Thursday, December 1, 2016

IJRUME: Peer-Assisted Reflection

You can now follow me on Twitter @dbressoud.

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).

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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. See also Rasmussen and Ellis (2013).

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

Rasmussen, C., & Ellis, J. (2013). Who is switching out of calculus and why? In Lindmeier, A. M. & Heinze, A. (Eds.). Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 73-80). Kiel, Germany: PME.

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,

Wednesday, June 1, 2016

The Role of Calculus in the Transition from High School to College Mathematics

This past March I ran a small workshop in Washington, DC to look at “The Role of Calculus in the Transition from High School to College Mathematics,” funded by the National Science Foundation (#1550484). As the introduction to the workshop, cited below, makes clear, this is a critically important issue. Regular readers of my column will recognize that I return to it often. Part of the problem is that we don’t even know what we know and what we need to know (echoes of the “unknown unknowns”). The stated purpose was to bring together stakeholders to
  1. Clarify the current state of knowledge of the effects on college performance and retention in STEM disciplines of acceleration into calculus at or before 12 th grade, as well as the effects of lack of access to calculus while in high school.
  2. Identify the most pressing research questions.
  3. Suggest strategies for answering these questions and identify the appropriate researchers or organizations to tackle them.
As it turned out, my hopes for the workshop were too ambitious. We were not able to leave with a research agenda, but we did make progress toward identifying what we know about the problems surrounding this transition and in clarifying the complex issues surrounding the movement of calculus into the high school curriculum.

The two days of the workshop provided an opportunity for a rich exchange. Participants included post-secondary mathematics faculty, high school calculus teachers, state and district supervisors of mathematics, researchers in both K-12 and post-secondary mathematics education, as well as representatives of The College Board, the National Council of Teachers of Mathematics, the National Research Council’s Board on Science Education, the National Math and Science Initiative, and Achieve. What is most gratifying is that all of these players recognize that there are serious problems around issues of preparation for post-secondary mathematics as well as equity and access that are being exasperated by the tremendous pressures to bring students into calculus ever earlier in their high school careers. My intention is that over the coming months it will be possible to build on the foundation laid by this workshop.

Before concluding with the preamble that I wrote for this workshop, I would like to send out an appeal for all readers who would like to contribute to this conversation to send me their thoughts at, subject line: Role of Calculus.

Last year, at least three quarters of a million U.S. high school students were enrolled in a calculus class.[1] This was three times the number of U.S. students who took their first calculus class in college. High school calculus enrollments are still growing at roughly 6% per year,[2] with increasing pressure on the most advantaged students to take calculus ever earlier. In 2015, over 120,000 students took the AP Calculus exam by the end of grade 11, and these numbers are growing by 9% per year. [3]
If we make the assumption that most of the students who enroll in calculus in high school will go on to matriculate as full-time students in a four-year undergraduate program—of whom there are 1.5 million each year[4]—then roughly half of these full-time, first-year students enter having already studied calculus. Calculus in high school is now commonly perceived as a prerequisite for college admission, with the result that high schools must start students with Algebra I in 8th grade high school if they are to be on track for calculus by grade 12, and calculus teachers find themselves under increasing pressure from parents and administrators to admit into their classes students they know are not adequately prepared. How do we provide support and alternatives for those students who cannot handle this accelerated progression?
We have learned from our study of Characteristics of Successful Programs in College Calculus (NSF DRL 0910240) that a quarter million of the students who study calculus in high school will retake mainstream Calculus I at the post-secondary level, and 40% of these quarter million will fail to get the A or B that signals they are prepared to continue in mathematics.[5] College faculty are very much aware that many of the students who enter with calculus on their high school transcript are, in fact, not ready for college-level mathematics. Of particular concern is that we know almost nothing about what happens to the remaining half million students who have studied calculus in high school: How many take advantage of advanced placement, and how well prepared are they? What are the mathematical trajectories of those who do not take calculus in college? How has the experience of calculus in high school shaped the aspirations and attitudes of these students and their ability to continue on toward mathematically demanding careers? 
At the same time, half of all U.S. high schools do not even offer calculus. [6] Students from underrepresented groups, even when they are in a high school that offers calculus, are often discouraged from enrolling in this course. Those who have not taken calculus in high school find themselves in competition against students with a much richer preparation. What does this do to their chances for admission to college or the pursuit of a STEM major? 
The questions are manifold. The goal of this workshop is to better understand these questions and to begin to develop strategies for answering them. Specifically, we seek to
  1. Clarify the current state of knowledge of the effects on college performance and retention in STEM disciplines of acceleration into calculus at or before 12th grade, as well as the effects of lack of access to calculus while in high school.
  1. Identify the most pressing research questions.
  1. Suggest strategies for answering these questions and identify the appropriate researchers or organizations to tackle them.


[1] Based on the NCES longitudinal study (HSLS:09) reporting that 19% of the four million students who were in 9th grade in 2009 had taken a calculus course in high school by the time they graduated.

[2] From the College Board’s AP Program Summary Reports from 2002 to 2015.

[3] Ibid.

[4] Higher Education Research Institute.The American Freshman: National Norms Fall 2015.

 [5] Data from Characteristics of Successful Programs in College Calculus project’s maalongdatafile. See David Bressoud. 2015. Insights from the MAA national study of college calculus. The Mathematics Teacher. 109 3:179–185.

[6] U.S. Department of Education Office of Civil rights. Issue Brief No. 3 (March, 2014). and-Career- Readiness-Snapshot.pdf

Sunday, May 1, 2016

Reflections on a Career in Teaching

Monday, May 2 is my last day of teaching. Not that I will never step into a classroom again, but it marks the end of my full-time employment by Macalester College. My responsibilities for the future will not require any teaching. I am looking forward to the freedom this phased retirement will bring to focus on my writing and other educational activities, but I also approach this date with some sense of loss. For forty years, I have defined myself as a teacher first. I am taking advantage of this column to reflect on a few of the lessons I have learned over these four decades in the classroom.

Class on Revolutions, taught at Penn State, Spring 1993, with John Harwood and Phil Jenkins (whose daughter Catherine is also in the picture). Taken on a field trip to the Joseph Priestley House, Northumberland, PA.

Students absorb far less than we think they do. As teachers, we must identify what is truly important and do all in our power to ensure that students are internalizing this knowledge.

The first time this reality hit home for me was a differential equations class I taught at the University of Wisconsin in 1980–81, before the days of computers in the classroom. As taught back then, it was all about finding exact solutions to certain limited classes of differential equations. I thought I did a pretty good job of explaining the motivation and usefulness of these techniques, but homework and midterm exams were focused on finding these exact solutions. For the final exam, I wanted to make it interesting by presenting situations that my students would have to model using one of the types of differential equations we had studied, and then they would solve it. Not only did the students totally bomb this exam, they were angry that I was asking them to do something—modeling real world situations with these differential equations—that they had never been asked to do during the semester.

They were absolutely right. Students learn by doing, not by listening. Knowing how to solve a differential equation is very different from knowing how to use a differential equation. It is far too easy to focus our instruction on the easily tested mechanical skills and assume that understanding and an appreciation for context will come as fortuitous byproducts.

My second major experience of this truth occurred in a minicourse run by Kathy Heid and Joan Ferrini-Mundy at the 1994 Joint Math Meetings (in frigid Cincinnati). The theme was the use of student interviews as a tool for deep assessment of student learning. I practiced on a volunteer, a student from a local university who had completed four semesters of calculus and was now a junior. Eight months after the last of these classes, I wanted to find out what he had retained. I was horrified. As much as I pressed, differentiation for him carried no connotations beyond a method for turning functions into simpler functions (in the sense that 3x2 is “simpler” than x3), and integration reverses that process. Nothing about tangents or rates or areas or accumulations had stuck. It drove home to me how little it is possible to learn while still passing a mathematics class.

These experiences explain why I am such a fan of Inquiry Based Learning (IBL). It uses class time to focus on what we really want students to learn and provides a means of constantly probing student understanding. In my columns from February and March of this year, What We Say/What They Hear and What We Say/What They Hear II, I described work in mathematics education that has validated my observation that what we think we are communicating is not what most students hear. I appreciate Stan Yoshinobu’s recent blog, A Practical Solution to “What We Say/What They Hear,” that illustrates how an IBL approach can address this problem of conveying meaning.

Very little of what is learned at the post-secondary level happens inside the classroom. As teachers, we have a responsibility to structure how students interact with the mathematics beyond the classroom walls.

This point really struck home in 1990–91, the year I taught AP Calculus at the State College Area High School. The leisurely pace through the AB curriculum meant that I could actually watch light bulbs coming on in my class, something that I had never witnessed in the large lecture calculus classes at Penn State. This has always been for me the greatest distinction between high school and college mathematics. For the latter, students have to know how to learn outside of class from notes and textbook and in exchanges with other students.

My last year at Penn State, 1993–94, I surveyed students in one of these large lecture sections to try to understand their experience. I found that most of them were very conscientious about studying, usually spending about two hours each evening that followed class going over that day’s lesson. But I also found that most of them had no idea how to study. They would read through their notes, paying particular attention to the problems that had been worked out in class, and then they would tackle problems from the end of that section, practicing the techniques that they had seen demonstrated. Anything that strayed too far from what had been worked by the instructor was considered irrelevant.

Uri Treisman has shown the importance of students working together to clarify understandings (see [1]). I have found that if I want students to think more deeply about the mathematics introduced in class and to share those understandings with others, I have to structure out-of-class assignments designed to accomplish this.

Final exams carry far too much weight. As teachers, we need to administer frequent and varied assessments that truly measure what our students are learning and that provide opportunities for students to learn.

I hate final exams. I have had colleagues who promise students that if they do really well on the final, any poor test results earlier in the term will be forgiven. I have even seen students who manage to pull off a superior performance on the final despite a record leading up to it that would not have predicted this. Over forty years, I can count them on one hand. On the other side, I have seen many of my students who were steady and successful during term completely fall apart at the final. It is a stressful time, not just in my class but in almost every class a student is taking. It is a time of late nights and cramming and incredible pressure. The final straw for me was about fifteen years ago when there was a major incident of cheating on my final exam. This was by good students who had been doing well and who I knew did not need to cheat, but they were overwhelmed by the fear of doing poorly on this major component of their final grade for which there would be no opportunity to overcome a poor result.

My policy now is—with rare exceptions—to never count a final exam for more than 15% of the total grade. As a complement to this, I test early, trying to get the first major assessment in by the end of the fourth week of class; I test often; and I assess student performance using a wide variety of measures: in class tests and quizzes that focus on procedural knowledge, take-home tests with problems that challenge students to apply their knowledge in unfamiliar situations requiring multiple steps, in multi-week projects that will be critiqued and returned for revision, in Reading Reflections—short answers to questions about the material read before class that help inform me before class begins of what students do and do not yet understand, and in written questions collected at the end of each class. Many of these have been inspired by Angelo and Cross’s Classroom Assessment Techniques, but there is nothing I have picked up from that book that I have not reshaped to fit my own style and needs.

I also believe in looking for ways to enable students to learn from the assessments I use. Every major project is turned in for feedback before the final submission. For each exam during the semester, students are allowed to earn back some of the lost points by explaining where they went wrong and how to do the problem correctly. By giving myself some flexibility on how much of the grade students can earn back, I find I that I can give very challenging exams without needing to grade on a curve. Grading on a curve is a practice I consider to be particularly pernicious because it communicates to students that they are competing against each other, that what matters is less how much you have learned than how much better you can perform than your neighbor.

I especially value group projects as a tool for teaching as well as assessing student learning. Most of my classes include several major projects. It was my last year at Penn State when I had the great good fortune to be able to teach the early Project CALC materials with David Smith present on campus (on sabbatical from Duke). I was able to meet with him weekly over lunch to talk about how the course was going. I have been able to watch how effective students are at teaching each other. It still never ceases to amaze me that I can say something in class without it registering with some student until the person next to them restates it as his or her own insight, though often verbatim, in a private conversation around solving a problem they are working on. I have found that group grades are problematic, and have experimented with a variety of techniques over the years to make them fairer. Now, as much as possible and for at least one project per class, everyone in the group is required to write up his or her own report of what the group has found. Clarity of exposition is every bit as important as the correctness of the results, a requirement that usually catches out those who simply attempt to reproduce the work done by other members of the group.

With many writing assignments, projects, and exams, my students always complain about how much work they have to do for my classes. But no single assessment counts for very much, and my students have many opportunities to learn and recover from a bad performance. I am proud of the fact that it has been many years since a student complained that my grading was not fair. I am also proud that, despite complaining about the amount of work, my students also note how much they have learned.

Most students get too few opportunities to appreciate the culture of mathematics. As teachers, our instruction should communicate the true nature of mathematics.

I find it deeply discouraging that so many students graduate from college without any appreciation for mathematics as a rich venue for discovery and innovation. Burger and Starbird have done an excellent job of communicating this side of mathematics in The Heart of Mathematics. I have tried to do it through the history of the subject. This is reflected in all of the textbooks that I have written, and my students will attest that I am constantly interjecting the history of the subject into my classes.

This spring, as a swan song, I am teaching a 100-level class on the history of mathematics for the first and last time. Fifty students are enrolled, most of whom are taking it to satisfy Macalester’s quantitative reasoning requirement. I am using Berlinghoff and Gouvêa’s Math through the Ages, a perfect book for my vision of this course. Their text consists of a stripped down history of mathematics, complemented by thirty short vignettes that survey topics from the development of negative numbers to non-Euclidean geometries to the rise of the computer. I use class time to tell the stories I love and introduce my students to the people who have been instrumental in the development of this vast subject. The real learning takes place in the writing students are required to do: short questions that must be submitted at the end of each class and from which I pick a few to answer at the start of the next, Reading Reflections in which students must tie what they have read to their own experiences, and many short papers in which they must pursue some of the many references provided by Berlinghoff and Gouvêa and explain to a younger version of themselves something about mathematics that they wish they had known earlier in their mathematical career. These have been especially insightful. I have seen so many of my students who were interested in, even excited about mathematical ideas early in their schooling, but had had that interest pounded out of them. It is emotional for me to see them, jaded as they now are about mathematics, reaching back to that younger self, trying to blow that ember back to life.

None of the insights I have presented here are particularly original. Others have described them much more eloquently. But for me they are hard-won truths achieved through years of constantly striving to be a better teacher. After all, that is how we learn, not by listening to someone else expound or by reading a book or column, but by observantly striving to master our chosen profession. What we hear or read can suggest fruitful directions in which to explore and grow. Ultimately, this is our challenge, to constantly seek to improve how we teach.

[1] Eric Hsu, Teri J. Murphy, Uri Treisman. 2008. Supporting high achievement in introductory mathematics courses: What we have learned from 30 years of the Emerging Scholars Program. Pages 205–220 in M.P. Carlson & C. Rasmussen (eds.), Making the Connection: Research and Teaching in Undergraduate Mathematics Education. MAA Notes #73. Washington, DC: Mathematical Association of America.

Friday, April 1, 2016

A Common Vision

Five major mathematical societies—AMATYC, AMS, ASA, MAA, and SIAM—have just released a joint report, A Common Vision for Undergraduate Mathematical Sciences Programs in 2025, authored by Karen Saxe and Linda Braddy and distributed by the MAA. This is a coordinated call for “modernizing undergraduate programs in the mathematical sciences.” For many years, all five of the societies have been proclaiming the need to improve the teaching and learning of undergraduate mathematics and statistics. For the first time, they have come together to identify their common concerns and to support shared recommendations. What follows is my own précis of the contents of this report. I strongly encourage you to read it for yourself. A full reference, with the address of the PDF file of the report, is at the end of this column.

The central message, repeated throughout this document, is that “The status quo is unacceptable.” Specifically, A Common Vision issues a joint appeal to
  1. Update curricula, 
  2. Articulate curricula across the critical divide between high school and college mathematics,
  3. Scale-up the use of evidence-based pedagogical methods, 
  4. Remove barriers at critical transition points, and 
  5. Establish stronger connections to other disciplines.
These are accompanied by a call to support those faculty engaged in these efforts. After explaining the need for these changes and summarizing the reports that have been issued by the five societies, A Common Vision describes in further detail the common themes that have emerged:

Curricula. The greatest concentration of themes reported in A Common Vision circles around curricular issues. These include calls for presenting key ideas from a variety of perspectives and motivating them through the use of applications to contemporary topics. The cited reports recognize the importance of providing multiple pathways into and through undergraduate mathematics, with particular concern that departments attend to the barriers that students often confront. Solutions should include entry points that emphasize modeling, statistics, and applications as well as programs that focus on the development of computational and statistical skills.

There is a recognized need for more statistics, computation, and modeling for all students within the first two years of undergraduate mathematics. And there is recognition of the need for closer cooperation with and awareness of the needs of other disciplines. The report includes a call for more attention to the development of the skills needed for effective mathematical communication, both orally and in writing. And, finally, this report highlights the common awareness among the five societies of the need to address issues of transition: from high school to college, in transfer between institutions, in issues of placement, and at critical juncture points such as the start of proof-based courses.

Course Structure. There is a consensus among all of the societies that instruction needs to move beyond simple lecture and embrace a variety of active learning approaches that engage students in grappling with the difficulties of mathematics. These include providing opportunities for collaboration and communication. In addition, all five societies advocate the use of technology in those situations where it can enhance student learning.

Workforce Preparation. Mathematics departments need to work with those in client departments within their own institutions as well as with the consumers of our graduates in business, industry, and government to understand the workforce skills that graduates will need. This must be done not as narrow technical training but in the recognition that our task is to equip students with a broad base of skills that will serve them in our rapidly evolving economy.

Faculty Development and Support. All of the five societies recognize the need to provide training opportunities for faculty to broaden their expertise in areas of the mathematical sciences where great needs have not been met. These include data analytics and computational science. We also must foster an institutional culture that encourages and values work on the issues raised in these reports.

Other Issues. In addition, other issues have arisen in one or more of the society reports. These include the need
  1. To attend to issues of student diversity, particularly the retention of students in at- risk groups,  
  2. To ease difficulties as students move between institutions, 
  3. To recognize the special needs of contingent faculty, 
  4. To devote energy toward the preparation of K-12 teachers, 
  5. To properly prepare graduate students for their contributions to the teaching mission of the department, 
  6. To recognize and address the issues that lead to high failure rates, 
  7. To look for ways of improving courses in developmental mathematics so that they retain and adequately prepare students, 
  8. To shape calculus instruction so that it responds to the reality that most students studying calculus in college have already experienced it in high school,  
  9. To be aware of technology-enabled models of delivery of course content and to critically consider when and where they might be beneficial, 
  10. To gather and use empirical data to refine programs and improve student learning, 
  11. To scale successful efforts by involving more faculty within each department, by increasing communication about these efforts within the mathematical sciences community, and by understanding the obstacles to effective transfer of successful programs.

This report, which is only intended to be a summary of the common themes of these five societies, is nevertheless an important first step in recognizing the commonalities in the messages they are all sending and in working toward coordinated efforts to improve undergraduate education in the mathematical sciences.


Karen Saxe & Linda Braddy. 2016. A Common Vision for Mathematical Sciences Programs in 2025. Forward by William “Brit” Kirwan. Washington, DC: Mathematical Association of America.

Tuesday, March 1, 2016

What we say/What they hear. II

In last month’s column, I introduced recent research by Kristen Lew, Tim Fukawa- Connelly, Juan Pablo Mejia-Ramos, and Keith Weber on the difficulties students encountered in picking out the points that the instructor wanted to emphasize. One of the lessons, of course, is that if you want to ensure that students note and remember a particular message that you, the instructor, wish to make, it is not enough to say it. You also need to write it. But something deeper is also at work. I appreciate that in response to my column Pat Thompson sent me copies of two of his articles on issues of meaning when teaching mathematics (see references).

Pat begins by describing the work of Dewey, Piaget and others who explained that the communication of meaning lies at the heart of effective teaching. However, communicating meaning is extremely difficult. As Pat says,

Figure [1] shows Persons A and B attempting to have a meaningful conversation. Person A intends to convey something to Person B. The intention is constituted by a thought that A holds that he wishes B to hold as well. The figure shows A not just considering how to express his thought, but considering how B might interpret A’s utterances and actions. It is worthwhile noting that A’s action towards B is not really towards B. A’s action towards B is towards A’s image of B. In a sophisticated conversation A’s action towards B is not just towards B, but it’s towards B with some understanding of how B might hear A. Likewise, B is doing the same thing. He assimilates A’s utterances, imbuing them with meanings that he would have were he to say the same thing. But B then colors those understandings with what he knows about A’s meanings and according to the extent to which A said something differently than B would have said it to mean what B thinks A means. B then formulates a response to A with the intent of conveying to A what B now has in mind, but B colors his intention with his model of how he thinks A might hear him, where the model is updated by anything he has just learned from attempting to understand A’s utterance. And so on. (Thompson, 2013, p. 63)

Figure 1. Summary of intersubjective operations involved in the communication of meaning. (Thompson, 2013, p. 64)

The problem is that just because each party has a mental image of the other as understanding their meaning is no guarantee that there is such a mutual understanding:

In Piaget’s and Glasersfeld’s usage, A’s and B’s conversation enters a state of intersubjectivity when neither of them has a reason to believe that he has misunderstood the other. They may in fact have completely misunderstood each other, but they have not discerned any evidence of such. (Thompson, 2013, p. 64)

I’d like to offer my own interpretation of what was happening in the class that Lew et al. observed. This is pure speculation, but it is based on more than forty years of teaching. I believe that the instructor and the students had attributed very specific and very different meanings to the proof that was presented in class.

To the instructor, this proof was an opportunity to showcase general approaches. The fact is that the theorem that was proven, “If a sequence {xn} has the property that there exists a constant r with 0<r<1 such that |xn–xn–1| < rn for any two consecutive terms in the sequence, then {xn} is convergent,” is not particularly important to the study of convergence. What is clear from the five points that instructor believed he had made was that this provided an opportunity to showcase the usefulness of the Cauchy criterion, the triangle inequality, and the geometric series. This was his meaning. The ease with which peers identified the majority of these messages signifies that they shared his image of the meaning of this example.

The student inability to recognize the points that the instructor had intended to convey suggests that their meaning for this proof was very different. They probably understood the instructor’s intention as one of communicating that this is a valid result worthy of being noted and remembered. Just laying out a formal proof immediately communicates this message to most students in real analysis. The fact that the only things written on the board were the steps in the proof of this result almost certainly reinforced their belief that it was the validity and significance of this statement that was the instructor’s meaning.

I suspect that, had the instructor written his five points on the board, that might have succeeded in shifting the understanding by some of the students of the instructor’s meaning for this proof. But I would be willing to wager that not all of them, probably not even a majority of them, would have seen these as being as important as the actual statement of the theorem. Their reluctance to even recognize that the instructor had made particular points when these were singled out from the lecture suggests that just writing them on the board would not have been sufficient.

Anyone who has probed student understanding has seen this miscommunication. This raises the obvious question: How do we manage to establish a shared understanding? Certainly, a lecture format with only occasional interaction between instructor and students is fertile ground for intersubjectivity that has nothing to do with mutually shared meanings. This is where clickers can help, especially in large format classes. But their effective use relies on a thorough understanding of the range of possible student understandings of the meanings of the lesson. And this understanding must be accompanied by careful construction of questions that can both identify miscommunication and create the cognitive dissonance that moves students toward understanding the instructor’s meaning.

Flipped classes can be even more effective in establishing common meanings, but they also are not easy to run effectively. The work that is done in class must be carefully tailored to identify student misinterpretations of the intent of the lesson, complete with leverage points for addressing these misunderstandings. It is too easy for a flipped class to degenerate into supervised practice. How much more instructive it would have been for the instructor to ask the students to work on a proof of the stated result, emphasizing the usefulness of each of the tools needed for the proof as students discovered—or were led to discover—them. And, of course, you do not just do this once. You need the students to encounter multiple instances where these tools are useful before they fully grasp their versatility and importance. This approach is not easy. It requires an instructor who is finely attuned to the knowledge and the ability to draw on that knowledge of each of the students. And it requires a considerable investment of time. Such an approach takes far more than the ten minutes the instructor actually spent on this theorem.

However, as the work in this study revealed, that ten minutes was largely wasted. The intended messages were never heard. If these were important messages, and I think that most of us who teach real analysis would acknowledge that they are, then they are worth the effort to communicate this importance. Inevitably, that will require “covering” less material. It forces the instructor not only to prioritize the understandings she intends that students carry away from this course, but also to prioritize her efforts to determine what students think she is saying.


Lew, K., Fukawa-Connelly, T., Mejia-Ramos, J.P., and Weber, K. 2016. Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey. Journal for Research in Mathematics Education. Preprint retrieved from on January 24, 2016.

 Thompson, P. W. (2013). In the absence of meaning… . In Leatham, K. (Ed.), Vital directions for research in mathematics education (pp. 57-93). New York, NY: Springer.

Thompson, P. W. (2015). Researching mathematical meanings for teaching. In English, L., & Kirshner, D. (Eds.), Third Handbook of International Research in Mathematics Education (pp. 435-461). London: Taylor and Francis.

Monday, February 1, 2016

What we say/What they hear

An important paper is about to appear in the Journal for Research in Mathematics Education, exploring why lecture is so ineffective for so many students: “Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey” by Kristen Lew, Tim Fukawa-Connelly, Juan Pablo Mejia-Ramos, and Keith Weber. The authors video-taped a portion of a lecture given in a junior-level real analysis course and performed a detailed analysis of the differences between what both the professor and his peers thought had been conveyed and what the students were able to take from it.

The study used a class by a professor at a large public university who is widely recognized as an excellent lecturer. It focused on a 10-minute stretch in which a proof was presented. The theorem in question is, “If a sequence {xn} has the property that there exists a constant r with 0 < < 1 such that |xn–xn–1| < rn for any two consecutive terms in the sequence, then {xn}is convergent.” The four authors of this paper and an additional instructor who teaches real analysis each observed the video and noted the messages that they saw the professor conveying. They then interviewed the professor who identified five messages that he was trying to convey during this lecture. These are listed below. All except the first had been noted by all of the other peer observers. A full transcript of what transpired during these 10 minutes is included in the appendix to the paper. You may want to check whether you can see these points.

  1. Cauchy sequences can be thought of as sequences that “bunch up”
  2. One can prove a sequence with an unknown limit converges by showing it is Cauchy
  3. This shows how one sets up a proof that a sequence is Cauchy
  4. The triangle inequality is useful in proving series in absolute value formulae are small
  5. The geometric series formula is part of the mathematical toolbox that can be used to keep some desired quantities small

Six students from this class agreed to participate in the extensive interviews required for the study. They were put into three pairs in order to encourage discussion that would help draw out and verbalize what they remembered.

About two or three weeks after the class in question, students were asked to review their notes about this proof and identify the points that the professor had made. These were compared with the professor’s five points. None of the pairs brought up any of the instructors messages. This is not particularly surprising. Students tend to restrict what they write in their notes to what is being written on the board, and all five of the professor’s points had only been made orally.

As a second pass, each of the students was given a transcript of all that had been written on the blackboard during this proof and then watched the 10-minute lecture, with the hope that they could now focus on what was being said rather than what had been written. They were again asked to identify the points that had been made. One pair did note the emphasis on the importance of the triangle inequality. Another pair noted the third point, that this was about how to set up a proof that a sequence is Cauchy. Nothing else from the list was mentioned.

At a third pass, the students were shown just the five short clips where these five points had been made. Two of the pairs now picked up the first message, two picked up the second, and two picked up the fourth. No one new picked up the third point, that the professor had been illustrating a general approach to proving that a particular sequence is Cauchy.

Finally, the students were told that these five messages might have been contained in the lecture and were asked whether, in fact, these points had been made. Now most of the students were able to see most of these messages, but one pair never acknowledged the second point, that one way to prove that a sequence converges is to show that it is Cauchy, and, even after seeing the clip in which this point was made, none of them acknowledged that the professor had made the fifth point: that the geometric series is part of the toolbox for approaching such proofs.

What I find particularly interesting is the sharp distinction between what was seen in this lecture by those who are familiar with the material and what was seen by those who are still struggling to build an understanding. This echoes much of the work of John and Annie Selden who have shown how difficult it is for undergraduate students to extract the significant features of a proof. This paper shows that it is not enough to accompany what is written on the board with oral indications of what is important and how to think about it. It is not even enough when these indications are repeatedly emphasized.

In the introduction, this paper presents the example of the Feynman Lectures, widely considered to be some of the finest scientific expositions ever made. Yet, the fact is that when they were given at Cal Tech, “Many of the students dreaded the course, and as the course wore on, attendance by the registered students dropped alarmingly.” (Goodstein and Negebauer, 1995, p. xxii–xxiii). There is no doubt that lectures have an important role to play in conveying information for which the recipients have a well-structured understanding in which to place it. However, as this study strongly suggests, lectures are not very helpful for students who are trying to find their way into a new area of mathematics and who still need to build such a structure of understanding.


Goodstein, G. & Negebauer, G. 1995. Preface to R. Feynman’s Six Easy Pieces. pp. xix–xxii. New York: Basic Books.

Lew, K., Fukawa-Connelly, T., Mejia-Ramos, J.P., and Weber, K. 2016. Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey. Journal for Research in Mathematics Education. Preprint retrieved from on January 24, 2016.

Friday, January 1, 2016

MAA Calculus Study: Building Networks

I am beginning this month’s column with the announcement of a conference and workshop that should interest readers of this blog. A discussion of the background and context for the conference will follow the announcement.

Announcing an NSF-sponsored MAA Conference on
Precalculus to Calculus: Insights & Innovations 
June 16–19, 2016  
University of Saint Thomas, Saint Paul, Minnesota 

To be followed immediately by a workshop on
 Curriculum, Instruction, and Placement in Algebra and Precalculus 
June 19–20, 2016
Same location

The conference will provide opportunities to learn from the MAA’s studies of precalculus and calculus, to hear what is happening at peer institutions, and to build networks of shared experience and practice. The two and a half days will be built around four themes:

Focus on Curriculum. Content of and alternative approaches to precalculus, articulation issues, preparation for downstream courses

Focus on Students. Placement, early warning systems and support services, formative and summative assessment, supporting students from underrepresented groups

Focus on Pedagogy. Active learning strategies, making the most of large lectures, use of Learning Assistants, assessing effectiveness of innovations

Focus on Instructors. Building communities of practice, training of graduate teaching assistants, working with adjuncts, getting faculty buy-in for innovative practices

The workshop will be an opportunity to learn from the work of Marilyn Carlson, Bernie Madison, and Michael Tallman on Using Research to Shape Instruction and Placement in Algebra and Precalculus (NSF #1122965).

There is no registration fee. Housing and meals are included at no cost to participants. Participants are responsible for their own transportation. Housing will be in the air- conditioned apartments in Flynn Hall. Each apartment consists of four single bedrooms, two bathrooms, and a kitchen and living room. The University of Saint Thomas sits on a bluff above the Mississippi River, six miles from the Minneapolis/St. Paul airport and midway between the downtowns of Minneapolis and Saint Paul.

The number of participants accepted to the conference and workshop will be limited. A link to the application to attend the conference and workshop is at

Review of applications will begin March 15. Those accepted will be notified by April 1.

This conference combines the efforts of the two studies on which I have been PI: Characteristics of Successful Programs in College Calculus (CSPCC, NSF #0910240) and Progress through Calculus (PtC, NSF #1430540). Part of its role is to disseminate results from CSPCC, many of which can also be found in the Notes volume Insights and Recommendations. But the more important task is to foster the building of networks of peer colleges and universities who are seeking to improve the effectiveness of their precalculus through calculus sequences. The four themes reflect the four areas of concern and ongoing work that have emerged from our surveys and from the meeting held in Washington, DC over the October 31 to November 1 weekend.

The DC workshop brought together representatives from 27 universities that either are now engaged in initiatives to improve this sequence or are seriously concerned about lack of student success in these courses and are looking to improve what they do. There were many common interests and concerns that emerged. I want to acknowledge the role of Naneh Apkarian, assisted by the other graduate students, who monitored the discussions and summarized the issues. These included:

  • Aligning precalculus/calculus courses to create more coherent programs based on student and client discipline needs (with an emphasis on the transition from precalculus to calculus)
    • What is “precalculus?” (content, purpose, function)
    • Aligning precalculus so that it is truly a preparation for calculus
    • Dealing with the multiple purposes for a variety of students (e.g., preparation, gen. ed., STEM, business)
  • Encouraging/Supporting/Implementing Active Learning
    • Especially when the institution insists on large classes 
  • Information about flexible and/or non-standard models for the precalculus/calculus 
  • GTA Training Programs 
    • Specifically with regards to issues surrounding active learning
  • Student skill retention within and across courses 
  • Making calculus accessible for students from varying backgrounds
    • Can it be done in one classroom, or are “flavors” needed? 
  • Placing students into appropriate courses and then supporting them
    • Establishing what various high school calculus courses really are
    • Early warning systems
    • Various pathways through calculus
  • Professional development/Increasing faculty buy-in
    • With respect to active learning
    • Identifying ways of supporting faculty interested in using active learning strategies
    • With respect to utilizing technology to support student learning
  • Strategies for increasing administrative support/handling administrative pressures 
  • Collecting and managing data 
The Saint Paul conference in June will be an opportunity to learn what is known about these issues, with examples of successful or promising interventions. In response to the request for networking opportunities to share information about materials, case studies, guidelines, and the experiences of peer institutions, we have established a website, the PtC Discussion Group, on a new platform, Trellis, managed by AAAS. To join this discussion, go to, register, then search for the PtC Discussion Group and request to join.