important theoretical and methodological issues for the learning sciences and for science and mathematics education that are raised by what might be called the complex systems framework of conceptual perspectives and principles. We use the term “framework” as it does not appear that there is a general “theory of complex systems” at this time. Rather, the multidisciplinary fields that study various types of complex systems use a set of conceptual perspectives or principles (e.g., multi-scale hierarchical organization, emergent patterning, dynamical attractors, scale-free networks) and methods of doing science (e.g., computational modeling, network analysis) that function as a shared framework for the discourse and representations used in the conduct of scientific inquiry. As such, various fields can formulate specific theoretical perspectives of relevance to the study of particular complex systems of interest that still share common elements due to their grounding in the complex systems framework.
As Jacobson and Wilensky (2006) argue, complex systems perspectives provide new methods and insights for learning science research related to how students come to understand challenging ideas. As an example, let us consider how complex systems perspectives may enhance or extend theory and research in the learning sciences and science and mathematics education through the use of computational modeling of learning and education systems.
It has been argued that there has been a recent major shift in what constitutes legitimate sources of scientific information (Jackson, 1996). The origins of modern science are often credited to Aristotle and his use of careful observations to obtain information upon which to make informed decisions rather than the logical argumentation of philosophical beliefs. The next metamorphosis in the conduct of inquires we now regard as “science” occurred with the intellectual contributions of Brahe, Galileo, Newton, Kepler, Liebniz, and Euler who not only advanced the field of mathematics, but who also demonstrated how new scientific discoveries could be made through the use of information derived from mathematical manipulations of observational data. The remarkable scientific achievements of the ensuing 300 years were predicated on these two sources of scientific information. Indeed, observational and mathematically derived information have been the norm in virtually all of the published research in the learning and cognitive sciences and in education to date.
However, Jackson (1996) has proposed that we are in the midst of a second historical metamorphosis in the conduct of science, one that involves the use of computational tools to generate a third legitimate source of scientific information. In addition, others, such as Pagels (1988), have observed how the use of computational tools in science allows dramatically enhanced capabilities to investigate complex and dynamical systems that otherwise could not be systematically investigated by scientists. These computational modeling approaches include cellular automata, network and agent-based modeling, neural networks, genetic algorithms, Monte Carlo simulations, and so on that are generally used in conjunction with scientific visualization techniques. Examples of complex systems that have been investigated with advanced computational modeling techniques include climate change (West & Dowlatabadi, 1999), urban transportation models (Balmer, Nagel, & Raney, 2004; Helbing & Nagel, 2004;
Noth, Borning, & Waddell, 2000), and economics (Anderson et al., 1988; Arthur, Durlauf, &
Lane, 1997; Axelrod, 1997; Epstein & Axtell, 1996). New communities of scientific practice have also emerged in which computational modeling techniques, in particular agent-based models and genetic algorithms, are being used to create synthetic worlds such as artificial life (Langton, 1989, 1995) and artificial societies (Epstein & Axtell, 1996) that allow tremendous flexibility to explore theoretical and research questions in the physical, biological, and social sciences that would be difficult or impossible in “real” or non-synthetic settings.
The typical approach used by researchers involved with computational science tools such as agent-based modeling is to articulate a model of the system of interest in terms of hypothesized rules that define the interactions between agents and between agents and their environment. In
scientific computational modeling work, as opposed to explorations of modeling by mathematicians, there generally is an existing body of observational and mathematical information about the system that allows (a) an initial specification of the parameters for the model and (b) a validity check of the articulated model with the real world data, generally with iterative revisions to the model in terms of the parameters or rules the agents in a model follow in their interactions in the synthetic world. Once the researcher has demonstrated a valid model for a particular system compared to available data, it is then possible to run “computational experiments” in which what-if scenarios about the behavior of the system may be explored to understand the system under different conditions than the observed data and to perhaps envision different possible futures for how the system might behave over time. It is important to understand, however, that nearly all examples of complex systems have important random or chaotic (i.e., sensitivity to initial conditions) factors that mean there is a high probability each run of the model may be different, sometimes in small ways but perhaps in dramatically large and chaotic ways (i.e., the “butterfly effect”).
Given the development of sophisticated computational modeling tools and their increasing acceptance in a wide range of scientific fields in the physical and social sciences, we argue that there is great potential to accept computationally generated information as part of research in the learning and cognitive sciences that explores complex learning, socio-cognitive, and educational systems. We believe that such work has enormous potential in four broad ways. First, the articulation of models, particularly those that are “bottom-up” such as agent-based models, often helps researchers distill their qualitative intuitions about critical factors that might be most responsible for the behavior of the system of interest. This “analytical catalyst” function of computational model building is often quite valuable when confronting systems of multi-dimensional and multi-level complexity. Second, complex systems models then become scientifically inspectable artifacts that, as mentioned above, may be compared to real world data and iteratively revised to improve the fit of the model. Third, models validated with one or more datasets may be used to explore the behavior of the system by varying model parameters (ideally with multiple runs involving all parameter combinations to investigate stochastic properties of the system). Fourth, such models may function as a tool to help generalize the findings from the observed and modeled system(s) to similar types of systems that probably have different specific local features.
In research into science and mathematics education and the learning sciences to date, there have been few examples of computational modeling along the lines discussed in the previous paragraph. For example, Lemke and Sabelli (2004) have proposed building “SimSchool” or
“SimDistrict” simulation programs that would not just model existing school or school district systems, but also could be used to create synthetic schools and district systems and to study their evolution over time in terms of needs, problems, and probable outcomes. Recently, actual systems have been developed along these lines. For example, researchers have done agent-based
simulations in areas of educational policy such as school choice where parents and school officials are agents in the simulation (Lauen, 2004; Maroulis & Wilensky, 2005, 2005).
Researchers are also using network analysis methods to study topics ranging from how social structure impacts technology adoption in schools (Frank, Zhao, & Borman, 2004) to the role of social structure on student achievement (Maroulis, Griesdorn, & Gomez, 2005). Overall, there would seem to be great potential for complex systems and computational modeling techniques to enhance science and mathematics educational and learning sciences research involving micro and macro levels of cognitive, learning, and educational systems, such as the evolution of cognitive representational networks, design experiments of technology interventions in classrooms, and social network analysis of collaborative interactions patterns.
Conclusion
The teaching and learning of science and mathematics faces two fundamental challenges that conflict with each other. First, the fragmented and superficial coverage of too many subjects is widely criticized for contributing to poor student learning in science. Second, there has typically been a decades long gap between the generation of new scientific knowledge and its integration into college and pre-college curricula. In this paper, it is argued that one way to address these challenges is to infuse knowledge from emerging multi-disciplinary scientific research, in particular work related to the study of complex systems, into K-16 curricula in the physical and social sciences. It is also argued in this paper that there is considerable potential for complex systems conceptual perspectives and methodological tools, such as agent-based modeling, to enhance research in science and mathematics education and in the field of the learning sciences. The overall goal, of course, is for students and citizens of the new century to understand many of these exciting new ideas and perspectives about how the world works, or, in the words of Nobel Laureate Herbert Simon (1996), “to make the wonderful commonplace: to show that complexity, correctly viewed, is only a mask for simplicity; to find pattern hidden in apparent chaos.”
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