- The need to teach ever more students, who often bring weaker preparation, using fewer resources.
- 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).
- 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