Thursday, April 25, 2019

IBL Blog Q&A: The TIMES Project, Karen Keene, Justin Dunmyre

This blog post is an Q&A session conducted via email with Dr. Karen Keene and Dr. Justin Dunmyre. They are sharing information about the TIMES project. Thank you Karen and Justin!

0. Please tell us about yourselves.

Karen Keene has her Ph.D. in Mathematics Education from Purdue University.  Karen was introduced to active learning in undergraduate mathematics education while she was a graduate student involved in the creation of the Inquiry-Oriented Differential Equations materials. She has been serving as a project leader on the TIMES project where inquiry-oriented instruction, one form of active learning since 2013. She is currently an Associate Professor of Math Education at North Carolina State University and is currently serving as a rotating Program Officer for the National Science Foundation.

Justin Dunmyre has his Ph.D. in Mathematics from the University of Pittsburgh, and is a Brown ’13 Project NExT fellow.  He is currently an Associate Professor and Chair of Mathematics at Frostburg State University. Through Project NExT, Justin got interested in active learning, and subsequently participated in the IBL Workshop.  This transformative experience led him to wonder what IBL would look like in his discipline (differential equations) and almost as soon as he had that thought he got an email through the IBL mailing list about this exciting TIMES project!

1. We'd like to learn about the TIMES project. What is the main idea behind this effort?

The TIMES project began as a collaboration of second-generation authors of varied inquiry-oriented (IO) classroom materials.  By second-generation we mean that Michelle Zandieh, Sean Larsen, and Chris Rasmussen wrote the original IO materials for Linear Algebra, Abstract Algebra, and Ordinary Differential Equations, respectively.  The TIMES Principal Investigators, Christy Andrews-Larsons, Estrella Johnson, and Karen Keene were each graduate students of these original authors and launched the TIMES project to study how they might support other faculty in using these materials.  That’s why TIMES actually stands for Teaching Inquiry Mathematics: Establishing Supports. These supports were three fold: providing the curricular material, a 3-day summer workshop on using the materials, and weekly online working groups. The three supported curricula were IOLA (Inquiry-Oriented Linear Algebra) and IOAA (Inquiry-Oriented Abstract Algebra formerly known as TAAFU - Teaching Abstract Algebra For Understanding) and IODE (Inquiry-Oriented Differential Equations).

Justin was in one of the first cohorts of TIMES fellows for IODE, and became involved in running the online working groups and became a coauthor on the materials, along with Nick Fortune whose dissertation research was supported by TIMES project.

2. What classes do you have materials for?

A first course in Linear Algebra, an abstract-algebra course focused on groups, and a first course in differential equations.

3. What is a typical day like in an IO class?

A typical day is centered around the guided reinvention of particular mathematical concept(s).  The tasks are based on the principles of the instructional design theory of Realistic Mathematics Education (RME), of which one of the tenets is that the material must be experientially real for the students.  The Inquiry Oriented materials are grounded in contexts that the students can initially understand and reason about, maybe from a less sophisticated viewpoint, even if they’ve never had that specific experience before.  For example, in IODE, one tasks early in the materials is focused on population growth of owls in a forest, which students may not know a lot about, but can understand. As in any IBL classroom, students are constructing the mathematics for themselves, and taking ownership of that mathematics. The students work through a series of tasks, often encountering the the tasks for the first time in class. Therefore, we rely on small group work for an initial translation from the context to concepts relating to the learning outcomes of the class.

The instructor facilitates this guided reinvention by primarily using four instructional components (Kuster, Johnson, Keene, Andrews-Larson, 2018) , not all of which may happen on the same day.  These four instructional components are general instructional goals: eliciting student thinking, building on student thinking, building a shared understanding of the mathematics reinvented in the classroom, and connecting the students’ mathematics to formal mathematics.

In any given day in our classes, one would see the students sitting in small groups, working on tasks. One would also hear whole class discussions where the instructor is soliciting student input, re-voicing, and, reshaping it, innocuously,  to guide the conversation with an eye on the instructor's mathematical agenda. Occasionally, the students would make presentations of their ideas, but this would not always happen. The whole class discussion and small group discussions do happen every day.  Finally, when the class has finished an idea, or perhaps developed a need for notation to express their ideas, the instructor connects their work with formal language and notation.

4. What are some of your best moments as a teacher in an IO class?

Justin:  There are so many!  One of my favorites came from the first time I taught our optional unit on bifurcation theory.  This unit starts with a task where students are challenged to model the introduction of a parameter that represents harvesting of fish from an otherwise logistic model.  After settling on a simple shift of the form dP/dt = 0.2P(1-P/25) - k, where k is the harvesting parameter, students are asked to come up with a one page report to explain to the owners of the fish hatchery the ramifications of varied choices in k.  The one page report is the trick! By requiring students to use space efficiently, they can actually invent the bifurcation diagram for themselves. What really surprised me was how many different forms this bifurcation diagram can take. I’ve seen students use spreadsheets that show if dP/dt is positive in green or negative in red (the bifurcation diagram then emerges as the change from green to red), carefully stacked phase lines, analytically drawn bifurcation curves using the quadratic formulas, and more.  When the students present these ideas to one another, they realize they’re all saying the same thing, and absorbing insights from other groups result in very deep understanding of this sophisticated concept. The first time I taught this unit I was giddy, I couldn’t believe that these bifurcation diagrams were emerging before my eyes, completely invented by my students!  We wrote about this task sequence in a PRIMUS paper here (Rasmussen, Dunmyre, Fortune & Keene, 2019). 

The “salty tank” problem is practically a rite of passage for students in my differential equations classes.  It’s developed its own legend here on campus, because the discussions are so robust, and it is the first time that students are really asked to develop their own equation.  The students really marvel at how they can have an 50 minute long debate over a prompt as simple as “A very large tank initially contains 15 gallons of saltwater containing 6 pounds of salt. Saltwater containing 1 pound of salt per gallon is pumped into the top of the tank at a rate of 2 gallons per minute, while a well-mixed solution leaves the bottom of the tank at a rate of 1 gallon per minute.” This problem is the seat of another favorite moment of mine. Ideas were flying all around the room, what does it mean to be well-mixed, what should the input term look like, what should the output term look like?  A student said something to her group, but she didn’t want to cut into the whole class discussion. Her group thought it was important though, so one of her more outgoing group members interrupted the discussion and he said “I think we should all hear what Sarah has to say.”  Of course, it was a critical insight that helped reframe the conversation in a productive direction! But the act of one student elevating the status of another student, that was a powerful moment that will always stick with me.

Karen:  When I taught this course to math and physics majors together, my favorite times were when we talked about the difference between instantaneous rate of change and rate of change over a "very very small time interval".  The conversations were always spirited and deep, with the Physics Majors declaring it doesn't matter if there is a difference and the Math majors wanted it to matter and try to understand what a "limit" really is. Of course, I was always rooting for the Math Majors, but it didn't really matter, as it was a situation where the students were engaged in thinking deeply about the math and taking the authority of learning on themselves.  Of course, ultimately, we had to agree to disagree and the physics majors usually could go along with the idea of instantaneous rate of change as that being the foundation of a differential equation.

I can think of other times that after we had small group discussions, two groups would present their ideas about a particular task- and they were not the same.  When that happened, I would encourage each side to state their case. Then I would send the students into small groups to continue the discussion and decide what they thought.  This might go on for much of a class. I know it took up a lot of time, but it was worth it--- they were not waiting on me to tell them- but making their own mathematical judgments.  Most of the time, it all seemed to move the agenda forward. I do remember one time that the whole class agreed on something that I knew was mathematically wrong. I made the decision (there was no test or assignment the next day) to let it stand.  On the next day, I brought of the decision with a question that led them to believe they were wrong-- and all was forgiven!

5. If you could give some advice to math instructors thinking about using active learning, who have not tried yet, what would you say to them?

Justin: There is evidence in the research that supports your decision to try active learning, so you should proceed with confidence.  For the IO curriculum, it is extremely exciting to be a sort of curator of the conversation. You don’t know what students are going to say, and you get the exhilaration of thinking on your feet to fit their ideas into your agenda.  So, my advice is: this is hard work! Be kind to yourself when you just can’t reshape their ideas the first time. We have found that, although active learning is our main mode of instruction, there is a “time to tell.”

Karen: Take it a step at a time— some instructors might go all out first time around, but trying one or two days or tasks and seeing how it works in your classroom is just fine. Ultimately, students will be more engaged and take ownership of the mathematics they learn in your new active learning classroom.

6. How can readers learn more about the TIMES project and get involved?

The TIMES NSF grant has essentially run its course, so we are no longer running workshops.  You can find the course materials, including instructor’s notes, at these websites:


When you begin to investigate these materials, please don’t hesitate to contact us; we are more than happy to help!  With sufficient interest, we may even run informal online working groups.

George Kuster, Estrella Johnson, Karen Keene & Christine Andrews-Larson
(2018) Inquiry-Oriented Instruction: A Conceptualization of the Instructional Principles, PRIMUS,
28:1, 13-30, DOI: 10.1080/10511970.2017.1338807

Chris Rasmussen, Justin Dunmyre, Nicholas Fortune & Karen Keene (2019) Modeling as a Means to Develop New Ideas: The Case of Reinventing a Bifurcation Diagram, PRIMUS, DOI: 10.1080/10511970.2018.1472160

Guest Post: Ed Parker on Graduate-Level Math Teaching and IBL

SY :  I’ve had several people sympathetic to IBL methods suggest that as IBL methods become more widespread at the undergraduate level, they will be unnecessary in graduate mathematics programs.  Do you have any thoughts on this?
Ed Parker:  Since I’ve never taught in a department that offered the PhD in mathematics, I’m probably not the right person to be asking.  But I’m certainly willing to respond.  You just need to understand that any expertise I may have is either historical or based on my limited teaching experience with master’s level students or my experiences as a graduate student.  It is somewhat ironic that the issue has arisen.  When I wrote Getting More from Moore back in 1988, IBL was, at least grudgingly, accepted as a viable option for graduate mathematics education while undergraduate IBL was pretty rare.  Several programs such as Auburn and North Texas had committed graduate programs and others such as Emory and the University of Texas had a visible presence.  When discussing IBL’s possibilities for undergraduates, I remember well being told by multiple persons from multiple places that IBL might work with well-prepared upper level majors, but that the students needed to be “ready for rigor” before it could have a chance to work.  And demonstrating and having the students reproduce what they had seen, then apply the theory to examples was apparently the way to make the students “ready”.  Readiness seems to be a given at the graduate level due to admission requirements.
First, I would suggest that anyone interested in the issue watch the video of John Neuberger’s talk on graduate mathematics education, delivered at the 2007 Legacy Conference. Rather than focusing on technical math education issues, he began with the challenge of turning curiosity into passion for the hunt and then proceeded to relate some of his experiences relative to benefitting from, and implementing, IBL.  Even an impartial judge (which I am not!) would likely judge his career as a teacher a successful one.  In 1977, I became his 13th PhD student and he continued to produce productive PhD’s after his move to the University of North Texas the following year.  A notable aspect of his talk is his view that graduate teaching is a natural adjunct to a mathematician’s research.
I entered Emory University’s graduate program in 1973 after a four-year hiatus following my graduation from Guilford College, during which time I had taught secondary mathematics at Bayside High School in Virginia Beach, Virginia.  My draft board had decided public school mathematics teaching was in the public interest and granted me an occupational deferment in lieu of processing my application for conscientious objector status.  (That draft board had not dealt with a CO hearing since my twin uncles during World War II.)  My undergraduate education at Guilford was decidedly IBL.  In the core courses, we mostly read stuff, talked in class about what we read, and then wrote about what we had read and talked about.  With the exception of the calculus sequence and differential equations, none of my mathematics courses had a textbook. Although Mr. Boyd had us buy Heider and Simpson’s Theoretical Analysis for analysis and Greever’s Theory and Examples of Point-Set Topology for topology and Mr. Walker had us buy Birkhoff and MacLane’s A Survey of Modern Algebra for algebra; all were used as reference points and problem sources.  We did not reproduce textbook proofs in any of these courses. 
At Emory, I took the algebra sequence and the analysis sequence my first year, taking only two courses since I had only a 2/3 assistantship.  David Ford’s introductory analysis course in Lebesgue measure and topological vector spaces was totally IBL.  Trevor Evans’s algebraic structures course was given from course notes.  He had us buy Herstein’s Topics in Algebra and the Schaum outline on group theory as resources.  Dr. Evans talked through the course notes three days a week, stating additional problems as he went, and the fourth day was student presentation day.  A typical presentation by me went like this: I presented.  Dr. Evans would stare at the board, stroke his chin, then put down his pipe and say,  “I suppose you are correct, Mr. Parker, but WHY DIDN’T YOU THINK OF THIS?”  Then he would take the chalk from me and show the class a “good” proof.
As a second-year student, I took the third first-year course, a topology course in Moore spaces given by William Mahavier through IBL.  I had come to Emory with the idea of studying foundations and Dick Sanerib was offering a course on Model Theory that year. However, Emory would not give me credit for the fall quarter due to a course in symbolic logic that I had done at Guilford, so I audited model theory which was done by straight lecture following Bell and Slomson’s text and took John Neuberger’s Functional Analysis and Differential Equations, which was given by IBL, for credit.  I solved three problems that quarter:
There is a single function, call it $f$, so that if $x$ is a number, then $f’(x) = f(x)$ and $f(0) =1$.
There is a single function, call it $f$, and a largest non-degenerate connected set containing $0$ that is the domain of $f$, so that if $x$ is a number in the domain of $f$, then $f'(x)=-f(x)^2$ and $f(0)=1$.
Suppose that $x$ is a number.  Then  $\Sigma_{n\in\mathbb{N}}\frac{1}{n!}*x^n=\Pi_{n \in \mathbb{N}}(1-\frac{x}{n})^{-n}$
Imagine, if you will, knowing that  expe was the answer to the first question but having no idea how to make it appear, or that you could solve the second differential equation by “separation of variables”, but realizing that assuming a solution existed begged the question.  Needless to say, I didn’t think I was doing very well.  Looking back on it, I’m kind of glad I didn’t think of producing a power series from thin air, then proving that it worked since the path I took led through the Fundamental Existence and Uniqueness Theorem.
Near the end of the fall quarter, Dr. Neuberger stated a list of eleven problems that I later found to be, if one took the collective hypotheses and conclusions, the Hille-Yosida Theorem.
At the end of the quarter, I had to decide whether to continue Model Theory or Functional Analysis and Differential Equations.  Neither professor recruited me and I still don’t know why I chose to continue FA&DE.  Did I mention that I didn’t think I was doing very well?
I finished Hille-Yosida in early March.  (It took my classmate only three weeks once he went to work on it!)  Within a calendar year of when I finished Hille-Yosida, I had the theorems that formed the core of my thesis although I had still not passed my algebra qualifying exam.
Heading into my third year, having passed my analysis and topology qualifiers and failing my algebra qualifier and having taken complex analysis in summer school, I was scheduled to take the second level topology course and Dr. Neuberger’s research seminar.   A note of comparison is in order here.  A student of Dr. Evans pursuing an algebra thesis was expected to spend his “year of preparation” reading the pertinent literature.  On the other hand, I didn’t even know I was beginning work on a thesis.  Dr. Neuberger gave me a paper of his on Lie Semigroups and a short paper of Tosio Kato’s that had distilled (brilliantly!) a very long paper of Miyadera’s which had originally proven the dense differentiability of non-expansive semigroups on Hilbert spaces to work through.  I was given no guidance of which I was aware about why or how.
Before continuing on this line, I should mention that I seriously considered dropping out after getting the news that I had failed the algebra preliminary exam.  The birth of our second child the day after news of having failed the algebra prelim rescued me psychologically, but it also added yet another level of family responsibility to my table.  I talked with Dr. Neuberger, who was teaching complex analysis and he said that it was fine for me to use the course time to write a master’s thesis and that Dr. Mahavier had described to him an example I had made in spring quarter of the first-year topology course that would likely provide the substance for the master’s thesis.  I talked to Dr. Mahavier and he agreed to supervise the thesis.  Ironically, Dr. Evans, with whom I had taken the first-year algebra sequence, whose second-level seminars I had attended, and who never seemed to like my proofs, suggested that I should continue.  That, together with my wife’s encouragement, won the day.  My assistantship was renewed and I embarked on my third year.
In the research seminar, I tried to work my way through the two papers.  I had never been good (as in quick) in following other persons’ arguments, but I dutifully slogged my way through, with a cognizance of the structures Dr. Neuberger had appropriated from Hille-Yosida.  The elegance of Kato’s argument made it easy (even for me) to follow, but I realized that I was just verifying details.  This caused me to set out on my own, mimicking Dr. Neuberger by thinking about Hille-Yosida structures in non-linear contexts.  The Cesaro mean (I later found out that was what it was called) was the vehicle to a theorem on non-linear semigroups that I formulated and proved.  In seminar, Dr. Neuberger listened to my argument without changing his expression.  When I finished, he gave me a copy of Glenn Webb’s landmark example of a non-expansive semigroup on a Banach space that contained an open set in its domain where it was nowhere differentiable and asked me to see if his semigroup satisfied the premise to my theorem.  That night my euphoria turned to despair; I could prove that the premise was satisfied, and once I understood Glenn’s example, the verification of the application was dirt simple.  Thus, because it was easy even for me, I was sure my theorem must be no good.  In seminar the next day, Dr. Neuberger asked if I had been able to do what he asked and I mumbled something like “It can’t be any good; it’s too easy,” and showed him my argument.  He became instantly animated and told me, “This is the sort of theorem that theses are built around.”
At this point, Dr. Neuberger gave me some entries into the literature through which I learned about the work of the Japanese school that Kato had consolidated and gained access to the (then) current work of Brezis, Pazy, Crandall, Martin, and Liggett, and Neuberger’s seminal paper which had ignited the Japanese school’s initial successes.  An ancillary aspect of learning to use the library was to make sure that my theorem and its application were original.
There was still the issue of passing my algebra qualifier.  At Emory, the rule was two strikes and you’re out.   I was auditing Mary Frances Neff’s first year algebra sequence which I continued for the year and passed the algebra qualifier on my second try.  The department was kind enough to expand the two 4-hour qualifiers from the year before to two one-day tests.  They kicked me out after 9 hours the first day and 8 hours the second.  Both days, there were still problems I thought I could do.
I finished the following year.
What inferences can I make if I add to the mix what I have learned from talking to colleagues about their graduate school experiences?
Broadly, IBL at the graduate level, if the goal of an advanced degree is to certify readiness for original problem solving and ability to pass the mathematical canon of one generation to the next, is a super-charged version of what happens at the undergraduate level.  In contrast to the undergraduate entry, at the graduate level a four-year baccalaureate mathematics degree is in place as well as an entry test result that gives comparisons with other such students and may show some level of breadth and recall of some curriculum.  The main question probably should be, “How can this base best be nurtured?”.  The traditional response has been to give a “graduate level” broadening and strengthening by “mastering” carefully selected texts and/or the arguments of the professors’ lectures to create a base, then to certify the students’ readiness with a battery of barrier examinations.  Those deemed worthy are then given a second dose of the program in more concentrated contexts, usually the advisors’ research specialties, and embark on their own research missions, often as colleagues of their advisors.   In IBL, the students recreate the canon by solving the problems fundamental to it as seen by their professors and realize some breadth as they are held accountable for the work of their peers.  The battery of barrier exams appears as an institutional commitment, but, according to what the students have demonstrated in their individual trips through the canons, the turn toward research is not much more than a continuation of what they were already doing, the major difference being that the questions are chosen closer to the frontiers of the subjects of the courses and the classes are smaller. [WARNING: The above characterization is the author’s and may not represent any consensus opinion!]   As results are achieved, the students are directed to the literature with the goal of broadening their knowledge of others’ efforts and increasing the effectiveness of the students’ abilities to find their own problems.
My prejudices in favor of the IBL model are likely too strongly held to give the traditional model a fair hearing.  However, a late ‘80’s/early 90’s tome out of our professional societies exhorted us to make mathematics education “a pump, not a filter”.  The traditional model, in its insistence on early graduate education being a preparation for its barrier examinations certainly looks like a filter to me.  A colleague (whom I greatly respect) who spent the bulk of his career at an urban state university in the same city as an elite private university once told me, “If we could just get the students that Elite U blows away, we would have a better graduate program than they do.”  Perhaps this was an idle boast; perhaps it was not.  The pump effect stands out in Lee May’s recounting of the “sheep and goats” parable in his book on IBL methods.  Lecture and test does not provide for the opportunity William Mahavier seized to split his topology class and recombine it two quarters later as a class of peers.  At a Legacy Conference a year or two after Robert Kauffman of University of Alabama-Birmingham had died, a former colleague spoke to the gathering.  Robert had fought what IBL practitioners might call “the good fight” for many years, standing on the principle of academic freedom to teach in the way he considered most effective.  The colleague, who admitted he was reluctant to become an ally of Robert at first, recounted how he and many of his peers, often after decrying the lack of preparation of their graduate students, would remark how lucky Robert was to get so many good students in his classes.
Stan has written thoughtfully and insightfully on the coverage issue, which is often used to justify criticism of IBL instruction at the graduate level.  Udayan Darji, at a Legacy Conference in the early 2000’s, used his time at the podium to remind the audience that, if there were gaps in what they “should” know, part of their research time should be spent in filling them.  If one looks at my experience in Neuberger’s Functional Analysis and Differential Equations course, it should be clear that anyone capable of doing graduate mathematics could “master” proofs of the three theorems I proved by reading them with the investment of less than a week’s work time rather than the two months it took me.   Similarly, one could likely slog through the Hille-Yosida treatment of the Hille-Yosida Theorem in less than a month rather than the three-plus months it took me.  But would the appropriation of other people’s ideas, at the expense of nurturing your own, get you to a thesis the following year?
Where then, might an IBL student get her/his breadth?  I would first point out that the library will always be there.  But budding mathematicians need not master its entirety before beginning to think on their own.  Considerable breadth is achieved in being handed the responsibility of verifying the veracity of classmates’ presentations.  I still remember Tom Pate’s proof of a theorem in Fourier Analysis, for which I had a “brute force” argument, using soft analysis.  I have not viewed linear algebra the same since that day.  I owe similar debts to Margaret Francel, Everette Mobley, and Terry McCabe, to name just three.  Lessons I learned from them gave me alternative outlooks when I would work through textbook proofs as I put together my own courses.   And, as one teaches with IBL, the students will direct you to “natural” lines of reasoning.  Accumulated experience as well as preparatory learning can also build a mathematician’s repertoire.
In conclusion, I return to Neuberger’s talk:  Your continued commitment to research will fire your teaching and your teaching will abet your research.  So let your students in on the hunt from the get-go.
No tome of mine is complete without a baseball analogy.  Cy Slapnicka became a legendary scout for discovering Bob Feller.  My question is, “Who could have seen Feller throw and not realize that he would become a star?”  In the modern game, the same could be said for Bryce Harper.  But they are the baseball equivalents of the students that Harvard, Duke, or Chicago recruits for its Putnam team and it is doubtful that any form of instruction in graduate school will keep them from succeeding.  There is another group of students that clearly has big-league possibilities and the minor league experience is expected to build into a body of players producing major league level play.  Certain organizations are known for “growing” these players while others let the cauldron of competition weed out the “weak”.  I would suggest that there is a strong analogy here with schools that admit only the testibly top students and then still blow many of them away.  But, in baseball, these two categories of players are not enough to fill all of the rosters.  Finding latent talent and nurturing it is responsible for developing the rest of the big-leaguers.  There are lots of mathematics majors out there with highly developable tools.  I suggest that IBL  does not inhibit the development of super-stars and is likely superior in the development of a far larger number of students.

Monday, April 8, 2019

Cliff's Column on Productive Failure: The Burden of Proof in Belonging

A second post by Cliff Bridges, CU Boulder

Alright, it’s time to get to the business at hand: what stories do I have of failure and how could they possibly soothe anyone else?
The failure which may resonate with the most students, in particular graduate students, is founded in my department’s introductory graduate exams. My department calls these “Preliminary Exams”, or prelims. These are written exams given in each of the first 5 semesters as a graduate student until you pass, and as the name suggests, is meant to test students in material which is requisite for their career as a mathematician. Most graduate math programs have an analogous exam, though names, timetables, and number of attempts may change. In any case, I have heard that failing these exams is the largest cause of attrition in math departments. And by now you may have guessed that I did not pass these exams at first.
Saying “at first” is something I’ve been conditioned to do; to soften the blow of failure and look forward to the future where the endless joy of success nullifies the sting of defeat. But describing endless joy is not the point of this column. The point of this column is to work through the sting. So, more earnestly, I did not just fail prelims at first, I failed at second, third, and fourth too. All of these attempts and failures affected my experience as a graduate student, so that is what I will focus on now.
Some of the more memorable parts of my prelim process were the comments I received, such as “Did you not study?” and “I passed, why didn’t you pass?” and “Maybe you shouldn’t be here. There are departments with lower standards than ours that you could think about joining.” In fairness to these department members, I sincerely don’t think their comments were ill-intentioned. I do, however, think their comments were callous and harmful. Despite this, I unnecessarily spent mental energy replaying their words and thinking about how to respond.
Initially, I took their comments as fairly innocuous. (Unless you are threatening my physical well-being, I usually don’t have much of an emotional response to words.) I thought, “Of course I studied, I studied a lot. That can’t be what you mean to ask.” But, there was no simple answer to the question “did you study?”, which was code for: “Do you really belong here?” Okay, I guess a simple answer was “yes, I belong”, but that felt pretty unconvincing after failing something so “introductory” it is called a prelim. For 3-6 months I had to sit with the unanswered question of my belonging until the next opportunity to provide evidence confirming my suspicions. Carrying a load like that weighed on me, and I think that this weight was the strongest force pushing me out of the department.
The comments of department members magnified this force urging me to leave, and this is why I say their words were harmful. But once I discovered this magnifying effect, I was very adamant that a scaling factor wasn’t going to be the thing that got me to leave. If I was going to leave, it would be on my own accord, based on my own feelings about belonging in this group. I would not be forced out by someone else’s emotional reaction towards my presence. I didn’t want anyone who would try to sabotage my career to have enough power in my life to carry out their plans.
Now, I still had to deal with the initial weight of the "do I belong" question. For me, there was no outside influence who could lighten this load, no support of a mentor or family member who could convince me of something I couldn’t convince myself of. So I had to figure out a way to prove to myself that I did belong, or I had to leave. However, for better or for worse, “prove” is a loaded word for a mathematician.
There was no formal way to prove that I belonged, and there is no formal way for anyone reading this to prove that they belong. Due to the implicit subjectivity of anything defined by society, belonging to a social structure is either trivial, completely subjective, or both. Because of this, there will always be people, a.k.a. gatekeepers, who want to exclude you from groups they think you shouldn’t belong to. I don’t know your struggle, and I won’t be able to help you prove to yourself that you belong. Honestly, I don’t even want to convince you of your belonging. I want to empower you to make this decision for yourself, despite what the gatekeepers are speaking in your ear, despite how much extra weight they are adding to the forces pushing you out, even despite the people who will talk you into belonging to a group you don’t like. I want you to be able to recognize that their words can make the decision heavier, but those words can’t determine whether or not you belong. Only you can do that.
Nowadays I intentionally make space to reflect on where my belonging has been questioned, and even where I have questioned someone else’s belonging. Practicing identifying those situations in my own life has shifted my mindset towards leaning into how failure can be productive. This has even helped me find words to encourage my students when they need another push. Overall, I can’t say that I see failure less, but I can say that I worry less when I see it.
Alright everyone, the bumpy journey through failure has begun. I hope you’re still hanging on looking forward to the next installment!

SY’s Editor’s Note Cliff hits many key issues in the critical period of time, when many students are considering staying vs. leaving graduate school. Thank you Cliff for sharing your story and for being your authentic self on a challenging topic.
One issue I would like to focus on is the role and responsibility of faculty. Speaking as faculty, I see my role as being a positive force for student. And this means mentoring students and being careful about how we speak and being attuned to the needs of students. Women and people of color are significantly underrepresented in Mathematics, and many carry extra burdens placed upon them by circumstances and society and have more obstacles to overcome. Many of these burdens and obstacles are not easily visible, and faculty can be quick to judge. I encourage all faculty from all backgrounds to consider how we can support students, especially when they are dealing with challenging circumstances, and to come into these situations from a position of generosity, compassion, and empathy.