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

Abstract

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.

Notes

¹ Movement was not important mathematically. However, if agents did not move the simulation would “stack” them in the visualization so that the population did not appear to grow. Therefore, we introduced a rule to make them move so that students could use the visualization to observe changes in the number of agents over time.

² The initial set of codes included eight categories: visuospatial, quantity, graphical, behavioral—agent level, behavioral—aggregate level, functional, programmatic, and systemic.

³ Raw agreement is the total percent agreement between coders on the presence and absence of each code for each response: . Agreement on presence is agreement only on the presence of a code and so adjusts for inflation when only a few codes might be applied to each response: .

⁴ The formula for calculating the probability of repeated independent events is P(e₁ and e₂ … e_n) = P(e₁) × P(e₂) … P(e_n), where e_x represents an event. In this case the probability for each event, individual reproduction during each successive tick in the simulation, is the same at 1%. This reduces to .01 × .01 × .01 … n times, or (.01)ⁿ—(.01)⁵ if one were to seek the probability of five births happening in a row.

⁵ A mathematical derivative measures how much a function f(x) will change as x changes. Here Sarah was proposing to find the derivative of a function that describes total population growth over time, which would reveal specifically how much the population is changing at a particular time.

Additional information

Funding

This work was supported by the National Science Foundation under Grant No. DRL-1020101 (principal investigator: Wilensky). Any opinions, findings, and conclusions or recommendations are our own and do not necessarily reflect the views of the National Science Foundation.