By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.
Editor’s note: This is the sixth and final article in a series devoted to active learning in mathematics courses. The other articles in the series can be found here.
How are mathematicians trained as teachers, what are the effects of this training, and what can we do to improve the quality of this training? We feel these questions are particularly important at this time, as a clamor of recent calls for the dramatic improvement of postsecondary education, made from both inside and outside of the mathematical community, has not abated. From the outside, we hear this call in venues ranging from opinion pieces in major newspapers [1,2,3] to federal advisory reports to the President of the United States  and beyond. The message has also been clearly conveyed by leadership from professional societies in the mathematical community: in early 2014, an article titled “Meeting the Challenges of Improved Post-Secondary Education in the Mathematical Sciences” was published in the AMS Notices, MAA Focus, SIAM News, and AMSTAT News through a coordinated effort by the professional societies — we urge any readers who have not already done so to read this statement.
Yet in order to be effective and achieve meaningful change, any actions taken by our professional societies and other leadership in the mathematics community must get buy-in from individual mathematicians who are in the classroom daily, working face-to-face with students. From our training in both mathematics and the teaching of mathematics, we each carry disciplinary habits, ways of thinking, biases, and strengths, many of which occur subconsciously as part of our mathematical culture, and all of which impact our teaching. In order to improve mathematical teaching and learning on a large scale, we must all work to better understand how mathematicians grow and develop as teachers, so that we may more thoughtfully respond to the educational challenges of our time. In this article, we focus our discussion on the topic of pedagogical training and development for graduate students and early-career faculty, with a view toward active learning.
Mathematical training in graduate school
To earn a graduate degree in mathematics, one must master a body of mathematics content through coursework, demonstrate a deep understanding of this through oral and written examinations, and (for doctoral degrees) complete a dissertation demonstrating original research. Thus, the primary focus of graduate students is on mastering advanced mathematical ideas and producing new mathematical results, and these qualities of graduate education are consistent across programs at different institutions, though specific program details can vary.
Less consistent across the graduate school spectrum is the preparation of students for positions where they will have teaching responsibilities. Duties for teaching assistantships in graduate school vary greatly, from providing grading support to professors, to working in a tutoring lab, to leading recitation sections, to having full responsibility to lead a course. Preparation for these duties is equally various, but an increasing number of graduate schools have explicitly attended to preparing their students for teaching by instituting or enhancing teaching programs for their TAs. More broadly, a look at the program for the 2016 Joint Mathematics Meetings reveals some promising developments, including a panel called Improving the Preparation of Graduate Students to Teach Mathematics: An NSF-Funded Project. While these are positive developments, it is not uncommon for graduate teaching assistants to be supervised by faculty members who are only familiar with a small set of teaching techniques, or who have had frustrating prior experiences with active learning methods, and for graduate students to receive little formal training as teachers.
Further, though this is slowly changing, many graduate students in mathematics have not personally experienced teaching environments that include active learning components. Thus, for many mathematicians and current graduate students, their first experience with active learning techniques will be as teachers rather than students. A consequence of this is that we cannot expect students to emerge from graduate programs prepared to be guided by their own classroom experiences where active learning is concerned. If we want graduate students to consider using active learning in undergraduate courses, we must provide them with some experiences, either as students or teachers, to help inform their practice.
There are many reasons to be optimistic that this can be accomplished. Researchers in mathematics education have begun to study training of teaching assistants in mathematics, which should lead to better information about effective practices [5,6]. Further, the gap between the reality of graduate school and the goal of producing graduate students who have a reasonable level of training as teachers, including some exposure to active learning methods, is not as wide as it might appear. One key is to recognize and promote the aspects of graduate programs that already have active learning embedded in them. Here are some examples.
Many universities with doctoral programs rely on their TAs to serve as recitation leaders, or to teach small sections of courses, roles which can easily incorporate active learning methods (avoiding, for example, the issue of scaling things up to large-lecture size). In departments such as the University of Michigan, University of Illinois Urbana-Champaign, University of Kentucky, and many others, graduate students lead recitations that are based on having students work in small groups through an activity built from a sequence of problems. In such settings, graduate students are already leading a class setting based on active learning, but they may or may not be receiving explicit training regarding how to effectively structure small group work, how to lead a discussion without directly providing an answer, etc. With a small amount of effort, course coordinators and TA supervisors can provide training in these areas for TAs, as long as they themselves are aware of how to do this effectively.
As another example, after initial coursework, many graduate students participate in formal or informal seminars where participants read through a paper or book and gather to discuss problem sets, sticking points in the reading, and general questions about the topic. This practice, which mathematicians would call “doing mathematics,” is active learning at the core. Most senior graduate students and mathematicians can reflect on times when they have struggled with an idea or topic, only to have it clarified through helpful conversation with others. If mathematics faculty and graduate students re-conceive these activities as examples of active learning, then it becomes easier to see how one might try to incorporate some small-group discussion in classes. In both the recitation model above, and in this example from the experience of many mathematicians, we recognize the benefits of conversing and communicating, doing mathematics with others. Active learning methods seek to bring this into the classroom, and it would be helpful for graduate students to be trained to make this connection explicitly.
As a final example, there are many outreach programs for K-12 students that are operated by mathematics departments with graduate programs, including Math Students’ Circles, Math Teachers’ Circles, math days, math camps, and more. Many of these programs are strongly based on active learning methods. Graduate students who serve as assistants for such programs might not make an explicit connection between these programs and their own teaching, though certainly many students do see connections. In any event, it would be a positive step forward if graduate students serving as assistants in these programs were explicitly encouraged as a part of their assistantship to consider and discuss ways in which effective techniques in extra-curricular K-12 outreach programs might be transferred into their own courses.
Even though most mathematicians consider graduate school the foundation of a mathematical career, it isn’t clear how much responsibility should be placed on the shoulders of doctoral programs for teacher training. It is unreasonable to expect that every doctoral student in math will emerge as an expert teacher, given the many demands of graduate school and the need for students to develop and defend a research dissertation. Yet it is clear that we can do more than we are at present, and that we have many strengths on which faculty and students can immediately build by being more explicit on the issue of effective teaching.
Training as early-career mathematicians
The training and mentoring of early-career faculty has long been recognized as important by the mathematics community, which has responded with a variety of efforts. As perhaps the largest single effort to date by a professional organization, since 1994 the Mathematical Association of America (MAA), through Project NExT (New Experiences in Teaching), has provided multi-year intensive mentorship and support for over 1500 early career faculty in the mathematical sciences. Other opportunities now abound as well. The Academy of Inquiry Based Learning provides weeklong summer training workshops, mentorship programs, and small grants to assist faculty with transitioning to an active-learning teaching style. The MAA also offers 4-hour mini-courses at each of the Joint Math Meetings and Mathfest, which vary greatly in topic as illustrated by the 2016 JMM offerings. Thus, at the national level there are many opportunities for professional development regarding teaching, a large number of which are focused on active learning methods; however, issues of access certainly exist for faculty at institutions with limited funds available to support participation in these programs.
At the local level, it is common for departments and colleges to have faculty mentoring programs, though as with graduate training, these programs vary widely across institutions. Unfortunately, at some institutions new instructors and assistant professors may go several terms before receiving feedback about their teaching (if they receive any at all). Other institutions do have forms of mentoring in place, such as regular classroom observations or meetings with a “master teacher” in the department. Still others have well-established, formal mentoring programs in which new instructors are paired with more experienced faculty. However, as we noted before regarding graduate school, these mentors may not have much experience with active learning techniques. The ways in which teaching is assessed also vary, with some departments emphasizing student evaluations and some prioritizing classroom observations. The variety of mentoring opportunities for new faculty, and the reality that many early-career faculty members do not receive sufficient mentoring and training, suggests that continued efforts are needed to improve the overall landscape of pedagogical training for early-career faculty.
Further, depending on departmental and institutional culture, junior faculty often have reasonable concerns regarding earning tenure or ensuring that a short-term contract is renewed. This can cause them to hold back from trying unfamiliar teaching methods for fear of negative student responses or of a classroom observation that is negative because the observer does not agree with the teaching method. This can sometimes cause early-career faculty to delay gaining experience with active learning methods that have been shown to have positive impacts on students. It is particularly important for department leadership and higher administrators to find clearly-communicated ways to support early-career faculty who wish to pilot the use of unfamiliar teaching methods, especially those active learning methods that have evidence supporting their effectiveness.
Even when quality support is available to early-career faculty, there remain inherent challenges for developing as a teacher. The term “expert blind spot” probably rings true for anyone who has tried to teach anything to a relative novice. It describes situations in which instructors’ advanced knowledge of content interferes with their ability to understand their students’ learning processes [7,8]. Within mathematics, our custom of proof-centered discourse does not always translate well to the classroom. Lower-level students may have weak backgrounds, including scant practice with the logic that now comes naturally to us. Even upper-level mathematics majors are not always ready for the presentations that have become second nature to their instructors with doctoral degrees. This is related to our previous post about telling in teaching mathematics; sometimes we as mathematicians can insist upon telling students facts over and over — \((a^2 + b^2)\) does not equal \((a+b)^2\) — facts which may be obvious to us, without fully acknowledging or accepting a student’s struggle to learn such facts.
This is not to say that the graduate school experience has nothing to offer an instructor. Having wrestled with an open problem in preparing a thesis, a PhD mathematician surely understands the value of struggle and occasional failure — recognized in current public discussions of education  — to the learning process. At the same time, experts tend to see the classroom as a place to organize ideas, while novices are better served if it is also an environment for discovery, error, and invention . Mathematicians who are teaching should consider the broader vision in order to reach the particular novices in their classes. This is not a trivial exercise.
Having come to the end of our series on active learning methods in mathematics, we wish to emphasize one last point. There is a fundamental way in which our training as mathematicians can help us develop as teachers: mathematicians are expert problem solvers. As a community of mathematicians and academics, we are in the process of solving the problem of how best to teach mathematics, and we are jointly working together toward that end. As with all complex, real-world problems, the challenge for us is that there is not an exact solution, but rather a collection of approximate solutions. Nevertheless, our mathematical training has prepared us as problem solvers to hone our intelligence, our diligence, our spirit of curiosity, and our love of learning in order to develop meaningful and effective ways of teaching. These qualities are directly related to who we are as mathematicians, and it gives us hope for success in our continued endeavor of improving mathematics teaching and learning for all.
 Is Algebra Necessary? Andrew Hacker. New York Times, July 29, 2012. http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=0
 Are College Lectures Unfair? Annie Murphy Paul. New York Times, September 13, 2015. http://www.nytimes.com/2015/09/13/opinion/sunday/are-college-lectures-unfair.html
 Lecture Me. Really. Molly Worthen. New York Times, October 18, 2015. http://www.nytimes.com/2015/10/18/opinion/sunday/lecture-me-really.html
 Engage to Excel. PCAST report, January 2012. https://www.whitehouse.gov/sites/default/files/microsites/ostp/pcast-engage-to-excel-final_feb.pdf
 Beisiegel, M. & Simmt, E. (2012). Formation of mathematics graduate students’ mathematician-as-teacher identity. For the Learning of Mathematics, 32(1), 34-39.
 Ellis, J. (2014). Preparing Future Professors: Highlighting The Importance Of Graduate Student Professional Development Programs In Calculus Instruction. Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3 (pp. 9-16). Vancouver, British Columbia: PME.
 Nathan, Mitchell J., Kenneth R. Koedinger, and Martha W. Alibali. “Expert blind spot: When content knowledge eclipses pedagogical content knowledge.” In Proceedings of the Third International Conference on Cognitive Science, pp. 644-648. 2001.
 Nathan, Mitchell J., and Anthony Petrosino. “Expert blind spot among preservice teachers.” American educational research journal 40.4 (2003): 905-928.
 Lahey, Jessica. The Gift of Failure. Short Books, 2015.
 Bransford, John D., Ann L. Brown, and Rodney R. Cocking. How people learn: Brain, mind, experience, and school. National Academy Press, 1999.