The below piece i wrote for a final year paper, its a bit long and probably a year out of date in regards to ecological dynamics and the language used around it. The whole thing is from the ‘ivory tower’ and is almost intentionally removed from the practical world of coaching. But hopefully prompts some interesting thoughts from the models used.
A focal point of understanding behaviour in sport is through the behaviours, actions and interaction conducted by individuals and collective groups. Existing models of observation in ecology (Madin, 2007), and computational models (McCellend et al, 1995) have focused on relatively static, homogeneous situations consisting of very few interacting components (Miller, 2009). However, in recent advancements in technology, qualitative/quantitative crossed study and broadening scopes across a vast array of scientific fields has allowed complex systems to become a recognised phenomenon. Yet it has been cited that a clear lack of empirical, cutting edge research continues to mystify this area (Goldenberg et al, 2001). The science underpinning complex happenings however allows practitioners, coaches and researchers the ability to explore the behaviours observed between dynamically interacting, competing and co-operating agents (Miller, 2009). This long read (bit of a ramble), before further discussion must address one particular problematic area in regards to system based perspectives that plagues all areas of understanding, from skill acquisition to performance analysis.
There are two conflicting systems; complicated versus complex, which can often create confusion, yet consist of hugely conflicting properties (Zadeh, 1993; Haken, 2006). As Miller (2009) discusses, complicated systems consist of characteristics that maintain a degree of independence from one another. If an element was removed from the system structure, there may not be any influence on the behaviour of the system unless the component constituted the outcome of the system in which the system would simply fail. In more complex systems, individual components mutually negotiate and (re)act/respond on the feedback and perception of the other components (Mason, 2008). Removal of an element will cause the system to adapt and continue to function with different structural properties due to its non-linear relationships.
Corning (1998) proposed three key characteristic of a complex system; firstly, a complex system should comprise of multiple agents; secondly, the agents negotiate the relationships with other agents to organise towards constraints within the environment. Thirdly, interactions will create novel behaviour, which may appear chaotic; this behaviour is mutually combined through synergetic relationships between the agents. The rest of this paper will answer brief discussions that have occurred over a career in coaching. This paper aims to cross-theorise to other fields to create arguments for their potential use in coaching and teaching of complex team behaviour. It is reminded that these theories will be discussed but not deeply unpicked, it becomes the purpose of the reader to search further for more detailed and informed discussions.
Why should we view coaching and team sports from a global perspective?
From a global understanding of complex systems, traditional discourses are rejected in regards to understanding system and team behaviour by reducing the system to a single occupant’s behaviour. The system must be viewed as a complex global phenomena opposed to a single local property. Thus, approaching a system on a global scale allows us to draw upon an ecological understanding towards ecosystems and patterns of behaviour in wildlife and nature. As noted in DeAngelis (1992), an ecosystem is often defined as “a group of interacting, interrelated, or interdependent elements… forming or regarded as forming a collective entity” (p. 17). This definition, crossed with the general understanding of complex systems from this paper, allows the proposal that game formats and learning environment constitute similar properties to complex ecosystems.
Dynamically Complex Eco-Systems
Rosser (2008) proposed Dynamic Complexity, suggesting that systems are dynamically complex if they are structurally and perpetually limitless, caused from internal structures of the negotiating agents. Rosser (2008) goes on, discussing how systems that are dynamically complex are open to fluctuations and complex chaotic behaviour due to multi-stability of the basins of attraction which create irregular pattern formations. Because a Dynamically Complex Ecosystem is in a constantly negotiable state, it is to be assumed that it is also an adaptive one (Bourbousson, 2010).
As Madin (2007) discussed, an adaptive system is one that has the ability to adapt its parameters to suit a new property and therefore constitute a new optimal behaviour. Holland (1995) furthers this, suggesting that a key feature of an adaptive system is that behaviour is negotiated by the actions of collective agents in the system, exhibiting their local behaviours to create a dynamically robust network of information, in which encompasses complex systems behaviour.
From this framework, the role that each agent possesses must be appreciated to analyse the macro behaviour of an aggregated set of agents without simply determining their single jobs. As Jennings (2001) discusses, agents hold a certain amount of control and observability allowing them to appropriately and socially interact effectively within their environment. The agents ability to manipulate their particular space also defines them as clear problem solving beings (Newell, 1972). Agents are naturally social beings, allowing them to constantly evolve, creating holistic behaviour (Light, 2014). These behaviours are determined by continued interaction with the environment that the agent is situated within, of which allows them to predict and respond. The social interactions that they possess exacerbate the actions as agent relations become more robust and information becomes shared and aggregated between a collection of social agents. Similarly to the discussion of Dynamic Complexity, social agents act in similar chaotic ways, demonstrating their complexity for which fuels the spontaneity of the systems behaviour (Sawyer, 2005).
How does each individual compromise as one being?
The Eight-fold Pathway to Agent Behaviour
The Eight-fold pathway, originated as a Budhist philosophy in which to live life by- the eight ‘rules’ demonstrate a framework to associate behaviour by (Truths, 2000; Marlatt, 2003). Miller (2009) attempted to associate this philosophy to the analysis of agent behaviour in complex systems. From the association of the two concepts, it must be approached with a broad lens, to allow simplicity in challenging complexity to better understand this phenomenon. The pathway will be briefly summarised (a more detailed mapping can be seen in Miller, 2009) to associate the rules of the eightfold pathway to the characteristics of the agents with in a complex system.
- Right view- The right view embodies an agents received information from the environment.
- Right Intention- The agent will have certain goals, by manipulating an agents intention a coach can afford particular behavioural outcomes.
- The communication network in which information is transferred to regulate action and which aggregates a set of shared local behaviour to constitute a global action.
- Right Action- Each individual agent possesses information from the environment, each agent will process and then create (non)action which will create further information to influence other local agents.
- Right Livelihood- The actions of the agents must aggregate towards a global system behaviour that will benefit the local agents in some way.
- Right Effort- Agents actions will form complex or simple patterns, the ability of a correlated number of agents to associate the most efficient and economic to the task will create more effective global behaviour.
- Right Mindfulness- The capability of the agent to adapt and learn based on the feedback from the environment towards a more optimal state.
- Right Concentration- The capacity of the global collection of agents to capture the phenomenon of interest in that particular state-space of the environment.
These properties of agent behaviour all demonstrate that agents are not individual components, but rather sophisticated, complex and collectively intelligent (Sawyer, 2005). In order for these local agents to dictate global behaviour, a set of aggregated affordances and values are required. It is thus proposed that in invasion game and team sport settings, teams and collective groups of individuals can be viewed as the Aggregated Individual. This aggregated behaviour evolves from the principle of emergence in which the behaviour forms from localised, individual agents (Choi, 2001).
The Law of Large Numbers and the Aggregated Individual
The Law of Large Numbers theorem details how the average of the results obtained from a large number of trials should be close to the expected value (Hsu, 1947). Originally digressed by S.D. Poisson, a mathematician in the 1500’s, the law deals with probability outcomes across large trial numbers on relatively stable properties; a regular example being that of rolling a dice (Artstein, 1975). This paper however, will approach this theorem from a different perspective. The Law will be used as a general set of parameters in which to quantify emergences of aggregated behaviour, within a dynamically complex ecosystem.
From this complex lens, it must be understood that according to the Law of Large Numbers, the probability that the mean behaviour will differ from the aggregated behaviour by a large fluctuation will decrease as the number of agents increase (Miller, 2009; Reichl, 1980; Doucet, 2001). Thus, because humans are extremely variable and non-linear, each will possess their own variable properties. To battle this issue, the theory of shared affordances can be drawn upon. Silva et al (2013) contested that team co-ordination is constructed on the development of a set of shared knowledge about the performance context, allowing for non-linear, complex patterns of behaviour to emerge as the aggregated individual negotiates the affecting constraints. If a correlated unit of individual agents are in an environment they will have a shared set of affordances that they will search for to initiate optimal behaviour in that moment on a global scale (Hmelo, 2000). Thus, from the Law of Large Numbers, each individual’s behaviour is summarised by a random variable, say X for example. X is mutually independent, has a common distribution and a mean equal to μ. If complexity of the system was changed (add more agents) from X(n=2) to X(n=10) variability decreases as the probability of the aggregate behaviour becomes the most centralised dependant. Because variable differences between agents; be it culturally, socially, mechanically but not limited to these are so variant that as the agents are increased, individual variance is so sparse and uncorrelated the most contextually stable variable becomes centralised and thus becomes the afforded aggregate of all the agents. Due to vast degrees of freedom possessed by dynamic systems, the most centralised behaviour must be exploited to aggregate the behaviour across a global scale. The Law of `Large Numbers Theory allows agents to be varied in all other characteristics and still the mean will maintain the most centralised and aggregated behaviour (Miller, 2009).
Uhlig (1996) offers further insight to the practicalities of this theorem, suggesting that if more independent agents are added to the eco-system, the variables of the stochastic elements will average out towards the aggregated behaviour. With single or very few agents it becomes impossible to predict behaviour because individual variation overwhelms predictable properties (Kahneman, 1972). Imagine a simple game task in which two people were working on, the variability in each of the learning systems may overwhelm the task at hand, the system is relatively simple and possesses little information in which to aggregate localised behaviours to a group affordance. It is to be conceded that this quite abstract framework may not appropriately consider the complexities of a neuro-biological system due to its spontaneous critical fluctuations and non-linear relations with the world (Kaneko, 1990). It thus becomes the responsibility of the coach to centralise aggregated behaviour across a variety of liberal components upon a variety of performance contexts. It is therefore proposed that the role of the coach is to unify a set of aggregated behaviours across a global scale, but also across dynamic sub-units within action. The key design of learning environments to constitute the development of behaviours and optimal states in which teams can globally operate becomes the most relevant property of learning.
How do we design training to allow for aggregated behaviour and shared affordances?
Predicting the Unpredictable: Applying The Boolean Function to Organised Decision Making
All complex actions can be coded through binary coding to analyse detailed perceptions and actions of complex agents (Miller, 2009). If each piece of information is considered as a single binary code, then a string of code can be formed to detail the actions of the agent. On a simplified scale, if a player in a football match received the ball at their feet they would automatically receive information from all the variables in the ecosystem at that given time. Each piece of information would be computed as a piece of binary code and computed as a binary string, which regulates behaviour. Something as complex as a 1 second state space becomes impossible to calculate without an equally complex computer model to create such a complex algorithm. However, it still holds precedence to allow coaches to not just analyse behaviour, but form environments that will allow for learners to negotiate certain binary formats. Much like the possibilities of a Rubik’s cube, certain patterns will solve the cube quicker than others, similarly to how certain patterns of play will create more effective economic solutions to the environment problem. From the potential millions of codes for a one second frame in football match, as a coach it would be desirable for them to aggregate certain codes across a collection of agents afforded to them by the individuals ability to process relevant information (Storey, 2013; Chow, 2015). If 0 codes principally take the low risk options and 1 codes are high risk, the coach will want to create a binary string consisting of similar coded patterns. For example, a binary string of decisions may look, on an extremely simplified case, like so; 000101010010000. This particular string may constitute a half second phase that the learner has processed yet the string shows predominantly 0 decisions, which are low risk. However, it the context changes and the team are now losing, the system may adapt on the basis of the new information, this new binary string may hold a more consistent pattern of 1’s as they look to play high risk football to score. If the team as an aggregated individual throughout a match created strings of predominantly 0 then a coach is then able to unpick and analyse the processes of the players and whether they correlate with the teams coaching principles. In the Boolean theorem, each function defines a problem that can be solved (Albert, 2000; Kauffman, 1984). Each code constitutes a different variable piece of information, which creates an ensemble of other variable outcomes. Hence, each function, or piece of code, will need to be processed by the agent locally and by the aggregated individual globally to create an optimal state of organisation that will benefit local agents to satisfy cooperating units of agents (Zeigler, 2000).
Miller (2009) asks the question ‘Do organisations just find solvable problems?’ this question is extremely important when discussing aggregated system behaviour. Within a dynamically complex ecosystem spontaneous yet still predictable behaviour can emerge. Yet, it is suggested here that the aggregated individual (or organisation) will centralise compatible behaviours to approach the problem. As discussed earlier, an agent is capable of learning from feedback and intuition centralised by information in the environment (Storey, 2013). From a contextual sense, an attacking team may deliberately exploit wide areas if they have received information to suggest that this is a particular weakness of the defence. Yes, the aggregated individual has exploited a solvable problem but this does not necessarily constitute a weakness of the system, rather it promotes a new adaptive quality of the system.
If the coach can create dynamically complex eco-systems which afford key binary strings that are desirable and disregard those that are undesirable then the aggregated individual can adapt to its performance environment and solve a multitude of tasks because of localised, stored information sources that can regulate action.
Calculating Power Laws to Regulate Affordances in Actions
Any complex, living system is subject to a power law when;
Probability is equal to (X=x)-x-k (Miller, 2009)
To give a limited but informed overview, if x is the number of occurrences of a particular size happening, then a power law is used to infer that the likelihood of the event size is proportional to the event size raised to the kth power. A simple statistical example being that if k=1, events of size 100 are 1/100 as likely of size 1. This basic equation demonstrates that systems that are governed by Power Laws are characterised by infinite number of small events and very few major incidents (Miller, 2009). In a meta-stable system the attractors would want to draw behaviour towards certain reoccurring events (Davids, 2013). A coach may want the team to be more effective at transitioning from defence to attack when possession is won. The practice design must then illuminate opportunities that inhibit lots of ball turnovers to allow for lots of exploration of this particular phase of play. The transitioning act would be the key attractor; a coach could therefore use a power law equation to calculate the probability of this event occurring in the particular environment design and thus manipulate accordingly. From this complex calculation a coach can attempt to manipulate the environment design in order to increase the number of occurrences of the event, thus affording more opportunities for the system to be drawn towards the key attractor.
The Edge of Chaos
Packard (1988) and Langton (1990) both proposed that any system that is poised at the edge of chaos possess the ability to create emergent stochastic behaviour. The edge of chaos becomes a minute space-time paradigm that is crucial for the system to enter an adaptive, complex and fragile state. Gladwell (2002) notioned this as a tipping point, a state in which abrupt shifts in system behaviour occur. Because all complex behaviour is deterministic, a small change in the system may constitute a global scale change in the aggregated individual, consequently causing a dynamic shift upon its binary string of information processes (See Davids, 2013 for a more focused discussion). If a system is too simple then the behaviour is accordingly static, on the other perspective any system that is too active will become inherently chaotic and unmanageable (Davids, 2003).
In the designing of dynamically complex eco-systems, a coach must consider this crucial principle, designing to ensure that the system is poised at a critical point situated at the edge of chaos where systems behaviour is observably random, but analytically predictable (Passos, 2008). If the designed eco-system isn’t capable of providing consistent questions then the system is too chaotic, if you placed novice invasion game players in to an environment with a large number of opponents and team mates with a multitude of performance outcomes the information on a localised level becomes too difficult to process and reorganise to suit the aggregated behaviour. Rather, if novice players were placed in to a reduced scale 3v1 situation, the information becomes much simpler to process and create an aggregated behaviour towards the answer to the question.
With system behaviour, from a computational perspective, the role of the system is to answer questions (Gee, 2005). Irrespective of the fact that these theoretical positions have been doctored to a sporting domain the principle remains the same, the questions now become less explicitly verbalised, propose a question to the unit of players and they will set out to answer it.
The Self-Organising System
Bak (1996) made empirical steps forward into self-organised criticality in systems. As Bak (1996) stated, agents within the system act aggregately to congregate towards critical states, the tipping point. These states that the system organises towards are fragile and influenced largely by small fluctuations that consequently cause global impacts (Light, 2014; Davids, 2013). In order to understand these critical periods, the system’s ability to self-organise must be understood first.
Self-organisation is inherently demarcated by the organisations ability to create organised decision making practices, the ability of the system to converge to a single critical moment inhibits a state in which a transitional moment of behaviour will occur (Paczuski, 1999; Gréhaigne, 1995). Cilliers (1998) defined self-organisation as a characteristic of any complex system, which allows them to reorganise its internal structure under adaptive conditions in order to maintain an optimal state in its environment. As discussed previously, individual agents have the capacity to understand, process and act upon local information; this information is aggregated by a number of units to organise internally towards the optimum state (Miller, 2009; Chow, 2015). A defensive unit in football has localised information that the attacking team attack through the middle of the field. On a global scale, the defending team will organise internally based on an aggregated knowledge; as a consequence of this, the team will squeeze out the middle of the pitch and isolate the central attacking areas. The defensive unit act upon a cue, responding to refined information, for example the team may have a shared knowledge that if the ball carriers head is up there is a high chance of a long ball being played. These localised individual responses of the agents are what initiate a global aggregated behaviour.
Hebbs rule of self-organisation (Linsker, 1988) fits comfortably with the practical example just discussed. Hebbs rule suggests that the more present that a certain particular information flow is, the system will start to recognise the properties of this information and begin to have a more refined response caused by the processing of this information on a more regular basis (Rubner, 1989). Certain patterns of information will begin to string together allowing the system to constitute a behavioural pattern in which to respond to the information afforded by the opposition. Bak and Chan (1995) very interestingly studied a sand pile. As more sand is poured onto the pile it reaches a critical height, a height that cannot be passed as avalanches of sand prevent the pile increasing in height. This is a state of criticality, challenging what has previously been discussed with self-organising systems, it is suggested that a system never reaches a true equilibrium, rather it jumps from one meta-stable state to the next as the system is constantly shifting and evolving as a process of adaptation (Nicolis, 1977; Chow, 2011). A system is inherently attracted to a sensitive state in which external inputs can have drastic system effects that will constitute a self-organised action (Davids, 2013). As discussed by Passos (2008) as two moving individuals begin to close in on each-others space there reaches a critical moment where the smallest of changes in either system can cause a global impact on this particular dyad (the attacker dropping a shoulder at the last minute will destabilise the defending system).
The state-space principle identifies how independent variables in a system create their own dimension of which upholds the dynamic of the system (Friedland, 2012; Siljak, 2011). The variables that form a trajectory (or an aggregated individual) will either lead towards or away from particular spaces in the system (Miller, 2009). If the trajectories lead towards, then the system is stable with one single determinant of behaviour; this space represents the role of the attractor. If the trajectory leads away from this particular space then this space acts as a repellor, spiralling the system into abundant chaos (Davids, 2013).
Due to the huge number of degrees of freedom that a system possesses, the state-space becomes limitlessly dimensional and complex (See Bernstein, 1967 for further study). If all the variables lead towards a single attractor the system becomes exceedingly rigid and incapable of tangible outcomes with such set and linear behaviours. An optimal system will be situated within meta-stability, attractor’s and repellors vary in weight allowing spontaneous behaviour caused by the constantly negotiating environment (Miller, 2009; Butler, 2013; Renshaw et al, 2010). In the role of dynamically complex ecosystems, their design must inhibit properties of meta-stability, allowing systems to pertain to certain system attractors yet still have the capacity to spontaneously deviate from any given course.
The Rubik Cube Model to Designing Dynamically Complex Eco-Systems
The design of dynamically complex eco-systems can be approached similarly to the task of a Rubik cube. Picture two learners, both blindfolded, using a new cube, one dis-organises the cube creating a problem. The other learner receives the cube and removes the blindfold and aims to solve the problem. There is one possible outcome; return the puzzle to its normal state of which constitutes success. Once this task has been repeated say, 15 times, the learner would have completed the task in 15 different ways all returning to the same outcome. This particular model is used, although a simple one, because there are 43 252 003 274 489 856 000 different ways in which to solve a Rubik cube (Davis, 1982; Bandelow, 2012). Contextualise this to a sporting environment, one team is defending a corner, the first trial will set a problem for the defensive side in which they organise to clear the ball from danger. Upon the second trial, the defending system may have learnt some behaviours from the attacking system, yet there are still all number of possible outcomes because of the complexity of the eco-system; weather, pressure, emotional stress, fatigue may all constitute to a slightly different pattern of behaviour, yet towards the same aggregated goal of the team. If coaches can then adopt a state of mind in which to approach the designing of games like the disorganisation of a Rubik cube, affording the same outcome through different task requirements. By layering affordances, triggers and cues the coach can manipulate the localised behaviour of the system towards centralised global behaviours to work as one system constituting of numerous sub-systems towards the task outcome. If a coach wanted their team to play quicker on the counter attacks they would need to create an environment that afforded game play that allowed the outcome and performance solution to be counter attacks, this way the team would develop a set of desirable aggregated behaviours.
For a more applied model, treating the environment as a dynamically complex eco-system gives coaches greater control on the design and manipulation.
Coaches can view their environments as a solved problem, like the Rubik cube. Then by rewinding and layering the problem the coach can create environments with meta-stable properties to inform organised decision-making,
Upon the construction of a dynamically complex ecosystem, the layering of environment affordances will infuse key attractors with more weighting of which will direct the focus of individual agents and global sub units. Drawing on the theoretical proposals from this paper, coaches can begin to create complex designs to allow for appropriate affordance driven pedagogy. If coaches identified with the below categories to framework there environment design, more robust and decision rich workspaces will be created.
- Aggregated individual and Shared Affordances: The Law of Large Numbers
- Event Occurrences: Power Law
- Behavioural Patterns: The Boolean Function
Gibson (1967, 1979, 1986) defined affordances as opportunities for action that are perceived by the individual. When assuming our aggregated individual would inhibit certain knowledge’s of predetermined information, the aggregated individual can be allowed to explore a playground of opportunity and seek answers to the environments questions. Highlighting Villar et al (2014), it was discussed that affordances are resources that open doors to new actions rather than shutting others off. Therefore, it is proposed here that the development of an aggregated knowledge can be created through structured affordances underpinned by defined coaching methods. From Silva (2013) shared affordances discussion, by highlighting appropriate perceptual cues and triggers learners can all respond to similar stimuli constructed by the information in the eco-system. By giving agents a shared localised meaning, for example, an opposition running with the ball at speed would mean that in a given context, the centre midfielder presses whilst the back four all squeeze the centre of the pitch and hold their line. Each individual possesses their own role in the system, but all correlate to the aggregated global behaviour of the team system.
This position paper suggests numerous cross-theorisations into the context of team sports. It is prompted that the reader maintains an open mind and to take the theorisations as abstract frameworks in which to apply to the design of dynamically complex eco-systems. The key outcome of this paper is that all spontaneous and unpredictable behaviour is in fact calculable and predictable. By approaching complex games as computational models, a structured approach can be taken in which to appropriately congregate the variables and complexities in sport yet still create an outcome in which to refine certain behavioural responses.
As discussed, by using the Law of Large Numbers, coaches can be assured that by installing the appropriate level of complexity into the environment, correlated and aggregated behaviour can naturally accrue across teams. By the emergence of relative information cues and triggers further development of organised decision making is fashioned to create more complex, robust and adaptable behavioural responses.
When designing environments for loaded attractors and affordances, using the Power Law determinant it becomes possible to be able to calculate the likelihood of certain events and thus manipulate and further afford environment design to focus system attention to certain parameter attractors.
The Boolean function offers further insight into calculating aggregated decision making. By creating binary strings, it becomes possible to aggregate certain behavioural responses to certain environmental parameters. As discussed, at each second of a game infinite amounts of binary codes are possible, it becomes the role of the coach to constrain the variability in the aggregated individual to attain certain binary strings that represent the most economical solution to the problem.
The final proposal is to approach environment design similarly to that of a Rubik cube, disorganising a solved problem to constantly create new possibilities of the final outcome. By structuring design through the process of affording the same outcome, the organising system is exploited as it contests with meta-stable states in a route to find its optimal state.
Albert, R. and Barabási, A.L. (2000). Dynamics of complex systems: Scaling laws for the period of Boolean networks. Physical Review Letters, 84(24), p.5660.
Artstein, Z. and Vitale, R.A. (1975). A strong law of large numbers for random compact sets. The Annals of Probability, pp.879-882.
Bak, P. (1996). How nature works: the science of self-organized criticality. Nature, 383(6603), pp.772-773.
Bak P, Chan K. (1995). Self-organized criticality. Sci Am, 264:46-54.
Bandelow, C. (2012). Inside Rubik’s cube and beyond. Springer Science & Business Media.
Bernstein, N.A. (1967). The co-ordination and regulation of movements.
Bourbousson, J., Seve, C. and McGarry, T. (2010). Space–time coordination dynamics in basketball: Part 2. The interaction between the two teams .Journal of Sports Sciences, 28(3), pp.349-358.
Choi, T.Y., Dooley, K.J. and Rungtusanatham, M. (2001). Supply networks and complex adaptive systems: control versus emergence. Journal of operations management, 19(3), pp.351-366.
Chow, J.Y., Davids, K., Hristovski, R., Araújo, D. and Passos, P. (2011). Nonlinear pedagogy: Learning design for self-organizing neurobiological systems. New Ideas in Psychology, 29(2), pp.189-200.
Chow, J.Y., Davids, K., Button, C. and Renshaw, I. (2015). Nonlinear Pedagogy in Skill Acquisition: An Introduction. Routledge.
Cilliers, P. and Spurrett, D. (1999). Complexity and post-modernism: Understanding complex systems. South African Journal of Philosophy,18(2), pp.258-274.
Corning, P.A., (1998). “The synergism hypothesis”: On the concept of synergy and its role in the evolution of complex systems. Journal of Social and Evolutionary Systems, 21(2), pp.133-172.
Davis, T. (1982). Teaching mathematics with Rubik’s cube. The Two-Year College Mathematics Journal, 13(3), pp.178-185.
Davids, K., Glazier, P., Araujo, D. and Bartlett, R. (2003). Movement systems as dynamical systems. Sports medicine, 33(4), pp.245-260.
Davids, K. (2013). Complex systems in sport (Vol. 7). Routledge.
DeAngelis, D.L. and Gross, L.J. (1992). Individual-based models and approaches in ecology: populations, communities and ecosystems. Chapman & Hall.
Doucet, A., De Freitas, N. and Gordon, N. (2001). An introduction to sequential Monte Carlo methods. In Sequential Monte Carlo methods in practice (pp. 3-14). Springer New York.
Friedland, B. (2012). Control system design: an introduction to state-space methods. Courier Corporation.
Gee, J.P. (2005). Learning by design: Good video games as learning machines. E-Learning and Digital Media, 2(1), pp.5-16.
Goldenberg, J., Libai, B. and Muller, E. (2001). Talk of the network: A complex systems look at the underlying process of word-of-mouth. Marketing letters, 12(3), pp.211-223.
Gréhaigne, J.F. and Godbout, P. (1995). Tactical knowledge in team sports from a constructivist and cognitivist perspective. Quest, 47(4), pp.490-505.
Gladwell, M. (2002). The Tipping Point. New York, NY: Little, Brown and Company.
Holland, J.H. (1995). Hidden order: How adaptation builds complexity. Basic Books.
Haken, H. (2006). Information and self-organization: A macroscopic approach to complex systems. Springer Science & Business Media.
Hmelo, C.E., Holton, D.L. and Kolodner, J.L. (2000). Designing to learn about complex systems. The Journal of the Learning Sciences, 9(3), pp.247-298.
Hsu, P.L. and Robbins, H. (1947). Complete convergence and the law of large numbers. Proceedings of the National Academy of Sciences of the United States of America, 33(2), p.25.
Jennings, N.R. (2001). An agent-based approach for building complex software systems. Communications of the ACM, 44(4), pp.35-41.
Kahneman, D. and Tversky, A. (1972). Subjective probability: A judgment of representativeness. Cognitive psychology, 3(3), pp.430-454.
Kaneko, K. (1990). Globally coupled chaos violates the law of large numbers but not the central-limit theorem. Physical review letters, 65(12), p.1391.
Kauffman, S.A. (1984). Emergent properties in random complex automata. Physica D: Nonlinear Phenomena, 10(1), pp.145-156.
Langton, C.G. (1990). Computation at the edge of chaos: phase transitions and emergent computation. Physica D: Nonlinear Phenomena, 42(1), pp.12-37.
Light, R.L., Harvey, S. and Mouchet, A. (2014). Improving ‘at-action’ decision-making in team sports through a holistic coaching approach. Sport, Education and Society, 19(3), pp.258-275.
Linsker, R., (1988). Self-organization in a perceptual network. Computer,21(3), pp.105-117.
Madin, J., Bowers, S., Schildhauer, M., Krivov, S., Pennington, D. and Villa, F. (2007). An ontology for describing and synthesizing ecological observation data. Ecological informatics, 2(3), pp.279-296.
Marlatt, G.A. (2003). Buddhist philosophy and the treatment of addictive behavior. Cognitive and behavioral practice, 9(1), pp.44-50.
Mason, M. (2008). Complexity theory and the philosophy of education. Educational Philosophy and Theory, 40(1), pp.4-18.
McClelland, J.L., McNaughton, B.L. and O’Reilly, R.C. (1995). Why there are complementary learning systems in the hippocampus and neocortex: insights from the successes and failures of connectionist models of learning and memory. Psychological review, 102(3), p.419.
Miller, J.H. and Page, S.E. (2009). Complex adaptive systems: An introduction to computational models of social life. Princeton university press.
Newell, A. and Simon, H.A. (1972). Human problem solving (Vol. 104, No. 9). Englewood Cliffs, NJ: Prentice-Hall.
Nicolis, G. and Prigogine, I. (1977). Self-organization in nonequilibrium systems (Vol. 191977). Wiley, New York.
Packard, N.H. (1988). Adaptation toward the edge of chaos. University of Illinois at Urbana-Champaign, Center for Complex Systems Research.
Paczuski, M. and Bak, P. (1999). Self-organization of complex systems. arXiv preprint cond-mat/9906077.
Passos, P., Araújo, D., Davids, K. and Shuttleworth, R. (2008). Manipulating constraints to train decision making in rugby union. International Journal of Sports Science and Coaching, 3(1), pp.125-140.
Reichl, L.E. and Prigogine, I. (1980). A modern course in statistical physics (Vol. 71). Austin: University of Texas press.
Renshaw, I., Chow, J.Y., Davids, K. and Hammond, J. (2010). A constraints-led perspective to understanding skill acquisition and game play: A basis for integration of motor learning theory and physical education praxis?. Physical Education and Sport Pedagogy, 15(2), pp.117-137.
Rubner, J. and Tavan, P. (1989). A self-organizing network for principal-component analysis. EPL (Europhysics Letters), 10(7), p.693.
Rosser, J.B. (2008). Econophysics and economic complexity. Advances in Complex Systems, 11(05), pp.745-760.
Sawyer, R.K. (2005). Social emergence: Societies as complex systems. Cambridge University Press.
Silva, P., Garganta, J., Araújo, D., Davids, K. and Aguiar, P. (2013). Shared knowledge or shared affordances? Insights from an ecological dynamics approach to team coordination in sports. Sports Medicine, 43(9), pp.765-772.
Siljak, D.D. (2011). Decentralized control of complex systems. Courier Corporation.
Storey, B. and Butler, J. (2013). Complexity thinking in PE: game-centred approaches, games as complex adaptive systems, and ecological values. Physical Education and Sport Pedagogy, 18(2), pp.133-149.
Truths, F.N. (2000). the Eightfold Path. The school is college preparatory, dedicated to peace education and.
Uhlig, H. (1996). A law of large numbers for large economies. Economic Theory, 8(1), pp.41-50.
Zadeh, L.A. (1973). Outline of a new approach to the analysis of complex systems and decision processes. Systems, Man and Cybernetics, IEEE Transactions on, (1), pp.28-44.
Zeigler, B.P., Praehofer, H. and Kim, T.G. (2000). Theory of modeling and simulation: integrating discrete event and continuous complex dynamic systems. Academic press.