By: Matt Stamps, Yale-NUS College
When Yale-NUS College reviewed the curriculum for its Mathematical, Computational, and Statistical (MCS) Sciences major in the autumn of 2018, I spent several weeks reading about mathematics programs at similar institutions. A common learning objective among many of the programs was a variation of “preparing students to become lifelong learners.” I really like this goal because, among many other reasons, it reminds teachers that students are human beings who have lives beyond their studies, and it reminds students that learning is not confined to the early years of one’s life. As I reflect on my life of learning thus far, I cannot help but notice how significantly the way I learn has changed since I was a student. Some of these differences arose naturally with changes in my circumstances over the years, while others could have been addressed while I was still a student.
In this post, I want to share some observations about how my approach to learning has changed since I started working as a professional mathematician, and how I have changed my approach to teaching with the hope of helping my students develop more effective and relevant learning strategies earlier in their mathematical journeys.
Reading has become my primary mode of knowledge acquisition.
When I was an undergraduate student, I rarely read mathematics. It wasn’t from a lack of interest in the subject. I remember being enthusiastic about my courses and the joy I felt from solving problems. I simply didn’t read much mathematics. Not really anyway. The closest thing I did to reading was scanning through a textbook for a proposition or theorem that could help me link two concepts that would allow me to solve a homework problem. That jigsaw-puzzle approach to reading mathematics lasted well into graduate school. It is dramatically different from my current situation, where the majority of new mathematics I learn, I get from reading. So what changed? Necessity. As a professor, I can go to seminars and conferences to learn more about certain subjects, but not to a degree that is comparable to taking a course. Instead, I spend a lot of time learning on my own and the available format is almost always written text.
When I was a student, the need to read simply wasn’t there. I was fortunate to study at an undergraduate program with many dedicated teachers, who prepared clear, accessible lectures and class activities, so I could successfully complete my coursework without doing the assigned readings. It didn’t become an issue for me until I was a graduate student when I had to look up details of proofs that didn’t fit into lecture notes and read lots of articles for my dissertation research. It was a difficult transition for me.
Learning new concepts and techniques becomes much easier when I need them to complete an ongoing project.
When I was a student, I spent a lot of time learning new techniques, diligently practicing them on problem sets… and then forgetting them almost immediately. I don’t think I was particularly unmotivated or lazy – and I completely trusted my professors when they said certain concepts were important – yet I forgot so much of what I learned shortly after learning it. What was going on? On one hand, it is natural to learn new things in stages, picking up a fraction of the content at the first encounter followed by pieces of new information with each subsequent exposure. At each stage of the process, we internalize a portion and forget the rest of what we observe.
On the other hand, I think my struggles were partially related to context. I remember my professors giving clear explanations for why different techniques were developed and how they were used in practice, but there was a disconnect for me because I didn’t have any personal experience developing mathematical techniques, nor did I have an application of my own in mind. Looking back over my career, my most productive learning experiences have come from working on a project where I didn’t have all of the tools I needed and had to learn them on the fly in order to complete the project. In those cases, I didn’t watch a tutorial or listen to a lecture about standard techniques and then practice them on a variety of examples; I started with the problem I was trying to solve, found a technique in the literature that was used to solve similar problems, and figured out how to apply or adapt the technique to my particular situation.
All of my best ideas have had humble beginnings.
When I was a student, I had a growth mindset about mathematical knowledge but a fixed mindset about mathematical creativity. I believed everyone could have positive, successful, and meaningful experiences with mathematics by learning new techniques but mathematical creativity was an inherent ability that could not be developed. I don’t know why I felt this way, and I can’t recall anyone ever telling me it was the case, yet I remember that impression weighing on me a lot. Whenever I worked on homework sets with other students and someone would figure out how to solve a problem I was stuck on, I always assumed it was because they had some amazing insight that I would not have been capable of finding on my own. I was so preoccupied with trying to figure out whether or not I had what it took to become a successful mathematician that it never occurred to me to ask them how they came up with their idea. Consequently, I spent a lot of time feeling frustrated, not being particularly productive, and waiting for inspiration to strike because that was where I thought creative solutions originated.
Now that I have more experience – and the confidence that comes with it – I can recognize that all my best ideas started with simple observations. And while there is no clear-cut recipe for creativity and innovation, there are concrete things I can do to cultivate situations that make those important kernels of ideas of possible. Instead of dwelling on what I don’t know how to do, I focus on exploring what I can do that might produce a new insight, such as writing out some examples, constructing a conceptual diagram, or drawing a picture.
All of my proudest accomplishments were made possible through the generous help of people whose experiences and perspectives are different from my own.
As a student, I found little satisfaction from working in teams, especially with unfamiliar teammates. Team assignments typically went one of two ways for me: either I was confident in my abilities and did the vast majority of the work or I was insecure about my abilities, didn’t want to look stupid, and held back my ideas thinking it was better to appear ignorant than open my mouth and confirm it. In the former scenario, I didn’t mind doing most of the work because I was confident in my ability to succeed, and it often seemed easier to do most of the work myself rather than try to coordinate my teammates’ efforts.
I didn’t see value in exploring different perspectives because there were never any consequences for taking a narrow approach. Like many who have the same privileges as I do (I am a heterosexual white male from North America), I had a limited understanding of how social identities affect group interactions, and I conflated inclusivity with civility. In the latter scenario, I was aware that teamwork required a lot of effort and collaboration. Even though I was willing to put in the work, my insecurities still got the better of me because I didn’t trust my teammates enough to share my ideas openly.
When I look at how the accomplishments I’m most proud of have come about, and how much I have learned in recent years working at an international college in Singapore, I can’t help but wonder how many opportunities to learn and grow I missed out on because I simply wasn’t looking or I didn’t appreciate how much effort goes in to building enough trust to open up a beneficial exchange of ideas.
Changing the Way I Teach
Here are a few ways I have changed my approach to teaching in response to these observations.
For starters, I no longer rely on lectures or video tutorials for presenting new ideas. Instead, the lion’s share of content delivery comes in the form of reading assignments. To support my students as they adapt to this model, I use the social annotation platform, Perusall, which allows them to highlight passages and ask questions, contribute or link alternative explanations, and propose solutions to “check your understanding” type exercises. They can also upvote annotations of their peers that they find helpful. In addition to developing technical reading skills, Perusall offers the valuable practice contributing to social media debates and online forums like StackExchange in a safe and controlled environment.
To offer my students an authentic learning environment that emulates the typical “on the job” learning that takes place in many technical professions, I have started to build each of my courses around three or four substantial team projects. Instead of asking students to master content and then apply what they have learned to a bigger project, I design the projects in a way that prompts students to learn the relevant material as they go. Each project is assigned on the first day of its respective segment of the course. The students are typically able to understand what the project prompts are asking but are not aware of any obviously relevant tools to get started.
To facilitate effective teamwork, I have adopted the Team-Based Learning (TBL) model, where each lesson has a reading assignment to be completed before class, individual and team readiness assurance tests at the start of each class, and a substantial problem-solving session that enables students to apply and extend their understanding of the tools they will need to successfully complete the project. Students take the readiness assurance tests and work together on the problem-solving sessions within their project teams throughout the duration of the project in order to develop a productive group dynamic.
To encourage and reinforce good habits for mathematical research and creativity, I have started acknowledging and giving credit to teams when they demonstrate important elements of a productive research process, such as generating examples, identifying patterns, asking questions and making conjectures, testing conjectures with new information, drawing connections between relevant topics in the literature to better understand the problem at hand, and re-evaluating an approach based on preliminary findings. Because many of these elements can be difficult to discern in a final written report, I have started asking each team to submit an activity log that documents their progress throughout the project. My rubric for the activity log was heavily influenced by the Creativity-in-Progress Rubric on Proving.
Finally, in addition to research and creativity, I have started to encourage and reinforce good habits for effective and respectful team interactions by asking each team to prepare a mission statement during the first week of the project where they agree on a team name, tentative work schedule, and initial plan of attack. I also ask each team to prepare a set of guidelines for how they will conduct their meetings and a set of criteria for how they will evaluate each other’s contributions to the project.
The idea for creating guidelines came from my experience facilitating Intergroup Dialogue (IGD) at Yale-NUS College. IGD is a structured conversation between members of different social identity groups that encourages participants to explore singular and intersecting aspects of their identities while critically examining dynamics of power, privilege, diversity and inequity in society. Because the dialogues can be difficult or contentious, a lot of the groundwork for IGD aims at building trust and creating a space in which people can share their ideas freely without judgment. For instance, at the beginning of each dialogue, the participants prepare a list of guidelines. I adapted those guidelines to fit a team-based learning classroom: The IGD guideline “We all recognize that participation in this dialogue is voluntary. Everyone who is here wants to be here.” became “We all recognize that this course is an elective. Everyone who is here wants to be here.” Most of the guidelines are common sense statements, but articulating them in a mission statement provides avenues for students to speak their discomfort and overcome obstacles in a responsible and respectful manner.
Sample Project Brief
Here is the first project brief from my Discrete Mathematics course, which is typically taken by second-year prospective MCS majors at Yale-NUS College whose primary interest is computer science. The course meets twice per week for 110 minutes at a time. Each lesson consists of a pre-class reading assignment (8-10 pages of text, approximately 2 hours of interactive reading), in-class readiness assurance tests (20 minutes), and an in-class problem-solving session (90 minutes). The project spans six class meetings, including one lesson each on the Pigeonhole Principle, mathematical induction, and basic enumeration, two lessons on combinatorial proofs and bijections with emphases on the Binomial Theorem and Fibonacci numbers, and one class meeting designated as work time so students have a full week free of reading assignments and problem sets to complete their reports.
The project description presents students with eight seemingly unrelated families of mathematical objects and asks them to find a formula for the number of objects in each family. It then asks them to describe how the families are related based on the formulas they find. Over the course of the project, the students discover that the families are all equinumerous. Indeed, they are all manifestations of the Catalan numbers!
While the project initially appears somewhat daunting, the students typically proceed by generating lots of examples of each family. From there, they tend to observe fairly quickly and conjecture that the number of elements in each family appears to be the same. This is a significant discovery for them since it means that, instead of finding the same formula eight different times, they only need to find the formula for one family and then argue why the different sets are in one-to-one correspondence with one another. That prompts them to review the reading assignments on mathematical induction, recursion, combinatorial proof, and bijections. The diversity of the objects themselves also makes the project well suited for teams made up of students with disparate backgrounds since finding all the connections requires a variety of perspectives.
The overall response to these changes has been positive. A number of students acknowledged the stated goals and embraced the project-based approach straight away. For instance one student wrote:
[The project] was actually a very fun and enjoyable experience, while also providing a good amount of challenge and difficulty. When we first received the project brief, we were genuinely stunned by what we had to do – we didn’t really know where to begin, and everything we tried seemed to be useless. But it was really nice to see us slowly progress, picking at the problem bit by bit, sometimes with no results, sometimes with huge chunks of the problem falling off. I really saw the advantage of having very different minds work on the same problem. I believe my teammates and I, having come from different backgrounds in terms of interests and experiences, approached the problems quite differently, and we were able to really complement each other and bounce off each other’s ideas. All of us contributed in big ways, and together we managed to come out with a closed formula pretty early into the project. Eventually, we managed to link the closed formula to one of the combinatorial objects, and quickly pieced bijections together. Even in the final moments of the project, the group shone through as we all picked on different parts of the project, trying to polish it off as well as we could.
Other students struggled at times, but eventually warmed up to the approach. For instance, a student wrote:
Initially, I felt rather excited about tackling the questions. We made some observations that turned out to be insightful and it felt like the project was going in the right direction. When the team and I got stuck at the later stages of this project, I became frustrated and lost motivation. But my teammates continued to encourage me and kept trying to develop new methods of solving the problems. Through this project, I learned that solving problems is not always a smooth path. It is helpful to acknowledge our frustration and to expect difficulties so that we are less anxious when we are stuck.
The most encouraging feedback I received, however, was from the many (more than 1 in 5 across my sections of Discrete Mathematics) students who explicitly described how empowering the experience was for them in their reflections. For instance, one student wrote:
Throughout this project, I have learnt a lot about how mathematical reasoning happens and [I] have changed a lot of my perceptions about how mathematics is done. Being used to the usual individual problem-solving method in high school, where there is only one right answer and a few preset methods that are best for determining this answer, I have come to love the collaborative approach taken in this project and in the whole Discrete Mathematics course in general. It is truly a vibrant environment for learning, and I am very grateful to have the support and knowledge of my team members. I have always felt my mathematical reasoning skills to be inferior to other upperclassmen or people who reason faster, sharper and more elegantly, but I have come to learn that the final polished product is not all that it appears to be – it is the process that is the most important, and there are lots of things I can contribute within the process while I am working on improving the skills I can use to refine the final solution.
Despite these successes, there is still a lot of room for improvement. A common piece of critical feedback I receive from students is that the reading assignments are very difficult, even with the added support from Perusall. There are a lot of factors at play here such as the choice of text, size of the class, and my (in)ability to effectively respond to the Perusall discussions in real time.
Revising my courses to emphasize reading, research, creativity, and teamwork has been a challenging but rewarding process. I am thankful to the many Yale-NUS students who worked diligently on the projects and offered their thoughtful, constructive feedback, and I am curious to see how my approach extends to other topics besides discrete mathematics and abstract algebra, which may, for one reason or another, be particularly well suited for this type of open-ended, collaborative, team project framework. I hope I have convinced you to try out some of these ideas and I look forward to hearing about the outcomes!