Last year I wanted to incorporate more quizzes and individual feedback into my college mathematics courses. My classes are inquiry-based, where students often work in groups solving challenging problems in a variety of ways. Most of my students were prospective elementary teachers – some in an introductory course, and some in courses for Mathematics for Teaching majors.
I’d seen the benefits of low stakes quizzes for helping students learn material, as well as for cutting through the self-delusion we all experience when we think we understand something better than we actually do. My main reluctance to giving more quizzes had been logistical. I didn’t want quizzes to take away too much class time from more interactive activities, but I also didn’t want to force students to rush. The solution I came up with, the “Mini-Quiz,” went well beyond alleviating logistical concerns and led me to question two of my fundamental assumptions about assessment: college students don’t turn in papers every class and being equitable and transparent requires many graded assignments.
A mini-quiz started with a quiz during the last five to ten minutes of class. When class was over, students took a photograph of their paper and then handed it in. As part of the homework for the next class, students used their photographs to write reflections on the in-class quiz, where they finished and/or improved their answer to the original question; found, corrected, and reflected on mistakes; and posed and solved a related new problem at the right level for them.
The mini-quizzes gave me formative assessment to help plan the next class. I used the information to set up pairs or groups, to find interesting mistakes the class could examine together, to choose students to present their thinking, or to decide not to discuss the quiz at all. When students didn’t do as well as I thought they might, it pointed the way toward the next class’s work. The quizzes helped identify students who needed extra help.
The mini-quizzes were usually easy to grade, and I almost always graded them by the next class (sometimes on the bus or subway). I was thrilled that I was able to provide students with frequent, prompt feedback; despite experiments with numerous time-management strategies, I typically get way behind on grading inquiry-based writing assignments, which creates a backbeat of guilt for much of the semester. I was always eager to see what students had done, which provided intrinsic motivation to look at the quizzes right away. Students got a lot of practice writing about mathematics in small chunks.
Part of what made the mini-quizzes easy to grade was that I used a simple two point scale, which was not connected to whether the answers were correct. A score of 2 meant students had done both the initial quiz and the reflection according to the instructions; a score of 1 meant they had done one part according to instructions.
The in-class part of the mini-quizzes promoted recall and consolidation of information, and the out-of-class parts were aimed at higher order thinking and improving students’ metacognition. Results of the out-of-class portion were mixed. The quizzes clearly worked well for many students, who thought deeply about the material. Some told me in final conferences how much they enjoyed the mini-quizzes, how they came to look forward to them as a non-scary way to check and improve their understanding, and how they would make up similar quizzes for themselves in future classes that didn’t include them.
However, I was sometimes stunned at students’ unwillingness to look at their own mistakes. First semester reflections often looked like, “I felt pretty good about this quiz. I think I got it right. I hope so.” I instituted a template second semester, which helped students better understand the expectations, but there were many mistakes that could be easily checked that students didn’t find on their own. In one class some students had made a mistake that led to an equation like 1 = (61 x 20) + (53 x 23). In class, I asked them to look at this equation, and they easily saw that it couldn’t possibly be true and found the mistake. I could put two students who made a similar mistake together, and they would find it in ten seconds, but neither would take the initiative to find the mistake on their own outside of class—even with permission to talk to other students, to use calculators, the Internet, etc.
When I asked about students’ previous experience with finding their own mistakes in math classes, some said there was no such thing as a “mistake” in their high school math classes, just a “wrong answer.” It was the teacher’s job to find “wrong answers,” and then to inform the students, via a lowered grade, and then the class moved on to another topic. For many students, finding mistakes wasn’t their job and wasn’t something that it would occur to them to do — even when explicitly assigned to do just that.
The scoring system also worked against metacognition for some students. The 2’s just indicated that the student had done all parts of the assignment, not how well they had done them. My intention was to keep the quizzes low-stakes, but grade-focused students could take the 2’s as markers that they were doing well, when they weren’t necessarily. I sometimes graded a quiz as a 1 because the reflection was shallow, but that didn’t help students do better reflections, and I didn’t keep track of enough specific information to always remember who needed to work on what.
The problem posing part of the reflection was also an important metacognitive strategy, and it often was a more concrete task for students than reflecting on their work. For many problems they could just change the numbers or the shapes involved to construct a new problem that seemed at the right level of challenge. Students found that they couldn’t always accurately predict how hard a problem would be, which is an important thing for future teachers to know. I thought this part of the reflection went well for most students and helped inculcate a habit of posing problems both when they were stuck on a problem and when they had finished a problem.
Sometimes the mini-quizzes didn’t feel meaningful to students who solved the in-class problem correctly and felt they could solve any problem they could pose by changing the numbers or shapes. For example, if students understand converting between number representations in base ten and other bases, instead of posing a problem like, “Convert 1122 from base four to base ten,” pose a problem like, “The last digit of a base four number is 2. What are the possibilities for the last digit of the number when it’s represented in base ten?” If this last question was the mini-quiz question, students can instead make up a question about a different pattern they notice or wonder about. I regularly seek to embed practice within higher order questions, but I realized that my process for developing interesting questions is mostly hidden from students, who don’t realize how many ideas I reject. In the future, I plan to do more explicit work in class to model problem posing and help students improve at it.
Many of my students have to take five professional licensure tests as undergraduates (MTELs), one of which is in mathematics. When I have these students in class, they are usually eighteen or nineteen years old, and the prospect of these tests can seem both frightening and far in the future. I incorporated several questions in the MTEL multiple-choice format into the mini-quizzes. For the reflection, students posed multiple-choice problems and thought about what’s involved in constructing the distractors for a test question. These quizzes and reflections led to discussions the next day of class that were often much richer than the actual test question.
It’s too early to tell whether the mini-quizzes actually helped students do better on the MTEL test, but I thought they were a good way to include test prep without changing the inquiry-based nature of the class. I also introduced students to my online practice questions and was pleased when some said they were doing them on their own. One student told me she kept the page on a tab in her browser and did a question or two when she had a spare moment; when she told me it was almost a year before she would actually take the test.
With the mini-quizzes, I found myself giving fewer graded assignments or handing back assignments that I had intended to grade with just feedback and asking students to redo them. I started getting nervous that I didn’t have enough grades to fairly assign final grades. I tried a variety of alternate assessment schemes, which I may post more about in the future. Well into the spring semester, I stumbled upon Arthur Chiravelli’s article, “Toward a Future of Growth not Grades,” and I joined the Teachers Going Gradeless Facebook group that he and Aaron Blackwelder started. Participating in the group helped me realize that there are other ways besides grades to be equitable and transparent, including plenty of things that I was already doing.
I also realized that even when I had a small number of students, an assignment that I wanted to read, and was managing time well, I often still didn’t want to grade papers, and the central issue was the grades themselves. No matter how you spin grades, they are fundamentally about comparing and ranking students. I needed large blocks of time and emotional space to try to compare complex assignments to each other in as “fair” a way as I could, all the while knowing just how much subjectivity remained, no matter how careful I was with my rubrics. The better and more interesting the assignment – with more student choice of topics, more pathways to solve problems, more creative ways to present the final product — the more impossible it was to pretend I was choosing letter or number grades objectively. The guilt about procrastinating was often easier for me than the guilt about participating in creating artificial hierarchies, although I usually couldn’t separate the two, as they merged into an unhelpful shame that I STILL had so much trouble with grading, after so many years of teaching.
In the spring I taught two small classes, where the students were very engaged, and most were not overly worried about grades. I incorporated many things I learned from the Facebook group into a final grading process that involved about a dozen “Learning Markers,” where students had to show me work to demonstrate how they had met them, and then propose their own grade (with meeting all the markers being an A). I was available the entire week after classes ended for help and conversation. I was pleased with the work students did on their own to improve their understanding and with how reasonable they were in the final grades they selected. I noted that I could see students better as individuals when I could set aside my background anxiety about assigning grades (and possibly assigning labels, such as “B-student”), and several students surprised me with much better work than I might have anticipated.
When I first started teaching, I tried to downplay grades as much as possible, and like a lot of novice teachers, imagined that I could be more like a friend than an authority figure. I tried to avoid taking responsibility for the power that I had in the classroom, which created more anxiety for students, and more concern about grades. Over the years, I learned to use my power in the classroom responsibly to create a learning environment, and I accepted the need to be as transparent as I could be about grades, because they were something students cared about, and something I reported. Participating in the Facebook group is helping me see a whole new way to structure assessment, one where I can fully accept responsibility for my power, without constantly needing to give grades and weigh and average them. Although tiring, it was wonderful to finish the semester talking to students about mathematics and about learning, rather than sorting a spreadsheet with one number for each and tweaking cutoff points. I even found myself a little disappointed that the start of the school year in September is so long after the end in May
Next year I will be teaching different, bigger courses than I did this year, and I won’t have as much time for individual conferences; I am looking into portfolios and other ways for students to demonstrate their learning. I intend to continue using mini-quizzes, within a structure with clear expectations, frequent feedback, and no grades besides the final grade. I plan to stop using the two-point scale, and just provide feedback, and to ask students to revise some reflections. We’ll do more work in class to support metacognition, particularly around finding mistakes and posing problems. I’m trying to work out how to keep records in a way that will support this structure but still enable me to return work promptly. I am open to questions, suggestions, and conversation – whether some small feature of the mini-quizzes seems interesting to you or whether you are also in a process of making bigger changes in how you assess students.
I’ve been teaching at Wheelock College for twenty-three years, experimenting all the while, and after all this time, few of the new things I try are radical departures. The mini-quizzes started with an idea for a tweak: the students all have phones, let them take a picture of a quiz, then they can both turn it in and take it home. That tweak led to more frequent feedback, which led to fewer grades, which is now leading to a mostly gradeless assessment system – a radical and exciting change that I’m making with the support of a wonderful new community of people, who live all over the world and teach subjects ranging from writing to computer science to biology and students ranging from elementary school to graduate school.