**Abstract**

This essay brings together two lines of work—that of children's cognition and that of complexity science. These two lines of work have been linked repeatedly in the past, including in the field of science education. Nevertheless, questions remain about how complexity constructs can be used to support children's learning. This uncertainty is particularly troublesome given the ongoing controversy about how to promote children's understanding of scientifically valid insights. We therefore seek to specify the knowledge–complexity link systematically. Our approach started with a preliminary step—namely, to consider issues of knowledge formation separately from issues of complexity. To this end, we defined central characteristics of knowledge formation (without considerations of complexity), and we defined central characteristics of complex systems (without considerations of cognition). This preliminary step allowed us to systematically explore the degree of alignment between these two lists of characteristics. The outcome of this analysis revealed a close correspondence between knowledge truisms and complexity constructs, though to various degrees. Equipped with this insight, we derive complexity answers to open questions relevant to science learning.

**Keywords:** cognitive development, science education, conceptual change, complex adaptive systems, thermodynamics, interdisciplinary theory

## **1. Introduction**

*We need to move toward a systems view that describes scientific concepts as complex.*

*– Andrea A. diSessa [1]*

It has long been accepted that children's knowledge formation defies straightforward processes of passive attention and associative learning [2, 3]. For example, rather than absorbing information indiscriminately, children will actively seek out some aspects of information, while ignoring others. Children can also imagine alternative realities, even fantasies that lack a grounding in reality [4]. This poses a problem when it comes to the question of how to improve children's knowledge. For example, it

remains unclear how to support the learning of abstract science concepts, especially when children hold incorrect naïve beliefs about the pertinent science phenomenon [5]. In the current paper, we seek to contribute to this conversation by systematically exploring links between knowledge formation and complexity constructs.

In order to offer a relatively unbiased discussion of the complexity of knowledge, we first identified central truisms about knowledge formation that are broadly supported by the literature. We then provided a glossary of complexity constructs that are potentially useful in understanding knowledge formation. Equipped with these two lists, we then evaluated whether facts about knowledge formation are anticipated by complexity constructs. In turn, this cross-tabulation served as a theoretical anchor to derive answers from complexity to open questions on children's science learning.

## **2. Established insights about knowledge formation**

Picture a child trying to balance a beam on a fulcrum. The principle of physics that matters in this task is that of weight distribution in the beam. While children are capable of detecting the beam's weight distribution, they sometimes focus on the beam's visual symmetry instead. The result is that children have trouble balancing beams with asymmetrical weight distribution; they try to balance them at their geometric center instead of their center of mass. This finding illustrates established facts about (1) the nature of knowledge, (2) the process of knowledge acquisition, and (3) the process by which knowledge is changed (see **Table 1** for an overview).

#### **2.1 Nature of knowledge**

A child who insists that a beam should balance at the geometric center is said to hold the mistaken belief that objects balance in the middle [6]. The nature of such a belief (or knowledge, more generally) is necessarily elusive, as it cannot be seen directly. For this reason, numerous models of knowledge have been proposed to rectify phenomenological and empirical findings (e.g., [7, 8]). Considered in the aggregate, the models largely agree on two characteristics of knowledge: (i) that knowledge is organized into structures, and (ii) that there are different kinds of knowledge structures. We elaborate on each of these characteristics next.

#### *2.1.1 Truism 1: Knowledge is organized into structures*

Rather than existing as encapsulated factoids, knowledge consists of interlined representations of experiences, also referred to as *schemas* or *mental models* [9–11]. Early evidence for such knowledge organization came from children's systematic errors in Piaget's classical conservation tasks [12]. Children spontaneously and consistently honed in on a particular variable to respond, suggesting the presence of mental structures that make one variable more salient than another. Numerous additional examples stem from errors in categorization tasks [13], causal-reasoning tasks [14], and learning tasks [15]. They suggest that knowledge needs to be conceptualized as an ordered set. A child's belief about beam-balancing is an example of such a knowledge structure.

#### *2.1.2 Truism 2: There are different types of knowledge structures*

Agreement also exists that there are qualitative differences in knowledge organization. A prominent distinction is between *implicit* and *explicit* knowledge: Only

**43**

nonlinearity).

tasks [22].

**Table 1.**

**2.2 Acquisition of knowledge**

*Central characteristics of knowledge formation.*

these characteristics next.

*2.2.1 Truism 3: Knowledge is construed*

*Exploring Links between Complexity Constructs and Children's Knowledge Formation…*

versus deep, or narrow versus broad.

of experiences.

their obvious shortcomings.

that can lead to conceptual change.

Structure Knowledge is organized coherently, rather than consisting of incoherent bits. Diversity Knowledge differs in various aspects, including implicit versus explicit, superficial

Holistic construal Knowledge is actively construed (abducted), rather than being a direct reflection

Context dependence The details of knowledge depend on various contextual factors, including the social context, the nature of specific tasks, and the available tools.

Persistence Mistaken beliefs often persist, even to the point of affecting perception, despite

Role of conflict Presenting children with the shortcomings of their naïve beliefs creates a conflict

explicit, not implicit knowledge, can be reported on [16, 17]. Another example is the distinction between *surface knowledge* and *deep knowledge*: Only deep knowledge, not surface knowledge, can be transferred to new situations [18, 19]. And yet another example is the distinction between *preconceptions* and *misconceptions* (e.g., [20]): While both types of knowledge lead to mistaken performance, only misconceptions, not preconceptions, persist [21]. These and other distinctions proved useful in capturing unexpected behavior, including that of balance-beam

The child presented with a balance-beam task will eventually realize the relevance of weight distribution and succeed in balancing off-center beams. That is to say, the child will eventually learn. This process of learning, like knowledge itself, cannot be seen directly [23]. Sure enough, there are numerous open questions and disagreements about how to best describe the process by which knowledge is formed [24]. There are, however, two characteristics of learning that are broadly agreed upon: (i) that knowledge is construed through the child's activity, and (ii) that aspects of the context strongly affect what is being learned. We elaborate on each of

At first glance, knowledge appears to reflect outside information, as if outside information was transported into the mind directly. There is indeed suggestive evidence in support of such passive learning [25]. On the other hand, however, there is widespread agreement that learning requires an active mind. Piaget coined the term 'constructivism' to capture this idea: The mind, rather than passively soaking up information, must actively build knowledge. As a result of such construal, knowledge structures might come into existence nonlinearly, reflected in the aha-moment of sense-making (see also *abduction*; [26]). DiSessa [1] captured this nonlinearity in the proposed trajectory from a naïve learner to the conceptually competent individual (see **Figure 1** for a schematic illustration of the suggested

*DOI: http://dx.doi.org/10.5772/intechopen.97642*

**Nature of Knowledge**

**Acquisition of Knowledge**

**Change of Knowledge**

*Exploring Links between Complexity Constructs and Children's Knowledge Formation… DOI: http://dx.doi.org/10.5772/intechopen.97642*


#### **Table 1.**

*Theory of Complexity - Definitions, Models, and Applications*

dren's science learning.

**2.1 Nature of knowledge**

such a knowledge structure.

remains unclear how to support the learning of abstract science concepts, especially when children hold incorrect naïve beliefs about the pertinent science phenomenon [5]. In the current paper, we seek to contribute to this conversation by systematically

In order to offer a relatively unbiased discussion of the complexity of knowledge, we first identified central truisms about knowledge formation that are broadly supported by the literature. We then provided a glossary of complexity constructs that are potentially useful in understanding knowledge formation. Equipped with these two lists, we then evaluated whether facts about knowledge formation are anticipated by complexity constructs. In turn, this cross-tabulation served as a theoretical anchor to derive answers from complexity to open questions on chil-

Picture a child trying to balance a beam on a fulcrum. The principle of physics that matters in this task is that of weight distribution in the beam. While children are capable of detecting the beam's weight distribution, they sometimes focus on the beam's visual symmetry instead. The result is that children have trouble balancing beams with asymmetrical weight distribution; they try to balance them at their geometric center instead of their center of mass. This finding illustrates established facts about (1) the nature of knowledge, (2) the process of knowledge acquisition, and (3) the process by which knowledge is changed (see **Table 1** for an overview).

A child who insists that a beam should balance at the geometric center is said to hold the mistaken belief that objects balance in the middle [6]. The nature of such a belief (or knowledge, more generally) is necessarily elusive, as it cannot be seen directly. For this reason, numerous models of knowledge have been proposed to rectify phenomenological and empirical findings (e.g., [7, 8]). Considered in the aggregate, the models largely agree on two characteristics of knowledge: (i) that knowledge is organized into structures, and (ii) that there are different kinds of knowledge structures. We elaborate on each of these characteristics next.

Rather than existing as encapsulated factoids, knowledge consists of interlined representations of experiences, also referred to as *schemas* or *mental models* [9–11]. Early evidence for such knowledge organization came from children's systematic errors in Piaget's classical conservation tasks [12]. Children spontaneously and consistently honed in on a particular variable to respond, suggesting the presence of mental structures that make one variable more salient than another. Numerous additional examples stem from errors in categorization tasks [13], causal-reasoning tasks [14], and learning tasks [15]. They suggest that knowledge needs to be conceptualized as an ordered set. A child's belief about beam-balancing is an example of

Agreement also exists that there are qualitative differences in knowledge organization. A prominent distinction is between *implicit* and *explicit* knowledge: Only

exploring links between knowledge formation and complexity constructs.

**2. Established insights about knowledge formation**

*2.1.1 Truism 1: Knowledge is organized into structures*

*2.1.2 Truism 2: There are different types of knowledge structures*

**42**

*Central characteristics of knowledge formation.*

explicit, not implicit knowledge, can be reported on [16, 17]. Another example is the distinction between *surface knowledge* and *deep knowledge*: Only deep knowledge, not surface knowledge, can be transferred to new situations [18, 19]. And yet another example is the distinction between *preconceptions* and *misconceptions* (e.g., [20]): While both types of knowledge lead to mistaken performance, only misconceptions, not preconceptions, persist [21]. These and other distinctions proved useful in capturing unexpected behavior, including that of balance-beam tasks [22].

#### **2.2 Acquisition of knowledge**

The child presented with a balance-beam task will eventually realize the relevance of weight distribution and succeed in balancing off-center beams. That is to say, the child will eventually learn. This process of learning, like knowledge itself, cannot be seen directly [23]. Sure enough, there are numerous open questions and disagreements about how to best describe the process by which knowledge is formed [24]. There are, however, two characteristics of learning that are broadly agreed upon: (i) that knowledge is construed through the child's activity, and (ii) that aspects of the context strongly affect what is being learned. We elaborate on each of these characteristics next.

#### *2.2.1 Truism 3: Knowledge is construed*

At first glance, knowledge appears to reflect outside information, as if outside information was transported into the mind directly. There is indeed suggestive evidence in support of such passive learning [25]. On the other hand, however, there is widespread agreement that learning requires an active mind. Piaget coined the term 'constructivism' to capture this idea: The mind, rather than passively soaking up information, must actively build knowledge. As a result of such construal, knowledge structures might come into existence nonlinearly, reflected in the aha-moment of sense-making (see also *abduction*; [26]). DiSessa [1] captured this nonlinearity in the proposed trajectory from a naïve learner to the conceptually competent individual (see **Figure 1** for a schematic illustration of the suggested nonlinearity).

#### *2.2.2 Truism 4: Knowledge formation depends on the context*

Evidence suggests that learning is strongly dependent on the context, even when the context appears irrelevant to the specifics of what needs to be learned. Illustrative evidence of such context dependence can again be found with Piaget's conservation tasks: Despite showing robust performance in the classical version of the task, the mere number of times children were asked for a comparative judgment affected their performance (e.g., [27]). Evidence also comes from the balance-beam task: Children managed to balance the beams better with their eyes closed than with their eyes open [6]. Overall, context (e.g., cultural, societal, physical) has the ability to influence how the information ends up being utilized within the system of knowledge [28, 29].

#### **2.3 Change of knowledge**

A child presented with the balance-beam task is unlikely to enter the situation without prior knowledge. It might pertain to general ideas about what to expect, or it can pertain to very specific ideas about how to solve a task (e.g., the belief that beams balance at their center). For science learning to take place, incorrect prior knowledge has to be replaced, a process also known as *conceptual change* [30]. Exactly how to promote conceptual change remains a challenge in science education [31]. At the same time, there are two characteristics of conceptual change that are broadly acknowledged: (i) that existing knowledge structures have a strong tendency to persist, and (ii) that the experience of conflict can prompt conceptual change. We elaborate on each of these characteristics next.

#### *2.3.1 Truism 5: Knowledge structures resist change*

There is wide-spread agreement that existing knowledge structures can persist despite strategic changes in the learning context. The domain of science learning is packed with examples of such persistence of mistaken beliefs [32]. It appears as though existing knowledge can affect how one perceives the surroundings, even to the point of inventing improbable experiences (e.g., [33]). The example of beambalancing illustrates this peculiarity: It is as if children's beliefs about beam-balancing impedes their ability to take in conflicting experiences. In fact, Karmiloff-Smith and Inhelder [6] described children who actively ignored the evidence of a beam tipping over when they attempted to balance it in the middle.

#### *2.3.2 Truism 6: Perceived conflict facilitates conceptual change*

Conceptual change is possible when a pedagogy is used that highlights the shortcomings of the existing belief [34, 35]. The power of conflict can be traced to the work of Piaget [36], Festinger [37], and Dewey [38]. The argument is that perceived contractions generate conceptual conflict, which, in turn, serves as a catalyst for deeper forms of cognitive processing [39]. Presumably, children who hold a naïve belief about beam-balancing can experience conflict as they continue to play with the beams, which, in turn, might prompt them to replace their naïve belief.

#### **2.4 Summary of central characteristics of knowledge formation**

In the first part of this preliminary section, we sought to systematize the vast literature on knowledge formation in a way that highlights central characteristics of this process. On the question of the nature of knowledge, we honed in on the

**45**

*Exploring Links between Complexity Constructs and Children's Knowledge Formation…*

ideas that knowledge is organized (rather than existing as an isolated fixture) and that several distinct types of organization exist (rather than differing merely in content). On the question of learning, we honed in on the ideas that knowledge emerges via the active involvement of the learner (as opposed to being transmitted passively) and that learning is affected by the context, whether relevant or not (rather than be affected merely by what matters most). On the question of conceptual change, we highlighted the persistence of mistaken beliefs and the power of

*Proposed illustration of knowledge at three stages of the learning process (adapted from [1]). Shapes are thought to be exemplars of experiences represented in the mind. They become organized as conceptually competent knowledge develops. Complexity constructs provide further suggestions to consider.*

There are excellent sources available to introduce complexity science (e.g., [40–44]). The field of complexity science can nevertheless appear unorganized, featuring constructs that are not fully integrated with each other. It is not immediately obvious, for example, how constructs such as attractors, scale-free patterns, or synchrony relate to one another (or differ from each other, for that matter). This hinders progress on how complexity theory could help with knowledge formation. For this reason, we provide a review of selected complexity constructs. We have organized the list by the type of system that best exemplifies the selected constructs: (1) non-living systems, (2) living systems, and (3) thermodynamic systems

There are several non-living systems that have been used as model domains to explore complex systems, including cellular automata [45], oscillators [46], or electricity grids [47]. Common to all of these systems is that their elements interact with each other. The nature of this interaction is fixed, as is the nature of the elements in these systems. Yet, despite this simplicity, non-living systems can behave in complex ways. Constructs that have been explored in these systems include selforganization, chaos, hysteresis, attractors, autocatalysis, self-organized criticality,

Arguably at the heart of complexity science, self-organization is the process by which global patterns form through local interaction of the system's elements

*DOI: http://dx.doi.org/10.5772/intechopen.97642*

conflict to prompt conceptual change.

**Figure 1.**

(see **Table 2** for an overview).

*3.1.1 Self-organization*

**3. Glossary of central complexity constructs**

**3.1 Constructs from the study of non-living systems**

and scale-free patterns. We describe these constructs next.

*Exploring Links between Complexity Constructs and Children's Knowledge Formation… DOI: http://dx.doi.org/10.5772/intechopen.97642*

**Figure 1.**

*Theory of Complexity - Definitions, Models, and Applications*

knowledge [28, 29].

**2.3 Change of knowledge**

*2.2.2 Truism 4: Knowledge formation depends on the context*

change. We elaborate on each of these characteristics next.

tipping over when they attempted to balance it in the middle.

**2.4 Summary of central characteristics of knowledge formation**

*2.3.2 Truism 6: Perceived conflict facilitates conceptual change*

*2.3.1 Truism 5: Knowledge structures resist change*

Evidence suggests that learning is strongly dependent on the context, even when the context appears irrelevant to the specifics of what needs to be learned. Illustrative evidence of such context dependence can again be found with Piaget's conservation tasks: Despite showing robust performance in the classical version of the task, the mere number of times children were asked for a comparative judgment affected their performance (e.g., [27]). Evidence also comes from the balance-beam task: Children managed to balance the beams better with their eyes closed than with their eyes open [6]. Overall, context (e.g., cultural, societal, physical) has the ability to influence how the information ends up being utilized within the system of

A child presented with the balance-beam task is unlikely to enter the situation without prior knowledge. It might pertain to general ideas about what to expect, or it can pertain to very specific ideas about how to solve a task (e.g., the belief that beams balance at their center). For science learning to take place, incorrect prior knowledge has to be replaced, a process also known as *conceptual change* [30]. Exactly how to promote conceptual change remains a challenge in science education [31]. At the same time, there are two characteristics of conceptual change that are broadly acknowledged: (i) that existing knowledge structures have a strong tendency to persist, and (ii) that the experience of conflict can prompt conceptual

There is wide-spread agreement that existing knowledge structures can persist despite strategic changes in the learning context. The domain of science learning is packed with examples of such persistence of mistaken beliefs [32]. It appears as though existing knowledge can affect how one perceives the surroundings, even to the point of inventing improbable experiences (e.g., [33]). The example of beambalancing illustrates this peculiarity: It is as if children's beliefs about beam-balancing impedes their ability to take in conflicting experiences. In fact, Karmiloff-Smith and Inhelder [6] described children who actively ignored the evidence of a beam

Conceptual change is possible when a pedagogy is used that highlights the shortcomings of the existing belief [34, 35]. The power of conflict can be traced to the work of Piaget [36], Festinger [37], and Dewey [38]. The argument is that perceived contractions generate conceptual conflict, which, in turn, serves as a catalyst for deeper forms of cognitive processing [39]. Presumably, children who hold a naïve belief about beam-balancing can experience conflict as they continue to play with the beams, which, in turn, might prompt them to replace their naïve belief.

In the first part of this preliminary section, we sought to systematize the vast literature on knowledge formation in a way that highlights central characteristics of this process. On the question of the nature of knowledge, we honed in on the

**44**

*Proposed illustration of knowledge at three stages of the learning process (adapted from [1]). Shapes are thought to be exemplars of experiences represented in the mind. They become organized as conceptually competent knowledge develops. Complexity constructs provide further suggestions to consider.*

ideas that knowledge is organized (rather than existing as an isolated fixture) and that several distinct types of organization exist (rather than differing merely in content). On the question of learning, we honed in on the ideas that knowledge emerges via the active involvement of the learner (as opposed to being transmitted passively) and that learning is affected by the context, whether relevant or not (rather than be affected merely by what matters most). On the question of conceptual change, we highlighted the persistence of mistaken beliefs and the power of conflict to prompt conceptual change.

### **3. Glossary of central complexity constructs**

There are excellent sources available to introduce complexity science (e.g., [40–44]). The field of complexity science can nevertheless appear unorganized, featuring constructs that are not fully integrated with each other. It is not immediately obvious, for example, how constructs such as attractors, scale-free patterns, or synchrony relate to one another (or differ from each other, for that matter). This hinders progress on how complexity theory could help with knowledge formation. For this reason, we provide a review of selected complexity constructs. We have organized the list by the type of system that best exemplifies the selected constructs: (1) non-living systems, (2) living systems, and (3) thermodynamic systems (see **Table 2** for an overview).

#### **3.1 Constructs from the study of non-living systems**

There are several non-living systems that have been used as model domains to explore complex systems, including cellular automata [45], oscillators [46], or electricity grids [47]. Common to all of these systems is that their elements interact with each other. The nature of this interaction is fixed, as is the nature of the elements in these systems. Yet, despite this simplicity, non-living systems can behave in complex ways. Constructs that have been explored in these systems include selforganization, chaos, hysteresis, attractors, autocatalysis, self-organized criticality, and scale-free patterns. We describe these constructs next.

#### *3.1.1 Self-organization*

Arguably at the heart of complexity science, self-organization is the process by which global patterns form through local interaction of the system's elements


*Note: While the complexity constructs are listed under only one type of system, they apply to other systems as well (marked by X in the last two columns of the table).*

**47**

*Exploring Links between Complexity Constructs and Children's Knowledge Formation…*

[48]. When ordered structures are caused by self-organization, there is no blueprint or central control. Instead, the observed pattern is an emergent property [49]. The marking of sand dunes is an example of such self-organization. It stems from the "interplay of windborne transport, collision-driven piling up, and slope-shaving avalanches" ([50], p. 1084). Another example is the synchronization of adjacent metronomes that are initially out of sync. Eventually, the metronomes settle on a synchronized rhythm by virtue of sharing the surface they are placed on [51]. In each of these cases, the interaction among individual elements gives rise to overarching patterns that could be reduced neither to the elements

In chaotic systems, future behavior is sensitive to the initial conditions [52]. Chaos can be illustrated with the butterfly effect as a metaphor: A butterfly fluttering its wings over a flower in China can, in principle, cause a hurricane in the Caribbean [53]. A simple system that exhibits chaotic behavior is the double pendulum: Small differences in the initial angles of the pendulum arms are amplified several orders of magnitude in the course of just a few seconds [54]. Chaotic behavior is the result of the coming together of various factors that allow a change to become amplified (or dampened) as the change reverberates through the system. The result is unpredictable behavior of the system, despite having fully determinis-

Hysteresis describes a sudden change in behavior that is modulated by the system's history. Relevant here is the direction in which an outside parameter changes (from low to high, or from high to low). A thermostat provides an illustrative example of this phenomenon: Its function is to detect the temperature of the surrounding to control whether the heat should be on or off. Importantly, the change in the system's on–off status is not necessarily the result of an absolute outside temperature. Instead, the thermostat might have a different temperature threshold for switching the heating on than for switching it off [55]. This allows the thermostat to avoid repeatedly switching the heating on and off when the temperature hovers around the set point. The mathematical branch of *catastrophe theory* provides further specifications of the patterns of hysteresis, including how the presence of an additional outside parameter can modulate hysteresis (see also

An attractor is a state to which the system returns after having been perturbed away from it. Attractors come in several forms, the simplest of which is a point attractor. Consider, for example, a damped harmonic oscillator. The behavior of the oscillator depends on its mass, the spring stiffness, and the damping coefficient—all of which are referred to as control parameters. These parameters determine the details of the oscillator's resting states. If the oscillator were to be pushed away from its resting state, it will eventually return to it, thus demonstrating the state as an attractor for the system [57]. Other forms of attractors are periodic attractors (i.e., the cyclical moving through several stable states; *limit cycle*) and strange attractors (i.e., the non-periodic or chaotic movement through

*DOI: http://dx.doi.org/10.5772/intechopen.97642*

tic links among its individual elements.

nor the outside.

*3.1.2 Chaos*

*3.1.3 Hysteresis*

*cusp-catastrophe*; [56]).

*3.1.4 Attractors*

several states).

#### **Table 2.**

*Overview of selected complexity constructs, separated by type of system that exemplifies them best.*

*Exploring Links between Complexity Constructs and Children's Knowledge Formation… DOI: http://dx.doi.org/10.5772/intechopen.97642*

[48]. When ordered structures are caused by self-organization, there is no blueprint or central control. Instead, the observed pattern is an emergent property [49]. The marking of sand dunes is an example of such self-organization. It stems from the "interplay of windborne transport, collision-driven piling up, and slope-shaving avalanches" ([50], p. 1084). Another example is the synchronization of adjacent metronomes that are initially out of sync. Eventually, the metronomes settle on a synchronized rhythm by virtue of sharing the surface they are placed on [51]. In each of these cases, the interaction among individual elements gives rise to overarching patterns that could be reduced neither to the elements nor the outside.

#### *3.1.2 Chaos*

*Theory of Complexity - Definitions, Models, and Applications*

Self-organization The emergence of spatiotemporal

Chaos Behavior is highly sensitive to

Hysteresis A nonlinear shift takes place at a

Attractors A behavior toward which the

Self-organized criticality A state of the system in which

Affordance Sense-making of the

Synchrony System elements mutually

Balance/Equilibrium The system settles on an

Dissipation pressure System elements organize

Autocatakinetics System elements become

*(marked by X in the last two columns of the table).*

Teleodynamics The coming together of mutually

**Brief Definition Living** 

patterns through the interaction

initial conditions because of the amplification of interacting

moment in time that is affected by the cumulative history of the

several behavioral options are

**Brief Definition Non-Living** 

**Brief Definition Non-Living** 

Patterns are composed of elements that look similar or identical to the patterns they

surrounding depends on the action of the individual.

constrain each other as they interact in a circular way.

organization that is most probable given the existing distribution of energy.

themselves into patterns to dissipate the gradient established

increasingly more organized in the service of the dissipation

constraining processes that perpetuate each other, seemingly bestowing agency to structures.

*Note: While the complexity constructs are listed under only one type of system, they apply to other systems as well* 

*Overview of selected complexity constructs, separated by type of system that exemplifies them best.*

by energy clusters.

pressure.

The system carries out processes that contribute to its own self-maintenance.

of system elements.

constraints.

system.

available.

make up.

system navigates.

**Systems**

**X**

**X**

**X**

**X**

**Systems**

**Systems**

**X**

**X X**

**X X**

**Thermodynamics**

**Thermodynamics**

**X**

**X**

**X X**

**Living Systems**

**Constructs from Non-Living** 

Self-similarity (e.g., scale-free patterns, pink noise, fractals)

**Constructs from Living** 

Self-preservation (e.g., autopoiesis, centripetality)

**Constructs from Thermodynamic Systems**

**Systems**

**Systems**

**46**

**Table 2.**

In chaotic systems, future behavior is sensitive to the initial conditions [52]. Chaos can be illustrated with the butterfly effect as a metaphor: A butterfly fluttering its wings over a flower in China can, in principle, cause a hurricane in the Caribbean [53]. A simple system that exhibits chaotic behavior is the double pendulum: Small differences in the initial angles of the pendulum arms are amplified several orders of magnitude in the course of just a few seconds [54]. Chaotic behavior is the result of the coming together of various factors that allow a change to become amplified (or dampened) as the change reverberates through the system. The result is unpredictable behavior of the system, despite having fully deterministic links among its individual elements.

#### *3.1.3 Hysteresis*

Hysteresis describes a sudden change in behavior that is modulated by the system's history. Relevant here is the direction in which an outside parameter changes (from low to high, or from high to low). A thermostat provides an illustrative example of this phenomenon: Its function is to detect the temperature of the surrounding to control whether the heat should be on or off. Importantly, the change in the system's on–off status is not necessarily the result of an absolute outside temperature. Instead, the thermostat might have a different temperature threshold for switching the heating on than for switching it off [55]. This allows the thermostat to avoid repeatedly switching the heating on and off when the temperature hovers around the set point. The mathematical branch of *catastrophe theory* provides further specifications of the patterns of hysteresis, including how the presence of an additional outside parameter can modulate hysteresis (see also *cusp-catastrophe*; [56]).

#### *3.1.4 Attractors*

An attractor is a state to which the system returns after having been perturbed away from it. Attractors come in several forms, the simplest of which is a point attractor. Consider, for example, a damped harmonic oscillator. The behavior of the oscillator depends on its mass, the spring stiffness, and the damping coefficient—all of which are referred to as control parameters. These parameters determine the details of the oscillator's resting states. If the oscillator were to be pushed away from its resting state, it will eventually return to it, thus demonstrating the state as an attractor for the system [57]. Other forms of attractors are periodic attractors (i.e., the cyclical moving through several stable states; *limit cycle*) and strange attractors (i.e., the non-periodic or chaotic movement through several states).

#### *3.1.5 Self-organized criticality*

Self-organized criticality combines the ideas of self-organization and attractors, stating that systems maneuver themselves into a specific state, referred to as the *critical state* [58]. In a critical state, small perturbations can lead to large-scale or catastrophic changes in the system (e.g., [59]). Bak and Chen [59] proposed that the systems attracted to a critical state exhibit 1/*f* noise. The spectral density of the system's response to perturbation can be approximated as: Df ≈ 1/*f* α (with 0.50 < α < 1.50). A well-studied example includes earthquakes, both simulated and real ones.

#### *3.1.6 Self-similarity*

Self-similarity is present when the elements the elements resemble the very pattern that they make up [60]. The geometric shape known as the Sierpiński triangle is a famous example: Upon zooming in, the parts of the triangle resemble the triangle itself. The relevance of self-similarity lies in the relation among hierarchically nested patterns. In a self-similar pattern, there are no unique 'starter' elements, as each element is itself composed of entire patterns. That is to say, there is no characteristic scale at which the behavior of a system resides, an idea captured in *scale-free patterns* (see also *cumulative advantage*; [61, 62]). Self-similar patterns are relevant in the understanding of *fractals* and *power-law* distributions, also referred to as *pink noise*. Common to these terms is the idea that there is a long-range dependence among the different levels of organization in a system.

#### **3.2 Constructs from the study of living systems**

Like non-living systems, living systems consist of interacting elements that give rise to patterns of organization. Obvious examples include systems of individual animals (e.g., a school of fish, a flock of birds, an ant hill, a group of synchronizing fireflies) or of entire species (e.g., ecosystem). There are also systems within an individual animal, like when cells organize into an organ system [63, 64]. Given the interaction among elements, all of the complexity constructs identified for non-living systems apply here as well. For example, the organizations observed in these systems (e.g., nest building, foraging routes, behavior of crowds) stem from processes of self-organization. There is also evidence of hysteresis (e.g., the switch from fight to flight) and the presence of self-similar patterns (e.g., the branching of trees).

There is, however, a crucial difference between living and non-living systems: Rather than being fixed, elements in a living system can change (see also *complex adaptive systems* vs. *complex physical systems*; [41]). In other words, "living" elements can learn, adapt, and evolve, which, in turn, changes the relation they have to each other. In an ecological niche, for example, entirely new elements can appear (e.g., a new individual in a group), yielding new interactions and configurations. For this reason, some complexity constructs pertain only to living systems. We consider the constructs of affordance, synchrony, and self-preservation.

#### *3.2.1 Affordance*

An affordance is the opportunity for action that is made possible by the environment. The construct was developed by James Gibson as an explanation to how animals make sense of and navigate their surroundings [65]. An example of an affordance is the optic flow, a vector field of the perceived motion of static objects

**49**

describe these next.

*Exploring Links between Complexity Constructs and Children's Knowledge Formation…*

that is established through the movement of an animal. The optic flow does not exist entirely in the surrounding, nor is it a process of internal mental symbol manipulation. Instead, it is caused by the relative motion between an agent and the scene. Many insects have visual systems that are specialized for extracting optic flow. For example, a bee flying through a tapering corridor would experience an increase in translational flow as the corridor narrows, unless the bee slows

Synchrony refers to the coordination that takes place among the elements of a system (see also *circularity*, *interdependence*, *coupling*). While it can be found in nonliving systems (e.g., coupled metronomes), it has been studied extensively in living systems, including in the behavior of molecules, plants, animals, neurons, muscles, bodily regulations, and human relations [67, 68]. There is, in fact, an entire subfield of mathematics focused on theories related to synchrony—namely, to capture the degree to which elements affect each other's behavior in interdependent ways (see also *coupling strength*). When a system is tightly coupled, its elements coordinate closely with each other. In contrast, when a system is loosely coupled, its elements

Living systems appear to perpetuate their own organization autonomously, what Darwin famously referred to as a "struggle for existence" [69]. There are a number of complexity constructs that can be used to describe this process of self-preservation. The concept of *agency*, for example, captures the tendency to act on one's own behalf, thus contributing to a system's ability to maintain itself [70]. The concept of *autopoiesis* is another example of self-preservation. An example is the process by which the cells of an organism are able to reproduce and maintain themselves via the production of and interaction between individual elements [71]. Some autopoietic systems can even undergo *recursive self-maintenance* in which the agent is able to select from a variety of processes, depending on their environmental circumstances [72]. Yet another construct that captures selfpreservation is that of *centripetality*. This refers to a system's capacity to produce and maintain its own complexity by attracting resources into its circular patterns

A third set of complexity constructs stems from thermodynamic systems—systems that illustrate the laws of thermodynamics [74–76]. These systems consist of an energy source, a set of elements that are sensitive to the outside energy source, and a mutually constraining coupling among elements. An illustrative example is a pot of water placed on a burner: The heat from the burner constitutes the energy source; the water molecules are the elements (sensitive to the heat); and the push–pull movement among the water molecules captures their coupling strengths. Another example is an ecosystem [77]: The resources available in the surrounding constitute the energy source; the species of the ecosystem are the elements (sensitive to these resources); and the relations among the species (predator–prey; symbiotic) capture their coupling strength. Relevant constructs from these systems are that of balance, gradient dissipation, autocatakinetics, and teleodynamics. We

*DOI: http://dx.doi.org/10.5772/intechopen.97642*

have little to no effect on each other.

*3.2.3 Self-preservation*

of self-organization [73].

**3.3 Constructs from the study of thermodynamic systems**

down [66].

*3.2.2 Synchrony*

#### *Exploring Links between Complexity Constructs and Children's Knowledge Formation… DOI: http://dx.doi.org/10.5772/intechopen.97642*

that is established through the movement of an animal. The optic flow does not exist entirely in the surrounding, nor is it a process of internal mental symbol manipulation. Instead, it is caused by the relative motion between an agent and the scene. Many insects have visual systems that are specialized for extracting optic flow. For example, a bee flying through a tapering corridor would experience an increase in translational flow as the corridor narrows, unless the bee slows down [66].

## *3.2.2 Synchrony*

*Theory of Complexity - Definitions, Models, and Applications*

among the different levels of organization in a system.

constructs of affordance, synchrony, and self-preservation.

**3.2 Constructs from the study of living systems**

Self-organized criticality combines the ideas of self-organization and attractors, stating that systems maneuver themselves into a specific state, referred to as the *critical state* [58]. In a critical state, small perturbations can lead to large-scale or catastrophic changes in the system (e.g., [59]). Bak and Chen [59] proposed that the systems attracted to a critical state exhibit 1/*f* noise. The spectral density

0.50 < α < 1.50). A well-studied example includes earthquakes, both simulated and

Self-similarity is present when the elements the elements resemble the very pattern that they make up [60]. The geometric shape known as the Sierpiński triangle is a famous example: Upon zooming in, the parts of the triangle resemble the triangle itself. The relevance of self-similarity lies in the relation among hierarchically nested patterns. In a self-similar pattern, there are no unique 'starter' elements, as each element is itself composed of entire patterns. That is to say, there is no characteristic scale at which the behavior of a system resides, an idea captured in *scale-free patterns* (see also *cumulative advantage*; [61, 62]). Self-similar patterns are relevant in the understanding of *fractals* and *power-law* distributions, also referred to as *pink noise*. Common to these terms is the idea that there is a long-range dependence

Like non-living systems, living systems consist of interacting elements that give rise to patterns of organization. Obvious examples include systems of individual animals (e.g., a school of fish, a flock of birds, an ant hill, a group of synchronizing fireflies) or of entire species (e.g., ecosystem). There are also systems within an individual animal, like when cells organize into an organ system [63, 64]. Given the interaction among elements, all of the complexity constructs identified for non-living systems apply here as well. For example, the organizations observed in these systems (e.g., nest building, foraging routes, behavior of crowds) stem from processes of self-organization. There is also evidence of hysteresis (e.g., the switch from fight to flight) and the presence of self-similar patterns (e.g., the branching

There is, however, a crucial difference between living and non-living systems: Rather than being fixed, elements in a living system can change (see also *complex adaptive systems* vs. *complex physical systems*; [41]). In other words, "living" elements can learn, adapt, and evolve, which, in turn, changes the relation they have to each other. In an ecological niche, for example, entirely new elements can appear (e.g., a new individual in a group), yielding new interactions and configurations. For this reason, some complexity constructs pertain only to living systems. We consider the

An affordance is the opportunity for action that is made possible by the environment. The construct was developed by James Gibson as an explanation to how animals make sense of and navigate their surroundings [65]. An example of an affordance is the optic flow, a vector field of the perceived motion of static objects

α (with

of the system's response to perturbation can be approximated as: Df ≈ 1/*f*

*3.1.5 Self-organized criticality*

real ones.

*3.1.6 Self-similarity*

**48**

of trees).

*3.2.1 Affordance*

Synchrony refers to the coordination that takes place among the elements of a system (see also *circularity*, *interdependence*, *coupling*). While it can be found in nonliving systems (e.g., coupled metronomes), it has been studied extensively in living systems, including in the behavior of molecules, plants, animals, neurons, muscles, bodily regulations, and human relations [67, 68]. There is, in fact, an entire subfield of mathematics focused on theories related to synchrony—namely, to capture the degree to which elements affect each other's behavior in interdependent ways (see also *coupling strength*). When a system is tightly coupled, its elements coordinate closely with each other. In contrast, when a system is loosely coupled, its elements have little to no effect on each other.

## *3.2.3 Self-preservation*

Living systems appear to perpetuate their own organization autonomously, what Darwin famously referred to as a "struggle for existence" [69]. There are a number of complexity constructs that can be used to describe this process of self-preservation. The concept of *agency*, for example, captures the tendency to act on one's own behalf, thus contributing to a system's ability to maintain itself [70]. The concept of *autopoiesis* is another example of self-preservation. An example is the process by which the cells of an organism are able to reproduce and maintain themselves via the production of and interaction between individual elements [71]. Some autopoietic systems can even undergo *recursive self-maintenance* in which the agent is able to select from a variety of processes, depending on their environmental circumstances [72]. Yet another construct that captures selfpreservation is that of *centripetality*. This refers to a system's capacity to produce and maintain its own complexity by attracting resources into its circular patterns of self-organization [73].

## **3.3 Constructs from the study of thermodynamic systems**

A third set of complexity constructs stems from thermodynamic systems—systems that illustrate the laws of thermodynamics [74–76]. These systems consist of an energy source, a set of elements that are sensitive to the outside energy source, and a mutually constraining coupling among elements. An illustrative example is a pot of water placed on a burner: The heat from the burner constitutes the energy source; the water molecules are the elements (sensitive to the heat); and the push–pull movement among the water molecules captures their coupling strengths. Another example is an ecosystem [77]: The resources available in the surrounding constitute the energy source; the species of the ecosystem are the elements (sensitive to these resources); and the relations among the species (predator–prey; symbiotic) capture their coupling strength. Relevant constructs from these systems are that of balance, gradient dissipation, autocatakinetics, and teleodynamics. We describe these next.

#### *3.3.1 Balance*

Thermodynamic systems move toward a state in which forces are balanced (also referred to as *homeostasis* or *equilibrium*). Grounded in fundamental laws of physics, balance exists when there is no longer any net change in forces, influences, and/or reactions. In that sense, thermodynamics offers a traceable endpoint to behavior (a purpose, so to speak), namely, in achieving balance. Outside of physics, balance is also used to indicate steady or stationary conditions in branches such as evolution, economy, and social sciences [78]. An example of balance is captured in the term of *ascendancy*, which is the degree of relative stability in an ecosystem, shown to increase over evolutionary timescales [79–81].

#### *3.3.2 Dissipation pressure*

In addition to endowing systems with the purpose of reaching a balance, thermodynamics also identifies the conditions necessary for systems to do so: The push toward balance comes from the presence of clustered energy. This is because the presence of clustered energy, in addition to affecting the system, also sets up a gradient that needs to be dissipated (captured in the second law of thermodynamics; [82]). For example, the mere presence of clustered heat in a cup of tea sets up a gradient to be dissipated (i.e., the heat clustered in the cup will eventually disperse to reach thermal equilibrium). This pressure to dissipate an energy gradient can push the system to create micro-clusters of energy. In boiling water, for example, water molecules organize themselves into vapor pockets that contain some of the heat (see also *morphodynamics*; [83]). Put differently, the pressure to dissipate an energy gradient provides opportunities for the system to organize itself (see also *antifragility*; [84]).

#### *3.3.3 Autocatakinetics*

Under some circumstances, systems become increasingly more ordered, seemingly going against the push for dissipation of clustered energy. Animals and plants, for example, appear to pursue the survival of their species, coming up with increasingly more efficient ways to harness and retain resources. These systems are known to be autocatakinetic [85]. **Figure 2**, adapted from Swenson [85], illustrates how the emergence of progressively more organized forms of a system is possible under the law of maximum entropy production. An external energy source (i.e., one that is outside of a local, open system) clusters to create an energy gradient that must be dissipated in order to reach entropic balance in the broader (closed) global system. In moving toward dissipation, a second cluster of energy emerges in the local system, composed of the self-organized behavior of the system's elements. This energy cluster, in turn, defines another energy gradient, hence another push toward dissipation that contributes to the entropic balance of the global system.

#### *3.3.4 Teleodynamics*

Teleodynamics is yet another principle that seeks to explain how elements of a system become increasingly more ordered, despite the push toward maximum entropy [83, 86]. The idea is that order is perpetuated when mutually supporting processes come together. A so-called *autocell* (or *autogen*) is a model system that can illustrate this idea. This model is based on two processes, that of autocatalysis (i.e., the mutual facilitation of two or more chemical reactions) and that of containment (i.e., the forming of enclosures from the biproduct of the autocatalytic reactions). The interaction of these two processes (i.e., autocatalysis and containment) allows

**51**

*Exploring Links between Complexity Constructs and Children's Knowledge Formation…*

each of them to continue, even as reactants are used up and the enclosures break apart. The outcome is a self-repair and self-replication of sorts (also see *hypercycles*,

*Illustration of autocatakinetic closure, adapted from Swenson [85]. The solid frame defines the boundary of a global (closed) system. The dashed circle defines the boundary of a local (open) system within the global one. The energy source E1 defines an energy gradient (ΔE1) that needs to be dissipated (F1) to reach entropic balance (ΔS). In moving toward dissipation, a second cluster of energy emerges (E2), which consists of the self-organized behavior of the system's elements. This second energy cluster, in turn, defines an energy gradient* 

In the second part of this preliminary section, we sought to review central complexity constructs in a way that facilitates the attempted link between complexity and knowledge formation. In total, we selected over a dozen complexity constructs, some of which apply to all systems (e.g., self-organization, attractors), and some of which apply to some systems exclusively (e.g., agency, hysteresis). For each of these terms, we offered an explanation at the level of phenomenology, bypassing mathematical advances. Emphasis was placed on providing a general sense of the concepts with explanations that were broad enough to subsume several complexity

The link between cognition and complexity is invoked often, as the quote at the top of the paper suggests (see also [90–94]). However, it is not always clear if the ideas are applied consistently, as neither the field of cognition nor the field of complexity is straightforward. Having provided an organization of both areas (Sections 2 and 3 above), we are in the position to address the link systematically.

There are several complexity constructs that anticipate knowledge being organized. *Self-organization* is one of those constructs—the idea that elements of a

*DOI: http://dx.doi.org/10.5772/intechopen.97642*

*autogenesis*, *negentropy ratchet*; [87–89]).

*(ΔE2) and, thus, another push toward dissipation (F2).*

**Figure 2.**

**3.4 Summary of central complexity constructs**

constructs (e.g., synchrony vs. coordination).

**4. Cross-tabulation of knowledge and complexity**

**Table 3** provides an overview of our cross-tabulation.

**4.1 Complexity links to the truisms of knowledge formation**

*4.1.1 Link 1: Complexity in the structural organization of knowledge*

*Exploring Links between Complexity Constructs and Children's Knowledge Formation… DOI: http://dx.doi.org/10.5772/intechopen.97642*

#### **Figure 2.**

*Theory of Complexity - Definitions, Models, and Applications*

increase over evolutionary timescales [79–81].

*3.3.2 Dissipation pressure*

*3.3.3 Autocatakinetics*

*3.3.4 Teleodynamics*

Thermodynamic systems move toward a state in which forces are balanced (also referred to as *homeostasis* or *equilibrium*). Grounded in fundamental laws of physics, balance exists when there is no longer any net change in forces, influences, and/or reactions. In that sense, thermodynamics offers a traceable endpoint to behavior (a purpose, so to speak), namely, in achieving balance. Outside of physics, balance is also used to indicate steady or stationary conditions in branches such as evolution, economy, and social sciences [78]. An example of balance is captured in the term of *ascendancy*, which is the degree of relative stability in an ecosystem, shown to

In addition to endowing systems with the purpose of reaching a balance, thermodynamics also identifies the conditions necessary for systems to do so: The push toward balance comes from the presence of clustered energy. This is because the presence of clustered energy, in addition to affecting the system, also sets up a gradient that needs to be dissipated (captured in the second law of thermodynamics; [82]). For example, the mere presence of clustered heat in a cup of tea sets up a gradient to be dissipated (i.e., the heat clustered in the cup will eventually disperse to reach thermal equilibrium). This pressure to dissipate an energy gradient can push the system to create micro-clusters of energy. In boiling water, for example, water molecules organize themselves into vapor pockets that contain some of the heat (see also *morphodynamics*; [83]). Put differently, the pressure to dissipate an energy gradient provides opportunities for the system to organize itself (see also *antifragility*; [84]).

Under some circumstances, systems become increasingly more ordered, seemingly going against the push for dissipation of clustered energy. Animals and plants, for example, appear to pursue the survival of their species, coming up with increasingly more efficient ways to harness and retain resources. These systems are known to be autocatakinetic [85]. **Figure 2**, adapted from Swenson [85], illustrates how the emergence of progressively more organized forms of a system is possible under the law of maximum entropy production. An external energy source (i.e., one that is outside of a local, open system) clusters to create an energy gradient that must be dissipated in order to reach entropic balance in the broader (closed) global system. In moving toward dissipation, a second cluster of energy emerges in the local system, composed of the self-organized behavior of the system's elements. This energy cluster, in turn, defines another energy gradient, hence another push toward

dissipation that contributes to the entropic balance of the global system.

Teleodynamics is yet another principle that seeks to explain how elements of a system become increasingly more ordered, despite the push toward maximum entropy [83, 86]. The idea is that order is perpetuated when mutually supporting processes come together. A so-called *autocell* (or *autogen*) is a model system that can illustrate this idea. This model is based on two processes, that of autocatalysis (i.e., the mutual facilitation of two or more chemical reactions) and that of containment (i.e., the forming of enclosures from the biproduct of the autocatalytic reactions). The interaction of these two processes (i.e., autocatalysis and containment) allows

*3.3.1 Balance*

**50**

*Illustration of autocatakinetic closure, adapted from Swenson [85]. The solid frame defines the boundary of a global (closed) system. The dashed circle defines the boundary of a local (open) system within the global one. The energy source E1 defines an energy gradient (ΔE1) that needs to be dissipated (F1) to reach entropic balance (ΔS). In moving toward dissipation, a second cluster of energy emerges (E2), which consists of the self-organized behavior of the system's elements. This second energy cluster, in turn, defines an energy gradient (ΔE2) and, thus, another push toward dissipation (F2).*

each of them to continue, even as reactants are used up and the enclosures break apart. The outcome is a self-repair and self-replication of sorts (also see *hypercycles*, *autogenesis*, *negentropy ratchet*; [87–89]).

#### **3.4 Summary of central complexity constructs**

In the second part of this preliminary section, we sought to review central complexity constructs in a way that facilitates the attempted link between complexity and knowledge formation. In total, we selected over a dozen complexity constructs, some of which apply to all systems (e.g., self-organization, attractors), and some of which apply to some systems exclusively (e.g., agency, hysteresis). For each of these terms, we offered an explanation at the level of phenomenology, bypassing mathematical advances. Emphasis was placed on providing a general sense of the concepts with explanations that were broad enough to subsume several complexity constructs (e.g., synchrony vs. coordination).

#### **4. Cross-tabulation of knowledge and complexity**

The link between cognition and complexity is invoked often, as the quote at the top of the paper suggests (see also [90–94]). However, it is not always clear if the ideas are applied consistently, as neither the field of cognition nor the field of complexity is straightforward. Having provided an organization of both areas (Sections 2 and 3 above), we are in the position to address the link systematically. **Table 3** provides an overview of our cross-tabulation.

#### **4.1 Complexity links to the truisms of knowledge formation**

#### *4.1.1 Link 1: Complexity in the structural organization of knowledge*

There are several complexity constructs that anticipate knowledge being organized. *Self-organization* is one of those constructs—the idea that elements of a

#### *Theory of Complexity - Definitions, Models, and Applications*

complex system organize themselves. There is indeed evidence of self-organization in cognitive activity. For example, the idea of self-organization has been invoked to address the origins of language (e.g., [70]), to observe the emergence of knowledge (e.g., [95]), to explain the systematic problem-solving behaviors of infants (e.g., [96, 97]), and to apply effective pedagogy [98]. Hence, it is reasonable to assume that knowledge is self-organized.

Another complexity construct that anticipates knowledge organization is *self-similarity*—the idea that an organized pattern repeats itself at various nested levels. Here too there is evidence that self-similarity applies to cognition. It was studied primarily by looking for *scale-free patterns* in cognitive behavior [99]. The signature of scale-free pattern is a *1/f scaling*, also known as *pink noise* (e.g., [100, 101]). Analyses of the variability in reaction time have revealed pink-noise patterns, indicating that the variability in a short time series is similar to that in a longer time series (e.g., [102]). Hence, it is reasonable to assume that knowledge is organized in scale-free patterns.
