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2015 Accumulation of experience in a vast number of cases: enactivism as a fit framework for the study of spatial reasoning in mathematics education

Khan, Steven; Francis, Krista; Davis, Brent

Springer

Khan, S., Francis, K., & Davis, B. (2015). Accumulation of experience in a vast number of cases: enactivism as a fit framework for the study of spatial reasoning in mathematics education. ZDM, 47(2), 269-279. http://hdl.handle.net/1880/51213 journal article

Downloaded from PRISM: https://prism.ucalgary.ca ZDM - The International Journal on Mathematics Education

Accumulation of experience in a vast number of cases: Enactivism as a fit framework for the study of spatial reasoning in mathematics education --Manuscript Draft--

Manuscript Number: Full Title: Accumulation of experience in a vast number of cases: Enactivism as a fit framework for the study of spatial reasoning in mathematics education Article Type: Issue 2015 - 2 Corresponding Author: Steven Khan, Ph.D University of Calgary Calgary, Alberta CANADA Corresponding Author Secondary Information: Corresponding Author's Institution: University of Calgary Corresponding Author's Secondary Institution: First Author: Steven Khan, Ph.D First Author Secondary Information: Order of Authors: Steven Khan, Ph.D Krista Francis, Ph.D Brent Davis, Ph.D Order of Authors Secondary Information: Abstract: As we witness a push towards studying spatial reasoning as a principal component of mathematical competency and instruction in the 21st century, we argue that enactivism, with its strong and explicit foci on the coupling of organism and environment, action-as-cognition, and sensory motor coordination provides an inclusive, expansive, apt, and fit framework. We illustrate the fit of Enactivism as a theory of learning with data from an ongoing research project involving teachers and elementary-aged children's engagement in the design and assembly, of motorised robots. We offer that, spatial reasoning, with its considerations of physical context, the dynamics of a body moving through space, sensorimotor coordination, and cognition appears different from other conceptual competencies in mathematics. Specifically, we argue that learner engagements with diverse types of informationally 'dense' visuo- spatial interfaces (eg. blueprints, programming icons, blocks, maps etc.) as in the research study, affords some of the necessary experiences with/in a vast number of cases described by Varela et al. (1991) that enable the development of other mathematical competencies.

Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Manuscript Click here to download Manuscript: Khan_Francis_Davis_Enactivism Robotics ZDM Paper_28_01_14.docx Click here to view linked References 1 2 3 4 Accumulation of experience in a vast number of cases: 5 6 7 Enactivism as a fit framework for the study of spatial reasoning in mathematics 8 9 education 10 11 12 13 14 15 16 17 Steven Khan, Krista Francis, Brent Davis 18 19 20 21 22 23 24 25 Abstract 26 27 28 As we witness a push towards studying spatial reasoning as a principal component of 29 30 mathematical competency and instruction in the 21st century, we argue that enactivism, 31 32 with its strong and explicit foci on the coupling of organism and environment, action-as- 33 34 cognition, and sensory motor coordination provides an inclusive, expansive, apt, and fit 35 36 framework. We illustrate the fit of Enactivism as a theory of learning with data from an 37 ongoing research project involving teachers and elementary-aged children’s engagement in 38 39 the design and assembly, of motorised robots. We offer that, spatial reasoning, with its 40 41 considerations of physical context, the dynamics of a body moving through space, 42 43 sensorimotor coordination, and cognition appears different from other conceptual 44 45 competencies in mathematics. Specifically, we argue that learner engagements with diverse 46 47 types of informationally ‘dense’ visuo-spatial interfaces (eg. blueprints, programming icons, 48 49 blocks, maps etc.) as in the research study, affords some of the necessary experiences 50 51 with/in a vast number of cases described by Varela et al. (1991) that enable the 52 development of other mathematical competencies. 53 54 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 [U]nlike the world of chessplaying, movement among objects is not a space that can be said to end 5 6 neatly at some point…successfully directed movement…depends upon acquired motor-skills and the 7 continuous use of common sense or background know-how. … Such commonsense knowledge is 8 9 difficult, perhaps impossible, to package into explicit, propositional knowledge – “knowledge 10 11 that”…since it is largely a matter of readiness to hand or “knowledge how” based on the accumulation 12 of experience in a vast number of cases. (Varela, Thompson & Rosch, 1991, 147–148) 13 14 15 16 17 18 Introduction 19 Awareness of the importance of spatial reasoning to mathematics education is 20 21 increasing. In North America, the NCTM intends to increase spatial reasoning in the early 22 23 years standards matching the focus on number (Gojak, 2012). Canadian curricula are likely 24 25 to follow. Contemplation is needed to determine what spatial skills are, how they might be 26 27 envisioned in educational settings and the characteristics of tasks that support their robust 28 29 development. 30 31 We open by inviting readers to consider a typical psychological measure used to 32 33 assess one aspect of spatial reasoning – visual rotation tasks (Figure 1). The final position 34 35 of a two-dimensional L-shape must be recognised from a series of similar gnomons that are 36 37 related by rotation and/or reflection. Such tasks are believed to isolate and measure a 38 39 singular dimension of spatial reasoning. The measure is also intended to be diagnostic of 40 41 spatial-rotational abilities and can be framed propositionally, i.e. IF agent correctly matches 42 43 gnomons THEN capable of rotational spatial reasoning. The task boundary is well defined. 44 45 46 47 48 49 50 51 52 53 54 55 Figure 1: Rotation Instrument (Kayhan, 2005) 56 57 Consider the difference in engagement illustrated by Eric, a 9-year-old participant in 58 59 a robotics summer camp (see Video 1). Beginning at 1:08, he looks at the L that is already 60 61 62 63 64 65 1 2 3 4 partially attached; he flips that one a few times to see its orientation; he moves it down to 5 6 the middle rod, puts that in and then he attaches the L-shape accurately (almost 7 8 immediately). This action/fitting/assembling involves more ‘real world’ complexity than 9 10 the visual rotation task in that it involves Eric moving among the (2D) visual 11 12 representation in the instruction booklet/guide, selecting the appropriate (3D) element 13 14 from a diverse/myriad collection of shapes (>80 different ones), then returning to the 2D 15 16 representation, all while manipulating the 3D element in order to figure out how to attach 17 it to the developing robot appropriately. 18 19 In his coordinations of the Lego pieces through his manipulations, the boundary of 20 21 the task, while constrained, is not as clearly defined – it shifts as the robot develops. The 22 23 ‘task’ in a sense can be seen as diagnostic in that he can either accomplish the task(s) or 24 25 not. We claim that the task(s) are also developmental in that he can and is learning how to 26 27 accomplish the task(s). Unlike the psychological rotation task feedback is provided by the 28 29 system in that the piece fits or it does not. Eric can make as many attempts as necessary 30 until the piece fits. The dominance of the psychologically influenced task in education over 31 32 educative tasks is still very much at play today in curriculum structure and content. Unlike 33 34 the abstract diagnostic psychological task, the educative dimension of the task(s) emerges 35 36 from the interplay among physical context, the dynamics of a human body moving through 37 38 space with other non-human bodies, cognitions and the coordination of these. 39 40 The two scenarios described above illustrate what we believe we do in education 41 42 based on that type of psychology that abstracts away from the complexity of the 43 phenomena, the relationship of the phenomena with the body, and what we observe 44 45 children as being capable of doing. Below we address the question, “What are some 46 47 necessary features to which a theory of human learning must deliberately and specifically 48 49 attend in order to make sense of the type of learning/engagement that we construe is 50 51 taking place in scenario 2?” Before presenting our response, we present an early concern 52 53 by John Dewey about the influence of psychology on education, in particular the metaphor 54 55 that was chosen to organize psychologists’ thinking about learning. 56 57 There’s Something about Dewey… 58 59 60 61 62 63 64 65 1 2 3 4 What shall we term that which is not sensation-followed-by-idea-followed-by-movement, but which 5 6 is primary; which is, as it were, the psychical organism of which sensation, idea and movement are 7 the chief organs? (Dewey, 1896, p. 358) 8 9 10 In The Reflex Arc Concept in Psychology, Dewey (1896) claimed that the 11 12 metaphorical image of the reflex-arc arising from neurology was transposed into and 13 14 satisfied a demand for an organizing principle for psychology. He argued that the metaphor 15 of the reflex-arc and its attendant principles of stimulus-response did not unseat previously 16 17 held dualistic conceptions in which sensation and idea, or body and soul were construed as 18 19 separate, but rather repeated them. He proposed instead the concept of “sensori-motor 20 21 coordination,”1 which unites, “sensory stimulus, central connections and motor 22 23 responses…as divisions of labor, functioning factors, within the single concrete whole now 24 25 designated the reflex arc” (p. 358). Dewey was elaborating the well-established 26 27 relationship between seeing and learning by identifying both seeing and learning as 28 29 instances of sensorimotor coordination. The metaphorical image of the reflex-arc 30 underpins schools of thought in 20th-century psychology, most notably behaviourism. It has 31 32 led to observational protocols of stimulus and responses, uni-dimensional measurement, 33 34 and “rigid distinctions between sensations, thoughts and acts” (p. 358), but it has ignored 35 36 or obfuscated such considerations as feedback and subjectivity. 37 38 Our opening contrast of tasks exemplifies the focus of Dewey’s critique – that is, a 39 40 narrow emphasis on diagnostic measures rather than a broad conception of action that 41 42 includes the organism itself and its activities. In short, Dewey drew attention to the lack of 43 a psychological framework of learning that is attentive to what he called sensori-motor 44 45 coordination. Following Cummins (2013), what is necessary for the study of perception and 46 47 action is a relational approach that is attentive to the limitations of “dualist mediated 48 49 epistemologies” (p. 178) – that is, theories of knowing in which human experience is not a 50 51 priori separated from the world. 52 53 We do not intend to suggest that Dewey’s pragmatist philosophy is an earlier form 54 55 of enactivism, but that the type of sensitivities and sensibilities that have come to be 56 57 58 59 1 We use the current spelling – sensorimotor – which in our opinion also serves to signal the juxtaposition of 60 sensory and motor coordination. 61 62 63 64 65 1 2 3 4 associated with enactivist approaches are also found in his work. While we would argue 5 6 that enactivism aligns with pragmatism in powerful ways, that is not our purpose here. 7 8 9 Enactivism: A fit framework 10 11 Enactivism is viewed as a “relatively young paradigm” (Villalobos, 2013, p. 159). 12 13 Despite increasing attention in philosophy, cognitive science, and education, the “Theory of 14 15 Enaction” has not yet managed to achieve significant traction in mathematics education 16 outside of a few established social and professional researcher networks. 17 18 In this section, we address the question, ‘What is enactivism?’ by asking ‘To what 19 20 does it attend?’ and ‘How does it attend to it?’ By way of initial, brief response, enactivism is 21 22 (1) a theory of engagement (2) that is simultaneously attentive to the coupling of 23 24 organisms and their environments, action as cognition, and sensorimotor coordination. (3) 25 26 It involves a methodological eclecticism that is concerned with inter-agent dynamics that 27 28 include feedback from the system and the organism’s responses. We work from the 29 position of Varela et al. (1991) that the enactivist approach comprises two principles, viz. 30 31 that, “(1) perception consists in perceptually guided action and (2) cognitive structures 32 33 emerge from the recurrent sensorimotor patterns that enable action to be perceptually 34 35 guided” (p. 173). 36 37 Enactivist theories of human learning attend explicitly and deliberately to action, 38 39 feedback, and discernment. Enactivism emphasises the bodily basis of meaning, 40 41 distinguishing it from most accounts of constructivism – which, while not denying the body 42 as ground and mediator of meaning, have not focused so intensely on the physicality of 43 44 knowing and being. Rather, constructivisms have tended to be more concerned with 45 46 conceptual understanding and propositional knowledge (Begg, 2013; Davis, 1996) – an 47 48 emphasis that perhaps inadvertently “[d]evelops rigid distinctions between sensations, 49 50 thoughts and acts” (Dewey, 1896, p. 358). 51 52 As is frequently noted, radical constructivism is a theory of how people assemble 53 54 ideas, not a theory of how teachers might direct the assembling of ideas. It is thus relatively 55 56 silent teaching practices, such as grading or distinguishing student interpretations as right 57 or wrong – noting only that while learning may be dependent on such teaching acts, it is 58 59 certainly not determined by them. In contrast, enactivism is attentive to the many feedback 60 61 62 63 64 65 1 2 3 4 structures in a greater-than-the-individual-learner system, and this quality prompts us to 5 6 regard enactivism as a much more educatively minded theory. More descriptively, 7 8 following Begg (2013), enactivism should not be thought of as “creating dichotomies 9 10 between non-cognitive and cognitive or between experiential and academic, but as 11 12 ensuring that complementary ways of knowing are all given attention and credit” (p. 93). 13 14 For us, this quality positions enactivism as a particularly useful frame for contrasting the 15 16 two scenarios presented above – that is, the paper-based rotation task and the physically 17 engaged robot building. Only in the second scenario does Eric receive feedback from the 18 19 system on his progress with L-shapes. 20 21 In recent work, Hutto (2013) argued that enactivism, with its starting assumption 22 23 that mental life can be understood as embodied activity, is a good candidate for “defining 24 25 and demarcating [psychology’s] subject matter” (p. 174) – that is, in his terms, for “unifying 26 27 psychology.” Traditional perspectives, he argued delimit psychological explanations to ones 28 29 that rely on inner representational states. He noted that enactivism, in its original 30 formulation by Varela et al., attended explicitly to organisms’ varied engagements with 31 32 contexts “not only of the biological kind but also of sociocultural varieties” (p. 177). The 33 34 mental-rotation task illustrated in Figure 1 could be interpreted in this way, as merely a 35 36 manipulation of an inner representation. Enactivism draws attention to the fact that similar 37 38 neural circuits in the sensory-motor cortex are engaged across three seemingly distinct 39 40 events: performing the actual physical rotation oneself, imagining the mental rotation, and 41 42 observing another perform the rotation (Bergen, 2012) although the subjective experience 43 would likely be different. This takes us to an enactivist re-framing of spatial reasoning. 44 45 46 How Enactivism Might Frame Spatial Reasoning 47 48 Much debate exists within and between communities of researchers on the precise 49 50 definitions and subdivisions of spatial abilities and spatial cognition, along with their 51 52 relationship to visualization, to experiences with problem solving in spatial contexts, and to 53 54 the curricular form of geometry. Drawing on Tepylo’s (2013) literature review, definitions 55 56 of spatial reasoning skills generally include: 57 58 59 60 61 62 63 64 65 1 2 3 4  visualizing part-whole objects (e.g., imagining how to put them together) 5 6 and mental rotation of part-whole objects (i.e., imagining how two- 7 8 dimensional and three-dimensional objects appear when rotated); 9 10  locating objects, recognizing shapes, their relations to each other, and their 11 12 paths of motion (Newcombe, 2010); 13 14  manipulating spatially presented information, which may involve multisteps 15 16 but not multiple solution strategies; rotating a two or three dimensional 17 18 figure rapidly and accurately (Linn & Petersen, 1985); 19  thinking about objects in three dimensions and being able to draw 20 21 conclusions about the object with limited information (Barnett, 2013). 22 23 These definitions may appear divergent, but they share some key assumptions. For 24 25 example, they all cast spatial reasoning as a sequential process of perceiving a separate- 26 27 from-actor object in the environment, encoding particular features of that object (e.g., 28 29 orientation), thinking about those features to generate motor actions and/or recognitions 30 31 (e.g., of orientation or similarity). In this sense, spatial reasoning would be analogous to 32 33 popular understandings of mental mathematics as, in Proulx’s (2013) terms, “solving of 34 mathematical tasks without paper and pencil or other computational/material aids” (p. 35 36 317). 37 38 Within an enactivist frame, the implicit separation of sensorimotor action from 39 40 cognitive process is likely inappropriate. For instance, young children’s fine motor 41 42 coordination and spatial reasoning have been identified as key to mathematics learning 43 44 and ability. In a longitudinal study that followed 213 three and four year-olds through to 45 46 the end of kindergarten, the ability to redraw designs or shapes was a predictor of the 47 ability to solve mathematical problems (Grissmer, Grimm, Aiyer, Murrah, & Steele, 2010). 48 49 Carlson, Rowe and Curby (2013) found that the fine motor skills associated with visual 50 51 spatial abilities was a predictor of mathematical problem solving for children aged 5 52 53 through18. In an intervening pilot study, Grissmer (as cited in Sparks, 2013) studied 54 55 kindergartners and first graders who played games that required them to copy designs and 56 57 shapes, cut and paste construction paper to make chains, and build models with clay or 58 59 Lego for seven months 4 days a week. At the end of the study, the children made significant 60 61 62 63 64 65 1 2 3 4 improvements to their mathematics skills. These studies point to the connections of 5 6 sensorimotor skills, spatial reasoning and mathematics ability. 7 8 As Cummins (2013) argued, “the reciprocity of perception and action is obscured in 9 10 a perception-then-cognition-then-action framework” (p. 183). Put differently, there is a 11 12 tendency to describe away most of the phenomenon of spatial reasoning in formal 13 14 definitions. We aim to recover some of the original complexity and dynamics by attending 15 16 to the presence of the knower and the materiality of the knower in the spatial reasoning. 17 The acts of isolating such aspect of spatial reasoning as rotation, orientation, and scale, and 18 19 of divorcing the knower from the context of knowing diminishes the complexity of the 20 21 construct of spatial reasoning. 22 23 We thus find it more productive to describe spatial reasoning as the constrained co- 24 25 occurrence of sensory flux (sensation), recognition/discrimination (perception), and 26 27 situated movement of a body. As might be illustrated in video clips of learner engagement 28 29 (Video 1 and Video 2), this tripartite set constitutes an act of spatial reasoning, rather than 30 either invisible cognitions or actual movements. This is the enactivist shift – in that, 31 32 following Dewey (1896), “both sensation and movement lie inside, not outside, the act” (p. 33 34 359). 35 36 37 Analysis: From sensorimotor coordination to sensory-motor control 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Figure 2: L-shaped (Video 1 Link) 61 62 63 64 65 1 2 3 4 In the video-linked episode , Eric was looking for an L-shaped piece to attach to his 5 6 robot-in-progress. (See Figure 2: the piece appears in two places – and in two ways – on the 7 8 page of instructions.) As we aim to illustrate, this engagement is an instance of perceptually 9 10 guided action that arises from perception involving visual, tactile, and sensorimotor 11 12 stimuli. Before delving into our analysis, we emphasize that we make no claims about 13 14 cognitive structures (which we cannot see), but we do talk about the recurrent 15 16 sensorimotor patterns that enable action. 17 The required L-shaped piece was in a section of the kit to Eric’s right, buried 18 19 beneath another shape, and differently oriented to both illustrations in the instruction 20 21 booklet. As Eric sought to find it, his gaze and hand acted in concert as he reached into the 22 23 orange container (see 1’15” of Video 1). Echoing Dewey (1986), the acts of seeing and 24 25 reaching were bound together. The eyes, the fingers, the wrist and the arms all worked in 26 27 unison to both see and grasp the L-shape. The seeing and reaching were part of a grander 28 29 coordination of recognizing and selecting the unique piece among an assortment of more 30 than 80 distinct shapes. 31 32 When Eric’s eyes, fingers, hands, wrist, and arm coordinated in the attempt to attach 33 34 the L-shaped piece, there was a new whole constituted in the cycle “which makes it 35 36 impossible to say which started first in the exchange of stimuli and responses” (Merleau- 37 38 Ponty, 1963, as cited in Cummins, 2013, p. 183). As Eric tried to fit the L-shaped piece to 39 40 the robot his perception was that it didn’t fit, which led him to manipulate the object and 41 42 try again (see 1’28” of Video 1). He continued to try many times, until his perception was 43 that it did fit. The perception (arising from sensorimotor information) guided (and arises 44 45 from) the cognition which then guided the action. As noted earlier, the task serves both 46 47 diagnostic and developmental purposes. 48 49 50 51 52 Structural Coupling & Uncoupling: The role/space of the social from an enactivist 53 perspective 54 55 56 In this section we illustrate the enactivist notion of structural coupling and point to 57 58 an important but often overlooked dimension to the phenomenon in a social system (in 59 60 contrast to a physical system), viz. uncoupling. The notion of structural coupling derives 61 62 63 64 65 1 2 3 4 from a biological perspective of an organism and environments co-adaption or evolution to 5 6 each other. The mutual interaction of the organism and the environment causes changes 7 8 and transformations in both. 9 10 We argue that, in an educational setting, learning is dependent on both socio- 11 12 cognitive coupling and uncoupling. The former serves as a trigger or a perturbation and the 13 14 latter provides opportunities for pursuing personal interest/focus necessary for individual 15 16 learning. Varela et al. (1991) drew on Darwin’s notion of evolution to describe structural 17 coupling of the co-adaptation of an organism and its environment, noting that the ability of 18 19 an organism to un-couple from its environment is also important for the organism’s 20 21 survival. Being too tightly coupled to a specific environment may lead to extinction if the 22 23 environment changes in even minor ways (e.g., should water levels drop substantially, a 24 25 fish has fewer options for survival than an amphibian). From an educational perspective, 26 27 the combined socio-cognitive coupling and uncoupling can provide opportunities for 28 29 learning that enable the organism to adapt learning to other environments. 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Figure 3: Coupling (Link to Video 2 clip) 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Figure 4: Uncoupling (Link to Video 2 clip) 25 26 27 An event of socio-cognitive coupling-and-uncoupling is presented through the video 28 29 links above (Figures 3 & 4). In these clips, Declan is working on his own robot, Christopher 30 31 approaches to find clarification of a problem he is having with robot construction (see 32 0’40” of Video 2). Declan observes, analyses Christopher’s current status, points, describes 33 34 where the error is, and identifies the stage where the error occurred. From and enactivist 35 36 perspective, this coupling of Declan and Chris triggers a number of processes. 37 38 Applying our definition above of spatial reasoning, as the constrained co-occurrence 39 40 of sensory flux (sensation), recognition/discrimination (perception), and movement, 41 42 Figure 3 (and the linked Video 2 clip) shows a coupling of two children whose object of 43 44 focus – the shared basis constraining the coupling – is one of their robots. The sensory flux 45 in this coupling involves speaking, pointing, looking, touching, and holding. The 46 47 recognitions of what is not right and when help is needed are the prompts for coupling. We 48 49 claim that spatial reasoning is occurring as a part of the coupling in this moment. Spatial 50 51 reasoning in this instance can be viewed in terms of either individual cognitive process or 52 53 as interpersonal social process – or, in more educationally productive terms, as a socio- 54 55 cognitive process. 56 57 On this count, we find dual-process theory (Kahneman, 2011) a useful supplement 58 to enactivism. Briefly, dual-process theorists posit that individual knowing arises in the co- 59 60 61 62 63 64 65 1 2 3 4 activity of two quasi-distinct knowing systems, the Automatic (System 1) and the Reflective 5 6 (System 2). System 1 is quick and intuitive, rooted in memory and rehearsed experience, 7 8 more given to analogy than to logic, and usually accurate in its reads and responses. System 9 10 2 is slow and deliberate, based in conscious though and concerned with novelty or 11 12 perceived incoherences. It tends to be much more logical and, at the same time, much more 13 14 prone to misreadings and unfitting responses. 15 16 Most of the time, the Reflective System 2 defers to the Automatic System 1. It is only 17 when a threshold of unfamiliarity or confusion is met that System 2 is triggered into action. 18 19 We see both systems in play for both actors in the clips above. To discern the 20 21 coupled and uncoupling of Declan’s and Chris’s Systems 1, we find it helpful to mute the 22 23 sound and focus on the fluid choreography of their mutually specifying actions. These 24 25 actions are smooth, precisely timed, exquisitely coordinated, and astonishingly free of 26 27 excess motion. Such are the hallmarks of automatized action – which, to our observation, 28 29 are appropriately characterized as embodied or enacted knowings. 30 Of course, we must be careful not to understate the roles and couplings of the actors’ 31 32 Systems 2 in this episode. After all, the event was triggered by Chris’s conscious recognition 33 34 of a difficulty. That is, in terms of dual-process theory, Chris encountered an instance of 35 36 insufficient and/or inadequately integrated experiences to evoke a routinized response in a 37 38 novel situation. Lacking that, another of his automatized responses appears: he calls on a 39 40 likely-to-know and proximate other. Once System 2 is oriented to this course of action, 41 42 System 1 appears to take over again, as suggested by the fluidity of the actions and 43 articulations. 44 45 Declan’s response is similarly interesting. Its immediacy indicates that the solution 46 47 he offers to Chris was drawn from his repertoire of rehearsed actions. But more interesting 48 49 to us is the seamless sequence of his couplings and uncouplings, starting with a cognitive 50 51 uncoupling from his own work, a socio-cognitive coupling with Chris, a socio-cognitive 52 53 uncoupling from Chris, and a cognitive recoupling with the original task. These subevents 54 55 occurred in just seconds, at a speed that exceeds the capability of System 2. They were 56 57 embodied. 58 Turning back to Chris, the instant of social uncoupling afforded him an opportunity 59 60 to process what Declan disclosed (see 0’56” of Video 2). He then returns to Declan to 61 62 63 64 65 1 2 3 4 explain what the problem was (see 1’05” of Video 2). The temporary uncoupling provides 5 6 the time and space for individual reflection, which within a social context that supports 7 8 accumulating experiences in a vast number of cases, serves as an occasion for sharing 9 10 learning. 11 12 To be clear, we are still talking about spatial reasoning here. Our point is that spatial 13 14 reasoning competence is not a solitary achievement, but one that arises in the main amid 15 16 such socio-cognitive couplings and uncouplings. Of course, the same might also be said of 17 other mathematical competencies. However, the particular advantage of the topic of spatial 18 19 reasoning is that understandings are typically much more available to observation. We can, 20 21 literally, see Chris’ and Declan’s understandings in their actions. 22 23 Moreover, and somewhat provocatively, when we watch an accelerated version of 24 25 the more complete recording of the extended engagement from which this episode is 26 27 extracted, we observe a distinct pattern of structured interactivity in which agents pull 28 29 together and move apart in a rhythmic pulse as they structurally couple and uncouple. In 30 our enactivist framing, we would be curious about those moments of coming together and 31 32 those moving apart, that pulse of complexity, as occasions for accumulating experience in a 33 34 vast number of cases for individuals and for the collective. 35 36 37 38 39 Implications 40 To understand mentality, however complex and sophisticated it may be, it 41 42 is necessary to start by appreciating how living beings dynamically interact 43 44 with their environments, both shaping and being shaped by those 45 encounters; ultimately there is no prospect of understanding minds 46 47 without reference to ongoing interactions between organisms and their 48 49 environments. (Hutto, 2013, p. 176) 50 51 52 We note that constructivisms, as adopted and adapted within the field of 53 mathematics education in the 20th and early-21st centuries, have been mainly concerned 54 55 with conceptual understanding of numerical and algebraic concepts. While some enactivist- 56 57 aligned contributions have highlighted the importance of taking into account the body in 58 59 efforts to make sense of these areas of mathematical competence (eg. akoff u n ez, 60 61 62 63 64 65 1 2 3 4 2000), it is telling that relatively little of the research into arithmetic and algebraic 5 6 competence delves deeply into the bodily basis of meaning. 7 8 In contrast, with the emergent recognition that spatial reasoning is a core element of 9 10 mathematics competence, it is apparent that theories of learning that are principally 11 12 focused on the evolutions of personal conceptual coherence are inadequate. Spatial 13 14 reasoning is much more obviously and directly anchored to one’s experiences, 15 16 situatedness, and intentions – in brief, one’s enactments. Returning to our title, what counts 17 as sufficiently vast for individual learners will not be the same for others. 18 19 When we consider this realization alongside the longstanding tensions between 20 21 knower-centered constructivisms and socio-cultural accounts of learning and knowing, we 22 23 are even more compelled toward an enactivist frame. Intricate dances of cognitive and 24 25 social coupling and uncoupling surpass perspectives that privilege one or another domain 26 27 of (inter)action. 28 29 Our enactivist framing above as the constrained co-occurrence of sensory flux 30 (sensation), recognition/discrimination (perception), and movement of a body, presents 31 32 one opportunity for the field to re-consider both the phenomenon of interest and the way 33 34 of studying it. However, we conclude that the phenomenon for which the concept/signifier 35 36 ‘spatial reasoning’ is used as a descriptor is a complex one. Enactivist perspectives we 37 38 believe offer fit frameworks for interpreting and investigating what it means to weave 39 40 one’s embodied and knowing self through the world. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 References 5 6 7 Barnett, T. (2013). What Is Spatial Ability? wiseGEEK. Retrieved September 26, 2013, from 8 9 http://www.wisegeek.org/what-is-spatial-ability.htm 10 11 12 Begg, A. (2013). Interpreting enactivism for learning and teaching. Education Sciences & 13 14 Society, 4(1). Retrieved from 15 16 17 http://riviste.unimc.it/index.php/es_s/article/view/699 18 19 Carlson, A. G., Rowe, E., Curby, T. W. (2013). Disentangling Fine Motor Skills’ Relations to 20 21 22 Academic Achievement: The Relative Contributions of Visual-Spatial Integration and 23 24 Visual-Motor Coordination. The Journal of Genetic Psychology, 174(5), 514–533. 25 26 27 doi:10.1080/00221325.2012.717122 28 29 Cummins, F. (2013). 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