Patterns, Probabilities, and People: Making Sense of Quantitative Change in Complex Systems

The learning sciences community has made significant progress in understanding how people think and learn about complex systems. But less is known about how people make sense of the quantitative patterns and mathematical formalisms often used to study these systems. In this article, we make a case for attending to and supporting connections between the behavior of complex systems, and the quantitative and mathematical descriptions. We introduce a framework to examine how students connect the behavioral and quantitative aspects of complex systems and use it to analyze interviews with 11 high school students as they interacted with an agent-based simulation that produces simple exponential-like population growth. Although the students were comfortable describing many connections between the simulation’s behavior and the quantitative patterns it generated, we found that they did not readily describe connections between individual behaviors and patterns of change. Case studies suggest that these missed connections led students who engaged in productive patterns of sense-making to nonetheless make errors interpreting quantitative patterns in the simulation. These difficulties could be resolved by drawing students’ attention to the graph of quantitative change featured in the simulation environment and the underlying rules that generated it. We discuss implications for the design of learning environments, for the study of quantitative reasoning about complex systems, and for the role of mathematical reasoning in complex systems fluency.