Three years ago I wrote a post about my transition to Boston University when I joined the Math/Stats department after Wheelock College, where I’d taught for 25 years, merged with BU. That post focused especially on the assessment system in Discrete Math, but with a wider lens to include broader thoughts about the transition. I taught Discrete Math in each of my first seven semesters at BU, and after just about every subsequent semester I started to update that post and got stuck partly because it was hard for me to talk about one course without also giving a whole lot of context. I’ve had two unusual academic jobs: At Wheelock I was the first mathematician ever hired at a college with no history of a traditional math program and with a deep history of promoting active learning, and at BU I have a secure position at an R1 institution without having had to engage with the institutional reward structure. It’s taken me a while to understand that this unusual trajectory is a strength, which gives me an insider/outsider view of many issues related to teaching, and thus it’s valuable as I write again about Discrete Math to also write broadly from this specific perspective and to let readers figure out what is useful to you.

This post has four sections. First is “Studenting,” which is about adjusting to teaching at a university that values rankings much more than I do and is full of students who are good at gaming rankings. Second is “Measurement,” which considers the distinction between using alternative assessments to measure learning more accurately and using them to prevent measurement from interfering with learning. Third is, “Ungrading in Discrete Math,” which looks at how some of my thinking described in the first two sections led to the ungrading assessment system I used last semester, with some more detailed reflections on how the system worked (spoiler: mostly I was very pleased, although there were some glitches). Fourth, in conclusion, I share a quote that I first read thirty years ago and that still articulates my feelings about grades and the unresolvable tensions for those of us who participate in grading systems we don’t believe in in order to keep our jobs.

Note: I would like to thank all of my colleagues and friends from the New England Community for Mathematics Inquiry in Teaching (NE-COMMIT), and particularly Chrissi von Renesee. Remote teaching allowed me much more interaction w/many wonderful teachers working in many different contexts, and they have helped me strengthen my voice.

## 1. Studenting

Last summer I learned the word “studenting” from Peter Liljedahl’s wonderful book,* Building Thinking Classrooms in Mathematics*, which Rebecca Mitchell suggested for a NE-COMMIT bookgroup. When I used the word with my students, mostly they understood immediately, chuckled, and readily came up with lists of examples of the ways they and other students “do school” that are sometimes about learning, but often are not: memorization, arguing for points, cheating, choosing the option that’s the least work when they have a choice, and buzzing in “Jeopardy” style before having a formed thought in order to get participation points in a Zoom class. One of my biggest challenges at BU is dealing with just how good the students are at studenting — BU selects for it. I miss the kind of diversity that we had at Wheelock, which came from including students who hadn’t done so well at school in the past, and I miss the opportunity to see many of those students blossom. Many very creative teachers at BU nonetheless rely on dangling a few points in front of students in order to motivate them to do something the instructor thinks is beneficial — using extrinsic motivation is a reflexive shortcut that is hard to see when you’re in an environment where it works so well. Saying that you need to do X to get an A is not particularly motivating to a student who had straight C’s in high school math, worries that not doing well in a math class might destroy their dreams, and throws up before the first class because they are so nervous. Some of my Wheelock students were that terrified of math, but many of those same students were creative mathematical thinkers, who tuned out of their rote high school classes instead of just playing along when, for example, they asked why something worked, and the teacher refused to answer or punished them for asking.

At BU I constantly feel the tension between transparency and expert studenting. I am the one with the power to assign the grades, and my students care about grades, so being responsible about power requires transparency. However, transparency also invites students to game whatever structure I am being transparent about, so I continually work to find ways to focus on learning, with grading somewhat off to the side, but nonetheless transparent, and sometimes that strategy works better than other times.

My students are embedded in a narrative of measurable merit, where at the highest level they see a school explicitly trying to raise its ranking. This narrative filters down to the smallest level; constantly ranking students is another reflexive assumption at BU (one student even told me that in one course — not in the math department — every week the students learn their rank in the class, and their final grades are determined by that rank). Math colleagues who are uncomfortable with the practice have told me that in large coordinated courses, they look for gaps in a spreadsheet to decide grade cutoffs, so perhaps some students averaged 89.3 and others 89.5 but no one averaged 89.4, so that’s the cutoff. This objectivity theater can have profoundly negative consequences; for example, meaningless decreases in grades contribute to women being more likely to drop a STEM major than men. My students talk about “grade deflation,” a term I’d never heard before coming to BU — where “too many” students could do well in the class, so as they perceive it, the instructor makes obstacles so they all don’t get A’s.

Fortunately, there’s another narrative at BU besides meritocracy: it’s a place with amazing people, who are doing interesting things, and want to learn and connect with each other and work together and be a positive force for the community and the world. Mostly, my students want what I have to offer and want to learn math. It’s so much fun to get to teach so many students who start the class liking math. Using alternative assessment systems works well as a first step toward building trust and opening students to try active pedagogies that they may not be used to. I find it tragic that we would bring together so many fascinating young people from all over the world and just have them write down what I say, rather than talk to each other. Students often start out skeptical about working together, but that usually fades quickly — they want to connect. When there is a clear, somewhat predictable path to the grade they covet, that allows us to focus more on learning and less on studenting, although the anxiety behind the grade-focus lies in wait, and sometimes takes center stage.

## 2. Measurement

In my second semester at BU I taught Linear Algebra, and the contrast between assessment in that class and in Discrete Math helped me better understand something about measurement, which I’ll call “topic definition,” although I think that’s not a great phrase and would welcome a better one. Linear Algebra is much more topic-defined than the way I teach Discrete Math: it’s easy to say we are going to study Chapters 1 to 5, and on this day, we’re doing this topic. Since Linear Algebra is a prerequisite for so many things, it’s somewhat standardized. I can’t just decide not to teach eigenvalues; that would be irresponsible. In a highly topic-defined class, part of the point of alternate assessment is to get a more valid measure of what topics students understand than traditional assessment with points and percentages and a few midterms and a final provides. Most of the descriptions of math courses in the catalog at BU and at most other colleges and universities consist of a lists of topics. Discrete Math is often taught in a highly topic-defined manner, classic Guided Inquiry Learning can be highly topic defined, and many seminars are also organized in a highly topic-defined way.

I had trouble getting my bearings in that first linear algebra course at BU. I was trying to use the department text along with some supplementary materials that required a lot of group work, and I had the quietest class I’d taught in 30 years. My pedagogical content knowledge for Linear Algebra was not strong at the time, I was sometimes surprised at what students struggled with, and I often didn’t allocate class time well. However, I used an assessment system which was mostly based on quizzes scored “Pass” or “Not Passed Yet,” corresponding to the topics, and this system not only made room for students to learn from mistakes, it also mitigated many of my pedagogical mistakes. That semester it was much harder for me to teach Linear Algebra than it was to teach Discrete Math, but I was happier with the assessment system in Linear Algebra. I thought I’d done one of the worst jobs teaching I had done in a while, but the students were very positive about the class: it was clear that I wanted them to learn, I’d convinced them that it was valuable for them to learn linear algebra, they weren’t competing against each other, and the assessment system was well aligned with the goals of the course .

Along with using grades for motivation and ranking students too often, topic-defined classes are something else that’s so ubiquitous in my new context that it can be hard to see that there are alternatives. At Wheelock, I initially explored some alternatives to topic-definition out of necessity, and then over time, my whole teaching style became a lot less topic-defined. There just weren’t many textbooks that worked for our math major for elementary teachers, so I often wrote my own activities and pieced together a wide variety of other materials. I like teaching classes that are less topic-defined: it’s easier to center the students instead of the topics, and I love writing activities, especially ones based on something interesting that happened in a previous class. With a small major at a small school, we weren’t able to build a very complicated prerequisite structure since most of our courses were offered once a year or once every two years; at a big university more coordination is necessary and more dependence is possible, which makes topic-definition make sense as an organizational structure.

I don’t think topic-definition is going anywhere, nor should it, but I think there is too much of it. Math classes simultaneously involve concepts and processes such as problem solving, reasoning, and communication, but in a topic-defined class, the processes are rarely highlighted. Students then get a distorted view of math, where they think all problems have solutions that “smart people” — possibly including them, possibly not — can find quickly and accurately, and where they think they need someone to show them a topic before they can think about it themselves. Then they forget a lot of the topics after the class ends, but they remember the unspoken lessons about mathematical processes and about their identity related to mathematics.

A few years ago on the last day of Discrete Math, one of my students told me that before taking the course he never realized you could solve a math problem by thinking about it; he had just “followed orders.” He was a junior math major, and I find his statement haunting — as do my colleagues when I mention it (although perhaps not with the “stop the curriculum today” urgency that I feel). Thus when I use some of the same course design and pedagogical ideas I developed at Wheelock in Discrete Math at BU, I am not just indulging myself, but addressing an actual need.

When I started at Wheelock in 1994, there was a required math course for all students called “Developing Problem Solving Skills,” which was developed by Rika Spungin, a math educator. The course helped students see that math classes could be creative, interesting, and interactive. As the curriculum at Wheelock changed over the years, we kept some of the elements from that early course, and our math for teachers sequence started with a unit on “Processes,” which focused on Problem Solving, Reasoning, and Communication. This unit was my absolute favorite to teach, since it was one where I never worried about “covering” the topics. Many of the activities I developed for that unit were related to combinatorics (my field in graduate school), and I’ve adapted some of them for Discrete Math at BU.

Students’ growth in Problem Solving, Reasoning, and Communication is hard to measure, especially if the goal is for all students to grow, not just to find a way to rank the students. Students start out in different places and need different things: I’m pretty sure that many of my students in the class I mentioned already knew before the course started that you could solve a math problem by thinking about it, but it’s also pretty clear that for the student who didn’t know that, the class was a success, even if he forgets every topic in a few years. Certainly in highly topic-defined classes we hope the students are also developing process skills, but if they’re hard to teach and hard to measure, it’s easy to put the responsibility on the student to just figure them out on their own.

Teaching Linear Algebra that first time helped me realize that in this highly topic-defined math course, my alternative assessment system was designed to help students learn more AND to have their grade accurately reflect what they learned. I am pretty confident that students who got an A in that class knew more linear algebra at the end than students who got lower grades. However, in my more process-focused, less topic-defined Discrete Math course, I’d initially worried about how to grade with integrity both the very clever student who could solve hard problems in ways that were new to me, but was disorganized and didn’t turn in work, and the persistent student who came to office hours every week and gradually built skills. It was a relief to be able to articulate that assessment in Discrete Math is tricky because the course as I teach it is somewhat topic-defined: there are parts where measurement is meaningful and can help students learn the topics better, but there are process-defined parts where measurement just interferes with learning and my goal is to get it out of the way.

## 3. Ungrading in Discrete Math

In my earlier post, I shared details of the assessment system in my Discrete Math classes during my first semester at BU, and here I’ll start with a quick overview of the most salient details of how that system evolved over the subsequent five semesters. The heart of the work students turned in were the “Challenge Problems,” which were difficult, non-routine problems that students wrote papers about. The papers were scored “Pass” or “Not Passed Yet,” and students revised them, often several times, until they passed. Pre-pandemic I required eight or nine passed Challenge Problems for an A, but when we pivoted to remote in March 2020, the students suggested we go down to six, which I agreed to somewhat reluctantly, but then thought was not a big difference — some of the problems were better than others and I cut the duds. However, in breakout rooms that semester, no matter what the class activity, the students worked on the Challenge Problems instead, because they were the ones that counted for the grade. I wasn’t that upset because they were still engaged in the class material, but I was a bit upset since they were missing activities and concepts, so when I taught fully remotely the next year, I reduced the number of Challenge Problems to four in the fall and then to three in the spring, made the problems a bit more elaborate, and allocated a week’s class time to work on each problem in class.

In the Fall of 2020, students did a “Final Showcase,” where they learned about a topic of interest and prepared materials to teach to their classmates. They could participate in the showcase as a creator, reviewer, and/or audience member, and they basically just had to do the work for each role to pass — I had no time to grade projects at the end of the semester, and if I had wanted to be able to grade them, I would have had to define the assignment in ways that would have created anxiety and increased studenting behavior. This addition to the course was a little bit of ungrading, and it worked very well: since students picked topics they were interested in, could work alone or with classmates, and had their peers as an audience, there was a lot of intrinsic motivation to do good work, which they did. Unfortunately, I had to cancel the showcase in the Spring of 21 because the loss of spring break due to the pandemic compressed the semester too much.

In the Spring of 21 I was also teaching my first very large class at BU: a remote introductory statistics course with 121 students, and a lot of my attention was focused on that. In Discrete Math I loosened things up a bit more — inviting students to make videos about their ideas on problem solving and to give little talks at the beginning of class about anything they were interested in, to try to build community when so many of my students were isolated due to the pandemic (with many in Asia, taking the class in the middle of the night). The students stepped up, and the dynamic in class was good, but the grading system felt out of sync — three Challenge problem and a few quizzes were elevated far above everything else in class. I also did not tweak the grades for any reason — if the goal is transparency, not measurement, then it doesn’t make sense to change the grade because I think a student “deserves” more. But there were a few students who engaged in ways besides rewriting the very last Challenge Problem, and were surprised about their final grades. Some of these students were paying attention to their learning in the way that I hoped they would, but my system didn’t make room for that and punished them for not studenting enough to know what counted the most. Also the Challenge Problems became high stakes, and I had too many revisions to read at the end. I ended up passing a lot of papers without reading them as carefully as I did earlier in the semester.

In my last year at Wheelock, I tried “going gradeless” in all of my classes (now mostly called *ungrading* in higher ed); I’d joined the Teachers Going Gradeless group the previous summer and rushed to try it before the merger. It was a great success in three of my classes and a disaster in the fourth. I took that disaster as a cautionary tale to step back when I went to BU. During my first semester at BU, I casually mentioned my Wheelock gradeless assessment system to one of my BU students who said, “Oh yeah, if you did that here, everyone would just give themselves A’s,” and that comment held undeserved weight in my brain for a while — not that I am opposed to everyone getting A’s, but I was scared of everyone doing no work and saying they should get an A, and then me having to deal with that situation — which is basically the reflexive response of many when they hear about ungrading; to which people respond, that that isn’t what happens.

Last summer there was great interest in ungrading in the NE-COMMIT group and in other circles I’m in, and I was asked to be on a panel with Candice Price, Erin Rizzie, Ileana Vasu, and Rachelle DeCoste. I focused on some of the ways I’d used ungrading at BU just for one assignment or project — ways to dip your toe in without taking a full plunge, as I had been doing. In my large stats class we’d done a final project that was just awesome — students read a research paper they were interested in, interpreted it as they could using what they’d learned in class, and shared on a virtual whiteboard with the class. The size of the class made for a breathtaking array of topics that showed students how applicable the class material was, how broad their classmates’ interests were, and that they could learn from a research paper even if they didn’t understand all of it. If I’d been grading every aspect of their work for correctness, I would have had to limit the assignment in ways that would have made it much less interesting and effective.

My co-panelists talked about using full ungrading in their classes, including in Discrete Math. One of my first responses was that it sounded like they talked about grades much more than I did, and I didn’t want to talk about grades that much in the service of ungrading. They shared ideas such as having students brainstorm what each grade means on the first day of class and having midterm check-ins about what the student thinks their grade is so far. At first I thought I didn’t want to do all of that, and I’d just stick to what I had been doing, but then when I decided I wanted to switch to ungrading, I went back to their materials first and got a lot of inspiration and ideas. The biggest reason I switched to ungrading was to better integrate the assessment system with the philosophy of the course, as these colleagues were doing.

I hoped that switching to ungrading would give students more choices in how to engage in the course and that it would also raise the ceiling: some students who found the Challenge Problems not that challenging were studenting and doing the minimum. I also was trying to grapple with a separate logistical issues: I accepted late papers, and students were prioritizing work from classes that did not accept late papers, and sometimes procrastinating on work for our class to the point that it seriously affected their work and also made the end of the semester brutal for me. Often students’ first suggestion for improving the course was more deadlines.

Balancing structure and flexibility is a teaching theme that’s always there, but it became especially prominent for me during the pandemic. I’ve always erred on the flexibility side, but during the first weeks when we were remote, it was clear that my classes were providing much needed structure for many students whose lives had been upended. I wanted more structure around deadlines, but I didn’t want to use something like a token system, which has never appealed to me.

The solution I settled on worked really well and could work within a variety of different assessment systems. I had been thinking of how to give students more choices on the Challenge Problems — I was thinking of having a structured option and an “Excursion” option, where students would follow-up on something we did in class, and use peer editing to make it possible for them to get feedback. I met with Chrissi von Renesee, who had organized the ungrading panel, when I was trying to sort out all the vocabulary, and she said “Excursion” should be the overarching name, with structured or unstructured options. Then I decided to name the assignments the September, October, and November Excursions. Students picked their own due dates within the months (more-or-less), and then rewrites were due the next month.

Not surprisingly, most students picked the last day of the month or close to it for the September excursion due date, but what was amazing was that almost all of them actually turned it in then. There were more extensions due to personal circumstances for the October and November Excursions, but within reasonable amounts. Just having the month in the name of the assignment made the intended schedule obvious, without having to put due dates in the calendar. Several students knew in September that they’d be traveling internationally for Thanksgiving break, and they were able to plan to turn in work before they left. One consequence of giving the assignments so early was that I included some options that ordinarily I would assign after we’d done a particular class activity. I suggested students wait until later in the month if they wanted to select that choice, but some tried the Excursion before the class activity, got as far as they could on their own, and then were quite primed for the in-class activity.

Students created a portfolio –their “December Excursion,” which was somewhat modeled on what I had used in one of my classes in my last semester at Wheelock. In order to make an assignment that students would understand with some degree of transparency, I realized that I needed to spell out Learning Markers for the course in more detail. I have always resisted creating long lists of Learning Objectives, or Learning Outcomes, or Learning Targets, even in highly topic-defined classes. I don’t like the idea of setting them before we even meet the students. I don’t like the finality of words like Objective, Outcome, or Target, as if we end there and stay there. I care more about what they’ll know ten years after the class, and we all know that the most measurable things in the span of one semester are also the ones they are most likely to forget. I also think the lists are kind of overwhelming and boring to students who don’t know what any of them mean at the beginning of the semester, and they feel constraining to me, locking me into a plan that makes it hard to be as present as I want to be in the classroom.

However, if I wanted to make my class more flexible in terms of how students could show their learning, I needed to define what they had to show better; I couldn’t just say do these assignments until they pass. Another irony: I needed to start with more structure to enable more freedom. At Wheelock I started using the phrase “Learning Marker,” as in a marker along the road, not a final destination, and I also used this phrase as part of ungrading in Discrete Math. In making the list, I grappled with the way the course had a medium level of topic-definition — it was not like my liberal arts math course where sometimes students work on completely different topics, but it had much more focus on problem solving, reasoning, and communication than a highly topic-defined class like linear algebra. I was also very happy with my approach to this tension.

I divided the Learning Markers into Concepts and Process Learning Markers. For the Concepts Markers, I chose topics we’d studied every semester in the past six semesters of the course and called them Core Learning Markers. Beyond this core, I have written many different activities that I’ve used sometimes, depending on the class, and I gathered together many of these activities in a folder on the Learning Management System and called them “Extensions.” I made completing the Core Markers equivalent to B, and completing the Core plus several Extensions of their choice an A. There were also Extensions based on the Excursions, and some students made up their own Extensions. I was interested to learn that the some writing instructors at BU adopted a similar ungrading frame last semester, with a contract for a core set of work to get a B+ and student choice for what to do for a higher grade.

Some examples of the Core Concept Learning Markers are, “I can use recursive equations to describe a variety of number sequences” and “I can distinguish between and solve one-step problems involving permutations, combinations, permutations with repetition, and combinations with repetition. I understand how to derive the formulas used to solve these problems.”

I made an uncategorized Process Learning Marker, “I make mistakes and learn from them.” Singling out this Learning Marker worked very well to communicate its importance and to help dispel anxiety: when students were afraid to present their work at the board because it might be wrong, I said if it was wrong, then they could take a picture of it and put it in their portfolio under Learning from Mistakes. I made Process Learning Markers for Problem Solving, Reasoning, and Communication, with the intention that students would choose a few for their portfolio based on their personal goals for the course, but also that they would be a guide throughout the course. At the beginning of the semester, students wrote an Introductory Assignment where they told me about their strengths as a mathematician and places they wanted to grow — with “mathematician” being defined as anyone who is engaging with math, just as a writing teacher may call them writers and a singing teacher singers. On the Midterm Reflection they had an opportunity to update their personal goals.

The Process Learning Markers were partly adapted from NCTM Process Standards and Common Core Mathematical Practices. Here are two examples for Problem Solving: “I have a repertoire of strategies to use to get started on a problem and when I am stuck,” and “I regularly pose my own problems, and when I solve a problem, I look for ways to generalize and extend my solution.” For more details see my list of Learning Markers.

The Extension assignments went well, although the word “Extension” was problematic: students constantly confused it with both the word, “Excursion,” and with the word they used when they asked for more time on an assignment. I’m thinking of using the labels “Core” and “MORE” next time. Changing the assessment system absolutely worked for raising the ceiling. There were some students who had a lot of background knowledge and who worked very quickly throughout many of the regular class activities, and I often asked them to ask more questions and make up their own Extensions, and they came up with some great ones that kept them occupied. One student decided to write a program to play a game that was the subject of one of the Excursion assignments. Several students made videos to share what they learned with classmates. The students had much more agency, and many of them went much further than in previous semesters. Very often the classroom felt relaxed and happy.

I also loved the way the Excursions were not quite as high stakes as in my old systems, which often made them richer. I made a bunch of ancillary handouts and videos to go with the Excursions — some were to explain common mistakes or to give hints when students were stuck in common places. I made some Extensions which students only got access to after they had solved key parts of the problems. I asked students not to share these materials with classmates yet, as their learning edge may be in a different place. When students had something non-trivial wrong with their paper, but not something that actually required an extra draft, I put on the rubric that that part “needs work,” but then just asked them to update their paper for their portfolio, rather than turn in a new draft. Sometimes I made openings for students to just stop working on an Excursion that they were struggling with unproductively and find something else for that part of the portfolio.

The number of students and the logistics of the portfolio were the most challenging parts of changing to ungrading. I expect the portfolio to go much more smoothly next time around, when I start out with more knowledge of how to use Digication, the electronic portfolio system BU subscribes to, which has some rather unintuitive features — e.g. uploading and publishing are separate steps, so I often couldn’t see students’ work if they had uploaded it but not published it. I had to make decisions early in the semester about a portfolio template for all students to start from, and I did pretty well for the first time around, but I was also relieved that students could modify the structure themselves.

Besides the 66 students in two sections of Discrete Math, I was also teaching a third class, which included final conferences, and which brought my total number of students to 91. Ideally I would have had several individual conferences with students throughout the semester, but I just didn’t have the time or energy for that, and I knew there was no way I could do 91 conferences at the end of the semester. I had planned to have group midterm conferences outside of classes, but I was overwhelmed at the thought when it got close to the time. Instead, I gave a brief midterm reflection assignment and scheduled individual 5-minute conferences in class, while other students worked on the October Excursion and other work.

These conferences took almost two weeks of class time, which was too much. Most of the students were positive about them, and it was good to see that students were engaging in the course in many different ways. I especially appreciated the opportunity to connect with the quieter students. Students reflected on their Introductory Assignments and discussed whether they wanted to modify any of their personal goals; common modifications were wanting to work more on oral communication and wanting to learn to write better proofs. However, we ended up spending too much of the conference time talking about technical details of the portfolio. These meetings helped me figure out a lot of things about the software, but that really wasn’t the purpose of them. It was grueling having so many conferences in a row, and I took notes on index cards, but otherwise remember very little of who said what.

I think conferences are very important, and how I organize them in the future will depend somewhat on what else is going on for me that semester. With constraints like last semester’s, I’d try spreading the conferences out over a longer time period of time and having more of them outside of class. I’d also give written feedback in the portfolio before the conference, to make the conference time more productive. If it were possible logistically, I might try group midterm conferences and individual final conferences — maybe having the group conferences in class during class period per week for about three weeks. Reluctantly, based on some issues with the final portfolio, I would probably discuss grades more explicitly at the midterm conferences.

I felt pretty good about the final portfolio assignment. It included a structured way to write a final reflection, with opportunities for modification, and a grade proposal at the end. In the last week of class students worked on their portfolios, and many said the portfolio gave them a real sense of achievement and showed them how much they learned. Mostly, the portfolio was for them, not for me. Once students learned how to use the software, creating the portfolio was not onerous. When they were ready for me to see their portfolio, they filled out an online form.

I had hoped to do what many of my ungrading co-panelists suggested if there wasn’t time for final conferences for everyone — message students if I thought their proposed grade was too low and schedule a conference if I thought it was too high. As everyone says with ungrading, the students didn’t all assign themselves A’s, and I raised several modesty A-‘s to A’s. However six students didn’t follow the guidelines for the final reflection assignment at all, but proposed A’s for themselves. Some of these students hadn’t turned in satisfactory work on any of the Excursions, and they said things along the lines of, “I tried hard and learned a lot so I think I should get an A.” There were others who didn’t do the final reflection as assigned but still included strong artifacts and had done well on the other work. I realized immediately that I should have made a template for the final reflection, as I did for the Excursion assignments. I had no time or energy to have conferences with these six students, and it was after the semester was over anyways. So I dealt with lowering their grade over email, and that felt kind of awful.

But overall the portfolio assignment worked very well for the vast majority of the students, and I will definitely continue with ungrading when I teach the class again next year. All semester I kept saying to my partner that this was the first time at BU when I went into the class and just felt like what I was doing was aligned with my values. I didn’t feel aligned and whole at the end of the semester turning in the grades — I never do — but for the rest of the semester those feelings were consistent. It was a difficult semester with wearing masks and having COVID hanging over our heads all the time, but it was also incredibly joyful to be in person, and to just spend time enjoying math and working together.

## 4. Conclusion and a Quote

For most of last semester, it felt good to have co-created a genuine space for learning with my students, and in the end, ungrading mitigated some of the distress of participating in grading, but that distress is related to structural issues that are much larger than one classroom. Having to feel the awfulness every semester goes with the privilege I get from being at a selective institution in a profoundly troubled society. I want to end with a quote from David Purpel, from *The Moral and Spiritual Crisis in Education*, which I stumbled on over thirty years ago, when I’d just started teaching, and which still helps me understand why no matter how creative I am in devising alternate assessment systems, I’m never fully satisfied.

I believe that most teachers regard grading (as opposed to evaluation) as an obstacle to the learning process. It is very difficult, and probably impossible, to develop procedures for giving grades that are valid, reliable, fair, and efficient; students come to worry more about grades than meaning; and both teachers and students respond to these problems by developing techniques (e.g., multiple choice tests, cramming, memorizing) which are at best distracting, and at worst counterproductive to serious learning. The concern for grading produces anxiety, cheating, grade grubbing, and unhealthy competition. However, even though grading may be at best of dubious value “educationally” it is absolutely vital to a culture that puts enormous stress on success, achievement, and individuality and to a system that requires social and economic inequality. The critical issues of grading are not primarily technical (though there are certainly such problems within the particular field of evaluation and grading) but moral and cultural. To value grading is to value competition and to accept a society of inequality and a psychology that posits external behavior rather than internal experience as more important. Grading is primarily a technique for promoting particular social, moral, and political goals, and it is those goals which should be debated rather than the technical and misleading questions about the value of essay vs. objective testing or whether to use grade point averages or standardized tests as the basis for college admission.

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