Education has long been driven by its metaphors for teaching and learning. These metaphors have influenced both educational research and educational practice (Leary, 1990). Since the late 1800s, three metaphors have dominated education, learning as the acquisition of stimulus-response pairs (behaviorism), learning as the processing of information (information processing), and learning as the construction of knowledge (constructivism) (Mayer, 1996). Currently, there is an opportunity within education to examine the essentials of a new metaphor, learning as self-organized adaptation (complexity theory), without the need for a comprehensive change in functional perspective and practice.

These changes in explanatory metaphors have resulted from, and have allowed for, new insights concerning the nature of learning and knowledge. As researchers began to see that complex learning was difficult, if not impossible, to explain using complicated chains of S-R pairs, and as the computer began to enter the academic consciousness, information processing theory emerged to explain how mental structures affect behavior. Then, after several years of productive research into the components of memory and cognition, it became apparent that context and culture influenced the representation of these components, and constructivism emerged to explain personal meaning. However, as these new metaphors have emerged, one perspective has remained constant, the idea that learning involves parts and wholes. For behaviorism and information processing, the part-whole relationship is such that the whole can be predicted from the understanding of its parts. For the behaviorists, the component parts are S-R pairs, while for information processing theorists the component parts are memory structures. For constructivists, the component parts are experiences; however, constructivists recognize that the global behavior of an individual is not directly predictable through the understanding of an individual's experiences.

In complexity theory, as in behaviorism, information processing, and constructivism, component parts are important. However, in complexity theory, what constitutes a "part," or an agent, depends on the level from which one views the learning process. An agent could be a neuron, a neuronal group, an experience, or even a whole person (in a social situation). What is of importance, regarding agents, is not the agents themselves, but rather, the interaction of these agents with each other (Holland, 1995). As with constructivism, a complexity perspective recognizes the difficulty in predicting global behavior from an understanding of the parts (Waldrop, 1992). This complexity-based alternative perspective to understanding the whole, by understanding its parts, is to understand the whole by understanding the interaction of its parts (Lewin, 1992).

The purpose of this paper is (a) to introduce educators to the essential elements of a complexity-based view of learning, (b) to demonstrate that the current emphasis in education on constructivism is compatible with a new perspective on education using complexity theory, and (c) to discuss how complexity theory may expand our view of the learning process. These aims will be addressed through the presentation of a generic constructivist model, a generic complexity model, the development of a hybrid complex constructivist model, and a discussion of the usefulness of a complex constructivist view of learning.

Constructivism, succinctly defined, is the belief that learners construct their own knowledge from their experiences. A more eloquent and inclusive definition is provided by Fosnot (1996),

Learning from this perspective is viewed as a self-regulatory process of struggling with the conflict between existing personal models of the world and discrepant new insights, constructing new representations and models of reality as a human meaning-making venture with culturally developed tools and symbols, and further negotiating such meaning through cooperative social activity, discourse, and debate. (p. ix)

Therefore, constructivism involves the active creation and modification of thoughts, ideas, and understandings as the result of experiences that occur within a socio-cultural context. It is this combination of learner autonomy and holistic perspective that has thrust constructivism to the forefront of learning science and education. Learner autonomy is the concept that learners are active participants in the learning process and ultimately responsible for their own learning. The holistic perspective is a non-reductionist approach that emphasizes learning in context.

The integration of learner autonomy and holistic perspective places constructivism at the nexus of psychology and philosophy. A foundational issue in this psychological and philosophical morass is the role of epistemology; that is, what is the nature of knowledge and how does the knower come to know (Ernst, 1995). From this perch, Lerman (1989), Kilpartrick (1987), von Glaserfeld (1984), and Wheatley (1991) all site the pillars of constructivist epistemology as:

1. Knowledge is not passively accumulated, but rather, is the result of active cognizing by the individual.
2. Cognition is an adaptive process that organizes and makes sense of one’s experiences. It is not a process to render an accurate representation of ontological reality.

These pillars, while illuminating, allow for great variability in what is typically called “constructivism" (Phillips, 1995; Prawat, 1996). Moshman (1982) helped to define this variability through a continuum of constructivism. Moshman defined the poles of this continuum as Exogenous Constructivism and Endogenous Constructivism, with Dialectical Constructivism in a center or moderate position. Moshman's constructivist continuum provides a rationale for the placement of other types of constructivism (see Table 1).

Exogenous constructivism emphasizes the external nature of knowledge (see Figure 1). Knowledge is seen as the internalization and reconstruction of external reality. Learning or knowledge acquisition is the process of building accurate internal models or representations of external structures in the “real” world. This view presupposes that “reality” is knowable. Exogenous constructivism is most associated with information processing and its component processes, including schemata, declarative and procedural knowledge, and propositional-networks (Derry, 1996; see Gagne, Yekovich, & Yekovich, 1993).

In Figure 1, the dark rectangle on the left represents some aspect of knowable reality that is to be constructed by the student. This knowledge is subdivided into discrete sub-skills by the teacher who then transmits this knowledge to the student. A successful teaching/learning event results when the student, after this transmission experience, has constructed an accurate representation (the dark rectangle on the right) of the original, knowable knowledge.

Exogenous constructivism represents one extreme of the constructivist continuum, endogenous constructivism represents the other extreme. Endogenous constructivism emphasizes the internal nature of knowledge (see Figure 2). Knowledge is constructed, not from external experiences, but from earlier mental structures. Learning or knowledge acquisition is the reconstruction and reorganization of old knowledge structures in light of new experiences. Thus, knowledge is not an accurate representation of external reality, but rather is an internally coherent and coordinated collection of processes and structures that provide for adaptive behavior. This view presupposes that “reality” is not knowable (von Glaserfeld, 1995). Endogenous constructivism is most associated with Piaget and his theory of cognitive development.

According to Piaget (1973), cognitive development is the result of invariant changes in internal mental structures, characterized by a continuum of qualitatively different reasoning skills, and caused by integrating and extending previous levels of cognitive development into new knowledge/cognitive levels. Piaget emphasized the role of discovery and exploration as activities or experiences that fostered these changes in mental structure.

In Figure 2, new knowledge structures are constructed from old or prior knowledge structures in a cyclical relationship. Old knowledge structures are reorganized through adaptive changes into new knowledge structures. These new knowledge structures are influenced by one’s experiences, but are not the direct result of these experiences and are not an accurate representation of these experiences. It should be noted, that the dark rectangle that represents external reality, that was present in Figure 1, is not included in Figure 2, as external reality is not knowable in endogenous constructivism.

Finally, dialectical constructivism lies somewhere between the transmission of knowable reality of the exogenous constructivists, and the discovery of personal/relativistic reality of the endogenous constructivists. Dialectical constructivism emphasizes the interactional nature of knowledge (see Figure 3). Knowledge is the result of the interaction between the learner (internal) and the environment (external). Learning or knowledge acquisition is the process of building internal models or representations of external structures as filtered through and influenced by one’s beliefs, culture, prior experiences, and language, based on interactions with others, direct instruction, and modeling. This view presupposes that “reality” is knowable only in light of one’s context (past and present). Dialectical constructivism is most associated with Vygotsky and his socio-cultural perspective.

According to Vygotsky (1978), cognitive development is based on a student’s ability to learn socially relevant tools (e.g., hammers, pencils, computers) and culturally based signs (e.g., language, writing, number systems) through interactions with other students and adults who socialize them into their culture. These culturally mediated activities provide social experiences that are internalized and which later become a part of the individual’s mental functioning. Thus, knowledge is the result of social experience, influenced by one’s socio-cultural history, and resulting in a modified representation of reality.

In Figure 3, students interact with knowledge (dark rectangle) within a socio-cultural environment. This external social experience results in the formation of internal mental structures (models) that are influenced by the presence of social, cultural, contextual, and activity-based factors. The student does not acquire an exact representation of this knowledge (light rectangle), but rather, a personal interpretation of the external knowledge. The accuracy of this newly constructed knowledge will be based on the student’s prior knowledge and the impact of the social, cultural, contextual, and activity-based factors.

These three types of constructivism, exogenous, endogenous, and dialectical, provide a continuum from which to view learning and cognition. Of the three perspectives, dialectical constructivism is the most general and inclusive, and exogenous and endogenous constructivism have been viewed as special classes of dialectical constructivism (Moshman, 1982; Pressley, Harris, & Marks, 1992). Given the inclusive nature of dialectical constructivism, the dialectical constructivist model is used throughout the remainder of this paper as a model that is representative of constructivism, realizing it's limitations as a generalized constructivist model.

Constructivism is a broad theory that lends itself to many interpretations. Under the guise of constructivism lay many theories of learning, including situated cognition, anchored instruction, cooperative learning, inquiry and problem-based learning, generative learning, exploratory learning, reciprocal teaching, cognitive apprenticeships, and information processing. Yet, from these constructivist theories and the constructivist models (Figures 1-3), the following principles of learning may be derived:

1. Knowledge is actively constructed and self-organized by the individual (Ernst, 1995; Fosnot, 1996; Prawat, 1996).

This single statement is perhaps the essential element of any constructivist perspective. According to Prawat and Foden (1994), "Constructivist learning theory is based on the now common place idea that knowledge is actively constructed by the learner" (p. 37). However, this consensus concerning knowledge construction is not as harmonious as it may seem. While there is agreement that knowledge is personally constructed, the nature of what is constructed and how these constructions are represented is far from agreed upon (Marshall, 1996; Prawat & Floden, 1994). These differences in what and how are, to a large extent, the reasons for the variability in constructivist theory. Another agreed upon, though not well defined, element of constructivism is the idea that these knowledge constructions are self-organized. It is individuals, through the history of their experiences, which provide the organization of their own knowledge. This self-organization is linked to the individual's adaptation, based on the individual's prior knowledge, current need, and the socio-cultural activities in which they are engaged. Lemke (1997), as quoted in Cobb and Yackel (1996), states "Learning can therefore be characterized as 'an aspect of self organization, not just of the human organism as a biological system, but of ecosocial systems in which the organism functions as a human being' " (p. 180).

2. Learning involves adaptation to the environment (i.e., social, cultural, and physical), based on an individual's experiences (Chinn & Brewer, 1993; DiSessa, 1993; Piaget, 1970).

Learning serves an adaptive purpose (von Glaserfeld, 1996). As an individual interacts with other people, objects, and organisms, his or her knowledge is continually evaluated. This evaluation may result in the construction of new knowledge and/or the validation or reconstruction of prior knowledge, in such a way, as to enhance the efficacy of one's knowledge and behavior. This cyclical process of experience-evaluation-construction/validation allows an individual to make subtle (and sometimes not so subtle) changes in knowledge that yields more effective behavior. These adaptive changes in knowledge serve to improve one's environmental/behavioral effectiveness and efficacy, based not on some external criterion reference, but rather, based on a personal reference that is related to a person's current view and understanding of the world. It is in this way that an individual may construct knowledge that is adaptive today, yet becomes maladaptive tomorrow (e.g., bad habits, phobias, neuroses). "To the constructivist, concepts, models, theories and so on are viable if they prove adequate in the contexts in which they were created " (von Glaserfeld, 1995, p. 7).

3. Prior knowledge is self-organized into schemas (Derry, 1996; Steffe & Gale, 1995), psychological tools (Newman & Holzman, 1993; Vygotsky, 1978), and internal mental models (Fosnot, 1996).

The self-organization of one's knowledge involves the construction of schemas, psychological tools, and mental models. Schemas, or schemata, are cognitive structures that represent generalizations or abstractions of experience (Bartlett, 1932; Rumelhart & Norman, 1985). Schema induction, abstracting the features that define a category from exposure to examples, and the resultant schemas, are inherently relativistic, that is, while two individuals may share similar experiences their self-constructed schemas will vary due to differing histories and prior knowledge. While the focus of schema induction tends to be the individual, the focus of psychological tools tends to be social interaction and social mediation. Psychological tools are culturally relevant signs or representations (e.g., words, images, labels) that are learned through social interaction and that allow an individual to think and behave in culturally relevant ways (Dixon-Kraus, 1996; Vygotsky, 1960; Wertsch, 1985). According to Vygotsky (1960), psychological tools, especially language, are first experienced when the individual is engaged in a culturally meaningful activity. Eventually, however, these external psychological tools turn inward to form mental structures that control one's mental processing (Wertsch, 1985). Finally, in addition to developing schemas and psychological tools, an individual constructs mental models that integrate schemas and psychological tools into a unified and coherent whole that represents one's understanding of the world. However, mental models are more than just self-organized schemas and psychological tools; mental models also include knowledge related to problem-solving, schematic and psychological tool viability, personal theories, and strategies (Derry, 1996). According to Marshall (1996), "...through continually testing the viability of these understandings in interactions with others and the world (cf. Von Glasersfeld, 1995), these understandings become more complex, coherent, and powerful (Marshall, 1995)" (p. 30).

4. Knowledge is a function of both the interaction of individuals, and the individual's prior knowledge (Cobb, 1989; Yackel, 1995; Cobb & Yackel, 1996).

Knowledge is the result of socio-culturally mediated meanings, community negotiated meanings, and personally constructed interpretations (Cobb & Yackel, 1996; Goodman, 1986). The society and local community within which an individual is living, experiencing, and learning, affects the knowledge that is learned. In addition, the prior knowledge, prior experiences, and history of an individual also affect what is learned. Cobb and Yackel (1996), however, emphasize that neither socio-cultural activity nor an individual's prior knowledge takes precedence over the other. Socio-cultural activity and an individual's prior knowledge are intimately intertwined in such a way that learning becomes the self-organization of an individual's prior knowledge as self-organized within the activities of the larger culture. That is, the individual's participation in a socio-cultural activity is self-organized according to cultural and individual norms, and this self-organized activity reflects back to influence the individual's interpretation of the socio-cultural activity, which subsequently influences the individual's participation in the socio-cultural activity. Thus, an individual's learning is the result of both socio-cultural mediation and personal construction. As stated by Anderson, Blumenfeld, Pintrich, Clark, Marx, and Person (1995), "...there is an interaction between the construction of meaning by individual learners and the situations in which the learning occurs" (p. 145).

These four principles encompass the essence of constructivism, that is, learning as the adaptive and self-organized construction of knowledge that is a function of both one's prior knowledge and experience, and one's current socio-cultural activity. This perspective on learning reflects the complexity of learning as involving adaptation, self-organization, interaction, and history.

The theory of complexity is not a theory of learning, memory, and cognition, per se; complexity is a broad-based theory concerning the evolution and functioning of non-linear systems that may be applied in many domains (e.g., evolution, immunology, economics), including learning, memory, and cognition (Coveney & Highfield, 1995; Morowitz & Singer, 1995). As a relatively new perspective on the world, complexity theory has not yet formally defined its own theoretical boundaries. This lack of definiteness makes defining complexity problematic, however the CTCS (Jacobson, 1997) has provided an excellent point of departure.

Complex systems or complex adaptive systems involve phenomena which may be characterized by the interactions of numerous individual agents or elements, that self-organize at a higher systems level, and then in turn show emergent and adaptive properties not exhibited by the individual agents. There are also ways that such systems take in data from their environments, find regularities in the data, and compress these perceived regularities into internal models that are used to describe and predict its future. Complex systems exhibit evolutionary processes in that these internal models are subjected to selection pressures in the context of specific environmental conditions resulting in changes to the internal models over time....Finally, the emergent characteristics of a particular complex system frequently form the individual agents in a higher level complex system. (p. 1)

As is evidenced by this definition, a new vocabulary is necessary to understand the essential aspects of a complexity perspective. A list of basic complexity theory terms includes adaptation, agents, complexity or complex adaptive systems, emergence, fitness, hierarchy, internal models, non-linearity, regularity and randomness, schemas, selection and selection pressures, self-organization, systems, and system dynamics. These terms and complexity theory, in general, are addressed using a school as an example of a complex system and Figure 4 as a conceptual model.

The study of complexity involves the study of complex systems that are inherently non-linear, open, and far from equilibrium (Thelen & Smith, 1994). A non-linear system is unpredictable, that is, if one is familiar with all the components of the system, one is still unable to determine exactly what will happen next (e.g., weather, human behavior, ecology). In addition, in a non-linear system, the whole is greater than the sum (or average) of its parts (Holland, 1995). While complex systems are unpredictable and non-linear, they are also open. An open-system is a system that needs and receives energy to maintain its order. This maintenance of order places the system in a state that is far from equilibrium, equilibrium being the degenerative state that the system would inhabit if there was no influx of energy. Thus, a complex system is greater than the sum of its parts, but requires energy to maintain this greater order. "These are called open, nonequilibrium systems: open in the sense that they can interact with their environment, exchanging energy, matter or information with their surrounds; and nonequilibrium, in the sense that without such sources they cannot maintain their structure or function" (Kelso, 1995, p. 4).

For example, a school is a non-linear, open, and far from equilibrium system. The school is non-linear because even if one was to know the position and direction of movement of all the students, teachers, and administrators at a given point in time, one would not be able to predict what would happen next. Students that were walking to the library may decide to go to their lockers instead and a teacher may suddenly decide to give a pop-quiz. Also, the activity in the school is far greater than the sum of the individual students, teachers, and administrators. As students, teachers, and administrators begin to collaborate, the whole becomes greater than its parts. Student and teacher teams emerge, interacting students learn more than they were capable of learning on their own, and special programs are formed to assist students with their special needs. All of this far from equilibrium activity and learning is made possible through an influx of energy into the school (an open-system), energy in the form of students, materials, and money. If there was not this influx of energy, or resources, the school would deteriorate into a state of disrepair and disorder.

A non-linear, open, and far from equilibrium system is a group of interdependent elements, or agents, that interact to form a composite whole, while system dynamics refers to the feedback structures, methods, and outcomes of these interactions (Brodnick & Krafft, 1997). A complex system is composed of agents, individual active elements of a system that possesses an internal state comprised of internal models, rules, and strategies that influence and guide the agent's behavior (Holland, 1995). A group of common agents is an agent type.

For instance, a school is a system that is comprised of several agents and agent types, such as students (a student is an agent, all of the students would constitute an agent type), teachers, and administrators. The system dynamics involving the interaction of the students, teachers, and administrators, is governed by explicit and implicit rules of conduct, order, need, and expectation.

In addition, each agent functions through the use of internal models or schemas (Gell-Mann, 1995; Holland, 1995). An internal model, or schema, is created or modified as the agent gains experience. As the agent gains experience, the agent abstracts the regularity from the randomness within the experience and begins to form internal models that describe these regularities. The agent may construct several internal models or schemas of a given experience, each internal model or schema providing a potential explanation of the experience. The process of changing recognized patterns of regularity into internal models occurs through compression. Compression results in abstractions or generalizations of experience, not a verbatim record. Often, internal models or schemas are described by a set of rules (see Holland, 1995, 1998). These internal models are then used by the agents to describe current events or behaviors, predict future events or behaviors, and prescribe subsequence behavior.

Continuing the school example, each student, teacher, and administrator has several internal models or schemas related to the school environment. A particular student, for example, will have internal models or schemas that relate to how she interacts with teachers, takes tests, or fits into the school social structure. This student, while having different internal models or schemas related to different topics, may also have more than one internal model or schema for the same topic. She may have several internal models or schemas of how to interact with teachers, such as a friend, as a subordinate, or as a mentor. This student will continually create and modify her internal models and schemas based on her continuing interaction with teachers. She may notice that the female teachers like to be referred to as "Ms." and not "Mrs.", and so she modifies her internal model or schema accordingly. This generalization, or compression, of addressing the female teachers as "Ms." will allow the student to anticipate the need to use the "Ms." title (prediction and prescription), and will provide a basis for understanding when a teacher is curt with her after she uses the title "Mrs." accidentally (description).

In a complex system, these interacting agents exist within a hierarchy of agents (see Figure 5) (Lewin, 1992). Agents at one level of the hierarchy interact with each other, and other agent types, and through this interaction an emergent global structure (Lewin, 1992), or aggregate system of meta-agents or behavior, emerges (Holland, 1995). A meta-agent is an assembly of lower agents that form a new agent at a higher level in the hierarchy (i.e., cells assemble to form organs, organs assemble to form organisms), while an aggregate behavior would be a behavior that is comprised of other more fundamental behaviors (e.g., playing basketball is comprised of walking, running, bouncing, etc). These emergent global structures and meta-agents, upon forming, then feedback to the lower level agents to influence the lower level agents' interactions. An essential aspect of this process of lower level agents giving rise to higher level agents is that the nature and formation of the higher level agents is not predictable from an understanding of the individual behaviors of the lower level agents, a process known as emergence (Casti, 1994; Crutchfield, 1994; Holland, 1998). Emergence is an enigmatic process whereby fundamental agents produce surprising and unpredictable meta-agents or behaviors. These meta-agents or behaviors are said to emerge from the interaction and collective properties of the lower agents (e.g., clouds emerge from water vapor and heat, life emerges from DNA, mind emerges from neurons). According to Thelen and Smith (1994), "These emergent organizations are totally different from the elements that constitute the system, and the patterns cannot be predicted solely from the characteristics of the individual elements" (p. 54).

Within a school setting, one potential hierarchy of agents is represented in Figure 5. In this hierarchy, the students, teachers, and administrators give rise to a particular school behavior or setting, that is, the specific nature, atmosphere, and environment of the school. The presence and interaction of these students, teachers, and administrators (the agents) leads to the emergence of surprising and unpredictable school behaviors, such as, racial tension, academic rigor, and/or drug acceptance. Emergence, in the case of academic rigor, may involve a school in which students consistently put forth effort, teachers continually challenge their students, and administrators actively support both the teachers and students. This academic rigor is not a function of any one student, teacher, or administrator, but rather, this academic rigor is a function of the interaction between the students, teachers, and administrators.

This process of emergence is deeply intertwined with the concept of self-organization. Self-organization refers to the spontaneous self-generation of order from within an open-system of agents (Capra, 1996; Kelso, 1997), or what Kauffman (1995) calls "order for free." A fundamental component of self-organization is that order arises from within the interactions of the agents and is not imposed on the agents from some external force. Thus, as agents interact, they organize themselves according to local parameters and self-interest, and from this self-organization a more global or higher structure emerges. In this way, self-organization and emergence are inexorably linked. According to Jacobson (1997), "Self-organizing phenomena are inherently decentralized due to the local interactions of many individual agents, with order 'emerging' without centralized control structures" (p. 3).

Self-organization within the school example, concerning academic rigor, might involve students meeting in study groups, teachers preparing academically challenging projects, and administrators purchasing extra equipment for students and teachers. These activities have not been organized outside of the school and imposed on the students, teachers, and administrators; rather, the students, teachers, and administrators have developed these activities themselves, to satisfy their own (agent-based) goals and needs. In addition, it is these self-organized activities that have led to the emergence of the academic rigor, and in turn, this academic rigor has influenced the further self-organization of activities, forming a feedback mechanism.

This process of self-organization-emergence-feedback forms the basis for selection pressure and adaptation (Gell-Mann, 1994). As agents interact with other agents and the environment, the agent's internal models and schemas self-organize and emerge. In this process, the agent's interactions with other agents and the environment serve as evaluations of the agent's internal models and schemas. If an agent repeatedly exhibits a behavior that is counter-productive, based on an internal model or schema, then the internal model or schema is modified, discarded, or ignored. If, however, the agent repeatedly exhibits a behavior that is productive, then the internal model or schema that is responsible is retained. Gell-Mann (1994) refers to this evaluative feedback, from instantiating internal models or schemas in the real world, as selection pressure. That is, the real world exerts pressure on the agent to select the internal model or schema that consistently produces favorable results. In a similar manner, from a Darwinian natural selection perspective, those agents that are able to generate and select effective internal models and schemas will be more likely to be selected for reproduction and survival. In addition, an agent that is capable of repeatedly selecting internal models and schemas that are favorable is considered fit, or to have fitness, in relation to the environment in which the agent exists. However, environments do not remain static, thus an agent's level of fitness is always in a state of flux. This state of flux requires the agent to continually monitor and modify their internal models and schemas as the environment changes, a process known as adaptation (Kauffman, 1993, 1995). Adaptation refers to changes in internal models or schemas that improve the performance of the agent, whether that performance is reproduction, survival, money, or knowledge. Holland (1995), in defining adaptation in complex systems, states "Roughly, experience guides changes in the organism's structure so that as time passes the organism makes better use of its environment for its own ends. Here we expand the term's range to include learning and related processes" (p. 9).

Maintaining the school and academic rigor example, selection pressures for student performance might involve grades, college admission, parental approval, peer approval, and/or personal satisfaction. As a student uses various study strategies, and succeeds or fails, and as this student watches other students use various study strategies, and they succeed or fail, the student constructs internal models and schemas related to study strategies and which work and which do not work, under various conditions. Indeed, this student may experience both success and failure using the same study strategy for two different teachers, reflecting two different levels of fitness for the same strategy. At this point, the student needs to adapt to the environment by using the appropriate study strategy with the appropriate teacher. Thus, retaining internal models or schemas that are fit and modifying internal models or schemas that are less fit leads to adaptation and better overall performance.

In summary, complex systems are non-linear, open, and far from equilibrium systems that are comprised of interdependent agents whose interactions, based on internal models and schemas, lead to self-organized and emergent behaviors that have dynamic fitness levels in response to selection pressures exerted by changing environmental conditions, thus facilitating the need for adaptation.

This broad-based theory has been developed as an inter-disciplinary theory, crossing any and all domain boundaries. Given this inter-disciplinary nature, the search for basic principles that underlie all complex systems is a major focus of complex systems research. However, there is not, as of yet, a core group of these basic complex systems principles (see Gell-Mann, 1995, and Holland, 1995, for examples). Given the nature of this article, the research base, and the complex systems model (Figure 4), the following principles have been derived:

1. Complex systems are non-linear, open, and far from equilibrium.
2. Complex system behavior involves adaptation to the environment, based on experience.
3. Complex system behavior is a function of internal models or schemas that are the result of perceived regularities in experience.
4. Emergent global complex system behavior involves the aggregate behavior of agents
5. Internal models and schemas are actively constructed, self-organized, and emergent.
6. Internal models and schemas are a function of both agent interaction and existing internal models and schemas.

Each of these principles has been discussed and exemplified in the preceding section. As general tenets of complexity, these principles may be applied to a wide array of complex systems. One such system involves human learning. Each of these principles may be applied to the process of learning, in general, and the theory of constructivism, specifically.

Complexity and constructivism are two theories that provide functional and robust metaphors. Complexity provides a metaphor for myriad phenomena, while constructivism provides a metaphor for learning. In the synthesis of these two powerful metaphors lies a new metaphor - complex constructivism. According to Lakoff and Johnson (1995), "New metaphors have the power to create a new reality" (p. xx).

The reality of complex constructivism is one in which the non-linear, adaptive, and constructive nature of learning is embraced. Complex constructivism views learning as the active construction and adaptation of one's internal models of reality based on the interaction between oneself and one's environment (including other persons), such that the functioning of one's internal models exceeds the sum of the models' components. This definition leads to five general principles of learning based on complex constructivist ideals:

• learning involves student adaptation to the environment
• learning involves the active construction of internal models by the student
• learning involves the self-organization of knowledge and experience into internal models
• learning involves the emergence of internal models as a natural consequence of a student's experience
• learning is a function of both student interaction and existing internal models

These general principles emerge from the synthesis of complexity theory and constructivist theory to provide links to a new perspective on learning, memory and cognition. These principles also provide a foundation upon which to build new ideas relating complexity and constructivism. Figure 6 is a graphical model of the relationship between these principles, based on the earlier models of constructivism (Figure 3) and complexity theory (Figure 4).

The complex constructivist model in Figure 6 represents the self-organized and adaptive nature of learning. Students enter an experience with existing internal (mental) models, or schemas, that allow the student to predict how the experience may transpire, to prescribe desired behaviors based on the predictions, and to describe the experience as it occurs. However, students with limited existing internal models related to the current experience will have only a limited ability to predict, prescribe, and describe, while students with more well developed internal models will be able to predict, prescribe, and describe more effectively.

These students' internal models are affected by not only the experience, but also interactions with other students and their internal models. As the students are engaged in the experience, and potentially interacting with other students, the students are determining which aspects of the experience are familiar, and which aspects of the experience are novel. The recognition of familiar experiences indicates the existence of internal models relative to the experience, while a lack of familiarity indicates the non-existence of a related internal model. An existing internal model connotes the student's prior identification and compression of regularities within this type of experience.

The process of recognizing and compressing regularities is paramount to students constructing internal models. Regularity is knowledge. As students encounter experiences, they begin to actively look for regularities and compress these regularities into an abbreviated form, that is, the students begin to actively construct knowledge from within the vast array of stimuli in the experience, based on existing internal models. This process of active regularity extraction, or active knowledge construction, is fundamental to making meaning from the experience. The student's make an experience meaningful by relating the experience to the regularities existing within their internal models (i.e., prior knowledge). If an experience contains many regularities that already exist within the student's internal models then the experience is highly meaningful.

As students gain experience and actively identify regularities, these regularities will self-organize to form internal models. That is, the way that a student organizes their knowledge is a function of the student, not the experience. The experience may indirectly influence or intimate as to how this knowledge might be organized, but the actual organization is dependent upon the student and their existing internal models. In addition, as related knowledge coalesces, or perceived regularities cluster, a more cohesive entity begins to emerge - an internal model. These internal models provide both a framework for knowledge and a formulation of the integration of the knowledge. Thus, internal models emerge from the abstraction of regularity and knowledge, and as such, are personal constructions of the student.

Finally, these internal models are continually re-engaged in various experiences to provide for adaptation. If the nature of familiar experiences change, then new regularities or knowledge will be abstracted, and old internal models will be modified or new internal models will be constructed.

The preceding description has been fairly linear due to the constraints of written language; however, it should be noted and emphasized that this process of learning and adaptation is dynamic, cyclical, and very non-linear.

Complex constructivism principles are based on the shared fundamental tenets of both complexity theory and constructivism theory. These principles provide a solid foundation for understanding the nature of learning in a complicated, dynamic, and culturally relevant world. Holland (1998) provides an excellent introduction,

Despite the perpetual novelty of the world, we contrive to turn experience into models of that world. We learn how to behave, and we anticipate the future, using the models to guide us in activities both common and uncommon. Somehow, through learning, these models emerge from the torrent of sensations that impinge upon us at every moment. (p. 53)

1. Learning involves student adaptation to the environment.

The concept that an organism alters its behavior to more effectively interact within its environment is common to both a constructivist and complexity perspective. For constructivists, this adaptation involves the construction of new mental structures, and the modification of existing mental structures, to facilitate students in interacting meaningfully and effectively within their sociocultural and physical environments. Eisenhart and Broko (1991) state the constructivist perspective well, "Learning occurs as [students] make sense of instructional events by using their existing cognitive structures to interpret environmental stimuli. It also occurs as they modify and elaborate their knowledge structures through a process of adaptation to the environment" (p. 142). This concept of constructing and modifying internal models in order to adapt to an environment is mirrored in complexity theory. Martin (1995), in delineating the essential characteristics of a complex system, states, "As a rule, these systems are adaptive; changes in their internal states occur in response to the environment" (p. 263; italics in original). Indeed, Waldrop (1992) bridges the gap, "In fact, you can think of internal models as the building blocks of behavior. And like any other building blocks, they can be tested, refined, and rearranged as the system gains experience" (p. 146).

2. Learning involves the active construction of internal models by the student.

There are many types of constructivism; however, their unifying theoretical tenet is the belief in an active learner, a learner that actively constructs knowledge from experience. This construction process is responsible for students "internalizing" their culture and making sense of their environment. This act of construction applies equally well from comprehending basic math facts to comprehending cultural mores. Comprehending and making sense of an environment, as a complex agent, involves the active search for regularities. These regularities reflect knowledge of the environment and allow agents to successfully adapt. Therefore, from a complex constructivist perspective, students construct internal mental models by actively searching for regularities in their experience.

3. Learning involves the self-organization of knowledge and experience into internal models.

Constructivism and complexity both emphasize that the organization of students' or agents' internal models is a process that is carried out exclusively by the student or agent. That is, the organization of a student's or agent's knowledge or internal model is not imposed on the student or agent by either an internal or external source. This does not negate the influence of society and the environment; rather, society and the environment have an indirect influence on self-organization, as an impetus for adaptation. Therefore, self-organization accounts for the individualistic or subjective nature of knowledge and "the view that learning is both a process of self-organization and a process of enculturation that occurs while participating in cultural practices, frequently while interacting with other" (Cobb, 1996, p. 45)

4. Learning involves the emergence of internal models as a natural consequence of a student's experience.

The complex constructivist perspective posits the idea that internal models are not "actively constructed", but rather, internal models are "naturally emerging." This statement is not as antithetical to a constructivist view as it may first seem. Students' active construction involves the active search for regularities in their experience and these regularities represent knowledge. However, the internal models that provide the relational organization of this knowledge emerge as a natural consequence of knowledge acquisition. Thus, the knowledge is actively constructed while the organization is naturally emerging. Furthermore, these naturally emerging internal models provide the non-linear foundation for the concept that the whole, the internal models, is greater than the sum of the parts, the knowledge. "The ascending levels of the hierarchy of complexity demonstrate emergent properties at teach level which appear to be nonpredictable from the properties of the component parts." (Cowan, 1994)

5. Learning is a function of both student interaction and existing internal models.

Within constructivism, this complex constructivist principle combines the social interactionist views of the sociocultural constructivists with the personal constructivist views of the radical and cognitive constructivists. This dualism also exists, although to a lesser extent, within complexity between the individual agent view of Gell-Mann (1994, 1995) and the aggregate agent view of Holland (1995, 1998). Cobb and Yackel (1996) provide support for this principle of combining the individual and the social by eschewing this duality and proposing an "emergent perspective" to constructivism that integrates these two views. Cobb and Yackel (1996) explain this relationship,

The characterization of learning as an individual constructive activity is, therefore, relativized because these constructions are seen to occur as students participate in and contribute to the practices of the local community....The link between collective and individual process in this approach is, therefore, indirect because participation enables and constrains learning but does not determine it. (p. 185)

Complex constructivism provides a new perspective on learning. A perspective that emphasizes both the active, self-organizing construction of knowledge, and the adaptive nature of those constructions. The complex constructivism perspective combines and addresses the common concerns of constructivism,

A psychology that decomposes...thinking into its elements in an attempt to explain its characteristics will search in vain for the unity that is characteristic of the whole. These characteristics are inherent in the phenomenon only as a unified whole. Therefore, when the whole is analyzed into its elements, these characteristics evaporate. (Vygotsky, 1987, p. 45)

and complexity,

Modeling the emergent characteristics of the mind presents probably the most difficult task for creating links between the hardware or software of human biology and the achievements of human consciousness. (Singer, 1995, p. 5)

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