I have never rehearsed the first five minutes of class as many times as I did last summer. Wheelock College, where I’d taught for twenty-five years, had just merged with Boston University. At Wheelock I led the math program and mostly taught future elementary school teachers and others preparing for human service careers. At BU I’d be losing tenure, taking on the awkward title of “Clinical Professor,” and teaching mostly Math and other STEM majors in bigger classes than I was used to. Many close colleagues lost their jobs in the merger, and the programs I’d helped build at Wheelock were dismantled. I channeled my grief and anxiety into constant tweaking of the wording of how I would first introduce myself and the course, as if those few minutes would determine the rest of my work life.

The BU Math Department tried to give me a soft landing, changing a 60-person section of Discrete Mathematics into two 30-person sections and expressing support for my using the active learning strategies I’d used at Wheelock, but we all recognized that the merger was hard. When I finally crossed Commonwealth Avenue on the morning of my first class, I was relieved to finally be done with the yearlong transition. After a few minutes in the hallway saying hi and introducing myself, I relaxed; the students were friendly, and I like teaching math. The first few minutes went according to script — after animatingly proclaiming that this was the first minute of my first class at BU and thanking the students for sharing it with me, I talked a bit about my background, asked them to count how many people in class they knew, had them write their names on paper so they could take photos of each other to help me learn their names (Americanization optional but not necessary, pronouns welcome), projected the day’s agenda, and promised I’d talk more about the syllabus at the end of class, after they played a math game in pairs.

Now the semester has ended, I’m preparing for the next one, and overall, I am pleased with how my classes went. This post is a reflection on teaching Discrete Math during my first semester at BU, with a special focus on the assessment system. Over the years I’ve used many non-traditional forms of asessment, and I’ve done some of my most radical experiments in the last few years. As the instructor with power in the class, the way I set up the assessment system communicates what I value to the students, but many of the things I value most are hard to assess — things like taking initiative and intellectual risks, problem solving, learning to make strong mathematical arguments, overcoming self-doubt, intrinsic motivation, appreciating the beauty of mathematics, finding unexpected connections, and being part of a classroom community — while things like factual knowledge, test taking skill, and solving problems like the ones in the text are easy to assess but often misaligned with these values.

The assessment system is where I navigate my power in the classroom. I would prefer that students started the class caring more about learning than grades, but that’s not the reality, and assigning the grades is part of my job. My overall goal is to get the grading out of the way as much as possible so that the learning can happen. In my last year at Wheelock, I tried using a gradeless system, where students got much feedback but no grades, until we had a conference at the end. This system was liberating in three classes, and a disaster in the fourth. For my first classes at BU I chose a system where students had ample, detailed information about how to earn each grade from the start, also with the hope that this system could help students focus on learning.

Planning the assessment system was linked to planning the structure of the course, and the way the discrete math course was initially set up didn’t particularly make sense to me. It had been decades since I was assigned a text that I wasn’t involved in choosing, and I didn’t like it as a basis for a class focused on active learning. There was far too much content in the course and too few connections between the content.

I decided that the overarching goal of the course was for students to become stronger mathematical thinkers; this was a coherent frame that linked the topics together. On the first day of class, I asked students if they wanted to become stronger mathematical thinkers; most said they did. I said that becoming a stronger mathematical thinker was a goal of any math course, but it wasn’t the overarching goal, of say, Calculus II. I referred to this frame throughout the semester; it allowed me to describe why a topic or assignment would help students become stronger mathematical thinkers rather than describing how it would help them get an A.

I told students from the start that they were not competing with each other, that they could all become stronger mathematical thinkers if they helped each other, and that I’d be delighted if all students earned A’s. I would have been even more delighted if we could have stopped talking about grades, but grades were clearly a big concern for many students in the class, and giving explicit details about exactly what they had to do to earn each grade did help students set them aside sometimes.

“Mastery Grading” and “Specifications Grading” are labels for the type of grading system I used; there are many people currently experimenting with different variations of these grading systems in college math classes and the labels are not yet fully defined. In preparing my system, I drew on my experiences from Wheelock, looked at exams from a previous semester of the course at BU, talked to BU faculty, read *Specifications Grading* by Linda B. Nilson, looked at blog posts and syllabi from Robert Talbert, syllabi from the Google Drive of material maintained by Rachel Weir, and I went to a summer workshop at the Academy for Inquiry Based Learning.

The assessment system for Discrete Math started with six components: Concept Quizzes, Proof Quizzes, Combination Problems, Challenge Homework Problems, Online Class Preparation Assignments, and Contributions to the Classroom Community. In all but the latter, every item was graded “Pass” or “Not Passed Yet,” and a matrix specified how many items students had to pass in each category to earn each grade. I left room for using my judgement so that if, for example, a student had done everything for an A except was in the C category for turning in preparation assignments, their grade would be an A- not a C. Over the course of the semester, I modified the requirements, but not as much as I thought I would need to. Here is the modified grid, which I gave to students in November, followed by more information about the structure of the course, a description of each component in the assessment system with reflection on how that component worked in practice, and thinking about what I will keep and what I will modify when I teach the class again this spring.

### More on the structure of the course:

I talked to people who had taught the class previously and asked what students found most challenging and they said counting (enumerative combinatorics), which was introduced in the second half of the course. I decided to introduce some at the beginning of the class and then cycle back later; this choice enabled me to start with activities like the ones that helped me build classroom communities at Wheelock, as well as giving students more time to struggle with some of the more difficult material. I started with three very loose units — Combinatorics I, Foundations and Proof, and Combinatorics II — these addressed most of the topics in the text, but not in order. Not knowing the level of the students, I didn’t want to be more prescriptive at the beginning, which was a source of stress for some students who were used to getting a complete outline.

The class included a mix of first years, sophomore, junior, and seniors. BU doesn’t have a transitions to proof class, and this course functioned as that for some students, but there were also many students who had already taken other proof-based courses (many said they wished they’d had this class first).

The course was set up with two 75-minute Lecture sessions and one 50-minute Discussion section per week. I ignored the distinction, and taught it as a three-day per week class (there were a few students who had to drop at the beginning because they had schedule conflicts and had initially planned to skip all discussion sections). At Wheelock we also had 200-minute per week classes, and the extra time is extremely helpful for an active course. Having three contact times per week is also much better for building community than two meetings per week.

I didn’t realize when I was designing the course that I would have a grader; without the grader, I would not have been able to keep up. At Wheelock I always got behind on grading, even if I only had 8 students in the class, but I found that with the grader and more importantly, with no more administrative responsibilities, I was able to keep up much better.

Below are descriptions of the components of the assessment system.

### 1. Quizzes

About 20 years ago, when we were revising an introductory sequence at Wheelock, we decided to include a set of “Baseline Proficiency” quizzes that all students had to pass to pass the course; these quizzes were scored “Pass” or “Not Passed Yet.” Generally there were five or ten questions, and students needed to get 80% correct to pass. Many of our students had struggled in middle and high school math and skated through getting C’s on tests, moving on, and never actually learning any topic well. Some of these students were absolutely up in arms when I’d first tell them they had to pass the quizzes to pass the class, but after acknowledging their anger and fear, I’d suggest redirecting their energy from complaining about the unfairness toward learning the material. When they did start passing (as all eventually did), they saw that they could do more than they thought they could, and that built confidence and momentum. An interesting question to ask about any assessment system is, “What does a student who gets a C in the class learn?”

At Wheelock I also found that the best way to teach some topics that sat on top of widespread misconceptions was to have students get questions wrong on a quiz (often several times) to get them to notice the problem in their thinking. The first chapter of *Small Teaching* by James Lang includes a summary of research on the benefits of quizzing, which aligned with my experiences with the baseline quizzes and also motivated me to incorporate more low-stakes quizzes into my courses for math majors.

Many people using mastery grading use clearly defined learning outcomes that test one skill, e.g. “I can take the derivative of a polynomial function,” which can lead to a very long list of outcomes for one course. I was concerned about how many quizzes I could manage with 60 students. I was also skeptical about chopping the material into too many measurable bits and risking missing the bigger picture of how the bits fit together; our baseline quizzes had worked well at Wheelock even though they sometimes could have been separated into more than one learning outcome. In practice, the difficulty levels of my quizzes were too disparate, and for next semester I will restructure the quizzes so that some will be more focused and straightforward.

It took the BU students a bit of time to get used to the quizzes, but as the semester progressed, I relished how students worked to understand why they had missed a question (even if they passed the quiz). I often walked into the room to see students helping each other prepare for a quiz or working to understand their mistakes on a quiz, and I enjoyed their feeling of satisfaction from passing after several attempts. The vast majority of the students in the class worked very hard to pass at least 14 of the 15 quizzes.

One of the quizzes that I found surprisingly effective was the one on Relations and Functions. It was essentially 25 true/false questions — is this function injective? surjective? Is this relation transitive? Students needed 21 correct questions to pass. I didn’t teach this topic much at Wheelock and underestimated how hard it is for students, even though most had seen the definitions in other classes. Many students took three or four iterations to pass, and taking the quiz, reflecting on mistakes, and retaking was the main way they learned the material. After the first pass, for example, they might realize that they had mixed up the definitions of a function and of a surjective function. On the next pass, they might realize that they had been always assuming the domain and codomain of a function were the reals and they needed to look more carefully.

Originally I separated the quizzes into 12 Concept Quizzes and 8 Proof Quizzes, but later in the semester I dropped the distinction in the grade matrix and removed or combined several quizzes. I had thought the Proof Quizzes would be harder than the Concept Quizzes, and of course I could have made them that way, but I ended up making them very basic problems about proof structures, e.g. prove directly that if a, b are odd integers then 5a + 4b is odd or prove by contrapositive that if 3a^2 is even then a is even. The first time through many students didn’t use complete sentences and had mistakes in their notation, but had the main idea. I had trouble deciding what should constitute a “pass” and ended up focusing on whether they used definitions correctly, had the right logic, and at least included some words, even if the sentences weren’t all complete. I didn’t want the grader to grade the proof quizzes, and I was mindful of how much work it was for me to ask the whole class to retake these quizzes. To raise the standard and make the grading more efficient, next semester I will include some specific criteria at the bottom of each proof quiz: Correct Structure, Correct Definitions, Correct Notation, Correct Logic, Complete Sentences, and will include an option for students who have most of these to just rewrite, rather than retaking the quiz.

There were overall problems in how I structured the unit on proof. The book section on proofs was only a few pages, and I expanded it to a four-week unit, which was still much shorter than a whole semester introduction to proofs course. I focused too much on the proof techniques and not enough on introducing enough content so we could actually do interesting proofs — that came later with combinatorial topics like the binomial theorem and derangements. Thus when students took the Proof Quizzes later in the semester, they were often confused about what they could or could not assume. Next time I will move some number theory earlier and introduce proof techniques along with the content. I don’t think quizzes are an optimal way for students to learn proofs, but I remain daunted by the logistics of managing something like proof portfolios, which I used at Wheelock with fewer students.

The quizzes that proved most problematic were the ones on counting. Initially I included three quizzes whose conceptualization was so muddled that I sometimes couldn’t decide which problems should go where. Many students were retaking the quizzes, making progress, but still not passing anything and getting discouraged. I replaced the three quizzes with two tiered quizzes: Counting I and Counting II. The Counting II quiz required more than one technique in the same problem, and it was much harder than the other quizzes, and many students never passed it. There were so many retakes of that quiz that I started repeating many problem types with different numbers, and many of the students who did pass just memorized how to do those types of problems. Next semester I’ve made three levels of counting quizzes: the first two are more straightforward than any quizzes I included last semester, and the last one is equivalent to last semester’s Counting I. I’ll give out the more complex problems one or two at a time on “MAP” quizzes (More Advanced Problems, more details below).

There was no official limit on quiz retakes, but there were logistical constraints. For most quizzes, students took them first at the end of class at the appropriate time in the semester; students were free to wait if they weren’t ready. Often at these sessions they also had the option of taking one or two of the most recent other quizzes. When there started to be a backlog, we’d have a quiz day in class where all previous quizzes were available; sometimes we had these days on the short class period and sometimes the long; we had about four of them during the semester. I usually made two or three versions of each quiz for the first pass, and then new versions when many students would be taking them again. Toward the end of the semester, I started recycling quizzes, and students were very responsible about choosing a version they hadn’t taken before.

I maintained a strict policy that students could not take quizzes during office hours — I joked that if I allowed that with 60 students it would destroy my life. Students understood my reasoning and did not complain about the policy. In October and in December I reserved classrooms and replaced my office hours with 3-hour sessions (two per week) outside of class where students could come take quizzes. These sessions went by surprisingly quickly, and I found them relaxed and full of learning. Students came and went as they pleased, and I talked quietly with individuals at the front of the room. Often they had a few questions before they took a quiz. Sometimes I had time to grade quizzes as they finished and go over them with them.

I gave each student a colored file folder and asked them to decorate it so that they could find it quickly on a table full of folders; the decoration could be as simple as scrawling their name across the front or could be more elaborate. I had loads of colored folders to recycle from cleaning out my office at Wheelock, and I bought some pastel folders so there would be more colors. This system worked very well for returning papers efficiently and privately. The grader and I could put the papers in folders pretty rapidly by first separating them into colors and then using a list of names and folder colors. Some students enjoyed decorating their folders, and I got another glimpse into their personality seeing what they did; at the end of the semester I gave them the folders to keep.

### 2. Challenge Problems

The Challenge Problems were the core of the course. The overarching goal was to become a stronger mathematical thinker, and these were the problems that required the most thinking. I was surprised when students told me they had never done such a hard math problem in their lives or that all their previous math courses had been about memorization, not thinking. I expected this response from first year elementary education students, but not from sophomore or junior math majors. Next semester I will build in more explicit work on strategies for getting started on a problem, ideas for what to do when you’re stuck, etc.

There were nine sets of Challenge Problems. Most had two choices of problems, and students were allowed to submit one. I didn’t want students to be able to do all their Challenge Problems too early in the course, but for the last set, they had four problems and could do as many of them as they wished. Many of the problems had extensions, and along the way I converted parts of some problems into extensions, when I found that they were much more difficult for students than I’d anticipated. Students needed to complete three extensions for an A and one for a B.

Each Challenge problem had a deadline, generally a week after students got the assignment, but late papers were allowed. This policy allowed students more time to think about difficult problems, and many students appreciated being able to postpone doing Challenge Problems in a week when they had a lot of work in their other classes. However, towards the end of the semester, it became a nightmare for me to keep track of all different problems. There were 258 submissions/resubmissions in December — over half of which were in the last week, when I didn’t have a grader. The lack of deadlines was difficult for some students who were used to relying on their instructors for providing structure.

Next semester I will allow submissions/resubmissions for four weeks after a set of Challenge Problems is assigned. That will give time for at least two revisions for students who turn in the initial paper on time, but it will also give students some flexibility and additional time to think. Toward the end of the semester, I will give some extra problems. These deadlines will better align the Challenge Problems with what we’re studying in class; students did learn a lot in class, and Challenge Problem 2 was less challenging in December than it was when it was originally assigned in September.

I’m also going to drop the requirement that students do extensions to earn A’s and B’s. There were a few extensions that proved a bit easier than I’d intended, and these ended up being the ones most students did. Grade-conscious students were most worried about the extensions, and for many they were a hurdle to cross, rather than an opportunity for deeper learning on topics they found most interesting. I will still include interesting extensions and encourage students to do them for their own learning and enjoyment, but I will remove them from the grading structure. Most of the Challenge Problems are considerably more difficult than the problems on old exams for the course, and the way I had configured the extensions ended up being more about ranking students than about learning.

Students were allowed to work together on Challenge Problems, and they were supposed to write a section explaining their problem solving process, including who they collaborated with and some description of each person’s contribution. This requirement was unfamiliar to many students, and they didn’t always do it, and to avoid having to grade another set of papers, I didn’t always force them to. Next semester I will include a very explicit template of questions about collaboration and not accept the first Challenge problem without a satisfactory description of their process.

At Wheelock our department supported writing and speaking across the curriculum, and all my courses included a lot of writing. Almost no matter what system I tried, I would always get behind on reading papers and giving feedback, so I was quite nervous about how to structure the logistics of the Challenge problems to keep myself from falling behind. At Wheelock we used Moodle, and BU uses Blackboard. I went to a workshop in June with faculty from all across BU, and they all told me how much they hated using Blackboard for assignments; some had switched to using Google Drive in various ways — since Google manages student email at BU, using these email addresses is FERPA compliant. Through the Teachers Going Gradeless Facebook Group and other resources from middle and high school teachers, I learned about Doctopus and decided to try it.

Doctopus is a Chrome Extension that’s a file management system for Google Docs. I distributed a template for each assignment and it made a spreadsheet with student names and links to each file. When it was time to read the files, I used Goobric to attach a simple rubric, and then I clicked on cells in the rubric, made regular comments in the document, and wrote a sentence or two of comments. I then clicked a button, and the software emailed the rubric and comments to the student, recorded the results on another sheet, and advanced me to the next student, avoiding the annoying downloading and uploading that usually goes with using a Learning Management System for assignments.

Without the grader I would not have been able to keep up, but with the grader, we usually were able to read all the papers within a week of the deadline. Sometimes I gave the grader one class and I took the other; sometimes I had the grader do one of the problems and I did the other. I made answer keys with stock phrases to cut and paste, such as, “Not passed yet. Please see the original document for comments. Please feel free to resubmit. Please do not respond to this email; if you have a question, please ask the professor.” During the last ridiculous pass, I just graded “Pass” or “Not Passed” (no “yet” at the end) and made a stock phrase to paste for those who didn’t pass saying that due to the high volume I couldn’t give feedback. The binary grading system with no feedback enabled me to decide very quickly whether students passed or not, and I was surprised by how efficiently I got through the massive e-pile of papers.

Students complained that they got more useful feedback from me than from the grader, and I said that I would hope so, since I have decades of experience teaching math and the grader is an undergraduate who took the course with someone else. I rotated so they did get feedback from me every other assignment or so. From now on I should be able to have graders who have had a class with me before, and I will work with the grader on how to give good feedback.

Getting math notation into Google Docs was challenging for many students. Some used the MathType add-on, but that could be cumbersome. Others wrote parts by hand and uploaded pictures. Many simply butchered the notation. As long as I could figure it out, I forgave problematic notation, grammar, and punctuation. Uploading handwriting worked, except when students uploaded a whole page, and then it was hard to write comments in the correct place, so in the future I will be more directive on that point. I knew LaTex well when I was in grad school, but I didn’t use it much at Wheelock; very few students in the class had ever used it, and given how rusty I was, I didn’t consider requiring LaTex as part of the class, but that is an option for the future.

The recordkeeping was problematic, as I had results from each assignment in two separate spreadsheets; Doctopus doesn’t handle more than 35 students per sheet. The main sheet showed when students last revised an assignment, but that didn’t necessarily mean they were ready for us to read it, and it was also easy to miss assignments. I had the grader going through each assignment every weekend to look for revisions, and that was fine when there were two or three assignments, but not when there were seven or eight. In December I made a form for students to submit/resubmit Challenge Problems. I also made another form to report mistakes in grading, since I had the Doctopus sheet, my own sheet, and then later in the semester I learned to upload my sheet to Blackboard (with a few bumps on that road). Once students heard the feedback that, “I turned in my problem yesterday and you haven’t graded it yet” was not a mistake, that system worked well (almost all reported recording mistakes were accurate).

### 3. Combination Problems

The parts of my initial design that didn’t work at all were the “Combination Problems,” which was also an especially terrible name for a type of quiz in a class where “combination” was a frequently used mathematical term that meant something else. I initially included them as a way to have an in-class individual assessment of more difficult problems that “combined” techniques from the class, where the issue was to decide what technique to use. I was thinking they would be like the medium-level questions on a more traditional final exam, whereas the quizzes would cover the routine questions and the Challenge problems would be more difficult than the difficult ones. However, the questions would be graded “Pass,” or “Not Passed,” and students could take a few in class and select from many more during the final exam period.

As it turned out, there was only time in class for one Combination Quiz problem, and it was a disaster. Students thought I was making them do Challenge problems in a half hour when usually it took them a week, because in their lexicon for the course, problems fell into two categories: quiz and challenge problem, with nothing in between. Many saw it as a bait-and-switch, where no matter what they had accomplished during the semester, they now had a high stakes final that could leave them failing the class. I realized that this scenario was considerably less fair than a traditional course, where at least the student has seen a midterm from an instructor before going into a final.

Fortunately, by then most students trusted me enough that when I told them that I knew the original format wouldn’t work, that I needed a little time to think about what to do, and to please stop sending me messages about the problem, most relaxed and agreed. I reflected on my own motivations for this assessment, which were varied and sometimes at cross-purposes. The first, and perhaps purest, motivation, was that there is value in reflecting on the material from a semester, making connections, and seeing how much you’ve learned. Next, I did have very real concerns that some students were riding on the coattails of other students, and I wanted some individual assessment, but the way I’d unskillfully set up a high stakes assessment risked undermining the integrity of the course for the many more students who had worked hard and appropriately. In setting up the course, I had been wary that someone would be looking over my shoulder with such a different assessment system, so the Combination Questions would be like final exam questions and would give me some cover. By December, however, I saw that others in the department were either indifferent to or supportive of my system. Finally and most pedestrian, if I didn’t give a final, my grades would be due a lot sooner, and I wouldn’t have time to read all the Challenge Problems.

I thought about how much fear is involved in the whole process of studying for and taking a final exam in math and wondered if there was a way for their to be some joy process — joy for making connections, joy for seeing how much you’ve learned, and joy for the amazing math we had been studying. I went back to our overarching goal of becoming a stronger mathematical thinker, and decided I wanted the final to be a learning experience too. I realized that it was too late to fix some of the mistakes I made in designing the course, but I was pleased with how many things had gone well, and I decided that catching a few hangers-on was less important than maintaining the positive elements of the course.

I designed the “Joyful Finale.” Here is the list of intentions I wrote to the students:

• I want the final to be consistent with the overarching course goal of helping you become a stronger mathematical thinker.

• I want you to consolidate learning from the semester, so that you make connections and so that you are more likely to retain what you’ve learned.

• I want our last time together to include interesting mathematics and for you to learn something from the testing experience.

• I hope to reduce stress so that you can focus on the interesting mathematics, not focus on anxiety about your grades. I hope you will enjoy the final!

The finale included five questions, all of which had a little something new for the students to learn, but they only required material from the course. Students could bring one sheet of notes. They worked for 55 minutes individually, and then there was a 10-minute interactive break, where they weren’t allowed to write, use electronics, or look at their test paper (they could look at a blank test), and they could talk to anyone in the class. After the break they had 55 more minutes to work individually. Each question was worth one point if it was mostly correct and half a point if it included a substantial start. An A was three points, a B two points, and grades could not go down more than two-thirds of a grade from the test.

The interactive break was absolutely electric in both classes. A few students chose not to participate, but most were enthusiastic and focused. Some students moved around the room, some huddled. There was spirited argument back and forth. One group high-fived each other and proclaimed, “We’ve got this!” When the students quieted and went back to work individually, the room felt different, so much tension had dissipated. The simple scoring system made grading a lot easier than I’d anticipated, and it only took about two hours per class. In the finale design I had been willing to use a little bit of fear to make sure students studied, but in the end, I didn’t lower anyone’s grade due to the final; it just didn’t seem right.

For next semester, I’m planning to replace the Combination Problems with a set of More Advanced Problems (MAPs) that students take throughout the semester; my crucial mistake in setting up the Combinations Problems last semester was leaving them all to the end. Students will need to pass twelve MAPs for an A. These problems will mostly be given one or two at a time, with limits on how many problems students can count on a certain topic or by a certain time, in order to spread them out. I can include group work and opportunities for revisions, require reflection on mistakes before taking another question, and require that students pass certain quizzes before taking particular MAP Questions. This change will mostly spread out the more difficult questions I was already giving on the counting quizzes and some other quizzes, as well as give me a place to include to include material that doesn’t fit into the Concept/Proof quizzes or Challenge Problems. The Concept quizzes will be more straightforward than last semester and should require fewer retakes. I’m considering allowing students to bring a sheet of notes for some MAP quizzes and letting them revise the sheet throughout the semester, which would give them an opportunity to reflect on what is actually helpful in creating what they often call a “cheat sheet.” All the problems on the Joyful Finale will also be MAP problems, and students will have to pass at least two MAP questions during the final to get an A in the class; I hope this minimal requirement will be enough to motivate students to review, but not to instill panic.

### 4. Online Class Preparation

Over the course of the semester, there were 23 Online Class Preparation Assignments, in which students filled out Google Forms that were due at 5 a.m. on the day of class. These forms were incredibly valuable for planning the next class, as there was so much I didn’t know about the population I was teaching; I often modified and sometimes abandoned my original plans based on the feedback.

The Online Class Preparation Assignments were loosely based on Robert Talbert’s Guided Practice assignments described in his book *Flipped Learning: A Guide for Higher Ed Faculty.* They included mathematical questions about the homework to help me gauge understanding, as well as many questions that helped me assess students’ backgrounds and attitudes. Some examples include, “Which of the problems we did last class was your favorite? Why?” or “What is your previous experience with mathematical induction?” with choices such as “Could pass a quiz on it now” and “Have seen it before but I’m not proficient.” I left space for them to ask questions or express concerns. I sometimes asked about how they were preparing for a quiz, and then after the quiz asked them to reflect on their preparation process. I could tell very quickly when the students were getting stressed in an unproductive way and respond. I used the forms to poll about best times for the out of class sessions and which quizzes they wanted to take.

I exported the forms to a spreadsheet when I got to them in the morning (usually after 5 am). Students asked me not to officially close the form, because even if they didn’t turn in the assignment on time, they wanted to be able to see the questions. They also asked me to set up the function that would email them a copy of their responses, as well as to keep the form open which I did. For record keeping, I used a spreadsheet lookup function to compare names on my roster to student submissions; this process required tweaking because several students had the same last names and some also mixed up the last and first name fields (including students who were not from cultures that typically write family names first). It would have been easier to use student ID numbers, but I decided that was too impersonal; overall the recordkeeping for these assignments was annoying, but manageable.

The 5 a.m. deadline was an issue. My second class didn’t meet until 2 p.m., and I didn’t realize until late in the semester that some students were doing the survey before attempting the homework assignment later in the day. I didn’t pay too much attention to the timestamps until writing this essay, but after perusing a few, it looks like many students were completing the forms at 2 or 3 a.m; I don’t want the structure of my class to promote such routine lack of sleep. However, the deadline did give me time to use the surveys to plan the class in the morning, when I typically needed to be done by 7:30 a.m. This semester I am going to make the deadlines an hour before class and not schedule anything else before class. At Wheelock I was in a leadership role and could not ever count on being able to shut my door and plan a class, but in my new math department culture its acceptable to shut your door. I will encourage students to turn the forms in earlier so I have some initial data to plan the class.

### 5. Contributions to the Classroom Community

I am personally adverse to participation grades that require me to track data while I’m in the classroom. This recording is routine for many instructors, but for me it feels like an imposition that takes me mentally away from being present in the classroom. One of my favorite things about the gradeless system I used at Wheelock was how much less monitoring and record keeping it required, both outside and inside class; I was a better teacher because I was more present. I worry that constantly monitoring and scoring students’ participation decreases intrinsic motivation, like the children who after getting certificates for doing art well, stopped choosing art projects unless a certificate was involved. Participation grades can also punish introverts and students who are learning English. I put Contributions to the Classroom Community on the grading matrix, but in the end, I didn’t try to quantify, and I raised the grades of students a little if they were active participants in class. Here’s how I presented this aspect of the course in the syllabus:

“A strong classroom community will help everyone learn, and you are part of creating that community. If you are a quick thinker or someone who likes to talk, that’s great, but that’s also not the only way to participate in creating community in the classroom. Here are some ways to help build community:

• Come to class prepared.

• Focused engagement in class activities and discussions

• Respecting and encouraging classmates

• Active and careful listening to classmates and the instructor

• Persistence when the material is challenging or you don’t understand

• Sharing mistakes, struggles, and successes, and thoughtfully responding when classmates’ share theirs.”

### Conclusion

Wheelock College started as part of the 19th century kindergarden movement. Active learning was part of its DNA, and the college had no history of a traditional math major. To the best of my knowledge, I was the first mathematician ever hired there, and it was a great place for me to learn to teach. When I started in the mid 1990’s, the work I was doing at Wheelock felt separate from what mathematicians were doing in the math community; my advisor once told me that mathematicians regard being interested in teaching as akin to being interested in golf — a separate hobby. Fortunately the math community has changed a lot over the years, but I didn’t always feel part of these changes. I was in a different context at Wheelock, teaching non-traditional courses with plenty of Wheelock colleagues to talk to. Much of my energy went toward running an underfunded program while dealing with a chronic illness, and it was often hard for me to travel or work on projects outside of Wheelock. My health is better now, and I have much less responsibility beyond teaching my courses.

At BU I am working to propose some non-traditional inquiry based classes (the first one was recently approved), but I am also teaching some of the standard courses. It’s wonderful to be able to connect with others who have thought a lot about these courses; I got many responses from the Mastery Grading Slack group to my request for Linear Algebra materials. I haven’t taught that class in twenty years, and I’m sure this help will allow some avoidance of wheel reinvention and make the class better.

The transition to BU has required my full emotional range. Sometimes I’m sad and lonely. Sometimes I’m infuriated. Sometimes I’m excited and interested. The best part of my first semester was teaching Discrete Math, and I am grateful to my students for welcoming me to BU, for their creativity which led me to learn a lot of math, for pushing me as a teacher, and for their willingness to try something new and different.

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