By Benjamin Braun, Director of Graduate Studies, University of Kentucky, benjamin.braun@uky.edu
For many mathematicians working in Master’s and Doctoral-granting departments, training and mentoring regarding graduate advising is done in ad hoc and informal ways. As a result, many new advisors work in some degree of isolation as they develop their advising styles, with only their own personal experience to guide them. In this article, I provide suggested practices that can make the advising process both more enjoyable and more effective. Knowledge of these ideas can also be helpful for graduate students, as they provide ideas for effective communication with their own advisors.
- Explicitly Agree on Expectations and Advising Style
Most graduate students do not have a clear idea what to expect from the student/advisor relationship. While I have heard some mathematicians complain about the ill-informed ideas or inexperience of graduate students, this is not a deficiency on the part of students. How are graduate students expected to acquire detailed knowledge of the graduate school experience if they are not part of families or communities that include academics or doctoral graduates? Further, how is anyone supposed to intuitively understand the expectations of a specific advisor or graduate program?
It is the responsibility of each graduate advisor to start a discussion with their students about expectations and advising style, and it is the responsibility of the student to be fully engaged in this discussion. A clarifying framework for conceptualizing advising styles can be found in the work of Gordon B. Davis in his article “Advising and Supervising” [5]. This framework describes advisor and student responsibilities for five styles: Strong master/apprentice style, Collegial master/apprentice style, Collegial development style, Guidance and suggestion style, and Passive Hands-off style. These styles are described in detail in the table provided below. It is reasonable to expect that at the start of the advising period, each advisor and each student will have a preferred style, and these might be distinct. It is important that both students and advisors are aware of these advising preferences and that they are openly and clearly communicated.
Rubric of Advising Styles [5, Table 1.1]
| Style | Advisor Role and Behavior | Student Role and Behavior |
| Strong master/apprentice style | Advisor is master. Advisor has a well specified domain of expertise and set of problems within it. | Student is an apprentice working for the advisor. Student works on advisor’s problems. |
| Collegial master/apprentice style | Advisor is expert who limits advising to problems that are within scope of his or her research skill set but will work on student’s problem. | Student develops a problem within advisor’s domain and skills and works under the advisor to develop the research plan and procedures. |
| Collegial development style | Advisor is senior colleague who will respond to student research problem and extend his or her advising domain to include new problems and new skills. | Student takes initiative to introduce new problem that requires new skill set and works as a junior colleague with advisor in joint development of new domain. |
| Guidance and suggestion style | Advisor is a senior colleague who gives good general guidance over a wide range of problems and methods but does not have personal skill in all of them. | Student is an independent, junior colleague who takes initiative for presenting problems and research plans for discussion and guidance. Student develops required skills. |
| Passive hands-off style | Advisor has quality control role and responds only to requests or documents and performs only general quality control review. | Student is an independent researcher who takes initiative for developing problem, developing skills, and presenting research plans for general review and approval. |
It is also important to observe that an advising style does not have to be fixed throughout graduate study for a student. For example, in my own doctoral advising, I tend to start out in the Collegial Master/Apprentice or Collegial Development style (as described in the Appendix) and shift over time to the Guidance and Suggestion or Passive Hands-Off style as my students mature intellectually and professionally. I do not use these phrases exactly with my students; instead, as my students make progress toward their degree requirements, I encourage them to develop the habit of operating more independently. I also make clear to them that developing this independence while they still have me as a mentor to fall back on is valuable experience before they complete their degree and start working independently.
One potential pitfall that advisors can fall into is assuming that their students will have the same needs they did as graduate students. Another pitfall is that advisors sometimes will forget that their students are advisees rather than employees, that at the end of the day the student is responsible for making their own academic and professional choices. These pitfalls can lead to miscommunication and tension between students and advisors. By initiating clear conversations about interpersonal expectations and advising style early in the advising relationship, and by continuing those conversations throughout the time of graduate study, advisors can proactively address issues that arise before they become a significant problem.
- Agree on a (Tentative) Plan and Review it Regularly
Just as it is helpful for tenure-track faculty to have clear expectations regarding their tenure requirements, it is similarly helpful for graduate students to have a sense of the “trajectory” that their advisor and program expect. The best way to achieve this is to lay out a few key goals in a target timeline that is developed in collaboration between advisor and student.
Here is an example of the type of timeline I might propose to a student. Suppose I have a new doctoral student who has completed their written exams by the end of their second year (which is common at my institution), and who has a career goal of getting a job at a teaching-focused academic institution with reasonable but not extensive research/scholarly expectations (this career goal is also common at my institution). In such a situation, I would use the following target timeline as a starting point for discussion.
| Overall goal | Graduate in May or August of fifth year (Note: the reason for this is that at my institution, support is generally given through the end of the sixth year. Thus, in case of delays of any type, e.g. unlucky research setbacks, poor outcome on job market, etc., this provides a “backup” year that can be used if needed.) |
| Fall of third year | Prepare for oral qualifying exam |
| Spring of third year | Complete oral qualifying exam, begin work on specific research problem (if not already started) |
| Summer of third year | Consider attending a summer school/program/workshop/internship if any are available |
| Fall of fourth year | Evaluate research progress, decide if original research problem is leading to adequate progress, change problem(s) if needed |
| Spring and Summer of fourth year | Complete enough research to constitute a dissertation, write and submit one or two papers prior to beginning job search |
| Summer of fourth year | Consider attending a summer school/program/workshop/internship if any are available |
| Fall of fifth year | Focus on job search, continue work on research |
| Spring of fifth year | Dissertation defense and conclusion of job search |
This particular timeline is not meant to be prescriptive for other new advisors, because it reflects my own advising style, the norms for my institution, the norms for my research area, etc. Rather, this timeline is meant to illustrate the level and depth of planning that I have found helpful to explicitly discuss with students. Laying out concrete goals (complete oral qualifying exam, complete first research project, etc.) gives graduate students a clear vision of what is expected from them, enabling them to better evaluate their progress through the program. Through revisiting and revising the tentative plan every 8-12 months, with input from both advisor and student, the planning process can be a positive and collaborative experience.
- Discuss Student Goals and Habits
It is critical for advisors to respect and support the long-term goals of their students, even when they differ significantly from what the advisor envisions as a successful career in mathematics. The range of career paths for PhD mathematicians is varied and growing, with connections to academia, industry, business, and government [1,7]. Further, the academic job market is in a state of major change and disruption [2,3], and this was happening long before the start of the covid pandemic. While many graduate students begin their studies with the goal of becoming a faculty member, and while an academic career remains a reasonable goal for many current students, it is important to make sure students are aware of the reality of the mathematical job market and are informed sufficiently to make a purposeful choice about how they want to approach goal setting. One resource worth passing along to students interested in jobs in business, industry, or government is the BIG Math Network [9].
The best way to handle these important issues is to begin the advising relationship by explicitly asking students about their career goals, discussing a range of possible goals (students do not have to pick only one possible goal!), and helping the students obtain access to any resources they need to pursue them. I have found it helpful to return to this conversation every 9-12 months, since goals often change as students gain experience and a broader vision of the mathematical community. As part of these discussions about goals, advisors and students should also discuss the habits that students need to put in place to reach their goals. For example, if a student is interested in working in government or industry, then they need to build the habit of spending time developing their coding skills. If they are aiming for a teaching-focused academic position, then they need to build the habit of spending time thinking about pedagogy and teaching. I have found that James Clear’s framework from his book Atomic Habits [10] is useful for facilitating conversations about goals and habits with students.
- Develop the Skills of Humble Inquiry
The suggestions I have provided so far rely on an advisor being skilled in the art of having clear, and sometimes challenging, conversations with students. The critical ingredient in such conversations is for advisors to ask their students a variety of questions to get a meaningful understanding of where students are coming from, and to respond to student ideas (even if they seem undeveloped or naive) without judgement. I have personally found it surprisingly difficult to do this well, because our cultural and professional training consistently underemphasizes this aspect of professional mathematical life.
The reality is that creating cultures and habits of authentic inquiry in our interpersonal relationships provides the social and emotional support that almost everyone needs to pursue mathematics professionally. One accessible and helpful resource to learn more about building such cultures and habits is Edgar Schein’s book Humble Inquiry: The Gentle Art of Asking Instead of Telling [8]. Schein describes Humble Inquiry as the type of inquiry that
“maximizes my curiosity and interest in the other person and minimizes bias and preconceptions about the other person. I want to access my ignorance and ask for information in the least biased and threatening way. I do not want to lead the other person or put him or her into a position of having to give a socially acceptable response. I want to inquire in the way that will best discover what is really on the other person’s mind. I want others to feel that I accept them, am interested in them, and am genuinely curious about what is on their minds regarding the particular situation we find ourselves in.” [8]
Schein is careful to distinguish this form of inquiry from others, such as diagnostic inquiry, confrontational inquiry, and process-oriented inquiry. He emphasizes that “the world is becoming more technologically complex, interdependent, and culturally diverse, which makes the building of relationships more and more necessary to get things accomplished and, at the same time, more difficult.” In other words, the challenge of building authentic interpersonal relationships is not only one for mathematical culture, but for society at large. Schein also emphasizes the importance of individuals in leadership positions, such as graduate advisors, learning to use and model authentic inquiry.
Humble inquiry can influence not only how you interact with students, but more generally how you view the possibilities within others. I share below some questions that we might not ordinarily consider when we are speaking with a student (or a colleague, employee, or peer). However, questions such as these can powerfully change our perspective toward those around us. I am not suggesting that we routinely ask these questions in ordinary conversation, but rather that we have these questions in our conscious mind, that we remain consistently open to the possibility that the people we interact with have interesting, complicated, inspiring, or difficult lives.
- Does my student have access to sufficient food and housing to meet their needs and the needs of their family?
- Does my student have a hobby that they are passionate about?
- Does my student have personal challenges or crises I don’t know about, such as a relative, spouse, or child with a serious health issue? Who is my student responsible for supporting financially?
- Was there a particular class or experience that made my student decide to pursue mathematics professionally?
- How many additional hours each week does my student have to work to pay for their housing and basic expenses?
- What is the most inspiring experience my student has had as a teaching assistant?
- Is my student suffering from PTSD due to military or other service?
- What would be the ingredients for a day filled with joy for my student?
These questions reflect both positive and negative realities of our students’ lives. While the answers to these questions would not necessarily change my academic expectations for my graduate students, they might inspire me to change some of my own attitudes, behaviors, or perceptions. By recognizing and affirming that we are often ignorant of meaningful aspects in the lives of those around us, we can be more empathetic, flexible, and ethical in our treatment of and relationships with others.
- Ask Others for Advice and Resources
Finally, to help support our students as advisors, it is important that we also have support. To the greatest extent possible, new graduate advisors should seek out trusted mentors and colleagues, whether at their own institution or elsewhere, for advice and suggestions. While some departments have effective mentoring programs, others have few or no formal support mechanisms in place, and junior faculty can be left in the position of needing to seek out help independently. For any faculty in that situation, reach out to others as much as possible; at every stage of our career, we each benefit from mentoring and support.
One resource to keep watch for is the ongoing series of workshops on graduate advising and mentoring in mathematics that have been offered over the past ten years. The two most recent such workshops took place in 2019 at Ohio State University and in 2023 at Rice University. For anyone who is interested in learning more about organizing such an event, see the AMS Notices article on this topic by Daniel J. Thompson [6]. Another source of support can often be found via university teaching and learning centers, which frequently offer workshops or professional learning communities on the theme of effective mentoring. Because many of the challenges of graduate advising transcend disciplinary boundaries, I encourage new advisors to seek support and guidance from both within and beyond the mathematics community.
References
[1] “Math PhD Careers: New Opportunities Emerging Amidst Crisis.” Yuliy Baryshnikov, Lee DeVille, and Richard Laugesen. Notices of the American Mathematical Society, Vol 64, No 3, March 2017, pp 260-264 http://www.ams.org/publications/journals/notices/201703/rnoti-p260.pdf
[2] “Survey on Math PostDocs.” Amy Cohen, Letter to the Editor, Notices of the American Mathematical Society, Vol 64, No 6, June/July 2017, pp 541 http://www.ams.org/publications/journals/notices/201706/rnoti-p540.pdf
[3] “Disruptions of the Academic Math Employment Market.” Amy Cohen. Notices of the American Mathematical Society, Vol 63, No 9, October 2016, pp 1057-1060 http://www.ams.org/journals/notices/201609/rnoti-p1057.pdf
[5] “Advising and Supervising.” Gordon B. Davis. In Research in Information Systems: A handbook for research supervisors and their students. Butterworth-Heinemann, 2005. Preprint at http://misrc.umn.edu/workingpapers/fullpapers/2004/0412_052404.pdf
[6] “Organizing a Graduate Advising Workshop in Mathematics.” Daniel J. Thompson. Notices of the American Mathematical Society, Vol 66, No 11, December 2019, pp 1829-1830
[7] National Research Council. The Mathematical Sciences in 2025. Washington, DC: The National Academies Press, 2013. http://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025
[8] Humble Inquiry: The Gentle Art of Asking Instead of Telling. Edgar H. Schein. Berrett-Koehler Publishers. 2013.
[9] BIG Math Network, https://bigmathnetwork.org/
[10] Atomic Habits. James Clear. Avery. 2018.
Note: This article includes updated versions of passages from two articles by the author published on 8 January 2018 and 14 May 2018 by the American Mathematical Society blog “On Teaching and Learning Mathematics”.