To seriously consider implementing long term and conceptually deep changes in the science and mathematics curricula is an exciting prospect. Properly infused into the curriculum, the cross-disciplinary concepts and methodologies emerging from complex systems research have the potential to form the basis of a new and principled scientific literacy for our student to learn, one that is powerful and appropriate for dealing with the problems and demands of the 21st century. However, in order for this vision to be realized, it will be necessary to seriously consider the learning and pedagogical challenges to be faced. In this section, we provide an overview of these challenges by focusing on six main areas:
These six areas are considered in turn.
There is reason to believe that many of the core ideas associated with new ways of thinking about complexity may be challenging for students to learn. Considerable research has documented a variety of difficulties students have with learning concepts relevant to understanding complex systems that are currently taught in existing science courses. For example, many students even at the college level believe that chemical reactions stop at equilibrium (Kozma, Russell, Johnston, & Dershimer, 1990) or that evolution is the result of trait use or disuse and that acquired traits are passed down from one generation to the next (i.e., Lamarckian view) (Bishop & Anderson, 1990; Samarapungayan & Wiers, 1997). In addition, it has been suggested that important concepts related to complex systems may be counter-intuitive or conflict with commonly held beliefs (Casti, 1994). Many people believe that there is a linear relationship between the size of an action and its corresponding effect: a small action has a small effect, while a large action has a correspondingly large effect. However, it is now commonly understood that in complex and dynamical systems, a small action may have interactions in the system that contribute to a significant and large-scale influence—the so-called "butterfly effect." Other researchers have proposed that people tend to favor reductive explanations that assume central control and deterministic single causality and that there are deep seated resistances towards ideas describing various phenomena in terms of self-organization, stochastic, and decentralized processes (Feltovich, Spiro, & Coulson, 1989; Resnick, 1994; Wilensky & Resnick, 1999). Consistent with these perspectives, recent research suggests that there may be a different "way of thinking" employed by individuals with an advanced understanding of complex systems (i.e., scientists working in this field) and novices (undergraduate university students) when solving problems dealing with familiar or "everyday" examples of complex systems phenomena (e.g., how do ants forage for food, can a butterfly in Brazil influence the weather in Alaska, or how to design a city such that goods and services were maximized) (Jacobson, 1999). For example, one novice solution proposed a city with centralized housing and food distribution; while a complex systems scientist described a solution that modeled the decentralized interactions of people in a city. Overall, the university students tended to solve problems using statements that were reductive, assumed central control, described a single source of causality, were predictable, and focused on objects, while the complex systems experts tended to solve the problems with statements that considered the overall system, described de-centralized control and multiple causal factors, noted probabilistic nature of solutions, and were process oriented.
Overall, the literature to date on the cognitive and learning challenges associated with complex systems is limited, and there is certainly a need for further research in these areas involving a wider ranger of students and grade levels. However, it is a reasonable assumption that the scientific knowledge about complex systems will be at least as difficult to learn as many science concepts in the existing curriculum. In addition, preliminary research suggests that there may be additional learning challenges imposed by counter-intuitive epistemological and ontological components of this knowledge. A critical issue, then, will be to identify suitable approaches for learning and teaching complex systems knowledge that address these challenges.
Over the past two decades, there has been a significant increase in our understanding of the developmental, cognitive, and social dimensions of learning. Older "instructivist" perspectives on learning tend to regard knowledge as a substance in the mind of individuals that is independent of context, to view learning as an activity for the individual, and to reductively structure learning in terms of the gradual accumulation of pieces of information. Unfortunately, it is not often obvious to the learner why a particular "piece" of information or fact is important to learn, what the purpose or meaning of isolated facts and information might be, or how the information might be applied. In addition, the inherent decontextualized nature of many traditional instructivist approaches has a negative impact on student motivation (Brown, Collins, & Duguid, 1989; Collins, 1996), and often results in knowledge that is "inert" and of little use to the student after the test (Bereiter & Scardamalia, 1985; Bransford, Franks, Vye, & Sherwood, 1989; Bruer, 1993; Cognition and Technology Group at Vanderbilt, 1997; Gick & Holyoak, 1983; Gick & Holyoak, 1987; Renkl, Mandl, & Gruber, 1996; Spiro, Vispoel, Schmitz, Samarapungavan, & Boerger, 1987; Voss, 1987; Whitehead, 1929).
In contrast, recent socio-cognitive or "constructivist" perspectives regard knowledge as an emerging characteristic of activities taking place among individuals in specific contexts, to view learning as a developmental process occurring first in an interpersonal domain (i.e., socio-cognitive or between people) and later in an intrapersonal domain (i.e., cognitively or within an individual), and to recognize that learning is a constructive activity that often requires active and substantial reorganization of existing conceptual structures. An increasing amount of research has been documenting how new constructivist models may be used to reconceptualize curricula, teaching practices, and learning activities, and to effect significant and rich types of learning gains (Bruer, 1993; Cognition and Technology Group at Vanderbilt, 1997). Many new constructivist models of learning utilize the affordances of new computational and communications technologies as part of learning environments in which students engage in challenging problem and project centered learning activities. Although more research into refining and extending these constructivist approaches in applied contexts of education is needed, there is beginning to emerge in the educational research and practice communities a sense that the refinement and systematic application of these new perspectives on learning may significantly enhance the educational process in the United States related to science and mathematics education (President's Panel on Educational Technology, 1997). For example, the recent report on learning commissioned by the National Research Council (Bransford, Brown, & Cocking, 1999) notes that:
Overall, the new science of learning is beginning to provide knowledge to improve significantly people's abilities to become active learners who seek to understand complex subject matter and are better prepared to transfer what they have learned to new problems and settings. Making this happen is a major challenge…, but it is not impossible. The emerging science of learning underscores the importance of rethinking what is taught, how it is taught, and how learning is assessed (p. 13).
It is the recommendation of Working Group 2 that learning and teaching approaches intended to help students learn about complex systems knowledge be informed by recent theory and research in the learning sciences.
In this section, we describe a set of general design principles for creating learning environments and tools for learning to help students understand scientific perspectives on complex systems. These design principles are informed by recent constructivist models of learning and by a consideration of the successes and challenges identified in recent education and complex systems projects.
3.1 Connecting With Learner's Passions, Interests, and Experiences
Students will need to understand that the concepts and methods related to complex systems research are powerful intellectual tools that are useful for understanding a wide range of phenomena. It will be important to develop activities for learning about complex systems that have immediate relevance to kids, such as how their body composition changes with diet and exercise, garbage generation and recycling, traffic jams, elections, and so on. Also, students often develop passions and interests when they are involved with authentic or "real world" activities. For example, students might use their knowledge of complex systems and multi-agent modeling tools for community related problems dealing with water quality and pollution, or as part of collaborations with actual scientific projects such as Project GLOBE (Means & Coleman, 2000).
3.2 Experiencing Complex Systems Phenomena
Students need opportunities to directly experience and to conduct systemic observations and experiments of complex systems phenomena. These phenomena might be commonly experienced in daily life, such as ants foraging for food or birds flocking, or they might exist along temporal or physical dimensions that challenge human sensory and cognitive capabilities, such as the big bang or quantum mechanics. Of course,"non-everyday" or impossible to directly experience phenomena have always represented special challenges for teachers to teach and for students to learn. But even"everyday" phenomena may have important characteristics that are not directly observable, such as the pheromone scent marker generated by ants when food is found. In the past, curricular materials intended to cover scientific phenomena relied almost exclusively on textual, linguistic, and pictorial representations to convey scientific perspectives on everyday and non-everyday phenomena, supplemented by experiments utilizing scientific instrumentation and techniques. However, new scientific visualizations and computational modeling and simulation tools are allowing scientists themselves to qualitatively experience representations of the"non-everyday" phenomena and thus to iteratively explore questions and hypotheses in the virtual representations and models. With the increasing power and decreasing cost of computational systems necessary for visualizations and computational modeling, it is now becoming possible for these tools to be adapted for use by students and teachers. In this way, students may now have"direct virtual" experiences of complex systems phenomena that are qualitatively similar to those of scientists. In addition, tools for providing quantitative representations and analyses of complex systems phenomena are also becoming available, and they may be either directly linked to computational models or import data generated by computational (not to mention"real world") experiments. Overall, there will be an increasing palette of opportunities for students to experience qualitative and quantitative dimensions of complex systems phenomena that span micro to macro scales of existence.
3.3 Make Core Concepts Explicit
A third principle for designing learning environments and tools dealing with complex systems is to make the core concepts explicit to the student. For example, even very young children have been seen ants moving about, carrying pieces of food, and just generally "milling" around anthills. Yet despite this rather detailed real world observational experience of ants, a case may be made that few young children—let alone older children or even adults—have developed an understanding of important core complex systems concepts such as random movements by ants in the environment, positive feedbackrelated to the generation of pheromone when food is found, self-organization as a characteristic of the ant colony, and so on. Thus it will be important that complex systems concepts relevant to various phenomena be made salient and explicit to the learner. Experts, in contrast to novices or intermediate learners, are able to "cognitively see" the relationship between the surface features of phenomena and problems, and underlying conceptual frameworks. One may regard techniques and tools that empower novice learners to become cognizant of the conceptual structures related to complex systems perspectives as a type of cognitive scaffolding that will help students as they construct more advanced conceptual understandings. There will be many opportunities to make concepts explicit as part of complex systems learning activities that highlight the unusual in the ordinary, such as the positive feedback of ants foraging for food, the emergence of traffic jams that move backwards although cars move forward, or that "mindless" bacterium may evolve to be resistant to advanced antibiotic medicine.
3.4 Encouraging Collaboration, Discussion, and Reflection
As noted above, contemporary views of learning acknowledge important ways that knowledge and beliefs about the world are shaped and constructed in situated and socially mediated contexts. Learning environments in which students come to experience and to construct their understandings about complex systems may be made significantly more powerful—not to mention more interesting, engaging, and motivating—by involving students with authentically interesting problems and projects which involve collaborative and cooperative interactions. These discussions may be peer-to-peer or peer-to-expert (where the notion of "expert" is an individual with more competence and experience in a domain), and may range from"face to face" to asynchronous computer-mediated comminations. Through collaborative interactions and the construction of shared artifacts, naturally occurring opportunities arise for students to articulate or reify their ideas and to reflect on the possible limitations to their ideas and theories. It will also be important as part of collaborations and discussions for students to have metacognitive scaffolds and to have complex systems specific questions to consider such as: "What underlying mechanisms might give rise to the observed behavior?" "How sensitive is the outcome to changes in the model's parameters or structure or assumed environment?"
3.5 Constructing Theories, Models, and Experiments
A central tenet of constructivist learning approaches is that a learner is actively constructing new understandings, rather than passively receiving and absorbing "facts." One way to implement this perspective as part of educational activities related to complex systems is to involve students with problems and issues dealing with such systems, to have them generate questions, theories, and hypotheses, and then to either run observational experiments and/or to create computational models related to their theories. In particular, there would seem to be great potential for students to construct and revise computational models with multi-agent or qualitative modeling software, and then (when possible and appropriate) to conduct real world experiments (Jackson, Krajcik, & Soloway, in press; Resnick, 1994; Wilensky, 1996). For example, a team of students might construct a computational model of a local ecosystem such as a lake, collect measurement data about the lake, compare their real world data with the initial model, and then revise their theories and models as necessary. The combination of model building and observational experiments would help minimize the danger that some students might view complex systems computational models as being"just" related to the computers—like computer games. Also, the explicit linking of model building (both conceptual models and computer models) and scientific experimentation should help students come to understand that modern scientific inquiry is fundamentally grounded on cycle of theorizing, model building, and experimentation, that in turn leads to theory and model revisions.
3.6 Learning as Trajectories of Deepening Understandings and Explorations
In this final principle for the design of learning tools and learning environments for complex systems, it must be stressed that the goal of learning is not about just "covering" various complex systems concepts. Rather, we believe that integrating complex systems knowledge and methodologies throughout the pre-college and college curriculum will foster trajectories of learning for our students that will lead to conceptual growth and deepening understandings over time, grade levels, and topics. Complex systems concepts learned in one class (e.g., core concepts such as multiple agents, feedback, self-organization, emergence, and so on) should form a conceptual toolkit that students will be able to use and to enhance in subsequent classes. Over time, students should begin to realize that complex systems knowledge applies in many science and even the social science disciplines, and that they may use their complex systems conceptual toolkit for both intra-disciplinary and inter-disciplinary knowledge explorations.
Many current science curricula have been criticized for superficially covering too many subjects, with the consequence that students typically fail to achieve a solid understanding of even a single domain (National Research Council, 1996). Consequently, it is vital that materials for complex systems not be developed that are just an "add-on" to an already bloated and over stretched science and mathematics curriculum. But how might this be done?
We believe there are ways that complex systems perspectives might contribute to a solution to this problem. The study of complex systems is starting to break down barriers that have existed between not only the "hard sciences" of physics, chemistry and biology, but also between these disciplines and the so-called "soft sciences" of psychology, sociology, economics, and anthropology (Bar-Yam, 1997). Thus concepts related to complex systems may function as curricular unifying cross-disciplinary themes that provide a conceptual grounding for students as they study different subjects in which different types of complex systems would be focussed on. For example, complex systems concepts such self-organization and positive feedback may be seen to apply in biological systems such as social insect colonies and in social science systems such as economics and income distribution patterns. Over time, with appropriate pedagogies, curricular materials, and learning tools, students should come to realize that in studying, for example, chemistry, they are looking at complex systems at a particular hierarchical level of nature, and that, in turn, contributes to emergent characteristics which support biological complex systems such as cell metabolic functioning or ecosystem niches of social insects at different hierarchical levels. A complex systems based curricula could function as a conceptual framework that would allow both for depth of coverage related to specific science and social science subjects and for cross-disciplinary conceptual "hooks" that could help students apply or transfer their knowledge to new situations and problems. Also, as discussed in section 3.3 Learning About Complex Systems: Design Principles for Learning Environments and Learning Tools, complex systems phenomena are well-suited to problem- and project-centered learning approaches that implement constructivist models of learning and teaching. Properly implemented, a complex systems centered curriculum would be focussed, conceptually principled, and scientifically current. In addition, such a curriculum could help address the unfortunate situation whereby many students come to view science as rote memorization of isolated and decontextualized facts for which they often see little use in their daily lives. Overall, it may be possible to exploit complex systems approaches in learning and instruction in order to help solve certain difficult curricular problems by making cross-disciplinary connections easier for students to appreciate.
Although the issue of assessment is also dealt with as part of Working Group 3's section of the report, it should be stressed here that the curricular and pedagogical changes recommended above would require different types of assessment approaches than are commonly used today. Not only would the "content" of traditional large scale paper and pencil tests need to be changed, but there is reason to believe that traditional summative assessment measures would not be likely to adequately measure the types of skills and knowledge students would be cultivating through constructivist learning activities dealing with complex systems (Frederiksen & Collins, 1989). Rather than just providing summative assessment after a period of instruction, it will be important for teachers to use formative assessment techniques continuously as a part of instruction in order to provide students with important feedback as to the degree of their understanding (Bransford et al., 1999). For example, assessment approaches such as student portfolios of projects dealing with complex systems or students constructing their own multi-agent models as part of inquiry oriented activities, would provide many opportunities for students to obtain feedback on their ideas, and for the students to evolve and refine their ideas over the course of their projects. Using assessment techniques such as these allows a natural shift away from excessive emphasis on memorization of facts and procedures towards pedagogical approaches that stress understanding and application of knowledge to problems.
In closing this section, we observe that there are a number of questions that need to be addressed related to learning, teaching, and complex systems, such as:
Answering some of these questions will require further research, while other questions relate to more general issues that are the focus of Working Group 3's efforts. These latter questions must be considered as part of discussions with various constituencies involved with the overall complex system of education in the United States.
In closing this section of the Report, we note that there is a vision of the importance of complex systems concepts and approaches as a means to help foster what might be a new and principled type of scientific literacy that will help students and adults both understand and to use important emerging scientific knowledge to address issues and problems of the 21st century. The cross-disciplinary themes and new ways of doing science related to complexity and complex systems provide an opportunity to present important dimensions of the increasing quantity of scientific knowledge in an interconnected and coherent manner that is both cognitive manageable, and perhaps more important, personally useful. Obviously, there are certain students for whom the sheer fascination of understanding new things will motivate them to learn about complex systems. However, for most students (and adults), there must be practical ways that what they learn may be applied to problems and issues they face in their daily lives. The applicability of complex systems concepts such as self-organization and selection, and methodologies such as multi-agent modeling, to a wide range of natural and social phenomena from gas laws to traffic jams offers a rich palette for educators to both reach students and to help them learn important scientific knowledge and skills.
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 Additional rationales for why students should learn perspectives and knowledge related to complex and dynamical systems are articulated in other areas of the report.
 Hence there will be critical need for longitudinal studies.
Video of Yaneer Bar-Yam's seminar on Complex Systems Principles and Education: Focusing on Universal Principles and Individual Differences
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