Saturday, December 1, 2012

Mathematics and the NRC Discipline-Based Education Research Report

This past spring, the National Research Council of the National Academies released its report, Discipline-Based Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering [1]. The charge to the committee writing this report was to synthesize existing research on teaching and learning in the sciences, to report on the effect of this research, and to identify future directions for this research. The project has its roots in two 2008 workshops on promising practices in undergraduate science, technology, engineering, and mathematics education.

Unfortunately, between 2008 and 2012 undergraduate mathematics education dropped out of the picture. The resulting report discusses undergraduate education research only for physics, chemistry, engineering, biology, the geosciences, and astronomy. Nevertheless, it is an interesting report with useful information—especially the instructional strategies that have been shown to be effective—that is relevant for those of us who teach undergraduate mathematics.

The studies that are described are founded on the assumption that students must build their own understanding of the discipline by applying its methods and principles, and this is best accomplished within a student-centered approach that puts less emphasis on simple transmission of factual information and more on student engagement with conceptual understanding, including active learning in the classroom.

The great strength of this report is the wealth of resources that it references and the common themes that emerge across all of the scientific disciplines.  A lot of attention is paid to the power of interactive lectures. Given that most science and mathematics instruction is still given in traditional lecture settings, finding ways of engaging students and getting them to think about the mathematics while they are in class is essential for increasing student understanding.

The recommendations of effective practice range from simple techniques, such as starting each class with a challenging question for students to keep in mind, to transformative practices such as collaborative learning. A common intermediate practice involves student engagement by posing a challenging question, having students interact with their peers to think through the answer, and then testing the answer. In some respects, this is more easily done in the sciences where student predictions can be verified or falsified experimentally. Yet it is also a very effective tool in mathematics education where a well-chosen example can falsify an invalid expectation and careful analysis can support correct understanding. But  most important is that it forces to try to use what they have been learning.

In large classes, this type of peer instruction can be facilitated by the use of clickers. The report does include the caveat, with supporting research, that merely using clickers without attention to how they are used is of no measurable benefit.

The greatest learning gains that have been documented occur when collaborative research is incorporated into the classroom. The NRC report includes many descriptions of how this can be accomplished in a variety of scientific disciplines. It also references the research that has established its effectiveness. Again, attention to how it is done is an important component of effective practice.

Two of the areas that are identified as needing more research are issues of transference (see my September column on Teaching and Learning for Transference) and metacognition. Usefully, the authors point out that there are two sides to transference: the ability to draw on prior knowledge and the ability to carry what is currently being learned to future situations. Metacognition is an important issue in research in undergraduate mathematics education, especially for those studying the difference between experts and novices engaged in activities such as constructing proofs. Experts monitor their assumptions and progress and are prepared to change track when a particular approach is not fruitful. Novices are more likely to choose what to them seems the likeliest approach and then ignore alternatives.

In sum, this is a useful and thought-provoking report. I wish that it had included undergraduate mathematics education research, but perhaps that omission can be corrected as we move forward.

[1] National Research Council. 2012. Discipline-Based Education Research: Understanding and ImprovingLearning in Undergraduate Science and Engineering. S.R. Singer, N.R. Nielsen, and H.A. Schweingruber, eds. Washington, DC. The National Academies Press. 

Thursday, November 1, 2012

MAA Calculus Study: The Instructors

One of the goals of the MAA Calculus Study, Characteristics of Successful Programs in College Calculus, was to gather information about the instructors of mainstream Calculus I. Here, stratified by type of institution, is some of what we have learned, refining some of the data presented in “The Calculus I Instructor” (Launchings, June 2011). Again, I am using Research University as code for institutions for which the highest mathematics degree that is offered is the PhD, Masters University if the highest degree is a Master’s, Undergraduate College if it is a Bachelor’s, and Two Year College if it is an Associate’s degree. These surveys were completed by 360 instructors at research universities, 73 at masters universities, 118 at undergraduate colleges, and 112 at two year colleges.

Calculus I instructors are predominantly white and male. Masters universities have the largest percentage of Black instructors, research universities of Asian instructors, and two-year colleges of Hispanic instructors. By and large, undergraduate colleges do not do well in representing any of these groups.

There is a dramatic difference between the status and highest degree of Calculus I instructors at research universities and those at other types of colleges and universities. At research universities, instructors are less likely to be tenured or on tenure track, or to hold a PhD. They are also less likely to want to teach calculus: One in five has no interest or only a mild interest in teaching calculus. The high number of part-time faculty at masters universities and two year colleges is troubling because of the evidence that such instructors tend to be less effective in the classroom and much less accessible to their students [1]. Not surprisingly, less than a quarter of the Calculus I instructors at two-year colleges hold a PhD.

Generally, calculus instructors consider themselves to be somewhat traditional in their instructional approaches, and they believe that students learn best from lectures. The greatest divergence from these views is at undergraduate colleges where almost half consider themselves to be innovative and 45% disagree that lectures are the best way to teach. The greatest variation among faculty at different types of institutions is over the use of calculators on exams. Close to half of the instructors at research universities do not allow them; 71% of the instructors at two year colleges do.

There also are institutional differences in beliefs about whether all of the students who enter Calculus I are capable of learning this material.

Finally, we look at the grade distributions by type of institution.

[1] Schmidt, P. Conditions Imposed on Part-Time Adjuncts Threaten Quality ofTeaching, Researchers Say. Chronicle of Higher Education. Nov 30, 2010.

The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.

Monday, October 1, 2012

MAA Calculus Study: Intended Careers

This is the first of what I intend to be a series of reports that delve into the data on Calculus I instruction that were collected in fall 2010 as part of MAA’s study Characteristics of Successful Programs in College Calculus (NSF #0910240). Some of the raw summative data was reported in earlier Launchings columns [1,2], but we have now separated the data by type of institution, characterized by highest degree offered in mathematics. Combined with knowledge of the number of students who took Calculus I in fall 2010 at each of the types of institutions—thanks to the Conference Board of the Mathematical Sciences (CBMS) 2010 survey—it is now possible to appropriately weight the data that has been collected.  A preprint of an article that summarizes the methods and instruments used for the surveys with selected results [3] has been posted on the website

For this month, I want to focus on the intended careers of students as they begin mainstream Calculus I. This is a course that took on its modern form after the Second World War, and in most places it still follows the curriculum as laid out in George Thomas’s Calculus and Analytic Geometry of 1951 [4].  The course was designed to meet the needs of engineers and those in the physical sciences. However, as illustrated in Figure 1, just under 35% of those taking Calculus I today are on one of these tracks. Today’s Calculus I student is more likely to be pursuing a career in the biological or life sciences than in engineering.

These percentages were calculated by determining the percentages for each of the four types of postsecondary institutions as classified by CBMS. We refer to universities that offer a doctorate in mathematics as research universities, those for which the highest degree in mathematics is the master’s as masters universities. If bachelor’s is the highest degree we call it an undergraduate college, and if associate’s is the highest degree we call it a two-year college. The distributions were then weighted according to the number of students who were enrolled in Calculus I that fall: 110,000 at research universities, 41,000 at masters universities, 82,000 at undergraduate colleges, and 65,000 at two-year colleges.

It is interesting to look at the data by type of institution. Only at research universities and two-year colleges do those heading into engineering outnumber those with an intended major in the biological or life sciences. Even there, the majors that normally require a full year of single variable calculus account for less than half of the Calculus I students. See Figures 2 and 3.

These are also the only types of institution where students going into science, technology, engineering, or mathematics (STEM) fields constitute over 75% of all Calculus I students. The biological sciences are much more dominant, and students are significantly less likely to be intending a STEM major, at masters universities and undergraduate colleges. See Figures 4 and 5.

The following graphs show the distribution of intended careers for women, Asian-American students, Black students, and Hispanic students. See Figures 6–9.

The variation in distributions is most dramatic for women. A women in Calculus I is three times as likely to be headed into the biological sciences as into engineering. It may come as a surprise to some that Asian-American students in Calculus I are almost twice as likely to be majoring in the biological or life sciences as engineering, but this follows a general trend of Asian-American students out of engineering and into biology. While Asian-Americans are still very well represented in engineering, making up over 12% of the bachelor’s degrees in engineering while they are only 7% of all bachelor’s degrees earned in the United States, Asian-Americans constitute almost 18% of the bachelor’s degrees in the biological sciences. See Figure 10.

Of course, these data are skewed by the fact that many students, especially many of those going into engineering or the physical or the mathematical sciences, never take Calculus I in college. They begin their college mathematics at the level of Calculus II or higher. Unfortunately, there are no good estimates for the size of this population. But that still leaves the question that we have addressed at Macalester: Why teach Calculus I as if it is the first half of a year-long course when—for most of the students who take it—the next calculus course is not required or even expected?

[1] Bressoud, D.M. 2011. The Calculus I Student. Launchings.

[2] Bressoud, D.M. 2011. The Calculus I Instructor. Launchings.

[3] D. Bressoud, M. Carlson, V. Mesa, C. Rasmussen. 2012. Description of and Selected Results from the MAA National Study of Calculus (pdf). Submitted to International Journal of Mathematical Education in Science and Technology.

[4] G.B. Thomas. 1951. Calculus and Analytic Geometry. Addison-Wesley. Reading, MA.

[5] National Center for Education Statistics (NCES). 2011. Digest of Education Statistics. US Department of Education. Washington, DC.

The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.

Saturday, September 1, 2012

Teaching and Learning for Transference

The question whether algebra deserves its prominent role in the high school curriculum was raised once again on July 28 by Andrew Hacker in a New York Times opinion column, “Is Algebra Necessary?’’  [1]. His piece echoes the argument made a year ago by Sol Garfunkel and David Mumford in the opinion pages of the New York Times, “How to Fix our Math Education” [2]. It also resonates with Mike Shaughnessy’s comments in his President’s Column for the NCTM newsletter, “Endless Algebra—the Deadly Pathway from High School Mathematics to College Mathematics’’ [3].

There have been many thoughtful responses to the Hacker article. For a clear explanation why algebra is important, the best is still Zal Usiskin’s piece from 1995, “Why is Algebra Important to Learn?” [4]. Daniel Willingham posted a good reply to Hacker on his blog, “Yes, algebra is necessary” [5].  In a private communication [6], Dan Kennedy of the Baylor School in Chattanooga describes the beauty of algebra and laments the fact that we are doing such a very poor job of communicating that beauty.

I want to focus this column on a theme raised in Lynn Steen’s response, “Reflections on Mathematics and Democracy” [7]. His article was conceived as a reaction to the Garfunkel and Mumford editorial, but he also discusses Hacker.  As Steen points out, Hacker’s argument is not that algebra is not important; it is that algebra is not working in the curriculum. The problem—as Steen distills it—is the fact that many if not most students never learn how to transfer the knowledge and skills that are taught in algebra. They are trapped within a perception of algebra as a system of arcane manipulations with no relevance to anything outside the mathematics classroom.

Steen’s solution focuses on the curriculum: embedding applications into mathematics courses, team-teaching cross-disciplinary courses, and employing project- or career-focused curricula. I fully agree that the curriculum can be and has been an obstacle to learning how to transfer one’s knowledge and skills. I also recognize that the problem is greater than just the curriculum.

Teaching for transference is one of the most difficult tasks we face as educators. I would like to share two personal stories that serve as touchstones for me.

In my last year at Penn State, 1993–94, I taught a yearlong honors course in calculus using the Project CALC materials developed by David Smith and Lang Moore. CALC stands for Calculus As a Laboratory Course. The course met five days a week, two of those days in computer labs where the students explored and applied the ideas of calculus. The classroom was used both to prepare students for the laboratories and to reflect on and distill what had happened there. Some students put up initial resistance to such an unconventional approach, and a few switched to a regular section at winter break, but most came to enjoy learning this way. They believed they were getting much more than they would have from traditional instruction.

Midway through spring semester, one of my students came breathlessly to my office. She had just completed an engineering exam where, as she told me, she had forgotten the formula needed to solve one of the problems. But then she remembered what she had learned in my course, and she figured out how to solve the problem with what she knew from calculus.

I do not know what the problem was or what tools she used, but she made it clear that she was relying on the ideas, on the conceptual knowledge she had acquired. I knew then that my job with her was done. Whatever specific information she might still learn from me, nothing would equal the power that came from recognizing that she did not have to rely on memorized procedures, that she was capable of applying the principles of calculus to derive solutions to problems that mattered to her. Not that she always would do this, but now she knew that she could.

The second story is about myself. During the spring of 7th grade, I met periodically after school with my math teacher, Mr. Checkley. He presented and challenged me with bits of mathematics. One of these consisted of the rules for determining divisibility by 3, 9, or 11 by considering the digits of the integer. He asked me to find an explanation why these rules work. I struggled, convincing myself that they are always valid but unable to frame a proof.

The following year I took Algebra I. I had some difficulty at first with this strange kind of mathematics, as everyone did, until I realized that what I was learning in this class was exactly the language that I needed to provide Mr. Checkley with his proof. Once I knew that algebra is simply a language for exploring and explaining mathematical patterns, a language that I could use to answer mathematical questions of interest to me, it became easy.

There are several lessons that I take from these experiences. First, transference need not be to a real-world application. Nor is it about the need to use one’s mathematical knowledge in a career. It is important because of the power that comes from discovering that one can rely on one’s own reasoning to recover a forgotten formula or uncover the logic behind an unexpected pattern.

Second, I have learned that what works in a particular setting with one instructor and a particular group of students will not necessarily work when these parameters are changed. My Penn State class consisted of University Scholars, selected from the top 3% of the student body. I was highly motivated to make it work. Project Calc was used at Duke with mixed results. No curriculum by itself will ever be sufficient.

Third and finally, learning for transference is a process that takes time. A student begins with new knowledge, either acquired through personal discovery or introduced and explained by a teacher. This is followed by an opportunity to apply this knowledge in a fresh context. The next steps are critical and far too often neglected. The student must now reflect on what worked and what difficulties were encountered, then distill the significant features that made it work.  Even now the process is not done. There must be a fresh attempt at application, followed once again by reflection and distillation. I have found that most students need to progress through this cycle several times before they have real control of a new piece of knowledge. I still struggle with the difficulties of engaging my students in this process while balancing the demands of the course. A continuing challenge for me is to decide which concepts deserve this much attention.

I work with privileged and highly motivated students. The difficulties inherent in accomplishing learning for transference in our public schools are far greater, but the goal should be the same. Lynn Steen has asked us to “organiz[e] the curriculum to pay greater attention to the goal of transferable knowledge and skills.” I would go beyond this. We need to organize the very way we teach so that we keep this end in mind.

[1] Hacker, A. 2012. Is algebra necessary? New York Times. July 28.

[2] Garfunkel, S. and Mumford, D. 2011. How to fix our math education. New York Times. August 24.

[3] Shaughnessy, J.M. 2011. Endless Algebra—the Deadly Pathway from High School Mathematics to College Mathematics. Summing Up. February 2. National Council of Teachers of Mathematics. Reston, VA.

[4] Usiskin, Z. 1995. Why is algebra important to learn? American Educator. Vol. 19. pp 30–37.

[5] Willingham, D. 2012. Yes, algebra is necessary. 7/30/2012.

[6] Kennedy, D. 2012. Response to Hacker’s editorial. (private communication posted with permission of the author)

[7] Steen, L.A. 2012. Reflections on mathematics and democracy. To appear in MAA Focus.

Wednesday, August 1, 2012

Barriers to Change

As promised in my last column (Learning from the Physicists, July 2012), this month I will look at work from the Physics Education Research (PER) community on the barriers to adoption of empirically validated teaching practices. I will conclude with three steps that the mathematics community needs to undertake if we are to improve the teaching and learning of undergraduate mathematics.

The primary source for this article is the report by Henderson et al. [1] of a survey completed by 722 physics faculty from the United States, but I also will draw on some of the other literature and I recommend the slides from two talks, one given by Melissa Dancy [2] at the conference on Transforming Research in Undergraduate STEM (Science, Technology, Engineering, and Mathematics) Education and the other given by Charles Henderson [3] at the American Society for Engineering Education, both given in June of this year.

For their study, Henderson et al. questioned faculty about their awareness and use of twenty-four different research-based instructional strategies (RBIS) for which there are published studies establishing their effectiveness.  These include a variety of materials produced in the 1990s and 2000s as well as Mazur’s Peer Instruction, developed at Harvard. The 722 respondents represent just over 50% of those who were contacted. While there may be some selection bias, only 12% had no knowledge of any of these strategies, and an impressive 72% had used at least one of them. But 32% of those who had used at least one strategy no longer used any of them. At least in physics, the issue is not getting people to try proven but innovative approaches to teaching and learning; it is enabling them to stick with it.

In trying to understand what influences faculty decisions about the use of innovative approaches, Henderson et al. looked at twenty possible explanatory variables that range from class size, type of institution, gender, rank, and number of years in the profession to research productivity (measured separately by presentations, articles, and research grants within the past two years), to departmental encouragement and discussions with peers about teaching, to goals for teaching (the importance of developing conceptual understanding and problem-solving ability) and interest in research in instructional strategies.

At the first level, simple familiarity with some of the instructional strategies, two of the strong predictors are attendance at talks or workshops on teaching and regular reading of journal articles about teaching. The strongest predictor is attendance at one of the New Faculty Workshops (NFW), 3½-day workshops held by the American Association of Physics Teachers (AAPT) for new physics faculty. The intent of these workshops is to introduce new faculty to the results of Physics Education Research, so it is not surprising that those who have participated in NFW are over ten times as likely to be aware of these strategies as the average faculty member.  Two other variables were also significant: level of satisfaction that one’s goals for teaching were being met and whether one’s position was full- or part-time. At the next level, actually trying one of these strategies, a new but not surprising variable becomes important: interest in using RBIS.

I am intrigued by the list of variables that did not come into play. These include class size, type of institution, faculty rank or years in position, or research productivity. None of these significantly impacted knowledge of or willingness to try innovative strategies.

The critical stage however, is not the decision to try RBIS but the willingness to stick with it. The only survey variables that were significant at this stage were a desire to try more RBIS and gender: Women were significantly more likely to continue the use of RBIS than were men. Henderson et al. cite four other studies that also showed that women are far more likely to have student-centered physics classrooms than are men. The authors speculate that gender may be serving as a proxy for a collection of beliefs about and attitudes toward teaching.

Many researchers have studied the phenomenon of trying and then abandoning innovative approaches to teaching. The Henderson et al. article includes an extensive bibliography. The hurdles that faculty face include student complaints, inability to cover the same quantity of material, and weaker than promised student outcomes. One of the issues is that implementation frequently is not faithful. As I have seen first-hand both at Penn State and at Macalester, if the course that students experience is not cohesive, if bits and pieces are poorly articulated or motivated, students will complain and resist.

That does not mean that faculty dare not try to modify a received instructional strategy. It does mean that whatever changes they incorporate into their classes must be reflected upon and monitored for effectiveness. In the Henderson, Beach and Finkelstein [4] review of the literature on facilitating change in undergraduate STEM education, one commonly observed phenomenon is the importance of evaluation and feedback, an aspect of change implementation that often is omitted.

Among the interesting findings of the survey by Henderson et al. were the variables that demarcated the distinction between high and low users of RBIS. Low users were defined as using one or two of these strategies, high as three or more. This is the first time that some of the traditionally assumed barriers come into play: research productivity, type of institution, and class size. Highly productive faculty at research universities teaching large classes of introductory physics are significantly less likely to be high users of RBIS. What is surprising is that this is the first time that these variables arise. These highly productive research faculty were no less likely to know about or try some of these innovative strategies, just less likely to be high users.

Also intriguing is the fact that age, as implied by rank and number of years in service, had nothing to do with knowledge of or willingness to try new approaches to teaching. Young faculty who have attended a New Faculty Workshop are more likely to be familiar with the education research and to have tried some of these approaches, but they are no more likely than older faculty to stay with them.

What are the implications for Undergraduate Mathematics Education?

First, we need to conduct a comparable study of mathematics faculty. I suspect that knowledge of what is being done and the evidence of its effectiveness is weaker among mathematicians than it is in the physics community. The combination of the near-universal acceptance of the Force Concept Inventory as a measure of student conceptual knowledge and the high profile work of Eric Mazur on the use of clickers for Peer Instruction make it far more difficult for a physicist to be ignorant of everything that is happening in Physics Education Research. Part of the study of awareness of and responses to Research in Undergraduate Mathematics Education will need to be a thorough assessment of the effect of Project NExT. We have mountains of anecdotal evidence of the importance and effectiveness of this program, but now we need to learn what impact it really has had. The New Faculty Workshops in physics do raise awareness of what can be done, but appear to have little long-term impact on adoption of these strategies. Project NExT is different in many respects, most notably providing a much broader introduction to what it means to be a professional mathematician in an academic position. It also builds communities and provides mentorships. Project NExT is almost twenty years old. We should be able to get good data on its long-term influence.

Second, MAA and the mathematics community need to work on expanding knowledge of what works, including the development of web resources comparable to AAPTs As I explained last month, this will be a much more challenging undertaking than it has been in physics. However, the next CUPM Curriculum Guide—now in its early stages of development and slated for publication in 2015—will address some of this lack.

Third, as the literature amply demonstrates, simply developing effective strategies and putting them out there for people to find is not sufficient. As I travel around the country and talk with faculty at a broad range of colleges and universities, I encounter a lot of dissatisfaction with the way undergraduate mathematics instruction is now conducted and frustration with the difficulty of maintaining quality in the face of budget cuts. I also see a lot of uncertainty about how to proceed and a fear of undertaking radical changes that might prove disastrous. MAA and the mathematics community need to develop mentorship programs that promote small, coherent steps and provide instruments for collecting feedback and monitoring the effectiveness of these efforts. This is part of the critical task of equipping faculty for continual development and refinement of their teaching. We also need to offer the kind of flexible guidance that enables each institution, ideally each instructor, to take ownership of these changes, allowing for modifications that meet local needs and preferences while signposting the dangerous mutations that would destroy the integrity and coherence of the strategy.

The third step will be complex and difficult, yet absolutely critical. I am optimistic that we can accomplish this. There is broad recognition that the teaching and learning of undergraduate mathematics needs to improve. Corporate, foundational, and governmental resources are lining up to bring about improvements. We also know much more than we ever have before about the barriers to change. In cooperation with the Physics Education Research community and all those working on STEM education, we can do this.

[1] Henderson, C., M. Dancy, and M. Niewiadomska-Bugaj. 2012. The Use of Research-Based Instructional Strategies in Introductory Physics: Where do Faculty Leave the Innovation-Decision Process? Accepted for publication in Physical Review Special Topics - Physics Education Research.

[2] Dancy, M. 2012. Educational Transformation in STEM: Why has it been limited and how can it be accelerated? Available at docs/TRUSE Talks 2012/Dancy TRUSE Talk .pdf

[3] Henderson, C. 2012. The Challenges of Spreading and Sustaining Research-Based Instruction in Undergraduate STEM. Available at

[4] Henderson, C., A Beach, and N. Finkelstein. 2011. Facilitating Change in Undergraduate STEM Instructional Practices: An Analytic Review of the Literature. Journal of Research in Science Teaching. Vol. 48, no. 8, pp. 952–984.

Sunday, July 1, 2012

Learning from the Physicists

This is a continuation of my personal responses to and musings upon the report Engage to Excel from the President’s Council of Advisors on Science and Technology (PCAST), this month focusing on the first recommendation:

  1. Catalyze widespread adoption of empirically validated teaching practices.

As the PCAST report points out, there are many techniques for improving classroom interaction that are known to improve student performance. Table 2 on page 17 of the report illustrates several of these and references the research literature. Their list includes small group discussion, one-minute papers, clickers, and problem-based learning.

The Physics Education Research (PER) community, through the American Association of Physics Teachers, has done a nice job of organizing a website of 51 Evidence-based teaching methods that have been demonstrated to be effective: The site is organized to make it useful for the instructor: a brief description and each method and six searchable cross-listings that describe

  • Level: the courses for which it is appropriate, usually introductory physics,
  • Setting: whether designed for large lecture, small classes, labs, or recitation sections,
  • Coverage: whether it requires studying fewer topics at greater depth,
  • Effort: low, medium, or high,
  • Resources: what is needed, from computer access to printed materials that must be purchased to classrooms with tables,
  • Skills: what students are expected to acquire, usually including conceptual understanding, but also possibly problem-solving skills and laboratory skills.

In addition, each of the methods includes a list of the types of validation that have been conducted: what aspects of student learning were studied, what skills the method has been demonstrated to improve, and the nature of the research methods.

I wish for a comparable site for mathematics. Inevitably, a site developed by MAA and the Research in Undergraduate Mathematics Education (RUME) community would need to be both more comprehensive and more complex. Introductory physics is a relatively straightforward course with clear goals, a restricted clientele, and only two flavors: calculus or non-calculus based. In addition, most of the content is new to most of its students.

Introductory college-level mathematics is far more diverse and serves a broad set of disciplines that place often quite specific and disparate demands on these courses. On top of this, we in the mathematics community are plagued by the fact that almost nothing commonly taught in the first year, even Calculus I and II, is completely fresh to these incoming students. At the same time, too many of these students enter without the conceptual knowledge of mathematics and skill in using it that are needed to thrive in this first college course. For those of us who teach college-level mathematics, pressures for coverage are greater and gaps in student preparation are more acute and problematic than they are for introductory physics.

Compounding the difficulties of conducting research in methods of undergraduate mathematics is the fact that the RUME community is only one small part of Mathematics Education Research, which studies the learning of all mathematical knowledge beginning with early childhood recognition of small counting numbers as cardinalities. RUME fights for dollars and publication space against well-established research programs with methods of validation that have been honed over decades but are often inappropriate for understanding the complexities inherent in the learning of higher mathematics.

Nevertheless, the mathematical community does have research evidence for instructional strategies that work. There is a long history of studies of Emerging Scholars Programs, active learning strategies, and computer-aided instruction (see my column Lessons for Effective Teaching, November 2008). Sandra Laursen and her group at UC-Boulder have studied Inquiry Based Learning and documented its benefits (see my column The Best Way to Learn, August 2011).

Unfortunately, the experience of the physicists demonstrates that the existence of research based instructional strategies together with documentation of their effectiveness is not sufficient to guarantee their widespread adoption. Why not?

Again, the PER community is ahead of the RUME community in this regard. At the recent interdisciplinary conference, Transforming Research in Undergraduate Science Education (TRUSE), held at the University of Saint Thomas in Saint Paul, MN, June 3–7, Melissa Dancy of UC-Boulder spoke on Educational Transformation in STEM: Why has it been limited and how can it be accelerated?

Much of the work described in Dancy’s talk can be found in the preprint by Henderson, Dancy, and Niewiadomska-Bugaj [1]. This paper will be the topic of my August column. The work that they have done via surveys of physics faculty demonstrates that the greatest problem is not in making faculty aware of what has been done, or even in getting faculty to try different approaches to teaching. The greatest problem is in getting faculty to stick with these strategies.

What Henderson et al. have to say resonates with my own experience. As I reported in Reform Fatigue, June 2007, the use of calculators, computers, writing assignments and group projects in Calculus rose during the 1990s, but dropped off sharply between 2000 and 2005. The 2010 survey conducted under MAA’s study of Characteristics of Successful Program in College Calculus showed a continuing but modest decline in the use of graphing calculators and computers. Yet there was a bright spot. The use of group projects has rebounded (see Graph 1). Much depends on the quality of the group projects and how they are used, but the data suggest that the mathematics community is not totally immune to empirically validated teaching practices.

Graph 1. Percentage of sections of Calculus I that use group projects for part of the course grade. Data for 1990 through 2005 from CBMS surveys. For 2010, data from MAA survey of Characteristics of Successful Programs in College Calculus.
[1] Henderson, C., M. Dancy, and M. Niewiadomska-Bugaj. 2012. The Use of Research-Based Instructional Strategies in Introductory Physics: Where do Faculty Leave the Innovation-Decision Process? submitted.

Friday, June 1, 2012

Response to PCAST

MAA has just made public its official response to the report to President Obama from the President’s Council of Advisors on Science and Technology (PCAST), Report to the President, Engage to Excel: producing one million additional college graduates with degrees in Science, Technology, Engineering, and Mathematics. This response to John Holdren and Eric Lander, the co-chairs of PCAST, is on MAA’s Science Policy page and can be downloaded here. One aspect of the response that I particularly like is the appendix, which lists many MAA activities that align with the PCAST report’s recommendations. Specifically, my synthesis of these recommendations is
1.   To draw on the available research and empirical evidence to improve undergraduate education in mathematics, science, and engineering;
2.   To improve attraction and retention of students by engaging them in activities where they get to discover the science or mathematics; and
3.   To encourage greater collaboration between mathematics and the science and engineering programs.
For this column, I will focus on the subheading to the title of the PCAST report: producing one million additional STEM graduates. As I explained in my March column, On Engaging to Excel, this impressive number is taken over a decade and includes associate’s degrees. Nevertheless, it translates into the still impressive-sounding goal of an additional 75­ to 80,000 bachelor’s degrees in STEM fields each year. How ambitious is that goal?

Graph 1 shows the number of full-time freshmen arriving each fall with the intention of majoring in engineering, as well as the number who graduated with a bachelor’s degree in engineering the previous spring. The most recent number of intended majors is for fall 2011. For actual bachelor’s degrees awarded, the most recent number is for spring 2010.

The most striking feature of this graph is the remarkable consistency in the number of intended Engineering majors from 1980 through 2007 and the dramatic increase since then, from 102,000 in fall 2007 to 184,000 in fall 2011. There are our extra 80,000 STEM majors, if we can keep them. [3]

It is too soon to be able to tell how well we are doing at retention. The first big increase, with the incoming class of fall 2008, has only just seen the graduation of those who completed their degrees in four years. It will be two more years before the US Department of Education releases the spring 2012 graduation numbers. But the situation in the physical and biological sciences can shed some light on what we might expect.

In both cases, the number of intended majors took off following the year 2000, more than doubling over the following decade. In the biological sciences, the annualized rate of growth in the number of bachelor’s degrees from 2005 to 2010 has almost exactly matched the rate of growth in the number of incoming students five years earlier, at about 6% per year. In the physical sciences, the percentage rate of growth in the number of degrees from 2005 to 2010, at 4% per year, is half of the 8% per year growth rate in the number of incoming physical science students five years earlier. It is reasonable to assume that engineering may be more similar to the physical sciences than the biological sciences.

In short, the problem is not with attracting students to STEM fields. The issue will be to retain them.

[1] Higher Education Research Institute. The American Freshman. UCLA.

[2] National Center for Education Statistics. Digest of Education Statistics. US Department of Education.

[3] The increased interest in engineering programs is almost certainly a result of the recession that began in 2008 and the high unemployment rate since then. I discussed this connection in an earlier column, A Benefit of High Unemployment, November 2010. Note that while enrollments in the biological and physical sciences began to increase following the class of 2000, the rates of increase accelerated after 2007.

Tuesday, May 1, 2012

Are Textbooks Better Online?

The March 30 issue of Science included a letter from psychologists David Daniel and Daniel Willingham [1] that provides an excellent overview of what is known about how students use online textbooks, including an account of what we know about the strengths and the weaknesses of switching from print to electronic delivery. I find that what they have to say is in line with my own experiences. For the past year, I have taught our Single Variable Calculus class using MAA’s online textbook Calculus: Modeling and Application, 2nd edition, by David Smith and Lang Moore. This is a direct descendant of their Project CALC materials that I have used and loved. The current edition is only available as an online textbook.

The overwhelming advantage of online publishing is the cost savings. The cost to my students is $25 apiece. This is charged as a lab fee. In exchange, my students get to download the entire textbook onto their own computers, a benefit that is relatively uncommon among online mathematics texts but which my students appreciate because they are not restricted to using their textbook only when they have internet access.

Smith and Moore’s book is written in html, using MathML for the mathematics (which effectively restricts the web browser in which it is read to FireFox). The authors have done a thoughtful job of separating the chapters into sections that each fit fairly comfortably on a single web page. Several of my students have commented on how much that helps with the readability of the text. But my students have found that reading an online textbook does require a period of adjustment.

I have yet to encounter a student who prefers reading web pages instead of printed pages. Daniel and Willingham cite three different studies that confirm that most students prefer traditional print books. Intriguingly, online texts work very well for young readers. In fact, those learning how to read often do better with online books. The difficulties seem to arise when students need to study and learn from the text, a characteristic that is especially true of mathematics books. For reasons that we do not fully understand but which are well documented, careful reading of an electronic text takes longer and is more fatiguing than trying to learn the same material from a printed text. The research also has found that this effect of greater difficulty with electronic texts is independent of the level of familiarity and experience with e-books.

One of the greatest benefits of online textbooks is the ability to embed links to definitions, animations, and software programs. Smith and Moore’s textbook is liberally sprinkled with links to explorations that are available in Maple, Mathcad, and Mathematica. There also are links to WeBWorK, where a library of problems linked to the sections of their book is available.

My own experience is that students frequently ignore the links to explorations unless I specifically assign them. This is in line with the findings reported by Daniel and Willingham: Students often find such ancillary material distracting at best, confusing at worst. Following these links often leads to loosing the thread of the conceptual development in the text. Another characteristic of e-textbooks that can cut both ways is the ability to link to networking sites where they can exchange thoughts about the mathematics and insights into each other’s difficulties. While that can be very beneficial, there is also the danger that these students will be tempted by the distraction of social media that is equally close at hand.

As Daniel and Willingham point out, online textbooks are easily corrected and updated. For the authors of such a text, that means that the job of working on the book is an ongoing task that is never completed, working against the ability to keep the cost down.

The bottom line is that we do not yet know how best to take advantage of online textbooks. Doing it right is clearly not as simple as putting the text on line and inserting links.

[1] David B. Daniel and Daniel T. Willingham. 2012. Electronic Textbooks: Why the Rush? Science. 335. 30 March, 2012. 1570–71.