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 Correspondence should be addressed to Color versions of one or more of the figures in the article can be found online at www.tandfonline. com/hmet. QUANTITATIVE CHANGE IN COMPLEX SYSTEMS 205productive 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.

Development of Mechanistic Reasoning and Multilevel Explanations of Ecology in Third Grade Using Agent‐Based Models

“…The change of populations of different species and their interdependence have been identified as emergent patterns in ecosystems (Dickes & Sengupta, ; Wilensky & Reisman, ; Wilkerson‐Jerde & Wilensky, ). The study of population growth and decay involves reasoning about the behaviors that cause populations to fluctuate, such as birth, death, immigration, competition for resources, or population density (NRC, ; Sandholm, ; Wilkerson‐Jerde & Wilensky, ).…”

“…The change of populations of different species and their interdependence have been identified as emergent patterns in ecosystems (Dickes & Sengupta, ; Wilensky & Reisman, ; Wilkerson‐Jerde & Wilensky, ). The study of population growth and decay involves reasoning about the behaviors that cause populations to fluctuate, such as birth, death, immigration, competition for resources, or population density (NRC, ; Sandholm, ; Wilkerson‐Jerde & Wilensky, ). Typically, mathematical models of population growth are part of most high school curricula (NRC, ; Wilkerson‐Jerde & Wilensky, ); however, we believe that it is possible for younger children (e.g., third and fourth graders) to begin to reason productively about at least some aspects of population growth and decay.…”

“…The study of population growth and decay involves reasoning about the behaviors that cause populations to fluctuate, such as birth, death, immigration, competition for resources, or population density (NRC, ; Sandholm, ; Wilkerson‐Jerde & Wilensky, ). Typically, mathematical models of population growth are part of most high school curricula (NRC, ; Wilkerson‐Jerde & Wilensky, ); however, we believe that it is possible for younger children (e.g., third and fourth graders) to begin to reason productively about at least some aspects of population growth and decay. Our previous work has shown that learners as young as fourth grade can, when appropriately scaffolded, reason about factors that affect changes in populations (Dickes & Sengupta, ).…”

“…While the later approach can provide a deeper pathway into computational modeling (e.g., Sengupta et al., ), even in cases where students interact with predeveloped agent‐based models (e.g., Danish, ), they can still engage in deep investigations of the relevant aspects of complexity by engaging in multilevel , model‐based explorations of, and reasoning about the relevant phenomena. Students typically accomplish this by (a) designing and conducting simulation experiments by selecting and testing agent‐level and environmental variables and (b) interpreting aggregate‐level outcomes in terms of the relevant agent‐level interactions (Danish, ; Dickes & Sengupta, ; Goldstone & Wilensky, ; Levy & Wilensky, ; Wilkerson‐Jerde & Wilensky, ).…”

“…Along the dimension of analysis of student learning, we make the following contributions. First, while several frameworks for analyzing student reasoning of complexity have been proposed (Chi, ; Hmelo‐Silver & Pfeffer, ; Stroup & Wilensky, ; Wilensky & Reisman, ; Wilkerson‐Jerde & Wilensky, ; etc. ), we present a more nuanced analysis of the progressive refinement of children's explanations about ecosystems using the lens of mechanistic reasoning (Machamer, Darden, & Carver, ; Russ, Scherr, Hammer, & Mikeska, ).…”