During my PhD program I have built up experience in teaching and mentorship. I have learned that balancing independence and friction with support and structure is crucial for student success. I have had the opportunity to explore several approaches to this balance: I have been a solo instructor for six sections of coordinated College Algebra and coordinator assistant for one year, solo instructor for two sections of an uncoordinated general education course, and run several programs where graduate students mentor undergraduates.
College Algebra
My typical lecture format is as follows: start with a five-minute warm-up, a short lecture explaining the technique or problem type of the day, group problem-solving time, and a concluding lecture period. When I taught the quadratic formula the warm-up was an exercise in solving quadratics by completing the square, and the lecture started with a discussion of those problems, which motivated the introduction and use of the quadratic formula. Group work solving quadratic formula problems followed from there, starting with mechanical application of the formula and building towards modeling problems. I concluded with a review of those problems and an explanation of the general approach.
In-class work leads students to engage with the material themselves, which builds self-reliance and student investment, but with insufficient structure, it is easy to lose students’ attention. By alternating between independent work and structured instruction, I have been able to challenge the students who prefer a faster pace without leaving behind the ones with less background. In my most recent teaching evaluation, done during a lecture where I took this approach, Dr. Jenssen, the faculty observer, concluded with “This was an excellent lecture, and Penn State is lucky to have Jacob teaching here!”
I later served as coordinator assistant for the same College Algebra course, where my responsibilities included hosting biweekly meetings to help other instructors with issues and take their feedback, writing course materials including exams and rubrics, and helping more junior instructors one-on-one as they adjusted to solo teaching.
Writing exams and rubrics taught me how to navigate the balance between accurately assessing student knowledge and the potential burden on other instructors when grading systems change. For instance, during my time in the position the rubric structure for exams was changed from a negative-scoring structure to a positive-scoring one. The instructors were divided on whether this made it more difficult to give detailed, accurate feedback or if it improved the efficiency of the grading process. In the end I facilitated dialogue between the coordinator and instructors on both sides, which helped us implement the change more smoothly. I hosted informal grading parties where new instructors could ask me how to handle particular exceptional cases that did not fall under the rubric, and met with instructors if they were having issues with particular students to help them brainstorm solutions in line with school policy.
Mathematics for Sustainability
In the uncoordinated setting, I taught Mathematics for Sustainability, which is attractive to journalism, political science, and other humanities majors because it emphasizes teaching students how to form quantitative arguments and how to question them. My students were mostly in journalism, so I focused on issues around communicating the mathematical models important to civil discourse: identifying assumptions and how conclusions were drawn from quantitative results. The textbook was pre-COVID, so I wrote materials from scratch for an infectious disease modeling unit.
My materials (most of which are available on my website) focused on collaborative problem solving, as well as writing and critiquing quantitative arguments. For the infectious disease unit, I wrote instructions for how the students should use a computer to find numerical solutions to models which included effects like increased spread from infected health professionals. I wrote the worksheet to guide them through implementing the model, stopping to ask key questions to prompt justifications or criticisms of its assumptions.
Thirty percent of this course is project-based: the students worked in groups to answer a quantitative sustainability question of interest to them. One obstacle to this part of the course was that students had generative tools which could do the work for them. To prevent my students from outsourcing all of their thinking I emphasized three parts of the project: elevator pitches, final presentations, and in-class working time. This encouraged students to learn to communicate their ideas to non-experts, set realistic, ambitious standards for their own work, and develop their ideas where I could see them in real time and provide feedback.
Advanced Content and Research Mentorship
Outside of the classroom, I led (with another PhD student) two research projects as part of outreach and research experience programs. In the Penn State EULER Summer Scholars program, we worked with upper-year mathematics students to use techniques from classical fractal geometry, harmonic analysis, and multifractal analysis to identify seizures in EEG data.
We led a related project as part of the STEM for All program at the University of Rochester, which is more outreach-focused, with an emphasis on the idea that anyone can contribute to mathematics research. Our group had mostly early-year non-math major students, including two high schoolers. Here we emphasized teaching the students quantitative research skills more broadly: building data analysis pipelines, writing code to implement them, and communicating their conclusions. At least one participant said she was considering graduate studies in the quantitative sciences because of the program. Collaboration continues with both groups, and their collective research contributions form the backbone of a paper currently in preparation.
This and last year, I have been the lead graduate student organizer for the Directed Reading Program at Penn State, and I have co-led three DRP groups on algorithmic game theory, mathematical physics, and currently on fractal analysis. In the mathematical physics group, my co-lead and I worked with two students for the full academic year and helped them to build background on, among other topics, the use of abstract algebra to encode symmetry in higher-level physics. Both students are pursuing further study in mathematical physics.
Conclusion and Future Goals
My teaching experience has prepared me to teach upper-division courses, from core courses like abstract algebra to electives such as fractal geometry, game theory, or math physics. DRPs, and programs like STEM for All, promote both engagement in research-level math and interactions between people at different levels of academic experience. In STEM for All, in particular, I saw students with very little background make real research contributions in a very short time, which underscores how impactful this kind of outreach can be. At my future institutions, I plan to participate in similar programs where they exist and found them where they do not.