# Autocatalytic Network Captures Cooperative Learning in the Classroom

“Right now I am teaching a class, and in teaching there is something called *cooperative learning*,” Meilin Huang of Saint Xavier University said to start off her minisymposium presentation. During the 2023 SIAM Conference on Applications of Dynamical Systems, which is taking place this week in Portland, Ore., Huang explained that student-to-student interactions and interpersonal relationships in the classroom comprise an important aspect of education that is often overlooked.

“This work has two parts: modeling and empirical investigation,” Huang continued. She recounted how she applied both approaches to explore the emergence, structure, and evolution of cooperation networks within her classroom. The dynamical systems modeling aspect was based on the *autocatalytic network*, which is a common framework in biology and chemistry. Each node in an autocatalytic network receives at least one positive link from another node. The model is based on two principles: (i) the law of mass action, in which the reaction rate is directly proportional to the product of the reactants’ activities, and (ii) natural selection based on performance, in which the least fit is eliminated.

**Figure 1.**A data example in which student \(C\) gives help to two people and receives help from three people. Figure courtesy of Meilin Huang.

Both the network structure and the individual performance in the autocatalytic model coevolve across short and long time scales. When Huang adapted this model for a classroom network application, she found that an initial seed of reciprocity led to an explosive growth in cooperation. After receiving some initial help, students adjusted both themselves and their behavior. “A major result of this modeling is that selection based on performance leads to the inevitable emergence of cooperation,” Huang said.

She then turned the talk to her empirical investigations of student networks within her own classes of nursing students at Saint Xavier University. She offered students a “helper’s credit” as an incentive to assist each other, and collected four surveys throughout the semester to ask the students who they helped and who they perceived to have helped them. In this context, “help” could include assisting with understanding the course materials, working together on homework or studying, or emotional support. Among the two sections of her course, Huang was able to collect 16 weighted directed network sets over one semester. Figure 1 provides a data example of the links of help given and received between students.

**Figure 2.**The amount of help that students in one section of the course reported giving to their peers. Figure courtesy of Meilin Huang.

To validate that students were not just randomly filling out the surveys, Huang plotted the amount of help that students claimed to have given others versus the help that student perceived getting. This check did indeed show that students were filling out the forms accurately. Overall, the survey results indicated that students both gave and received an increasing amount of help over the course of the semester (see Figure 2). Using students’ final grades as a proxy for their learning outcomes, individuals with higher grades also generally tended to give and receive more help.

**Figure 3.**The grades predicted by the autocatalytic model compared to the grades that students actually received. Figure courtesy of Meilin Huang.

There are naturally some limitations in the application of the autocatalytic network to classroom cooperation. Bias and noise can appear in the survey data and grades, and students might not accurately remember the help given or received or may perceive help in different ways. Furthermore, students might receive assistance from people outside of the class, such as a friend or partner who was already familiar with the course material. The model also makes various assumptions that are not accurate to classroom conditions.

“It’s a toy model, but I hope it can help make some more sense of teaching,” Huang said. Overall, she is still developing this system to further combine empirical investigation with modeling and hopes to look more into reciprocity in the future. But even at present, the model is able to predict how mutual help networks will emerge and grow, which could provide insights for improving teaching and learning.

Jillian Kunze is the associate editor of SIAM News. |