students have been conditioned to assume that when presented with a mathematical problem, they should be able to identify immediately which tool to use. Among the possible activities that students might engage in while solving a problem—read, analyze, explore, plan, implement, and verify—most students quickly chose one approach to explore and then “pursue that direction come hell or high water” (Figure 1).

Macintosh HD:Users:macalester:Dropbox:Personal:Launchings:2016:2016-12:Schoenfeld fig3.tiff
Figure 1: Time-line graph of a typical student attempt to solve a non-standard problem.
Source: [4, p.356, Figure 15-3]
In contrast, when he observed a mathematician working on an unfamiliar problem, he observed all of these strategies coming into play, a constant appraisal of whether the approach being used was likely to succeed and a readiness to try different ways of approaching the problem. He also found that mathematicians would verbalize the difficulties they were encountering, something seldom encountered among students (Figure 2). Note that over half the time was spent making sense of the problem rather than committing to a particular direction. Triangles represent moments when explicit comments were made such as “Hmm, I don’t know exactly where to start here.”

Macintosh HD:Users:macalester:Dropbox:Personal:Launchings:2016:2016-12:Schoenfeld fig4.tiff
Figure 2: Time-line graph of a mathematician working on a difficult problem.
Source: [4, p.356, Figure 15-4]
In the conclusion to this section of his chapter, Schoenfeld wrote, “Developing self-regulatory skills in complex subject-matter domains is difficult.” In reference to two of the studies that had attempted to foster these skills, he concluded that, “Making the move from such ‘existence proofs’ (problematic as they are) to standard classrooms will require a substantial amount of conceptualizing and pedagogical engineering.”

One of the problems with the early attempts at instilling self-reflection was the tremendous amount of work required of the instructor. Reinholz implemented PAR in Calculus I, greatly simplifying the role of the instructor by using students as partners in analyzing each other’s work. The study was conducted in two phases over two separate semesters in studies that each semester included one experimental section and eight to ten control sections, all of whom used the same examinations that were blind-graded. There were no significant differences between sections in either student ability on entering the class or in student demographics. The measure of success was an increase in the percentage of students earning a grade of C or higher. In the first phase of the study, the experimental section had a success rate of 82%, as opposed to the control sections where success was 69%. In the second phase, success rose from 56% in the control sections to 79% in the experimental section.

Reinholz observed a noticeable improvement in student solutions to the PAR problems after they had received peer feedback. From student interviews, he found that many students in the PAR section had learned the importance of iteration, that homework is not just something to be turned in and then forgotten, but that getting it wrong the first time was okay as long as they were learning from their mistakes. Students were learning the importance of explaining how they arrived at their solutions. And they appreciated the chance to see the different approaches that other students in the class might take.

What is most impressive about this intervention is how relatively easy it is to implement. Each week, the students would be given one “PAR problem” as part of their homework assignment. They were required to work on the problem outside of class, reflect on their work, exchange their solution with another student and provide feedback on the other student’s work in class, and then finalize the solution for submission. The time in class in which students read each other’s work and exchanged feedback took only ten minutes per week: five minutes for reading the other’s work (to ensure they really were focusing on reasoning, not just the solution) and five minutes for discussion.

The difficulty, of course, lies in ensuring that the feedback provided by peers is useful. Reinholz identifies what he learned from several iterations of PAR instruction. In particular, he found that it is essential for the students to be explicitly taught how to provide useful feedback. By the time he got to Phase II, Reinholz was giving the students three sample solutions to that week’s PAR problem, allowing two to three minutes to read and reflect on the reasoning in each, and then engaging in a whole class discussion for about five minutes before pairing up to analyze and reflect on each other’s work.

Further details can be found in [2] and [3]. For anyone interested in using Peer-Assisted Reflection, this is a useful body of work with a wealth of details on how it can be implemented and strong evidence for its effectiveness.


[1] Reinholz, D.L. (2015). Peer-Assisted Reflection: A design-based intervention for improving success in calculus. International Journal of Research in Undergraduate Mathematics Education. 1:234–267.

[2] Reinholz, D. (2015). Peer conferences in calculus: the impact of systematic training. Assessment & Evaluation in Higher Education, DOI: 10.1080/02602938.2015.1077197

[3] Reinholz, D.L. (2016). Improving calculus explanations through peer review. The Journal of Mathematical Behavior. 44: 34–49.

[4] Schoenfeld, A.H. (1992). Learning to think mathematically: problem-solving, metacognition, and sense-making in mathematics. Pp. 334–370 in Handbook for Research in Mathematics Teaching and Learning. D. Grouws (Ed.). New York: Macmillan.

Tuesday, November 1, 2016

IJRUME: Measuring Readiness for Calculus

You can now follow me on Twitter @dbressoud.

In 2015, the International Journal of Research in Undergraduate Mathematics Education (IJRUME) was launched by Springer with editors-in- chief Karen Marrongelle and Chris Rasmussen from the U.S. and Mike Thomas from New Zealand. It was established to “become the central, premier international journal dedicated to university mathematics education research.” While this is a journal by mathematics education researchers for mathematics education researchers, many of the articles are directly relevant to those of us engaged in the teaching of post-secondary mathematics. This then is the first of what I anticipate will be a series of columns abstracting some of the insights that I gather from this journal.

I have chosen for the first of these columns the paper by Marilyn Carlson, Bernie Madison, and Richard West, “A study of students’ readiness to learn calculus.” [1] It is common to point to students’ lack of procedural fluency as the culprit behind their difficulties when they get to post- secondary calculus. Certainly, this is a problem, but not the whole story. Work over the past quarter century by Tall, Vinner, Dubinsky, Monk, Harel, Zandieh, Thompson, Carlson and many others have led the authors to identify major reasoning abilities and understandings that students need for success in calculus. This paper describes a validated diagnostic test that measures foundational reasoning abilities and understandings for learning calculus, the Calculus Concept Readiness (CCR) instrument.

The reasoning abilities and conceptual understandings assessed by CCR require students to move beyond a procedural or action-oriented understanding of mathematics. Whether it is an equation such as 2 + 3 = 5 or a function definition, f(x) = x2 + 3x + 6, students are introduced to these as describing an action to be taken, adding 2 to 3 or plugging in various values for x. To make sense of and use the ideas of calculus, students need to view a function as a process (defined by a function formula, graph, or word description) that characterizes how the values of two varying quantities change together. Listed below are four of the reasoning abilities and understandings assessed by CCR and which the authors highlight in their article.

  1. Covariational Reasoning. When two variables are linked by an equation or a functional relationship, students need to understand how changes in one variable are reflected in changes in the other variable. The classic example considers how the rates of change of height and volume are related when water is poured into a non-cylindrical container such as a cone. At an even more basic level, students need to be able to interpret information on the velocities of two runners to an understanding of which is ahead at what times. Another example, which involves covariational reasoning as well as understanding rate as a ratio, considers the height of a ladder and its distance from a wall (Figure 1). When the authors administered their instrument to 631 students who were starting Calculus I, only 27% were able to select the correct answer (c) to the ladder problem.

    Figure 1. The ladder problem.

  2. Understanding the Function Concept. Too many students interpret f(x) as an unnecessarily long-winded way of saying y. They see a function definition such as f(x) = x2+ 3x + 6 as simply a prescription for how to take an input x and turn it into an output f(x). Such a limited view makes it difficult for students to manipulate functional relationships or to compose function formulas. Carlson et al. asked their 631 students for the formula for the area of a circle in terms of its circumference and offered the following list of possible answers:
          a. A = C2/4π
          b. A = C2/2
          c. A = (2πr)2a     
    A = πr2   
    A = π(C2/4)
    Only 28% chose the correct answer (a). As the authors learned from interviewing a sample of these students, those who answered correctly were the students who could see the equation C = 2πr as a process relating C and r which could be inverted and then composed with the familiar functional relationship between the area and radius.
  3. Proportional Relationships. Too many students do not understand proportional reasoning. When Carlson et al. in an earlier study [2] administered the rain-gauge problem of Piaget et al. (Figure 2) to 1205 students who were finishing a precalculus course, only 43% identified the correct answer (as presented in Figure 2, it is 4⅔). Many students preserve the difference rather than the ratio, giving 5 as the answer. Difficulties with proportional reasoning are known to impede student understanding of constant rate of change, which in turn underpins average rate of change, which is fundamental to understanding the meaning of the derivative.

    Figure 2. The rain gauge problem (taken from [3], [4])

  4. Angle Measure and Sine Function. As I described some years ago in an article for The Mathematics Teacher [5], the emphasis in high school trigonometry on the sine as a ratio of the lengths of sides of a triangle—often leading to the misconception that the sine is a function of a triangle rather than an angle—can lead to difficulties when encountering the sine in calculus, where it must be understood as a periodic function expressible in terms of arc length. An example is given in Figure 3, a problem for which only 21% of the Calculus I students chose the correct answer (e). Student interviews revealed that difficulties with this problem most often arose because students did not understand how to represent an angle measure using the length of the arc cut off by the angle’s rays.

What lessons are we to take away from this for our own classes? Last spring, in What we say/What they hear and What we say/What they hear II, I discussed problems of communication between instructors and students. The work of Carlson, Madison, and West illustrates some of the fundamental levels at which miscommunication can occur and identifies the productive ways of thinking that students need to develop.


[1] Carlson, M.P., Madison, B., & West, R.D. (2015). A study of students’ readiness to learn Calculus. Int. J. Res. Undergrad. Math. Ed. 1:209–233. DOI 10.1007/s40753-015- 0013-y.

[2] Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment (PCA) instrument: a tool for assessing reasoning patterns, understandings and knowledge of precalculus level students. Cognition and Instruction, 28(2):113–145.

[3] Piaget, J., Blaise-Grize, J., Szeminska, A., & Bang, V. (1977). Epistemology and psychology of functions. Dordrecht: Reidel.

[4] Lawson, A.E. (1978). The development and validation of a classroom test of formal reasoning. Journal of Research in Science Teaching, 15, 11–24. doi:10.1002/tea.3660150103.

[5] Bressoud, D.M. (2010). Historical reflections on teaching trigonometry. The Mathematics Teacher. 104(2):106–112.

Saturday, October 1, 2016

MAA Calculus Study: Women in STEM

It is nice to see that the national media has picked up one of the publications arising from the MAA’s national study, Characteristics of Successful Programs in College Calculus (NSF #0910240). It is the article by Ellis, Fosdick, and Rasmussen, “Women 1.5 times more likely toleave STEM pipeline,” that was published in PLoS ONE on July 13 of this year. The media coverage includes: well as a host of blogs and regional news sources.

The article was an outgrowth of the “switcher” analysis that Jess Ellis and Chris Rasmussen had begun, using data from our 2010 national survey to study who came into Calculus I with the intention of staying on to Calculus II but then changed their minds by the end of the course. You can find a preliminary report on the Ellis and Rasmussen switcher analysis in my column for December 2013, MAA Calculus Study: Persistence through Calculus and a further analysis of the differences between men and women in the November, 2014 column, MAA Calculus Study: Women are Different.

The 2013 column reported that women were about twice as likely as men to switch out of the calculus sequence, but those data were compromised by several lurking variables, most significantly intended major. Women are heavily represented in the biological sciences, much less so in engineering and the physical sciences. Since the biological sciences are less likely to require a second semester of calculus, some of the effect was almost certainly due to different requirements.

The study published in PLOS One controlled for student preparedness for Calculus I, intended career goals, institutional environment, and student perceptions of instructor quality and use of student-centered practices. They found that even with these controls, women were 50% more likely to switch out than men. As I discussed in my 2014 column, while Calculus I is very efficient at destroying the mathematical confidence of most of the students who take it, it is particularly effective for women (see Figure 1). As Ellis et al. report, 35% of the STEM-intending women who switched out chose as one of their reasons, “I do not believe I understand the ideas of Calculus I well enough to take Calculus II.” Only 14% of the men chose this reason.

Figure 1: Change in standard mathematical confidence at the beginning of the Calculus I semester (pre- survey) and at the end of the semester (post-survey) separated by career intentions, gender and persistence status, [N = 1524] doi:10.1371/journal.pone.0157447.g004

The last figure in the Ellis et al. article is enlightening (see Figure 2). If we could just raise the persistence rates of women once they choose enter Calculus I to match that of men, we could get a 50% increase in the percentage of women who enter the STEM workforce each year.

Figure 2: Projected participation of STEM if women and men persisted at equal rates after Calculus I. The dotted line represents the projected participation of women. doi:10.1371/journal.pone.0157447.g005

I believe that this issue of women’s confidence is cultural, not biological. It fits in with all we know about stereotype threat. When the message is that women are not expected to do as well as men in mathematics, negative signals loom very large. Calculus—as taught in most of our colleges and universities—is filled with negative signals.


Ellis, J., Fosdick, B.K., and Rasmussen, C. (2016). Women 1.5 times more likely to leave STEM pipeline after calculus compared to men: Lack of mathematical confidence a potential culprit. PLoS ONE 11(7): e0157447. doi10.1371/journal.pone.0157447

Thursday, September 1, 2016

CBMS and Active Learning

I have just accepted the position of Director of the Conference Board of the Mathematical Sciences (CBMS) and will be taking over from Ron Rosier at the end of this year. Most mathematicians, if they have heard of it at all, know of CBMS for its national survey of the mathematical sciences conducted every five years or for its regional research conferences. A few may know of CBMS through its forums on educational issues, its series on Issues in Mathematics Education, or the Mathematical Education of Teachers (MET II) report.

These have emerged from the core mission of CBMS, which is to provide a structure within which the presidents of the societies that represent the mathematical sciences [1] can identify issues of common concern and coordinate efforts to address them. This is exemplified in the joint statement on Active Learning in Postsecondary Mathematics [2] that was released this past July. This statement explains what is meant by active learning, presents the case for its importance, points to some of the published evidence of its effectiveness, lists society reports that have encouraged its use, and urges the following recommendation:

We call on institutions of higher education, mathematics departments and the mathematics faculty, public policy-makers, and funding agencies to invest time and resources to ensure that effective active learning is incorporated into postsecondary mathematics classrooms.

Ben Braun led the society representatives who drafted this position paper [2]. The presidents of all of the member societies with strong interest in mathematics education have signed onto it [3].

I see this statement as an example of what can be accomplished when the mathematical societies look to issues of common interest, and I am looking forward to working with them to coordinate efforts that will help colleges and universities identify and implement locally appropriate strategies for active learning.

I also hope to use my position to assist these societies in addressing the issues of articulation that so plague mathematics education. These include the transitions from two-year to four-year institutions, from undergraduate to either graduate school or the workforce, and from graduate school to either academic or non-academic employment. But the transition on which I am currently focusing my attention is from secondary to postsecondary education. This point of discontinuity is rife with difficulties for many of our students who would seek STEM careers as well those who have struggled with mathematics. It is especially problematic for students from underrepresented groups: racially, ethnically, by socio-economic status, by gender, and by family experience with postsecondary education.

The solutions—for there will be many pieces to be addressed if we are to succeed in ameliorating the problems—will require strong and coordinated efforts from both sides of the transition from high school to college. I am very encouraged by the clear messages of support for this work that I have received from NCTM, NCSM, and ASSM on the secondary side of the divide as well as AMS, MAA, AMATYC, ASA, and SIAM from the postsecondary side. CBMS is uniquely situated to bridge their work.

While I expect my primary focus to be on educational concerns, CBMS has and must continue to work on all matters of common interest including public awareness of the role and importance of mathematics, advocacy for programs that improve opportunities for underrepresented minorities, and issues of employment in the mathematical sciences.

I want to conclude by acknowledging the tremendous debt that the mathematical community owes to Ron Rosier and Lisa Kolbe who have been the entire staff of CBMS for roughly three decades. They have made this an effective organization. Under their direction, it has run smoothly and accomplished a great deal. They have left me with a very strong base on which to continue to build.


[1] The seventeen professional societies that belong to CBMS can be grouped into those that are primarily focused at the postsecondary level:

[2] Active Learning in Postsecondary Mathematics, available at
The writing team was led by Ben Braun and included myself as well as Diane Briars, Ted Coe, Jim Crowley, Jackie Dewar, Edray Herber Goins, Tara Holm, Pao-Sheng Hsu, Ken Krehbiel, Donna LaLonde, Matt Larson, Jacqueline Leonard, Rachel Levy, Doug Mupasiri, Brea Ratliff, Francis Su, Jane Tanner, Christine Thomas, Margaret Walker, and Mark Daniel Ward. The presidents of the member societies undertook the final wordsmithing.

[3] The presidents of INFORMS and SOA were the only ones who were not engaged in the formulation or signing of this position paper.

Monday, August 1, 2016

MAA Calculus Study: Placement

In November’s column, MAA Calculus Study: A New Initiative, I described a survey that MAA has conducted of practices for and concerns about the precalculus through calculus sequence at departments of mathematics that have graduate programs. The initial summary of the survey results is now available as Progress through Calculus: National Survey Summary, which can also be accessed through the Publications & Reports under Progress through Calculus on the web page Universities were distinguished by whether the highest degree offered in mathematics was a Masters or a PhD.

As I reported in November, placement was the number one issue among mathematics departments when comparing self-evaluation of importance to the program with confidence that the department is doing it well. Figure 1 shows that most PhD-granting departments rely on internally constructed instruments for placement.

Figure 1. Percentage of respondents using specific placement tools for precalculus/calculus.
Respondents could select more than one.
It is discouraging that a majority of Masters-granting departments and almost half of the PhD- granting departments use ACT or SAT scores for placement, instruments that are particularly ill suited to this purpose, even when only used to distinguish between placement into precalculus versus a previous course. It is also discouraging that so few PhD-granting universities use high school grades in determining placement. While not sufficient on their own, the study of Characteristics of Successful Program in College Calculus did reveal that including these grades improved departmental satisfaction with its placement decisions (see [1]). One of the striking results of the survey is that the number of PhD-granting departments using ALEKS increased from 10% in our 2010 survey to 28% in 2015. This may be somewhat misleading because the 2010 question only asked about placement into Calculus I, while the 2015 question asked about placement into precalculus or calculus, but from my own experience, the past several years have seen strong growing interest in and adoption of ALEKS.

Figure 2 shows the overall degree of satisfaction of the department with their placement procedures. Note that the bars above the placement tools represent degree of satisfaction with the entire placement procedure among those institutions that include this particular tool. Thus it does not necessarily reflect the degree of satisfaction with that particular instrument. Nevertheless, this does indicate that there is no single instrument that guarantees satisfaction.

Figure 2. Number of universities (out of 223) using each placement, with degree of
overall satisfaction with placement procedures.
Among all of the surveyed universities, 9% were not satisfied with their placement procedures, and 39% considered them adequate but could be improved. Even though 52% were generally satisfied, we found that there is a lot of churn in placement procedures: 30% of the universities had recently replaced or were currently replacing their placement procedures, and an additional 29% were considering replacing these procedures.

Perhaps the most interesting and potentially alarming result is that only 43% of respondents (45% of PhD-granting departments and 41% of Masters-granting departments) reported that they regularly review adherence to placement recommendations. It is hard to know how well your placement is working if you do not monitor it.

 [1] Hsu, E. and Bressoud, D. 2015. Placement and Student Performance in Calculus I. pages 59–67 in Insights and Recommendations from the MAA National Study of College Calculus, Bressoud, Mesa, and Rasmussen, editors. MAA Notes #84. Washington, DC: MAA Press.

Friday, July 1, 2016

MAA and Active Learning

There is a general perception among both research mathematicians and those working in our partner disciplines that—with a few exceptional pockets such as the community of those promoting Inquiry Based Learning (IBL)—the mathematical community is only now beginning to wake up to the importance of active learning. In fact, the MAA’s Committee on the Undergraduate Program in Mathematics (CUPM) began to promote the use of active learning in 1981 and has never ceased. It is a cry to which many have responded, but which has recently been rediscovered and promoted with urgency as chairs, deans, provosts, and presidents have come to realize that the way mathematics instruction has traditionally been organized cannot meet our present needs, much less those of the future. I was reminded of the origins of MAA’s support for active learning after encountering a particular piece of misleading data in Andrew Hacker’s The Math Myth, an unpleasant little book filled with half-truths, deceptive innuendo, and misleading statistics.

Hacker argues that the “math mandarins” cannot even attract students to major in mathematics and supports his argument with the fact that the number of Bachelor’s degrees in mathematics earned by US citizens dropped from 27,135 in 1970 to 17,408 in 2013. As a percentage of the total number of Bachelor’s degrees, the drop is even more impressive: from 3.4% to 1.0%. These numbers are symptomatic of Hacker’s deceptive use of data.

The first thing that is deceptive about these numbers is that they suggest a steady erosion of interest in mathematics. In fact, as Figures 1 and 2 show, the drop was precipitous during the 1970s, with the total number of Bachelor’s degrees in mathematics bottoming out in 1981 at 11,078, showing some recovery in the ‘80’s, followed by a steady decline until 2001 when it dipped back below 12,000, only 0.94% of Bachelor’s degrees. Since then, the growth has been reasonably strong, rising to 20,980 (of whom 2,438 were non-resident aliens) in 2014. That was back up to 1.12%.

(Note: The sharp increase in the early 1980s is almost certainly due to the high unemployment the United States was then suffering. Similarly, the noticeable increase in slope around 2010 is most probably a product of the unemployment rate that peaked in 2009.)

The other thing that is deceptive is the choice of when to start. The year 1970 came at the end of a strong national push for young people to enter mathematics and science. We had begun that decade in 1960 with only 11,399 mathematics degrees, though admittedly that was 2.9% of the total. Much of the loss during the ’70’s may be attributed to the creation of computer science majors. Bachelor’s degrees in computer science rose from 2388 in 1971 to 15,121 in 1981. Much, but not all. In fact, many members of the mathematical community were alarmed by this drop. Therein begins the story that is far more important than Hacker’s data.

CUPM was established in the early 1950’s to bring order to the chaotic assortment of courses that constituted mathematics majors across the country. In 1965 this committee of leading mathematicians published A General Curriculum in Mathematics for Colleges, which codified what by then was becoming the standard undergraduate major, beginning with three semesters of calculus and one semester of linear algebra. CUPM’s concern was almost entirely what to teach, not how to teach it. That changed in 1981 when Alan Tucker’s CUPM panel published Recommendations for a General Mathematical Sciences Program.

Concerned about the precipitous fall in the number of majors as well as enrollments in upper division courses, the attention in this report was focused on the goals of an undergraduate major and how they could be achieved. It laid out a five-point program philosophy that included an appeal to use active learning:
  1. “The curriculum should have a primary goal of developing attitudes of mind and analytical skills required for efficient use and understanding of mathematics … 
  2. “The mathematical sciences curriculum should be designed around the abilities and academic needs of the average mathematical sciences student … 
  3. A mathematical sciences program should use interactive classroom teaching to involve students actively in the development of new material. Whenever possible, the teacher should guide students to discover new mathematics for themselves rather than present students with concisely sculptured theories. (My italics.) 
  4. “Applications should be used to illustrate and motivate material in abstract and applied courses… 
  5. “First courses in a subject should be designed to appeal to as broad an audience as is academically reasonable …” 
 In the 1991 CUPM report, The Undergraduate Major in the Mathematical Sciences, chaired by Lynn Steen, the third point from 1981 was expanded to a clearer articulation of active learning.
III. Interaction. Since active participation is essential to learning mathematics, instruction in mathematics should be an interactive process in which students participate in the development of new concepts, questions, and answers. Students should be asked to explain their ideas both by writing and by speaking, and should be given experience working on team projects. In consequence, curriculum planners must act to assure appropriate sizes of various classes. Moreover, as new information about learning styles among mathematics students emerges, care should be taken to respond by suitably altering teaching styles.
The next report, CUPM Curriculum Guide 2004, chaired by Harriet Pollatsek, continued to build on the theme of how we teach. In this iteration, CUPM expanded its vision to all of the courses taught by departments of mathematics, insisting that “Every course should incorporate activities that will help all students progress in developing analytical, critical reasoning, problem-solving, and communication skills and acquiring mathematical habits of mind.”

This emphasis continues in the most recent CUPM guide, 2015 CUPM Curriculum Guide toMajors in the Mathematical Sciences, co-chaired by Carol Schumacher and Martha Siegel and edited by Paul Zorn. It begins with four “Cognitive Recommendations:"

  1. Students should develop effective thinking and communication skills. 
  2. Students should learn to link applications and theory. 
  3. Students should learn to use technological tools. 
  4. Students should develop mathematical independence and experience open-ended inquiry.

Throughout these decades, MAA has done more than issue recommendations. All of these reports have been backed up by MAA Notes volumes that have pointed to successful programs and explained how such instruction can be implemented within specific courses. (For a list of all Notes volumes, click here.) The Notes began in 1983 with Problem Solving in the Mathematical Sciences, edited by Alan Schoenfeld. MAA has run workshops as well as focused sessions and presentations at both national and regional meetings. Project NExT, MAA’s program for new faculty now in its third decade, has always had an emphasis on introducing newly minted PhDs to the use of active learning strategies.

It is hard to say whether these measures have been responsible for arresting and reversing the slide in the number of majors. Economic factors have certainly played a role. But MAA publications and activities have established a depth of experience and expertise within the mathematical community. Now that there is broad recognition of the importance of active learning strategies in the teaching and learning of undergraduate mathematics, we are fortunate to have this foundation on which to build.


Duren, W.L. Jr., Chair. 1965. A General Curriculum in Mathematics for Colleges. Berkeley, CA: CUPM.

Hacker, A. The Math Myth: and other STEM Delusions. New York, NY: The New Press.

Pollatsek, H., Chair. 2004. CUPM Curriculum Guide 2004. Washington, DC: MAA. departments/curriculum-department- guidelines-recommendations/cupm/cupm-guide- 2004

Schoenfeld, A.H., Editor. 1983. Problem Solving in the Mathematical Sciences. Washington, DC: MAA.

Schumacher, C.S. and Siegel, M.J., Co-Chairs, and Zorn, P., Editor. 2015. 2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences. Washington, DC: MAA. departments/curriculum-department- guidelines-recommendations/cupm

Steen, L.A., Chair. 1991. The Undergraduate Major in the Mathematical Sciences. Washington, DC: MAA.

Tucker, A., Chair. 1981. Recommendations for a General Mathematical Sciences Program. Washington, DC: MAA. Reprinted on pages 1–59 in Reshaping College Mathematics, L.A. Steen, editor. Washington, DC: MAA,