Visualizing Stochastic Processes

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Categories: GSI Online LibraryTeaching Effectiveness Award Essays

by Ella Hiesmayr, Statistics

Teaching Effectiveness Award Essay, 2021

One effective way of making content accessible to a wide range of people is to present the material in a variety of formats. It is common to teach mathematical courses by relying mainly on material in text form, but some mathematical areas provide ample opportunities to also use visualizations to convey the important notions. I used this possibility to create alternative entryways to a course and thus address the problem that students lack visual imagery of mathematical concepts, which might make it hard for some students to effectively apply those concepts.

During my first few semesters at Berkeley, I was glad about the many incentives I had to rethink my teaching methods. Student evaluations, the teaching preparation course, and discussions with other graduate students made me realize that presenting the abstract mathematical material in diverse formats to students would be beneficial to their understanding, and more specifically lead them to have better conceptual ideas of the notions in question. In particular, I believe that mathematics can be perceived as a subject that is pleasurable for a few and a necessary evil for most. By conferring the material in an additional, more visual way, I think that mathematics can become more inclusive and enjoyable for a larger range of people.

Once I knew that I would be a GSI for a course on stochastic processes, I immediately wanted to find a way of providing students with other channels for learning this mathematical content besides the classical text-based format. While there are several options to do so, I opted for simulations and visualizations. The former is a great way to emphasize the probabilistic content of the objects of study and the latter allows one to highlight the fact that these objects are viewed as evolving. I had the luck to be part of the Graduate Remote Instruction Innovation Fellows Program over the summer, which allowed me to start creating Jupyter notebooks that simulate stochastic processes. I continued developing more of these notebooks during the semester. Each focused on a different topic that students were currently learning about from a theoretical point of view and that they could explore in the notebooks in a more interactive way. I think that randomness is sometimes a hard mathematical concept to grasp, so I hope that such simulations broaden the audience that can be effectively reached.

To make the notebooks accessible to everyone, I wrote the code such that students could easily tweak it in order to see how models change according to different parameters, even without a background in programming. Moreover, these notebooks can be run in a browser, so they do not require technology beyond what is already necessary for the course. By being able to view different realizations of the same random process, for different initial configurations and governing mechanisms, students could then make conjectures about the behavior of the stochastic process, and the reasons behind it. Every notebook included a low-stakes assignment that was easy to complete and mostly meant as an incentive to go through the notebooks.

While these assignments were easy to complete, the answers students gave showed that they were thinking about the reasons behind what the visualizations showed. In some cases, this would lead them back to the proofs of certain theorems and thus also enhance their theoretical understanding. In other cases, this would make them realize that their initial guess was wrong, and the surprise would increase their curiosity and motivation. Further evidence that these simulations helped enhance students’ conceptual understanding was the feedback that students could give anonymously at any time during the semester. This feedback was entirely positive and generally voiced that the visualizations were increasing their intuitive understanding of the underlying concepts. In addition, several students mentioned in the teaching evaluations that the notebooks were detailed, well-organized, and accessible, as well as helpful for building intuition.