One of the courses in my regular teaching assignment that I enjoy most is Math 104, Penn’s largest entry-level calculus course. People who know the course are usually surprised when I say this because there are a host of challenges in 104 that are much easier to face in more advanced courses. Most students in 104 are freshman Wharton, Engineering or College students; many experience 104 as a “gateway course,” which can seem mainly like an obstacle to overcome on the way to some preferred course of study. Grading across all sections is curved to roughly 30% As, and most sections are rather large compared to a typical high school class. In practice, this means that students experience substantial academic and personal growing pains during the semester. For instructors, there is a constant threat of triggering quick frustration in the face of challenges, skepticism around the value of new ideas, grade anxiety or resistance to raised expectations; this threat means that far more of the instructor’s mental energy is devoted to finding ways to increase engagement and manage expectations than to the actual “content” of the course. It’s also expected that an instructor’s effort on this front be “seamless,” i.e., effectively invisible to the students. In spite of all this, I still find it rewarding to be among the first Penn faculty members to engage with incoming students and to show them new levels of intellectual rigor and depth in material that is deceptively familiar to many of them.

In recent years, I have been teaching my section of 104 in an active learning (SAIL) format, which, in practical terms, means that I do very little lecturing and instead reserve nearly all class time for students to work on structured group activities. My experience has been that this makes it much easier for me to quickly and thoroughly assess student progress than I could in a traditional lecture. One thing I learned about my own teaching after switching to SAIL was that I was not doing as well as I thought to foster the development of critical and reflective thinking skills or to create a cooperative learning environment. There are, of course, many forces in the classroom that make it hard for students to think deeply and work cooperatively, but this semester, influenced heavily by the Center for Teaching and Learning’s Inclusive Teaching Seminar and the article “Psychological insights for improved physics teaching’’ by Aguilar, Walton and Weiman (Physics Today 67, 5, 42 (2014)), I decided to work on student feelings of social belonging in the classroom.

There is substantial research to support the idea that students belonging to negatively-stereotyped groups (in math, there are prominent stereotypes directed against women and non-Asian ethnic minorities) face extra stress and distraction because of stereotypes, which can substantially impede classroom performance. However, research also suggests that there are relatively simple and effective ways to work against negative stereotypes. The main activity I used this semester to help put students more at ease was a simple, short writing assignment based on an activity described by Aguilar et al. In late October, just after the second midterm, I asked students to reflect on their growth in the class for the purpose of using these reflections to help others. The prompt I used was the following:

Many new students in Math 104 find themselves facing anxiety about things like their previous academic preparation, difficulty feeling like they belong in the course, shifting understandings of grades and academic standards, or concerns about keeping up in such a fast-paced course. As time goes on, most find that such anxieties are far more common among peers than it initially appeared and that there is a natural adjustment period during which they come to feel more capable and confident in their own ability to succeed.

The purpose of this assignment is to use your unique perspectives as recently new students to generate meaningful, personal insights that can be compiled and shared as a valuable resource for next fall’s incoming Math 104 students.

Reflect your own personal experience in Math 104 so far and write about the ways that you see it illustrating the general processes of intellectual growth, learning to understand and manage obstacles and anxieties, and gaining confidence in your own abilities.

You should aim to keep responses relatively short (1-2 paragraphs maximum). Submissions will be graded anonymously in Canvas. All submissions received before the deadline will receive full credit. 

Normally such an assignment would clearly stand out as unusual in 104 and consequently fail the seamlessness test; in this case, I relied on having already established a pattern since I had also asked students to reflect on their performance after the first midterm, albeit in a more academic, impersonal way. I received a truly impressive number of deeply thoughtful responses. Two in particular that stood out are as follows:

“When I started this course in August, I was very nervous about how I would do in the class. I had taken AP Calculus AB in high school but I quickly found out that many of my classmates had taken Calculus BC instead, which made me think that I was not as prepared as they were for the class. In addition, I unknowingly registered for a SAIL class, so the format was very unfamiliar to me and there was definitely a bit of a learning curve associated with that. After the first midterm, I was very upset because I had never gotten below a C on a test, and I briefly considered dropping down to Math 103. However, as the semester continues I am getting used to consulting outside resources (Nakia Rimmer’s Math 104 video lectures are a lifesaver) and teaching myself the material and then reinforcing it in class. In a way, teaching myself the material instead of relying solely on a professor has made me more confident in the class over time. There are still times when I doubt myself and my abilities in this class, but my advice is to always take a step back and look at the bigger picture. You are at this university because you belong here, and one low grade will not ruin your life or even your GPA. This class is not an easy class, but if you have enough patience to work through the struggles and challenges then you will be fine.”

“Based perhaps a little too much on my diagnostic test score, my advisor urged me to try math 114 for my first semester at Penn. Looking at the 104 syllabus, though, I noticed that there was plenty of material I had never encountered before. I needed—and wanted—a better foundation in math, partly as a way of challenging myself. Plus, as someone who gravitates more towards the humanities or at least fields that aren’t math-heavy, I did not feel confident in my knowledge and skillset. During the first few weeks of 104, I experienced some highs and lows: I found the material engaging, the class format refreshing and the new ways of thinking a good step outside what I have been used to. But I also had some struggles—among them getting a much lower score on the first midterm than I would’ve hoped. I wasn’t as organized or as disciplined as I could’ve been beforehand, catalyzing my anxiety to the point where I suddenly freaked out during it. But it got better: I took a step back and realized this isn’t a crazy battle I’m fighting. It’s just something I have to really dig into and stay focused on.  

The class can feel like a really competitive, high-stakes environment at times (I guess it is to some extent), but that doesn’t change the fact that we’re here to learn, here to explore the language and reasoning of math together. The help and resources (e.g., great minds and great practice problems) are here for us, so we just have to make use of them! (How else would you get the maximum output you want from the class?) And here’s a final word of advice: do not be afraid to ask questions when you’re stuck—it’s the only way you can truly put your own abilities into context and improve.”

One theorized benefit to the students of an exercise like this is that it increases feelings of social belonging inside the classroom, which then decreases the mental obstacles present during learning. There was such widespread thoughtfulness to the responses I received that I find it reasonable to suspect that there was at least some temporary benefit of this kind for my students. It was also encouraging to see that my students did very well on the final exam this year, with more than half earning an A. As an instructor, I felt like it was also extremely beneficial to me to hear directly from students about their concerns and the ways in which they have (or have not yet) managed to overcome them; in 104, people don’t expect to talk about feelings, so I don’t get many opportunities for this kind of feedback. My plan for next time in 104 is to share these student-generated insights with the new class early in the semester. I hope that these future students get a head start on reinforcing the ideas that everyone belongs and that early setbacks are common and can be overcome with time and growth. By giving my students this head start, I hope to be able to devote more of my class time to new challenging classroom activities.

Philip Gressman is professor of mathematics in SAS.

This essay continues the series that began in the fall of 1994 as the joint creation of the College of Arts and Sciences, the Center for Teaching and Learning and the Lindback Society for Distinguished Teaching.

See https://almanac.upenn.edu/talk-about-teaching-and-learning-archive for previous essays.