Calculus via problem sets with a revise and resubmit process
How I’ve moved away from timed tests to focus on growth and learning from homework
This month we welcome Brendan W. Sullivan as our guest poster. Brendan is an Assistant Professor of Mathematics at Emmanuel College in Boston, MA. He earned a Doctor of Arts from Carnegie Mellon, where his dissertation was an Introduction to Proofs textbook, and he recently published a book on voting methods, as well. You may contact him at firstname.lastname@example.org.
Emmanuel is a small liberal arts college with around 2,000 undergraduates and – given our history rooted in the Sisters of Notre Dame de Namur – a strong focus on social justice and community. Popular majors include biology, business, and psychology, so we in the math department teach courses from College Algebra through Vector Calculus that are populated mostly by various science students and a handful of math majors. These courses are capped at 25 students, with typically one or two sections of each course offered every semester.
In the last few years, I’ve reworked the courses I’ve taught in this sequence – namely Precalculus and Calculus II – to move away from high-stakes, timed assessments and towards more authentic mathematical tasks. While I’ve had to create many new assignments and rewrite my syllabi and course materials, this work has been worth it because I see students learn so much not only about calculus but also about themselves as learners.
In this blog post, I will use the context of a Calculus II course to describe how I use Problem Sets with a Revise & Resubmit process. I’ll explain the overall grading system, which combines the structure of standards-based grading with some of the philosophy and practices of ungrading. Afterwards, I’ll share the details of one Problem Set and use it to help explain why I’ve moved away from timed tests.
Course design and assignments
The course is organized by a list of course learning objectives (CLOs). Check out my list for Calculus II in this link, or see below for a screenshot.
Each week of the course covers two or three objectives, and all class materials and assignments are organized in weekly modules on our LMS (example below). In addition, each week has two kinds of assignments: one Practice Homework (via WebWork) and one or two Problem Sets.
Practice Homework assignments are low-stakes learning opportunities that help students develop fundamental skills. Questions are similar to examples from that week’s class notes, with an occasional challenge question or extra application. They’re graded instantly, with a total of 100 points per assignment. I encourage students to work together and discuss their problem-solving process, and they’re allowed unlimited attempts on each question. These details are restated in each assignment’s instructions, along with a list of which CLOs are covered: see here for an example.
Meanwhile, Problem Sets are opportunities for students to apply the skills and knowledge they’ve developed. Each Problem Set has a list of questions organized around a central idea, usually an application of course content or a challenging task. And instead of with points, Problem Sets are graded as Complete or Incomplete.
“Complete” means that a student demonstrated the skills and knowledge required for that assignment. To assess their work, I look at the central idea of the assignment, as well as the relevant “I can…” skills in each associated CLO, and ask myself:
Is the submitted work mathematically correct?
And is that work communicated clearly?
If I can comfortably answer “Yes” to these questions, then I mark the assignment “Complete” and the student can move on. Importantly, though, I am not looking for 100% perfection! A submission might be marked “Complete” even with a minor arithmetic error, for instance, or with an explanation that’s good but not great. (I always leave a comment, though, so the student is aware of the issue but knows it doesn’t require a resubmission.)
Otherwise, there’s something about the student’s work that gives me pause. Perhaps there’s a more significant issue with their mathematical work, like incorrect or missing steps. Or, perhaps the work is correct but something about their written explanations indicates that they haven’t fully grasped the relevant concepts. But instead of letting the student just move on, I want them to learn from those mistakes. This is where the Revise & Resubmit process begins!
It then becomes my responsibility to provide helpful feedback so the student knows what to fix. In the LMS grading tool, I highlight their work and type detailed comments, often using leading questions to help students identify errors and think about how to fix them. Sometimes, especially for issues with communication, I’ll ask followup questions for the student to respond to in the comments. Throughout, I make sure to use encouraging language, reminding students that they’re not done yet and that revising their work is an essential part of the learning process. (Early in the semester, these comments can also help onboard students to the grading system.)
After receiving my comments, students revisit their work and turn in a new version whenever they’re ready. If their resubmission now meets the standards of the assignment (by addressing the comments I had made), I mark it “Complete.” Otherwise, we go through the process again: I leave detailed comments about what still needs fixing, and they revise their work and submit again.
Students may resubmit each Problem Set at least once (except for those in the last week of the semester). During my weekly grading of new submissions, I check for revisions of past assignments and mark those. (They’re typically much faster to grade because my comments on the prior submission tell me what to look for in the revision.) I also remind students often that late submissions are acceptable for course credit, but they may not get feedback from me in time to complete a revision. I admit that this is probably a much looser policy than other instructors have, but it has worked well for me and my students.
Final letter grades
By the end of the semester, students have been engaging with course content regularly via both kinds of weekly assignments, requiring frequent review and application of prior knowledge and skills. I believe that working on roughly twenty challenging Problem Sets and fixing mistakes is plenty of evidence of their growth and progress. So, instead of a final exam or project, students write a reflective essay that describes their “learning journey” and advocates for a final letter grade. Here are the details of that assignment.
This is where some ungrading principles come into play. Unlike some courses with a standards-based grading approach, I do not have set requirements for how many Problem Sets must be “Complete” or what Practice Homework scores must be. (We do have “points” because of WebWork, but I remind students that “there is no official ‘cutoff’ and your overall grade will depend on many other assignments that do not have scores, too.”) Instead, I provide qualitative descriptions in the syllabus of overall A-level work, B-level work, etc., and I ask students to “describe how your work this semester aligns with a particular letter grade.”
There’s a final semester deadline during exam week by which students must submit this reflective essay and any Problem Set revisions. To choose a final letter grade, I read a student’s essay, then look at their Practice Homework scores and Problem Set completions, often consulting the comments history on some submissions. I reread the description of their learning journey and what grade (or range) they chose, and use my best professional judgment to make a final decision. Students know that “I may end up choosing a grade that is not what [they] argued for,” but, for the most part, I find their reflections honest and informative, and sometimes quite detailed. Indeed, I learn much more overall about what and how my students learn from these essays than I ever did from final exams.
Example problem set
Check out this assignment from the Applications of Integrals theme: Week 7: What "average value" means visually.
The central idea of this Problem Set is that the average value of a function can be interpreted as a balancing act of areas between curves. (Explore the idea with this interactive graph, if you’d like.) I’ve always loved explaining this concept in class, but exam questions about the topic were necessarily limited in scope and difficulty. Students had to state the formula or apply it to a simple example with “nice” numbers, but I never felt like this captured anything about their deeper, conceptual understanding.
Instead, I use this Problem Set to guide students into discovering that central idea for themselves. They’re asked to apply knowledge from recent class sessions and communicate their understanding in written explanations, all while constructing a graph that illustrates the main result. In assessing student work on the detailed steps of this assignment, I believe I get a much better understanding of both what they’re doing and learning outside of class, and how they’re progressing on the Course Learning Objectives. Moreover, creating and sharing a graph helps students learn how to use technology appropriately and encourages a sense of agency over their mathematical work.
In short, without the constraints of a typical exam, I find great freedom in crafting assignments that ask my students to complete specific tasks or grapple with important ideas. I wish I had allowed myself this freedom earlier in my teaching career.
I first used this exam-free grading system in Fall 2020, hoping it would make a challenging course more accessible and engaging during a trying time, and that it would lead to better student learning. I’ve been refining the system ever since, and I’m committed to developing grading systems like this one for every course I teach.
The reason I’m so committed is because, on the whole, what I hoped for did happen: students have found the course more accessible and engaging, and I’ve witnessed a tremendous amount of learning! Writing the Course Learning Objectives and Problem Sets forced me to decide what student outcomes really matter, and then identify how to help students reach those outcomes. Meanwhile, without worrying about how to recall information and solve problems quickly, students can focus on conceptual understanding and take the time they need to figure things out. And the Revise & Resubmit process is key because it reminds them that learning is a journey, and we all stumble on the way. This student comment sums it up well:
“The weekly problem sets challenged me to think deeply and connect several different concepts together into one problem, but in a low-stakes way that allowed me to make mistakes and take academic risks. It was new for me to not be memorizing a whole bunch of information and spitting it out on a test, full of anxiety about how I performed. I feel like I learned to learn, not to get a grade.”
However, there are important features of my teaching context that may not always apply. I have small classes, typically around 20, which makes the grading more manageable. In our close-knit college community, I trust my students with out-of-class work, they trust me as an evaluator of their work, and my colleagues trust me as an educator. Moreover, I had already taught this course before and had plenty of materials to draw on for inspiration with assignments. If any of these features were different, I’d have to think more carefully about how best to go exam-free.
Whatever your teaching context, I hope this post at least convinces you to reconsider what your assignments ask students to do, and how those tasks align with your course objectives and what you want students to remember. I don’t think Problem Sets, or anything else, are the perfect solution that everyone should use. But, I do worry about how timed exams limit our ability to probe student understanding, and how they seem to convince students that being a mathematician means being really good at taking a test. At a certain point, I realized that I didn’t want myself or my students to be limited in those ways, so I had to try something else. It was totally worth it.
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