**4.1 Thinking non-linearly**

Linear thinking tends to result in linear expectations, which for teachers means the assumption that tangible evidence of pupil learning will follow shortly after teaching input. This mechanistic reckoning about learning is a key part of the simplification problem presented at the start of this chapter and is ubiquitous in classrooms across the world. However, it does not reflect how many teachers experience classroom learning on a day-to-day basis. Thinking non-linearly about teaching and learning means understanding that learning is a consequence of multiple, networked factors which may not covary linearly or occur in the same moments or places. Learning, when it emerges tangibly, is the result of inputs from multiple sources, including direct teaching, autonomous activity, peer influence, social dynamics, personality traits, prior conceptions, understandings, misunderstandings, environment, and chance, from a range of temporally diverse events. In short, learning (including individual learning) results from system-wide factors, not simply the linear mechanisms of teaching. Instruction may be the dominant influence on pupil learning, but is far from the only influence and its effects are filtered through multiple influences both within, and outside of, the classroom system. Awareness of the networked and recursive nature of learning enables teachers to notice, attend to and even plan for the pluriform influences which may assert themselves in, and beyond, the classroom. For example, awareness that pupils' mental models, their understandings, and the connections they make are substantially influenced by experiences outside of school may lead a teacher to elicit information about pupils' wider experiences of a phenomenon before planning to teach it. Similarly, knowing that new knowledge may emerge through decentralized and distributed interactive structures may lead teachers to facilitate episodes of autonomous, or semi-autonomous group activity expecting pupils to create or evolve ideas which they themselves had not planned for. Thinking non-linearly can encourage teachers to not only ask conditional "if-then" questions about teaching and learning but also consider "what if …" questions which can produce sensitivity to teachable moments. Thinking non-linearly changes expectations about how, when, and from where learning might emerge and can result in teachers noticing potential and future learning in its infancy. Sensitivity and responsiveness to "soon-to-be" learning as it appears are at the heart of effective pedagogy. Understanding that pupils, their ideas, utterances, moods, similarities, differences, and personal histories interact in non-linear ways can help teachers to look for potential learning in places and at times not typically explored when expectations follow singular, linear logic. This includes finding learning, or the conditions for learning in unlikely places, including classroom disruption, in social conflict, or in incidental, unintended moments.

Thinking non-linearly is not a typical habit for most teachers, however, trained as they often are to view teaching and learning in a unidirectional causal relationship in which learning is singularly the product of teaching. This is essentially the central presumption of the "what works" educational paradigm [88] and there are understandable justifications for this habit of mind; not least because it is accurate to some extent. Learning is a consequence of teaching. The problem is not that teaching has no part to play in learning, it plays a significant part, of course. The problem is the failure to understand teaching and learning as existing in a recursive, mutually influencing relationship and failure to see learning as an emergent phenomenon that surfaces because of bottom-up dynamics as much as from top-down instruction. Non-linear

thinking, which attempts to accommodate these concepts, will require some effortful reimagining on the part of teachers well-versed in the "what works" paradigm.

#### **4.2 Attunement to networked influences on learning**

Non-linear thinking about teaching and learning has the potential to unlock awareness of the different sources which influence learning and how they interact with one another. Once aware that instruction is only one of several influences and that the multiple influencing factors are mutually interactive, teachers may view common classroom scenarios and occurrences differently. For example, when pupils articulate their understanding of a concept or process it is likely that the new understanding was constructed in an uneven trajectory involving bursts, plateaus, and stops over time. In addition to teaching input, the conceptual understanding will be a result of their interactions with other pupils, the collision of ideas those interactions permit, interaction with the environment and with aspects of their own and others' personalities, learning habits, social status, intellectual and non-intellectual pre-requisites. Sensitivity to these factors can support teacher judgments about pupil grouping, sequencing, and duration of classroom activities, whether and when to intervene, scaffolding questions to ask, and the importance of helping pupils to link experiences. My research with 8 and 9-year-old pupils in a British Year 4 classroom [62] suggests that learning is always a process of organizing, reorganizing, constructing, reconstructing, and refining existing knowledge and understanding. This was illustrated in instances where pupil learning was the consequence of errors and misunderstandings, off-task as well as on-task behaviors, and because of, rather than in spite of, social conflict. There was also evidence that asymmetries in pupil social hierarchies produced discussion and knowledge sharing which more symmetrical dynamics almost certainly would not have. In one such example, the disruptive influence of one pupil in a practical small group activity and the frustration it caused other group members actually drove reconfigurations in pupil roles and articulation of perspectives, arguments, and explanations. The atmosphere was not calm and cooperative, but ideas, demonstrations, and later articulation of learning almost certainly resulted in part because of this. Behaviors which a teacher might instinctively want to prevent unlocked learning for some pupils, to some extent. Attunement to these factors and possibilities, particularly the knowledge that learning can emerge out of social conflict (which teachers routinely, and understandably, aim to suppress) has the potential to shift teachers away from expecting evidence of learning during or shortly after instruction, towards a more authentic appreciation of and attention to the range of antecedents, including unexpected ones.

#### **4.3 The unexpected emergence of learning**

Attunement to the CAS-like features of classrooms and non-linear thinking about learning processes opens the door for teachers to re-evaluate their intuitions about where and how learning can, or might, emerge. Acknowledging that some antecedents of visible learning originate from outside of the linear mechanisms of direct instruction means acknowledging that learning is caused and supported by more than simply teacher explanation and demonstration or planned classroom activities. It may also emerge incidentally during the non-lesson time or out of instances of social disruption, which teachers typically aim to discourage. Evidence from numerous classroom studies [89–93] suggests that when pupils collaborate autonomously, or

#### *Classroom as Complex Adaptive System and the Emergence of Learning DOI: http://dx.doi.org/10.5772/intechopen.101699*

semi-autonomously, on shared activities some degree of social conflict is inevitable. This may range from minor disagreements about turn-taking or resources that do not interrupt the flow of activity, to larger arguments that disrupt purposeful activity, cooperation, and group productivity. What these studies do not acknowledge, however, is that whilst there are sound reasons to discourage an atmosphere of conflict in the classroom, clashes of personality, asymmetries of social dominance, disagreements about pupil roles, and the socially engendered necessity to resolve these can actually produce moments of novelty, innovation, knowledge sharing or motivation which lead to learning. It is no surprise that nowhere in the literature is there any support for the idea that social conflict might also support curriculum learning in unexpected ways, however, some of the data in my recent research [62] suggests that learning does not just survive episodes of social conflict, but can be occasioned by it. I am not here advocating a laissez-faire approach to classroom management or an "anything goes" attitude to pupil behaviors. Calmness, cooperation, and mutual respect are necessary and desirable qualities to encourage in classrooms. However, I am (somewhat tentatively) suggesting that when teachers suspend, even temporarily, their linear expectations about what produces learning (and perhaps even stop thinking about learning as a product altogether), they may notice teachable moments worth capitalizing upon in some unexpected places.

According to Davis and Simmt [63] the network of mutual self-influence between pupils, environment, and classroom climate, and the resulting randomness, opens the door to novel moments of teaching and learning. An example of this in my recent research included instances in which arguments between two pupils prompted periods of increased productivity and "on-task" behaviors from others. In another example the repeated ignoring of a low social status pupil by his small group caused him out of frustration to share his ideas with the teacher, resulting in him eventually explaining it to the whole class, whereafter most groups adopted his ideas. The learning in these incidents, or the conditions for it, emerged bottom-up rather than top-down. What emerged was a consequence of the situated, unique dynamic choreography of multiple pupils' wills, personalities, social standing, and knowledge states, among other factors. Sensitivity to the ways these and other factors can interact to produce conditions for learning where one might not expect to find it can be advantageous to teachers. How then might teachers use this knowledge and capitalize on it in the interest of learning? The knowledge that whilst excessive pupil autonomy, interactive license, or social conflict is likely to obstruct learning, appropriate degrees of freedom from centralized teacher control can invite novelty, could lead teachers to reevaluate their centralizing instincts and look for a balance between Radford's "clockish" and "cloudish" organization [40] in their classrooms. Such re-evaluations can lead to teachers pausing, observing, and assessing for potential teachable moments. If there are potential benefits in allowing social conflict to play out, enabling pupils to find their own resolutions, this may lead teachers to re-assess how and when they intervene. Judging the line between tolerable and intolerable degrees of social conflict or noticing signs of potential learning benefits is unlikely to be easy, however there is evidence, albeit tentative, that there may be payoffs for teachers brave enough to try. The "edge of chaos" [1] will always be a double-edged sword, inviting innovation but also risking havoc. Semetsky [76] acknowledges that radical visions of open and unrestrained classroom systems, whilst opening doors to possibility, might also be counterproductive. Accepting that interactive learning is by nature open and generative (and therefore susceptible to disorganization), Biesta [72] has argued in favor of enabling constraints, structures which facilitate autonomous pupil interaction,

encourage novelty and originality but without courting havoc. A useful question for teachers therefore might be, how can I implement limits on interactive learning which unlock and enable, rather than dampen, novelty and originality?

#### **4.4 Occasioning emergent learning**

CAS-framed classroom research and discussion [33, 61, 62, 64] suggests that CAS-like behaviors do not necessarily occur naturally in all classrooms, but depend to a large extent on balance between sources of coherence and disruption; between centralized control and decentralized autonomy. Having looked at some of the opportunities which CAS-like behaviors might offer teachers and pupils in pursuit of learning, it is worth considering ways teachers might occasion emergence by locating and exploiting "sweet spots" between rigid order and all-out chaos; in which pupils benefit from what autonomous interaction and teacher scaffolding have to offer. This is what Radford [40] refers to when describing systems, which are a balance between "clockish" and "cloudish". This would include knowing how, and judging when, to centralize or decentralize the organization and autonomy, and understanding the consequences. Much like a jazz leader, knowing when to allow improvisation and when to return the ensemble to the main theme. "Sweet spots" exploit the most useful products of centralized, decentralized, and distributed classroom structures, whilst avoiding the unproductive excesses of each.

In tightly controlled, centralized classroom structures ideas tend to flow linearly from the teacher to the pupils and back from individual pupils towards the teacher. Under this organizing principle, ideas do not collide, they simply travel along straight lines, often in just one direction emanating from the teacher. Those ideas become known by everyone but are not interrogated, trialed, experienced, regurgitated, or challenged. The principal casualty of this tendency in highly centralized classrooms is the cross-fertilization of pupil perspectives and emergent learning. Newly emerging understandings, partially articulated thinking, and challenge to pupil assertions tend to arise more freely in distributed peer-to-peer interaction than in centralized, didactic scenarios. Too much centralized communication is likely to stifle this "soonto-be" learning. At the opposite extreme, in classrooms principally characterized by distributed structure, ideas tend to collide but are rarely refined, sustained, or linked coherently. The main casualty here is the spread of ideas since ideas tend to remain local, petering out without dissemination, and coherence, because thinking remains atomized. Somewhere between these extremes lie sweet spots where pupils experience sufficient freedom and autonomy to encourage elements of self-organization and bottom-up novelty, accompanied by sufficient central teacher coordination for emerging ideas to be shared, digested, shaped, and sustained. Factors that teachers can manipulate to locate such sweet spots include (among other things) physical space, time periods, activity types, and groupings. Activities with sufficient openness undertaken for appropriate durations by autonomous or semi-autonomous pupil groupings with considered classroom organization and sensitive teacher intervention have the potential to create space for novelty, creativity, and innovation to emerge. Timely shifts between distributed, decentralized, and centralized structures may enable learning to emerge and progress from emergent states towards elaborated, more secure states from which it can be more easily redistributed.

In order to create conditions in which partially formed ideas may grow into articulated explanations or productive applications, a range of pedagogical prerequisites are necessary, beginning with an understanding of what centralized,

#### *Classroom as Complex Adaptive System and the Emergence of Learning DOI: http://dx.doi.org/10.5772/intechopen.101699*

decentralized, and distributed modes of classroom organization offer the pursuit of learning. Confidence and competence in noticing signs that classroom structures need to be loosened or tightened follow logically from this, along with knowing how and judging when to do this. Creating open learning activities and opportunities that are fertile ground for ideas to collide, which are encouraging autonomous interaction and structured sufficiently to elicit novelty and originality, whilst avoiding excessive teacher control takes courage and practice. The starting point, surely, is sensitivity to potential teachable moments, attunement to the networked interacting influences at work in the classroom, and openness to the possibility of learning emerging from unexpected places.
