Education as a Complex System: Conceptual and Methodological Implications

Education is a complex system, which has conceptual and methodological implications for educational research. In this article, an overview is first provided of the Complex Systems Conceptual Framework for Learning (CSCFL), which consists of a set of conceptual perspectives that are generally shared by educational complex systems organized into two focus areas: collective behaviors of a system and behaviors of individual agents in a system . Complexity and research methodologies for education are then considered, and it is observed that commonly used quantitative and qualitative techniques are generally appropriate for studying linear dynamics of educational systems. However, it is proposed that computational modeling approaches being extensively used for studying nonlinear characteristics of complex systems in other fields can provide a methodological complement to quantitative and qualitative educational research approaches. Two research case studies of this approach are discussed. We conclude with a consideration of how viewing education as a complex system using complex systems conceptual and methodological tools that can help advance educational research and also inform policy.

Scientific study of the behavior of complex physical and social systems over the past three decades has led to significant insights about the world that classical approaches tended to over simplify or to ignore (Bar-Yam, 2003). However, the application of complexity perspectives to educational research is at a relatively early stage, although there is increasing use of complex systems conceptual perspectives (e.g., Jacobson & Wilensky, 2006;Wilensky & Jacobson, 2014). For example, Bereiter and Scardamalia (2005) noted this influence in the use of complexity concepts in the educational research literature: "self-organization and emergence … [in] mainstream educational psychology, … [make it] increasingly apparent that there are no simple causal explanations for anything in this field" and "learning itself, at both neural and knowledge levels, has emergent properties" (p. 707). (Italics added for complex systems concepts in the quotes. ) We are also seeing suggestions that complexity perspectives provide important ways to understand more deeply educational change as well as having the potential to inform educational policy (Mason, 2008). Lemke and Sabelli (2008) have noted that the "education system is one of the most complex and challenging systems for research" (p. 128). They further recommended combining conceptual perspectives about complex systems with computer modeling capabilities to inform policymakers about proposed interventions and their potential impact.
The main purpose of this article is to consider education as a complex system and to discuss conceptual and methodological implications. We then review two recent studies for which complexity conceptual perspectives and methods allowed insights that may not have been revealed by conventional educational research techniques. We conclude with a consideration of how using complex systems conceptual and methodological tools can help advance educational research that also informs policy. Wing coloration of peppered moth changed (evolved) from mainly whitish/mottled to mainly darkish brown from pre-to post-industrial age Great Britain Young children often have "flat earth" mental models, primary-age children often have synthetic "hollow earth" mental models, and older students have "globe earth" mental models.

Education as a Complex System
Scientific views of complex systems (sometimes referred to as the field of complexity) primarily come from research in the physical sciences, mathematics, and computer science (Gell-Mann, 1994;Holland, 2006;Kauffman, 1993;Wolfram, 2002) as well as in social science research (Byrne, 2013;Mason, 2008;Sawyer, 2005). 1 To consider what it means to view education as a complex system, Jacobson, Kapur, and Reimann (2016) have proposed a Complex Systems Conceptual Framework for Learning (CSCFL), which consists of a set of conceptual perspectives that are generally shared by complex systems relevant to education (see Table 1).
The CSCFL organizes these conceptual perspectives in two focus areas: collective behaviors of a system and behaviors of individual agents in a system. Key conceptual perspectives in the first focus area are: (a) interactions of individual agents or components of the system that often may be described in terms of simple rules; (b) feedback interactions between agents that may occur within or across system levels; (c) self-organization of agents in a system that typically result from the two previous conceptual perspectives; (d) sensitivity to initial conditions or chaos where there is an amplification of initial state differences in a system (often based on positive feedback interactions) that may contribute to major behavioral changes in a system; and (e) emergence, regarded by many scientists as the most important complexity conceptual perspective (Bar-Yam, 2003;Gell-Mann, 1994;Holland, 2006;Kauffman, 1995;Mitchell, 2009). There is a somewhat counter-intuitive aspect of emergence, which is described by Jacobson et al. (2016) as the: formation of collective properties at a macroscopic level of a system from simple behaviors of the parts, with those properties frequently are not found in the parts. For example, in a traffic system the macro-level formation of a traffic jam propagates backwards even though the individual cars at the micro-level general move forward as they speed up or slow down, with some lateral lane changes-but rarely do the cars move backwards in traffic. (p. 211) This example includes conceptual perspectives (a) -(d) of complex collective behaviors of a system while also illustrating key features of emergence, which are that the whole of a complex system is not merely the sum of parts (i.e., cars move forward) , but also often different than those parts (i.e., the traffic jam goes backwards) in key and perhaps even surprising ways (Casti, 1994).
Our reading of the complex systems and education literature is that in general the conceptual perspectives in the CSCFL focus area complex collective behaviors of a system such as nonlinearity and emergence have been emphasized. However, conceptual perspectives in the focus area behaviors of individual agents in a system have received less attention, even though educational systems, in common with complex systems, "involve many components that adapt or learn as they interact" (Holland, 2006, p. 1). Holland proposes several important characteristics of how individual elements or agents behave, of which three are currently included in the CSCFL as being the most relevant for educational systems: (a) parallelism, (b) conditional actions, and (c) adaptation and evolution.
First, parallelism is exhibited when agents in a complex system have simultaneous interactions with each other by sending and receiving signals. For example, students on a playground will be doing a variety of things simultaneously while talking and listening to each other (sending and receiving signals), some riding a swing or perhaps pushing a friend, others throwing a ball back and forth, playing hopscotch, and so on.
Second, conditional actions are how an agent might respond to received signals, often described with rules such as IF a certain signal is received, THEN act in a certain way. For example, if a soccer ball is close to a player (i.e., an agent in the system), then she would try to kick it, unless IF the player is the goal keeper and THEN she would try to catch or deflect the ball. An important characteristic of complex systems is that the combination of relatively simple agent rules and parallelism of many agents simultaneously acting based on these rules can result in very complex and dynamically changing behaviors.
Third, adaptation and evolution is a particularly important complex systems conceptual perspective of relevance to educational systems in that the agents themselves change over time; that is, they learn. 2 Gell-Mann (1994) has described learning as changes in an agent's internalization of perceived regularities in its environment, which in turn increases the agent's potential for adaptive behavior in its environment. For example, students in a classroom may be regarded as agents in an educational complex system who, at a given time, have certain internal cognitive structures and affective knowledge related to a subject, and who over time at school will (hopefully) construct (i.e., evolve) new or modified cognitive structures from their learning activities.
In closing this section, we note that Jacobson et al. (2016) do not claim the CSCFL is exhaustive in terms of its currently included complexity conceptual perspectives. There are, of course, many, many more complexity concepts-such as autocatalytic systems (Kauffman, 1995), activation and inhibition (Bar-Yam, 2003), bifurcations (Mitchell, 2009), and so onthat also can have relevance for understanding various aspects of education as a complex system. Still, we believe the CSCFL includes a reasonable core of complexity conceptual perspectives relevant to educational and learning systems and that these can be useful analytical tools for educational researchers, such as providing a principled way to reconcile the long-running debate between cognitive and situative theories of learning (Jacobson et al., 2016). In the next two sections, we consider how the CSCFL also has relevance for research and methodological issues concerning educational complex systems.

Complexity and Research Methodologies for Education
We now shift our focus from the CSCFL to considering implications for methodologies used for educational research as well as to inform policy about educational systems. But how are areas of educational research and policy connected so that complexity perspectives can be valuable analytical tools to each? One key way is that the information flows available to inform policy decisions are constrained by the types of methodologies that have been developed and validated by educational and social sciences researchers.
Broadly speaking, existing methodological approaches for educational research fall into two main categories: quantitative and qualitative (Firestone, 1987). Quantitative approaches (including experimental and quasi-experimental) are pervasively used in educational research (Kapur, Hung, Jacobson, & Voiklis, 2007;Suthers & Hundhausen, 2003). Rooted in a positivist philosophical tradition, quantitative methods typically seek to establish causal or quasi-causal explanations of design or intervention effects versus control or comparison conditions. In contrast, qualitative approaches have a phenomenological philosophical basis that seeks to describe and to understand educational contexts and environments. Although there are educational researchers who exclusively use one or the other of these methodologies, since the late 1980s it has become increasingly common for educational researchers to use both quantitative and qualitative methods in a complementary manner to understand the educational issue being investigated in terms of the different types of information generated by these two methodological perspectives (Firestone, 1987).
However, there is an important question that must be asked. Are the existing quantitative and qualitative methodologies-whether separately or in combination-in fact sufficient for providing appropriate information and understandings of the dynamics of educational systems viewed from the complexity perspectives outlined in the previous section?
Unfortunately, the answer is "no." Most mathematical tools commonly used in quantitative research (e.g., differential equations, statistical modeling) are linear tools that work by breaking a system into its components or parts, studying the parts individually, and then adding the parts together to form the whole. However, emergent phenomena in an educational complex system generally have nonlinear properties, which cannot be analyzed by "adding up the parts" since the patterns at the macro-level of a complex system generally have different properties than the constituent parts at the micro-level of the system. Holland (1995) argues "Nonlinearities mean that our most useful tools for generalizing observations into theory-trend analysis, determination of equilibria, sample means, and so on-are badly blunted" (p. 5).
There is another important limitation to quantitative and qualitative approaches: they are best suited to explaining and understanding what has already emerged (Epstein & Axtell, 1996). For example, opinions, norms, convergence in group discussions may be viewed as intra-or inter-personal patterns. Once these emerge, then quantitative methods may explain aggregate-level relationships and qualitative methods may provide rich descriptions of these opinions, norms, or group interactions. However, as Kauffman (1995) observes, the same trajectory of interactions may not have occurred even with similar initial conditions. In contrast, to more fully study emergent phenomenon in complex systems of education (and other domains), one needs to understand and explain both what patterns actually unfolded as well as the space of possible trajectories of what could have unfolded.
For policy purposes, the space of possible trajectories for an educational system is of particular importance, as we discuss further below. Still, the two predominately used methodological approaches available to educational researchers and policy makers have fundamental limitations for understanding two key components of the CSCFL-nonlinearity and emergence-in complex systems of education.
We certainly acknowledge that quantitative and qualitative approaches each have value for educational research, as well as approaches that integrate or blend these methods (Firestone, 1987), in order to study linear characteristics of educational systems. However, complex systems have regions where system behaviors are in fact linear and nonlinear. Jacobson and Kapur (2012) have argued that there is a "dialectical co-existence of linearity and non-linearity in terms of feedback interactions within and across levels of the system so that collective properties arise (i.e., emerge) from the behaviors of the parts, often with properties that are not exhibited by those parts" (p. 310). Currently used quantitative and qualitative techniques are generally appropriate for studying linear dynamics of educational systems, but what techniques are appropriate for studying nonlinear dynamics of educational complex systems? Jacobson and Kapur (2012) note that scientists conducting research into nonlinear dynamics in other complex systems areas (e.g., physics, biology, economics) have been developing and using a range of computer modeling techniques. They also propose that modeling methods such as agent-based models (ABMs) can function as a methodological complement to quantitative and qualitative approaches.
Briefly, there are two main types of computational modeling: agent-based models (ABMs) and equation-based models (EBMs) (Parunak, Savit, & Riolo, 1998). These two approaches have a similar goal, which is to create a computer model of a system, but they differ in two fundamental ways. First, they use different assumptions to define relationships between entities in the model. EBMs typically use quantitative formalisms such as algebraic or partial differential equations to express how entities in the system are related over time. In contrast, ABMs use algorithmical formalisms to represent the behaviors of the individual entities (i.e., agents) such as wolves eating sheep or teachers interacting with students, and then "turns them loose to interact" (Parunak et al., 1998, p. 10).
Second, ABMs and EBMs differ fundamentally in terms of their respective direction of focus on levels. 3 ABMs are often referred to as being "bottom up" in that they algorithmically model the behaviors of agents or component elements at a particular level of the system and then allow a focus "up" at emergent behaviors at a higher (i.e., less granular) level. In contrast, EBMs are often viewed as being "top-down" in that they also start at a particular system level but use equations to model component behaviors at a lower (i.e., more granular) system level.
EBM methods are best suited if the interest of the modeler is at a macro-level of system where the aggregate properties are reasonably well understood to the degree that they can be captured by equations and are used to explore different "what-if" scenarios, such as a reduction in tax revenue during a recession that leads to a reduction in a school district's budget and options such as increasing class size or reducing extra-curricular activities to balance the budget. In these examples, the macro level relationships between "tax revenue," "school budget," "class size," and "extra-curricular activities" might be linear. But also, note that EBMs do not consider micro-level interactions such as specific individuals who are out of work and thus pay little or no taxes, individual school staff having to make decisions about whether to purchase a greater number of chairs and other classroom supplies for larger classes or cut popular extra-curricular classes such as art, music, and sports, and so on. In general, if the behavior of a system is linear, then normalized assumptions about micro-level behaviors that contribute to the macro-level properties (such as we described) may be sufficient to generate a model that can be useful for certain types of educational research or policy decisions.
However, what if the micro-level interactions and possible emergent properties at the macro-level are not necessarily well-understood or cannot be anticipated because of nonlinearity in the educational system of interest? In such circumstances, ABM approaches can be effective because they can focus on micro-level interactions-for which there is often quantitative and/or qualitative data to inform the specification of agent-based rules-and then to allow model runs (i.e., "turn them loose to interact"). This will, in turn, likely generate macro-level system behaviors that may or may not have been anticipated as well as information about interactions between micro-and macro-levels of the system. It is also possible to explore the model through multiple runs in order to gain insight into possible trajectories of what could have unfolded (Kauffman, 1995), such as by identifying attractors in a high dimensional space that may influence system behaviors (Gick, 1987). We also note that the use of ABM methods are increasingly being used not only in the natural sciences (Wilensky & Rand, 2015) but also in economics (Arthur, Durlauf, & Lane, 1997;Testfatsion, 2006), business (Rand & Rust, 2011), sociology (Squazzoni, 2012;Watts & Strogatz, 1998), and socio-cultural psychology (Axelrod, 1997;Epstein, 2006), just to name a few areas. Grounded in complexity theory, ABM is providing important theoretical and empirical insights into the dynamics of complex systems in the social sciences (Eidelson, 1997) .
We believe that ABMs, when integrated with quantitative and qualitative methods, can potentially reveal insights about the dynamics of complex systems of education across the range of levels and time scales, such as those discussed by Lemke and Sabelli (2008), that may not be possible through the use of any single methodology. We view the integration of complexity modeling with quantitative and qualitative methods as being overlapping and complementing (see Figure 1), with each method providing analytical tools for gaining different types of insights into the dynamics of the educational system issue being explored while also providing analytical focus when used together (as suggested by the Venn diagram overlap in the center of Figure 1). Further, we are beginning to see examples of educational and educational policy research in which modeling methods such as ABMs are being productively used as an important methodological complement to quantitative and qualitative approaches, which are discussed in the next section.

Studying Education as a Complex System: Two Research Case Studies
In this section we have selected two research case studies to illustrate the use of complexity conceptual perspectives and computer modeling tools. We believe that certain findings may not have been identified with more commonly used quantitative and qualitative educational research methods and analytical perspectives. We discuss these two programs of research in turn.
Our first research case study is the work of Maroulis, Bakshy, Gomez, and Wilensky (2014) that involved the use of ABMs to study initiatives to provide parents with school choice in the United States. Briefly, proponents of school choice reform argue that competition introduced by allowing parents to select the schools their children attend will lead to better schooling and incentives for school reform. In contrast, opponents of this type of reform claim resources are drained away from schools and that school quality is thus hurt, not helped, by such a reform. Research into this issue since the 1990s had employed standard quantitative and qualitative methods, but these studies have provided inconclusive and even conflicting findings. Maroulis et al. (2014) investigated this policy debate by creating ABMs of a school district's transition from a local neighborhood school "catchment area" system to a school choice system. The agents in the system were schools and students. School agents varied in terms of the quality and building capacity of existing schools, and new schools that entered the system by imitating top existing schools. Student agents varied in their ability and background, and they would rank schools in terms of achievement and geographic proximity. The academic achievement of the student agents combined individual traits and the "valueadded" by the quality of the school they attended. Real data from Chicago Public Schools was used to initialize the model (see Figure 2).

Figure 2. Visualization from an agent-based model of school choice in Chicago,
Illinois. Small dots represent students, large circles represent schools, circle size represents academic performance, and dark red and dark green colors show high and low poverty areas respectively (Maroulis et al., 2014).
The use of these ABMs helped identify dynamics-such as CSCFL conceptual perspectives of micro-macro levels, nonlinearity, and emergent properties-that had not been revealed in previous quantitative and qualitative research. Specifically, model runs demonstrated that the timing of new schools entering the system was a critical factor. The overall system improves because new schools entering the system imitate the top existing schools. However, a high emphasis on achievement at the schools leads to new schools entering the system earlier, which resulted in lower achieving new schools. Thus, there was a paradoxical mismatch between the macro-level and micro-level behaviors of the system in that increasing the emphasis on school achievement at the household level did not generally lead to increasing achievement at the district level. From a policy perspective, results of using ABMs suggest that the critics of school choice reform were correct that school achievement in the overall system would not rise. However, the reason proposed by the critics-draining of resources away from existing schools-was not actually the causal factor; rather, it was the timing of new schools entering the system. This ABM of the also provided insights into other implications of the school choice policy, such as the unintended transfer of top students to private schools where vouchers issued by the government were used to pay for the private schooling, which was an emergent property of the changes in the Chicago Public School system (Maroulis et al., 2010).
Another unexpected dynamic of the Maroulis et al. (2014) model was being a "toprated" school (based on mean achievement levels of its students) was an unstable (i.e., nonlinear) state: the top-rated school attracted many new students, some of whom did not achieve as highly, thus bringing down the school's achievement rating, so that another school becomes a "top-rated" one. This unexpected insight from their modeling has policy implications for the domain being modeled. Many "choice" schools avoided this issue by being selective, but if "school choice" is really implemented in the "free market" form that advocates sketched out, then this instability will become a reality. That is, if students and parents really have "choice" and base that choice on the level of achievement by students at a school, then the highest achieving school will get the most applications from a range of students, which, if they have to accept all or a random selection of those students, will lead to that school no longer being a top-school. Or to put it differently, a top-school is often in fact the top-school precisely because of its selective admission policy, which in fact is counter to a free choice model.
Our second research case study involves the work of White and Levin (2016) and Levin and Datnow (2012) in which they used computer simulation models based on complexity theory to better understand and guide educational change initiatives. In a study at a continuation high school (a school of "last resort" for students having difficulties in regular high schools), several complex systems conceptual perspectives, such as self-organization, feedback loops, equilibrium, nonlinearity, and emergence were used to guide the implementation of a reform to provide access to higher education for these students. These complexity concepts were also used as a means for understanding the ways that the reform unfolded, and to provide a guide (i.e., inform policy) for implementing similar reforms in other high schools.
Changing a stable complex system (i.e., one at equilibrium) requires a perturbation to how the agents interact with each other in order to shift to a different stable state. In their research, White and Levin developed the concept of a "purposeful perturbation," a change in the everyday operation of education that both makes sense locally and moves the stable educational system away from the status quo, through a "tipping point" or nonlinear change, and then to a new desired stable state in which the educational reform becomes routine practice (i.e., a new equilibrium emerges). Several of these purposeful perturbations that were identified in the school reform design experiment research were captured in agent-based models by White and Levin (2016), within a modeling framework called multi-mediator modeling (MMM).
One of the MMM models they developed is shown in Figure 3. The labeled orange circles represent the key concepts in the model, and the blue "globes" represent the impact of everything outside of the model on the concepts in the model. Green lines show positive impact that one concept in the network has on another concept, and red lines show the negative impact that a concept in the network has on another.
This model captures the initial effort of teachers involved in the reform (called ACCESS) to raise student expectations of their own capabilities for success in college-level academic work, effort that was opposed by their own low self-expectations that were reinforced by low expectations of these students held by other staff members at the school.
These expectations were raised in a non-linear way, in part by the ACCESS teacher expectations and in part by their own improved college placement test scores. However, this change alone did not lead to a school-wide implementation of the reform. Protection from the Principal of the school along with the improved student scores over time was found to be necessary in order to lead to a tipping point at which the ACCESS reform replaced the previous status quo programs at the school.  (White & Levin, 2016).
The White and Levin research demonstrates how complex systems conceptual perspectives can inform and help analyze the changes in the school practices over time. The multi-mediator models that were developed provided "runnable representations" of the key agents (e.g., teachers, students, school staff) and factors changing the school environment (e.g., ACCESS reform) that resulted in outcomes from various runs of these models that aligned in key ways to the qualitative research findings. In particular, the ability to model the tipping-points-the nonlinear changes-that were found illustrates our assertion of how computer modeling of complex systems can be synergistically combined with more standard educational research methods, such as a qualitative design experiment in this example.
These two projects represent proof-of-concept research that illustrates how the use of computer modeling, in particular ABMs, in conjunction with complexity conceptual perspectives such as those from the CSCFL, can provide useful and sometimes unique research and policy insights about educational complex systems. Also, these two projects demonstrate that complexity-based computer modeling approaches can provide analytics and information that go beyond traditional quantitative and qualitative educational research approaches. As Jacobson and Kapur (2012) suggested, these projects use modeling methods to complement and extend traditional educational research methodologies, not to replace them. Future work is now needed to further develop and validate modeling approaches that would meet the needs of educational researchers and policy makers.
For those who are interested in exploring and extending approaches such as we have discussed, then we recommend consulting other research in areas that have employed modeling approaches and complex systems conceptual perspectives. Mitchell (2009) provides a balanced discussion of the conceptual and methodological issues related to research involving complex systems in a wide range of areas in the physical and social sciences. Wilensky and Rand (2015) discuss both general techniques for developing agentbased models as well as an historical overview of computational modeling and a range of case examples. But given these are still early days in the use of such approaches for educational research, we recommend examination of high quality research in other social sciences fields that have used various computational modeling approaches (e.g., Epstein, 2006;Epstein & Axtell, 1996;Testfatsion, 2006). Overall, we believe such modeling approaches can be effectively adapted and employed in educational research and can inform educational policy as well.

Conclusion
Viewing education as a complex system has important implications both for educational research and for educational policy (Lemke & Sabelli, 2008). Combining new conceptual tools, such as the Complex Systems Conceptual Framework for Learning (CSCFL), with new methodological tools for complex system analysis, especially agent-based modeling, can provide educational researchers with new insights into the dynamics of complex educational systems. We also believe these complexity oriented conceptual and methodological tools can inform educational policy by showing different "possible futures" that various efforts at systemic educational reform might follow, especially because these tools allow us to examine the often-nonlinear dynamics of educational complex systems. We hope this overview of conceptual perspectives and computer modeling methods will stimulate further awareness of these approaches by educational researchers and policy makers as they engage the wide range of critically important challenges in 21st century education.