Strategies for Change

One of the most striking findings from the MAA’s survey of university mathematics departments undertaken this past spring (see last month’s column) is the almost universal recognition that current practice in the precalculus through single variable calculus sequence needs to be improved. Many such efforts are now underway, but many of them lack understanding of how institutional change occurs as well as recognition of the importance of this understanding.

Much of the literature on institutional change lies too far from the contexts or concerns of mathematics departments to be easily translatable, but an important paper appeared a little over a year ago in the Journal of Engineering Education that provides an insightful framework for understanding change in the context of undergraduate STEM education: “Increasing the Use of Evidence-Based Teaching in STEM Higher Education: A Comparison of Eight Change Strategies” by Borrego and Henderson (2014). This paper takes the framework distilled by Henderson, Beach, and Finkelstein in 2010 and 2011 from their literature review of change strategies and applies it to eight different approaches to bringing evidence-based teaching into the undergraduate STEM classroom. This short column cannot do justice to their extensive discussion, but it can perhaps whet interest in reading their paper.

Henderson, Beach, and Finkelstein have identified two axes along which change strategies occur (Table 1): those whose focus is on changing individuals versus those that focus on changing environments and structures, and those that they describe as prescribed, meaning that they try to implement specific solutions, versus those they describe as emergent, meaning that they attempt to foster conditions that support local actors in finding their own solutions. This results in the four categories shown in Table 1.

Table I: Change theories mapped to the four categories of change strategies. The italicized text lists two specific change strategies for each of the four categories. Reproduced from Borrego and Hnderson (2014).

Within each of the four categories, they identify two strategies that have been used. For example, under a prescribed outcome focused on individuals, Category I, they identify Diffusion and Implementation as two change strategies. Diffusion describes the common practice of developing an innovation at a single location and then publicizing it in the hope that others will pick it up. Implementation involves the development of a curriculum or specified set of practices that are intended to be implemented at other institutions. For each of the eight change strategies, they describe the underlying logic of how it could effect change, describe what it looks like in practice, and give an example of how it has been used, accompanied by some assessment of its potential strengths and weaknesses. Diffusion, in particular, is very common and is known to be capable of raising awareness of what can be done, but it often runs into challenges of incompatibility together with a lack of support for those who would attempt to implement it.

At the opposite corner are the emergent strategies that focus on environments and structures. Here Borrego and Henderson consider Learning Organizations and Complexity Leadership Theory. Learning organizations have emerged from management theory as a means of facilitating improvements. They involve informal communities of practice that share their insights into what is and is not working, embedded within a formal structure that facilitates the implementation of the best ideas that emerge from these communities. In management-speak, it is the middle-line managers who are the key to the success of this approach. In the context of higher education, these middle-line managers are the department chairs and the senior, most highly respected faculty.

The effectiveness of Learning Organizations resonates with what I have seen of effective departments. They require an upper administration that recognizes there are problems in undergraduate mathematics education and are willing to invest resources in practical and cost- effective means of improving this education, together with faculty in the trenches who are passionate about finding ways of improving the teaching and learning that takes place at their institution. The faculty need to be encouraged to form such communities of practice, sharing their understanding and envisioning what changes would improve teaching and learning. Some of the best undergraduate teaching we have seen has been built on the practice of regular meetings of the instructors for a particular class. The role of the chair and senior faculty is one of encouraging the generation of these ideas, providing feedback and guidance in refining them, and then selling the result to the upper administration, conscious of how it fits into the concerns and priorities of deans and provosts. Throughout this process, it is critical to have access to robust and timely data on student performance for this class as well as for the downstream courses both within and beyond the mathematics department.

Complexity Leadership Theory is based on recognition of the difficulties inherent in trying to change any complex institution and calls on the leadership to do three things: to disrupt existing patterns, to encourage novelty, and to make sense of the responses that emerge. Borrego and Henderson could not find any examples of Complexity Leadership Theory within higher education, but, as I interpret this approach as it might appear within a mathematics department, it speaks to the responsibility of the chair and leading faculty to draw attention to what is not working, to encourage faculty to seek creative solutions to these problems, and then to shape what emerges in a way that can be implemented. In many respects, it is not so different from Learning Organizations. The strategies of Category IV highlight the key role of the departmental leadership, which must involve more than just the chair or head of the department.

In their discussion, Borrego and Henderson emphasize that they are not suggesting a preference for any of these categories, although they do note that Category I is the most common within higher education and Category IV the least. My own experience suggests that the strategies of Category IV have the greatest chance of making a lasting improvement. Nevertheless, anyone seeking systemic change will need to employ a variety of strategies that span all of these approaches. Their point is that anyone seeking change must be aware of the nature of what they seek to accomplish and must recognize which strategies are best suited to their desired goals.

Bibliography

M. Borrego and C. Henderson. 2014. Increasing the use of evidence-based teaching in STEM higher education: A comparison of eight change strategies. Journal of Engineering Education. 103 (2): 220–252.

C. Henderson, A. Beach, N. Finkelstein. 2011. Facilitating change in undergraduate STEM instructional practices: An analytic review of the literature. Journal of Research in Science Teaching. 48 (8): 952–984.

C. Henderson, N. Finkelstein, A. Beach. 2010. Beyond dissemination in college science teaching: An introduction to four core change strategies. Journal of College Science Teaching. 39 (5): 18–25.

MAA Calculus Study: A New Initiative

With the publication of Insights and Recommendations from the MAA National Study of College Calculus, we are wrapping up the original MAA calculus study, Characteristics of Successful Programs in College Calculus (CSPCC, NSF #0910240). This past January, MAA began a new large-scale program, Progress through Calculus (PtC, NSF #1430540), that is designed to build on the lessons of CSPCC. I am continuing as PI of the new project. Co-PIs Chris Rasmussen at San Diego State, Sean Larsen at Portland State, Jess Ellis at Colorado State, and senior researcher Estrella Johnson at Virginia Tech are leading local teams of post-docs, graduate students, and undergraduates who will be working on this effort.

CSPCC sought to identify what made certain calculus programs more successful than others but was limited in its measures of success to what could be learned about changes in student attitudes between the start and end of Calculus I and to what could be observed from a single three-day visit to a select group of 20 colleges and universities. PtC is extending its purview to the entire sequence of precalculus through single variable calculus, and it will take broader measures of success, including performance on a standardized assessment instrument, persistence into subsequent mathematics courses, and performance in subsequent courses. It also is shifting emphasis from description of the attributes of successful programs to analysis of the process of change: What obstacles do departments encounter as they attempt to improve the success of their students? What accounts for the difference between departments that are successful in institutionalizing improvements and those that are not?

We began this past spring with a survey of all mathematics departments offering a graduate degree in Mathematics, either MA/MS or PhD. This is a manageable number of institutions: 178 PhD and 152 Masters universities. These are the places that most often struggle with large classes and with the trade-off between teaching and research. We had an excellent participation rate: 75% of PhD and 59% of Masters universities filled out the survey.

Data from this survey will appear in future papers and articles, but for this column I want to focus on the most important information we learned: what these departments see as critical to offering successful classes and how that compares to how well they consider themselves to be doing on these measures.

CSPCC identified eight practices of successful programs. These are listed here in the order implied by the number of doctoral departments in the PtC survey that identified each as “very important to a successful precalculus/calculus sequence.”

1. Student placement into the appropriate initial course 
2. GTA teaching preparation and development 
3. Student support programs (e.g. tutoring center) 
4. Uniform course components (e.g. textbook, schedule, homework) 
5. Courses that challenge students 
6. Active learning strategies 
7. Monitoring of the precalculus/calculus sequence through the collection of local data 
8. Regular instructor meetings about course delivery.

The graphs in Figures 1 and 2 show the percentage of respondents who identified each as “very important” (as opposed to “somewhat important” or “not important”), as well as the percentage of respondents who considered themselves to be “very successful” with each (opposed to “somewhat successful” or “not successful”).

Figure 1. PhD universities. What they consider to be important versus how successful they consider themselves to be.

Figure 2. Masters universities. What they consider to be important versus how successful they consider themselves to be.

What is most interesting for our purposes is where departments see a substantial gap between what they consider to be very important and where they see themselves as very successful. These are the areas where departments are going to be most receptive to change. If we look for large absolute or relative gaps, five of the eight practices show up as areas of concern (Table 1). The biggest absolute gap is for placement; approximately half of all universities consider placement to be very important but do not rate themselves as very successful. The largest relative gap is for active learning, where only 27% of doctoral universities and 36% of masters universities that consider this to be very important also consider themselves to be very successful at it.

Table 1. Departments that consider themselves to be very successful as percentage of those that consider the practice to be very important.

The next stage of this project will be the building of networks of universities with common concerns and the identification of twelve universities for intense study over a three-year period. This stage has begun with a small workshop for representatives of 27 universities, a workshop that will begin building these networks and is ending as this column goes live on November 1. It will be continuing with a larger conference in Saint Paul, MN, June 16–19, 2016. Watch this space for more information about that conference.

Evidence for IBL

Special Note: The AMS Blog On Teaching and Learning Mathematics has started a six-part series on active learning.

Over the past decade, the Educational Advancement Foundation has supported programs to promote Inquiry-Based Learning (IBL) in mathematics at four major universities. IBL is not a curriculum. Rather, it is a guiding philosophy for instruction that takes a structured approach to active learning, directing student activities and projects toward building a fluent and comprehensive understanding of the central concepts of the course. Ethnography & Evaluation Research (E&ER) at the University of Colorado, Boulder has studied the effectiveness of these implementations. Several research papers have resulted, of which the paper by Kogan and Laursen (2014), discussed in my column Evidence of Improved Teaching (October 2013), presented very clear evidence that IBL prepares students for subsequent courses better than standard instruction and that IBL can result in students taking more mathematics courses, especially when offered early enough in the curriculum. Two recent papers document the benefits of IBL in preparing future teachers and in building personal empowerment.

In Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers (Laursen, Hassi, and Hough, 2015), the authors focused on the development of Mathematical Knowledge for Teaching (MKT), a term coined by Deborah Ball to describe the kind of knowledge that teachers must draw upon to teach mathematics well and that reflects understanding of how ideas and concepts relate to one another as well as the common difficulties and misunderstandings that students are likely to encounter. Being prepared for teaching requires more than being able to find solutions to particular problems. A good teacher must have at her or his disposal a variety of approaches to a solution and the ability to take a student’s incorrect attempt at an answer, recognize where the misunderstanding lies, and build on what the student does understand.

In theory, IBL should help develop MKT because it focuses on precisely those characteristics of practicing mathematicians that teachers most need, the habits of mind than include sense-making, conjecture, experimentation, creation, and communication.

E&ER studied students in thirteen sections of seven courses for pre-service teachers at two of the four universities, courses that collectively spanned preparation for primary, middle school, and secondary teaching. They used an instrument developed by Ball and colleagues, Learning Mathematics for Teaching (LMT) that has been validated as an effective measure of MKT for practicing teachers. The results were impressive. The students had begun the term with LMT scores that averaged at the mean for in-service teachers across the country. Each of the IBL classes saw mean LMT scores rise by 0.67 to 0.90 standard deviations. In line with the results of the 2014 report, all students experienced gains from IBL, but the weakest students saw the greatest gains.

The second recent article is Transforming learning: Personal empowerment in learning mathematics (Hassi and Laursen, 2015). In Adding It Up (NRC 2001), mathematical proficiency is recognized as consisting of five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. They are each critically important. This paper investigates the effect of IBL on both strategic competence, what the authors term cognitive empowerment, and productive disposition, which they separate into self-empowerment and social empowerment, the last of which also incorporates effective communication.

The study was conducted through interviews with students who had taken a class at one of the four universities using IBL. An overwhelming majority of students reported gains in each of the three areas of personal empowerment. Among women 77% and among men 69% reported an increase in self-esteem, sense of self-efficacy, and confidence from their IBL experience. For general thinking skills, deep thinking and learning, flexibility, and creativity, 77% of the women and 90% of the men described improvements. For ability to explain and discuss mathematics as well as skills in writing and presenting mathematics, 79% of the women and 76% of the men saw gains.

When pressed for what made the IBL experience special, students identified their own role in influencing the course pace and direction, the importance of combining both individual and collaborative work, and the fact that they were faced with problems that were both challenging and meaningful. They appreciated that they were given responsibility to think on their own. Such experiences were especially important for women and for first-year students.

In the very discouraging reports on the effects of Calculus I instruction in most US universities (Sonnert and Sadler 2015), we see courses that accomplish exactly the opposite of personal empowerment, courses that sharply decrease student confidence and sense of self-efficacy. It does not have to be this way.

References

Hassi, M.-L., and Laursen, S.L. 2015. Transformative learning: Personal empowerment in learning mathematics. Journal of Transformative Education. Published online before print May 24, 2015, doi: 10.1177/1541344615587111.

M. Kogan and S. Laursen. 2014. Assessing long-term effects of inquiry-based learning: A case study from college mathematics. Innovative Higher Education 39 (3), 183–199. http://link.springer.com/article/10.1007/s10755-013-9269-9

Laursen, S.L., Hassi, M.-L., and Hough, S. 2015. Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers. International Journal of Mathematical Education in Science and Technology. Published online before print July 25, 2015, doi: 0.1080/0020739X.2015.1068390

National Research Council (NRC). 2001 Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Washington, DC: National Academy Press.

Sonnert, G. and Sadler, P. 2015. The impact of instructor and institutional factors on students’ attitudes. Pages 17–29 in Insights and Recommendations from the MAA National Study of College Calculus, D. Bressoud, V. Mesa, and C. Rasmussen (Eds.). Washington, DC: Mathematical Association of America Press.

Calculus at Crisis V: Networks of Support

Special Notice: The MAA Notes volume summarizing the results of Characteristics of Successful programs in College Calculus (NSF #0910240), Insights and Recommendations from the MAA National Study of College Calculus, is now available for free download as a PDF file at www.maa.org/cspcc.

This is the last of my columns on Calculus at Crisis. In the first three, from May, June, and July, I explained why we can no longer afford to continue doing what we have always done. Last month I described some of the lessons that have been learned in recent years about best practices with regard to placement, student support, curriculum, and pedagogy. Unfortunately, as those who seek to improve the teaching and learning of introductory mathematics and science have come to realize, knowing what works is not enough.

There are many barriers to change, both individual and institutional. Lack of awareness of what can be done is seldom one of them. In recent years, leaders in physics and chemistry education research, especially Melissa Dancy, Noah Finkelstein, and Charles Henderson have studied these barriers and begun to translate insights from the study of how institutional change comes about in order to assist those who seek to improve post- secondary science, mathematics, and engineering education.

One of the best short summaries describing specific steps toward achieving long-term change is Achieving Systemic Change, a report issued jointly by the American Association for the Advancement of Science (AAAS), the American Association of Colleges and Universities (AAC&U), the Association of American Universities (AAU), and the Association of Public and Land-grant Universities (APLU) that I discussed this past December. Its emphasis on creating supportive networks within and across institutions is reflected in our own findings in the MAA’s calculus study.

There has always been lively interest from individual faculty members in improving mathematics education. Heroic efforts have often succeeded in moving the dial, but without strong departmental support they are not sustainable. As I have explained over the past months, deans, provosts, and even presidents now realize that something must be done. I have yet to meet a dean of science who is not willing—usually even eager—to fund a proposal from the mathematics department for improving student outcomes provided it is concrete, workable, and cost-effective. (Just hiring more mathematicians does not cut it.) The key link between eager faculty and concerned administrators is the department chair, together with the senior, most highly respected faculty. Without their support and cooperation, no lasting improvements are possible.

The department chair is essential. This is the person who can take an enthusiastic proposal and massage it into a workable plan whose benefits are understandable to the upper administration. This is the person who can take a request from the dean, understand the resources that will be required, and find the right people to work on it. Unfortunately, appointment as chair does not automatically confer such wisdom. Part of what is needed is an understanding of what is being done at comparable institutions, how it is being implemented, what is working or failing and why. This is where the mathematical societies have an important role to play. AMS does this through its Information for Department Leaders, the work of the Committee on Education, and its blog On the Teaching and Learning of Mathematics. The MAA’s CUPM, CTUM, and CRAFTY committees provide this information through publications, panels, and contributed paper sessions. SIAM, ASA, and AMATYC also embrace this mission. Common Vision began this year as an effort to coordinate these activities across the five societies.

But a supportive department chair is not enough. The lasting power center in any department consists of senior faculty who are highly respected for their research visibility. The most successful calculus programs we have seen in the MAA study Characteristics of Successful Programs of College Calculus involved some of these senior faculty in an advisory capacity: monitoring the annual data on student performance, observing occasional classes, mentoring graduate students not just for research but also for the development of teaching expertise, and providing encouragement and a sounding board to those—usually younger faculty—engaged in trying new methods in the classroom. It will be the chair’s responsibility to identify the right people for this advisory group, but once it is in existence it can help ensure that future chairs are sympathetic to these efforts.

Finally, any mathematics department seeking to improve undergraduate education must remember that it is not alone within its institution. Similar efforts are underway in each of the sciences as well as engineering. Deans and provosts can help by formally recognizing those who serve in these senior roles across all STEM departments and encouraging links between these groups of faculty. They can draw on support and advice from consortia of colleges and universities such as AAU, APLU, and AAC&U, as well as multidisciplinary societies and consortia such as AAAS and the Partnership for Undergraduate Life Science Education (PULSE), all of whom are working to promote networks of educational innovation that cross STEM disciplines. Joining with other departments within the institution can dispel the perception of mathematics as insular and unconcerned with the needs of others as it strengthens individual departmental efforts. All STEM departments are facing similar difficulties. This crisis presents us with an exceptional opportunity to work across traditional boundaries.

Calculus at Crisis IV: Best Practices

In my last three columns I explained the reasons that college calculus instruction is now at crisis:

1. The need to teach ever more students, who often bring weaker preparation, using fewer resources.
2. The fact that most Calculus I students have already studied calculus in high school (this past spring 424,000 students took an AP Calculus exam, an increase of 100,000 over the past five years).
3. The pressures from the client disciplines to equip their students with the mathematical knowledge and habits of mind that they actually will need.

As I have traveled this country to meet with mathematics departments, I have seen that there is a general recognition on the part of chairs, deans, provosts, and occasionally even presidents that the past solutions for calculus instruction are no longer adequate. I am encouraged by the fact that the mainstream calculus sequence is so central to all of the STEM disciplines that, even in these tight budget times, many deans and provosts can find the resources to support innovative programs if they can be convinced these efforts are sustainable, cost-effective, and will actually make a difference.

There are four basic leverage points for improving the calculus sequence so that it better meets at least some of these pressures: placement, student support, curriculum, and pedagogy. We know a lot about what does work for each of these. Much of this knowledge—relevant to the teaching of calculus—is contained in the new MAA publication Insights and Recommendations from the MAA National Study of College Calculus, the report on a five-year study of Characteristics of Successful Programs in College Calculus undertaken by the MAA with support from NSF (#0910240). I briefly summarize some of the insights.

Placement. Placement can have a huge impact on student success rates. However, given the demands of the client disciplines and the fact that remediation is usually of doubtful value (see The Pitfalls of Precalculus), just tightening up the requirements for access to calculus is unlikely to make a dean or provost happy. We do have evidence of the effectiveness of adaptive online exams such as ALEKS that probe student understanding to reveal individual strengths and weaknesses, especially when combined with tools that can help students address specific topics on which they need refreshing. But there is no one placement exam or means of implementation that will work for all institutions. Further elaboration on what we have learned about placement exams can be found in Chapter 5, Placement and Student Performance in Calculus I, of Insights and Recommendations.

Student Support. Programs modeled on the Emerging Scholars Programs can be very effective for supporting at-risk students (see Hsu, Murphy, Treisman, 2008). Tutoring centers are virtually universal, but not always as useful as they could be. The best we have seen put thought into the training of the tutors, require classroom instructors to hold some of their office hours in the center, and are located conveniently with a congenial atmosphere that encourages students to drop in to study or work on group projects even if they do not need the assistance of a tutor. In addition, quick identification and effective guidance of students who are struggling with the course is essential. More on these points can be found in Chapter 6, Academic and Social Supports, of Insights and Recommendations.

Curriculum. This is the toughest place at which to apply leverage. Most faculty are fine with changes to placement procedures and support services but are appalled at the very thought of touching the curriculum. The pushback against the Calculus Reform movement of the early 1990s was strongest where curricular changes were suggested. Yet this is where we are most likely to be successful in meeting the needs of students who studied calculus in high school, and it must be part of any strategy for meeting the needs of the client disciplines. Research coming out of Arizona State University and other centers of research in undergraduate mathematics education has revealed the basic wisdom of many of the Calculus Reform curricula that approached calculus as a study of dynamical systems. Curricular materials are now being developed that have a much firmer basis in an understanding of student difficulties with the concepts of calculus (for an example, see Beyond the Limit).

Pedagogy. Another aspect of the Calculus Reform movement that was poorly received was the emphasis on active learning. The evidence is now overwhelming that active learning is critical, especially important for at-risk students and essential for meeting the needs of the client disciplines. We have learned a lot in the intervening quarter century about how to do it well and cost-effectively, and this is one of the places where new technologies can be particularly helpful. There are now many models for implementation of active learning strategies, spanning classrooms of all sizes, student audiences at varied levels of expertise, and faculty with different levels of commitment to changing how they teach (see Reaching Students). Evidence for the effectiveness of active learning and recommendations of strategies for implementing it can be found in Donovan & Bransford, 2005; Freeman et al., 2014; Fry, 2014; Kober, 2015; and Kogan & Laursen, 2014.

The bottom line is that we do have knowledge that can help us face this crisis. There is no universal solution. Each department will have to find its own way toward its own solutions. But it need not stumble alone. As I will explain next month in the fifth and final column in this series, making meaningful and lasting change requires networks of support both within and beyond the individual department. Here also our knowledge base of what works and why has expanded in recent years.

References

Bressoud, D., Mesa, V., Rasmussen, C. (eds.) (2015). Insights and Recommendations from the MAA National Study of College Calculus. MAA Notes. Washington, DC: Mathematical Association of America (to be available August, 2015).

Donovan, M.S. & Bransford, J.D. (eds.). (2005). How Students Learn: Mathematics in the Classroom. Washington, DC: National Academies Press. www.nap.edu/catalog/11101/how-students-learn-mathematics-in-the-classroom

 Freeman, S. et al. (2014). Active learning increases student performance in science, engineering, and mathematics. Proc. National Academy of Sciences. 111 (23), 8410–8415. www.pnas.org/content/111/23/8410.abstract

Fry, C. (ed.). (2014). Achieving Systemic Change: A sourcebook for advancing and funding undergraduate STEM education. Washington, DC: AAC&U. www.aacu.org/pkal/sourcebook

Hsu, E., Murphy, T.J., Treisman, U. (2008). Supporting high achievement in introductory mathematics courses: What we have learned from 30 years of the Emerging Scholars Program. Pages 205–220 in Carlson and Rasmussen (eds.). Making the Connection: Research and Teaching in Undergraduate Mathematics Education. MAA Notes #73. Washington, DC: Mathematical Association of America. www.maa.org/publications/books/making-the-connection-research-and-teaching-in-undergraduate-mathematics-education

Kober, N. (2015). Reaching Students: What research says about effective instruction in undergraduate science and engineering. Washington, DC: National Academies Press. www.nap.edu/catalog/18687/reaching-students-what-research-says-about-effective-instruction-in-undergraduate

Kogan, M. & Laursen, S.L. (2014). Assessing long-term effects of Inquiry-Based Learning: A case study from college mathematics. Innovative Higher Education 39(3) 183–199. link.springer.com/article/10.1007%2Fs10755-013-9269-9

Calculus at Crisis III: The Client Disciplines

In my last years at Penn State, I worked with faculty in the College of Engineering on issues of undergraduate education. They had two complaints about the mathematics department. First, we were failing too many of their students. Second, the ones we passed seemed incapable of using the mathematics we presumably had taught them when they got to their engineering classes. Chief among their specific gripes was an inability among their students to read a differential equation, to understand its assumptions of the relationships among the quantities being measured.

A decade later, in the Curriculum Foundations Project workshops with engineering faculty brought from across the country, we heard the same concerns about what their students should be learning from the mathematics department:

Students “should understand the reasons for selecting a particular technique develop an understanding of the range of applicability of the technique, acquire familiarity with the mechanics of the solution technique, and understand the limitations of the technique.” (from civil engineers, p. 59)

“There is often a disconnect between the knowledge that students gain in mathematics courses and their ability to apply such knowledge in engineering situations … We would like examples of mathematical techniques explained in terms of the reality they represent.” (from electrical engineers, p. 66)

“In an engineering discipline problem solving essentially mean mathematical modeling; the ability to take a physical problem, express it in mathematical terms, solve the equations, and then interpret the results.” (from mechanical engineers, p. 81)

From the current ABET (Accreditation Board for Engineering and Technology) Criteria for Accreditation, all of the references to mathematics under Curriculum talk about “creative applications,” building “a bridge between mathematics and the basic sciences on the one hand and engineering practice on the other,” and the use of mathematics in the “decision-making process.” As ABET moves into the criteria for specific programs, again the emphasis is entirely on the ability to apply knowledge of mathematics, not on any list of techniques or procedures.

In the biological sciences, the other big driver for calculus enrollments, the American Association of Medical College and the Howard Hughes Medical Institute have dropped the traditional lists of specific courses that students should take in preparation for medical and instead list the competencies that students will need. First among these is mathematics. Of the seven specific objectives within this competency, six speak of quantitative reasoning and the use of data, statistics, modeling, and logical reasoning. The seventh comes closest to calculus, but what they actually ask for is the ability to “quantify and interpret changes in dynamical systems,” a far cry from the usual calculus course. (For more on this report, see my column on The New Pre-Med Requirements.)

In the influential Vision and Change document crafted by the biological sciences with assistance from AAAS, six core competencies for undergraduate biology education are identified. Two of them are mathematical: quantitative reasoning and the ability to use modeling and simulation. The report goes on to specify that “all students should understand how mathematical and computational tools describe living systems.”

These examples can be multiplied in other client disciplines. What we see is a universal need for students to be able to use mathematical knowledge in the context of their own disciplines. In the case of calculus, the challenge is to understand it as a tool for modeling dynamical systems. This is why calculus is required by so many disciplines. But this is an understanding of calculus that is achieved by very few of our students because their focus has been narrowed down to learning how to solve the particular problems that will be on the next exam.

None of this disconnect between what we teach in calculus and the needs of the client disciplines is new. It now rises to the level of a force that is bringing us to crisis because these client disciplines are themselves under the same increased pressure to have their students succeed. There may have been a time when there was a sufficiently rich pool of potential engineers that we could afford the luxury of allowing the mathematics department to filter out all but the most talented, the ones who would succeed in spite of how we taught them. If it ever existed, that time has passed. Our client disciplines now have higher expectations for what and how we teach their students.

Nothing has driven this point home more clearly than Engage to Excel, the Report to the President from his Council of Advisors on Science and Technology (PCAST). (See my columns On Engaging to Excel, Response to PCAST, and JPBM Presentation to PCAST.) The frustration of the scientists in PCAST with calculus instruction that does not meet the needs of their disciplines is evident in their call for “a national experiment [that] should fund … college mathematics teaching and curricula developed and taught by faculty from mathematics-intensive disciplines other than mathematics, including physics, engineering, and computer science.” (Recommendation 3-1, p. vii)

While there was one particular physicist who was the driver behind this report, it did reflect the concerns of all of PCAST’s members. These are scientists and leaders in technology who deplored the fact that “many college students … often are left with the impression that the field [of mathematics] is dull and unimaginative.” (p. 28)

I have yet to find physicists, engineers, or computer scientists who want to take over our calculus instruction. They have better things to do. But some have been forced to do so, and others are contemplating undertaking it as a necessary correction to mathematical instruction that is not meeting their needs.

This completes my triad of forces that constitute the reason we are at crisis. It is the nature of a crisis that the solution is not readily apparent. Nevertheless, there are actions that can be taken to improve the situation. Next month, I will explore the first of these: drawing on knowledge of best practices for effective teaching and learning.

Calculus at Crisis II: The Rush to Calculus

I began this series last month by explaining how recent economic conditions are sending more students into the primary STEM fields (engineering and the physical, biological, mathematical, and computer and information sciences) while constricting the resources available to meet the needs of educating them. This is just one of a triad of phenomena that are pushing college calculus toward crisis. This month, I will discuss the second of these forces: the rush to calculus.

Nothing illustrates the relentless growth of high school calculus better than the graph of the number of AP Calculus exams taken each year (Figure 1), surpassing 400,000 in 2014. According to NCES data [1], 53% of the students who study calculus in high school take an AP Calculus exam, implying that roughly 750,000 U.S. high school students studied calculus this past year. By comparison, this past year only 250,000 students took their first mainstream calculus class at a 2- or 4-year college or university [2].

Figure 1: Total AP Calculus exams and fall enrollments in mainstream Calculus I. Sources: The College Board and CBMS Statistical Abstracts.

What happens to the students who study calculus in high school? We know from the MAA study Characteristics of Successful Programs in College Calculus that about one- third of them retake Calculus I when they get to college. Based on AP scores and common policies for granting credit, roughly 200,000 students accept credit and/or advanced placement for their high school work. From a limited study [3], a clear majority of these students, probably three-quarters or more, do continue on to further courses that build on calculus.

All of these patterns intensify at research universities and elite colleges, where at least 70% of Calculus I students are retaking a course they have already seen in high school, and large numbers of students heading for math-intensive majors skip over Calculus I.

The result has been a dramatic change in the make-up of Calculus I. At most colleges and universities, it makes little sense to teach this course as if students are encountering calculus for the first time; few of them are. It also makes little sense to teach this course as if the students are heading into the mathematical or physical sciences. Nationwide, only 6% of Calculus I students intend such a major [4]. Finally, it makes little sense to teach this course as if this is where we see our best-prepared students.

This last point is clear if we consider how many of the best-prepared students skip Calculus I, but it also is a consequence of what the rush to calculus has done to the middle and high school curricula in mathematics.

In fall 2014 there were just over 1.6 million full-time first-year students enrolled in 4- year undergraduate programs in the U.S. [5]. Assuming that most of the 750,000 who take calculus in high school are traditional college-bound students who will enroll as full- time students in 4-year programs, these high school calculus students will constitute 40–45% of traditional first-year college students. The result is a common belief among parents, guidance counselors, and administrators that every college-bound student should, if at all possible, study calculus before high school graduation. I hear this from college students whose reason for taking calculus in high school was that it was expected of their peer group, and I hear it especially from high school teachers who complain of the tremendous pressure they are under to expand calculus classes and admit students they know are not ready for it.

Because high school calculus by itself has become such common coin, those students who aspire to an elite college or university try to take calculus, preferably BC Calculus, before 12th grade. Figure 2 shows the exceptional growth in the number of students who take an AB or BC Calculus exam before grade 12.

Figure 2: Number of AP Calculus exams taken by students in grade 11 or earlier. Source: The College Board.

We do not know the full effect of this movement of calculus into ever earlier grades, but there is strong anecdotal evidence from teachers at both the high school and university level that many of these students are short-changing their preparation in middle and high school mathematics to join the fast track to calculus. Again anecdotally, this appears to be a significant problem when students attempt a math-intensive major where weaknesses in precalculus material can be disastrous.

We can deplore the rush to calculus in high school, but the forces that are sustaining it are formidable. We have neither the authority nor the certain knowledge that would enable us to halt or reverse it. For the foreseeable future, we will have to live with it.

Just in the past ten years, the preparation and aspirations of our college calculus students have shifted significantly. We cannot afford to assume that curricula and methods of instruction that were sufficient for the past will be adequate for the future.


[1] National Center for Education Statistics (NCES). (2012). An overview of classes taken and credits earned by beginning postsecondary students. NCES 2013-151rev. Washington, DC: US Department of Education. nces.ed.gov/pubs2013/2013151rev.pdf

[2] By “mainstream” we mean a calculus course that can be used as part of the pre- requisite stream for more advanced mathematics courses. It usually does not include business calculus, but may or may not include calculus for biologists. The figure of 250,000 is an estimate based on data from the CBMS Statistical Abstracts and the MAA study Characteristics of Successful Programs in College Calculus. Approximately 500,000 students began mainstream Calculus I at the post-secondary level at some point in the past year, and roughly half of them had studied calculus in high school.

[3] Morgan, K. (2002). The use of AP Examination Grades by Students in College. Paper presented at the 2002 AP National Conference, Chicago, IL.

[4] Source: MAA National Study of College Calculus, www.maa.org/cspcc.

[5] Source: HERI, The American Freshman. www.heri.ucla.edu/tfsPublications.php