Making sense of movement in embodied design for mathematics learning

Raising the question of design: proximal versus distal movement

When we say that students participating in designed educational activities learn mathematical content by first learning to move in a new way, we must clarify the mediating role of technology in acculturating this capacity to move in a new way. Very often, our physical actions are in direct contact with the objects we manipulate, such as when we lift a ball and throw it. Some educational designs leverage these naturally available, unmediated manipulation processes as sources of learning. For example, students may learn about physics principles of kinematics by experiencing and reflecting on the greater physical effort demanded in throwing a ball across greater spatial distance. Such design is both governed and limited by universal laws. However, our contact with the objects we manipulate can be moderated by intervening tools, such as when we use physical utensils (e.g., a fork) or electronic appliances (e.g., a remote control) to extend, augment, distribute, scale, and transform our physical actions over and through space, time, media, cultural forms, and fellow participants (Hutchins, 2014; White, 1984). As such, technology creates enactive distance between intention and effect. Learning to control the environment in instrumented situations is the process of removing this enactive distance by assimilating and mastering the instruments (Morgan & Kynigos, 2014; Pratt & Noss, 2010; Vérillon & Rabardel, 1995). For example, the blind and visually impaired learn to navigate space using a cane, where the cane becomes through practice a sensorimotor extension of the body into a thus expanded “enactive landscape” (Kirsh, 2013).

This principle of sensorimotor augmentation is well known among cognitive scientists of neuroplasticity who build sensory substitution technological systems (Bach-y-Rita, Collins, Saunders, White, & Scadden, 1969). Similarly, designers of educational activities capitalize on this neuroplasticity principle (Siu, 2016). Following principles of the embodied design framework, we build tools whose operatory function is engineered specifically so as to demand, and therefore cultivate, the development of particular sensorimotor schemes as a condition for masterful control of the environment in accord with task demands. In so doing, we target specific sensorimotor schemes that, through instructional guidance, will come to ground the mathematical concepts we want these students to learn (Abrahamson, 2014; Abrahamson & Trninic, 2015).

When a tool mediates an individual’s action on the environment, we often witness differences between proximal and distal action, that is, between the hand motions of manipulating the tool and the extended result of this manipulation in the domain of action. For example, compare the bimanual motion of operating a pair of gardening shears and the effect of these mechanically mediated motions on a branch. Notably, competent shearing is oriented on the branch, not on the hands, unless we experience physical or mechanical breakdown (Koschmann, Kuuti, & Hickman, 1998). In analyzing the dynamics of these activities and the locus of a student’s attention, it is therefore helpful to clarify whether we are referring to the proximal or distal movements.

Whereas the proximal/distal distinction is quite straightforward and probably uncontested, we worry that the distinction is sometimes obscured in discussions of technological design for embodied–interaction mathematics learning. In particular, we are concerned that insufficient distinction is being made in the literature between, on the one hand, the primary manipulations that students enact and, on the other, the effects of these technologically extended manipulations within the domains of action, whether these be mechanical or virtual media. This lack of distinction between proximal and distal actions, we submit, may implicitly hamper our community’s conversation about learning processes and design principles. For example, the distinction could help us implicate where students attend as their teacher guides their work (Shvarts & Krichevets, 2016).

In the next sections, we aim further to clarify the proximal/distal distinction by complexifying the relation between a person’s sensorimotor orientation toward a situation and the technologically moderated environmental effects of their actions. We will be looking at cases where actions are mediated by a computational platform, specifically at tablet-based activities designed for students to learn through solving interaction problems. The cases were selected to offer a two-stepped salvo, as follows.

The next section, a thought experiment, will treat a dissociation between how we move our hands on the tablet interface and what trace the software generates on the screen as the mediated result of our manual motions. We then present this dissociation as bearing potential for educational design. In the subsequent section we discuss an eye-tracking study of how students solve challenging interaction problems leading to mathematical notions. Drawing on empirical findings, we will argue for a systemic conceptualization of proximal and distal movement as task-driven, situated, distributed sensorimotor schemes oriented on emergent perceptions of the environment.

Interpolating technology into the agent–environment relation mediates new sensorimotor schemes supporting learning objectives

Consider the following activity involving two hypothetical task scenarios. In both Condition 1 and Condition 2, (1) you are presented with a tablet interactive application; (2) on the screen you see a black circular line; and (3) you are asked to draw a red line on top of the black line. In Condition 1, a “finger painting” task, you use your index finger to trace directly along the existing circle perimeter. As you do so, virtual red paint oozes from under your fingertip to cover the black line. In Condition 2, you cannot trace directly on the black circle. Instead, you are asked to imagine this circle as plotted in a Cartesian space. Now you must move your left hand index up/down along the y-axis to the left of the circle simultaneous with moving your right hand index right/left along an x-axis below the circle. The application plots red points at spatial locations corresponding to the ordered-pair [x, y] Cartesian intersection of your fingers’ respective measured distances from the origin point. You are thus asked to graph a circle manually.²

Whereas the two conditions share a task objective of generating a red circle on top of the black circle, clearly Condition 2 is more difficult that Condition 1. Condition 2, unlike Condition 1, presents you with a problem, and more so if you have never before worked in the Cartesian field. To solve this problem, you must engage in inquiry. You actively explore the new space to discover its embedded functions and determine their utilities relevant to the task objective. In so doing, you enter a cycle of tight, rapid, recursive sensorimotor feedback loops, where you attempt various manipulations, attend to their coinciding instrumented effect, and constantly tune your movement pattern. Perhaps you infer heuristics for action and even articulate them verbally. Through practice, you progressively accommodate your operatory schemes so as to assimilate the tool’s discovered affordances for action. Condition 2 thus presented you with a new constraint that initially impeded your capacity to fulfill the task. However, through figuring out how to cope with this constraint you developed new subjective affordance for the environment (Forman, 1988; Greeno, 1994; Newell, 1986, 1996). You have learned to move in a new way through a representational system that has become central in mathematics, the Cartesian system of perpendicular x- and y-axes.

In this case, we demonstrated two different ways of generating the same geometrical figure. In a sense, the red circle of Condition 1 is not the same as the red circle of Condition 2. At least, from your subjective perspective, Condition 2 demands of you to develop a new phenomenological and conceptual construction of what a circle is from a mathematical perspective (Abelson & diSessa, 1986; Papert, 1980; Piaget & Inhelder, 1969; Wilensky & Papert, 2010). More generally, the available means of production we have learned to access and use in fulfilling an activity task are instrumental in forming our conceptions of the objects we produce and perceive (Baird, 2004; Chase & Abrahamson, 2015; Meira, 1998; Rosenbaum & Abrahamson, 2016).

As educational designers, we target particular concepts by creating both the task objectives and the means of accomplishing those objectives. The designer’s goal ultimately is not about having a student produce a circle per se but having a student struggle to produce a circle with the given means. In the case of Condition 2, the technological interaction conditions are designed to foster opportunities for students to develop a particular sensorimotor scheme – a bimanual coordinated motor action oriented on a geometrical figure. Still, a new coordination is not yet new mathematical knowledge. For manual know-how to become conceptual know-that, students need to use disciplinary frames of reference to re-describe their own actions (Bartolini Bussi & Mariotti, 2008). In so doing, students shift into disciplinary ways of seeing and talking (Abrahamson, 2009; Bamberger & diSessa, 2003; Sfard, 2002).

In this subsection, we offered a hypothetical study of technologically mediated action. For the purposes of this essay, our objective was to differentiate conceptually between proximal and distal movement insofar as this differentiation bears on the practice and theory of educational design for mathematics learning. As such, we contrasted two technologically mediated activity conditions where the goal distal movement was identical (drawing a red circle on a screen) but the proximal movements were dramatically different (tracing with a finger vs. simultaneously operating two orthogonal axes in a Cartesian field). Whereas the distal movement constituted a necessary task goal, the activity’s educational potential was largely determined by the proximal movement required to generate the distal movement. Thus, a study of mathematics learning from an embodiment perspective should focus on the development of proximal movement, not just its distal product.

In the next section we “recede” into proximal movement to demonstrate that it, too, can be usefully differentiated. Looking at empirical data, we will show that different sensorimotor schemes can achieve the same proximal movement. The case will demonstrate variability, both intra-personal and inter-personal, in how students orient sensomotorically toward objects on the tablet screen so as to generate a particular pattern of hand movements.

Design background

In the previous sections, we contrasted two scenarios to clarify the distinction between proximal and distal movement as well as direct versus indirect approaches to embodied design. A direct approach was to draw a red circle on top of a black one. An indirect one was to move two hands along an x- and y-axis in such a way that the same thing happens. We aimed to convince the reader that, while the former scenario would engage students in embodied activity, the latter is more likely to involve them in embodied mathematical activity and thus help them develop sensorimotor schemes that are mathematically relevant to the concept of circle.

In this section we continue by thinking through and illustrating what embodied design for learning about proportion could mean. A direct scenario could be to ask students to point at halfway up a bar, or make a bar twice as long. Again, we maintain such embodied tasks would have little potential for learning about mathematics. In our collaborative work, initiated by the Embodied Design Research Laboratory at Berkeley, we have explored many different designs. An early mechanical design involved a student holding two pulleys that were moved up and down in a particular ratio. From an observer’s perspective, the student was moving her hands in a mathematical proportion. But was she learning? Very little. There was no need for reflection (Trninic, Reinholz, Howison, & Abrahamson, 2010).

The next design iteration was centered on using Wii devices to remote-control two cursors on a screen (Fig. 1). The task was to move the cursors up and down in parallel with the objective of keeping the screen green; this was the case if the cursors’ heights above the bottom of the screen instantiated a particular ratio, regardless of whether the students initially knew or understood this. We call this pedagogical activity architecture a Mathematical Imagery Trainer (Howison et al., 2011).

The Mathematical Imagery Trainer is an interactive technological system designed to foster opportunities for students first to develop targeted sensorimotor schemes and then model these schemes in particular forms that lead to understanding targeted concepts. Now in its tenth year of research, the system has been implemented in a variety of media, including a mechanical apparatus, Wii, Kinect, and iOS touchscreens; it has been evaluated in a variety of settings, including laboratory clinical interviews with individual or paired students and school sites with groups and whole classrooms; and it has served as a context for evaluating an artificially intelligent pedagogical agent. Here, we focus on findings relevant to this theoretical essay. In particular, we will discuss data gathered at Utrecht University suggesting variability in students’ sensorimotor schemes for enacting proximal movement solutions.³

Attentional anchors as mediating proximal and distal movement

The activity task presents students with the motor-action problem of moving their hands in a motion pattern where both hands rise in parallel but at different speeds, for example, the right hand must rise twice as fast as the left hand. (For the sake of readability we will only focus on the 1:2 ratio in the remainder of this paper). We have studied the process by which students learn to move in this new way. Our analyses have drawn on data that include audio–video recordings of students’ actions and multimodal explanations to the researcher, logs of interface actions, and eye-tracking of foveal vision (Abrahamson, Shayan, Bakker, & Van der Schaaf, 2016). These analyses indicated that students were using attentional anchors, as we now explain.

As students manipulated the two cursors, they attended to particular loci on the screen. These loci included not only the two cursors but also “non-stimuli”, that is, particular locations on the screen where there were no discernable contours (Fig. 2). For example, while moving the two cursors the students would stare at a point on the screen background between the two cursors. To an objective viewer, there was nothing there, and yet the students focused on those points. It appeared that the students were somehow using these loci to facilitate their competent simultaneous manipulation of the two cursors. When students explained their solution strategy in words and gestures, they referred to objects they perceived on the screen, for example, they explained that they were looking at the spatial interval between the two cursors. They would say that this interval should increase as the hands rise and decrease as the hands descend (Fig. 2) or that this interval should move to the right in the Orthogonal Pluses task variant (Fig. 3).

We realized that these imaginary objects emerged into students’ interaction dynamics through the process of solving the control problem of keeping the screen green – the imaginary objects were goal-oriented sensorimotor objectifications of the interaction space. The objects emerged from empty space to help the students perform a challenging motor-control coordination. That is, instead of attending to two hands separately, students could attend to a single object and control it. We further realized that attending to these objects helped students learn the new mathematical ideas, because students referred to these objects in modeling and describing their sensorimotor schemes, first qualitatively and then, once we introduced mathematical tools onto the screen, also quantitatively. Borrowing from Hutto and Sánchez-García (2015), we called these imaginary objects attentional anchors (Abrahamson & Sánchez-García, 2016; Abrahamson et al., 2016).

The empirical context of these studies enables us to track the emergence of a sensorimotor scheme as the integration of two components – a new gestalt in the environment (the “sensori-” component, e.g., a new attentional anchor) and a way of moving relative to this perceptual invariant (the “motor” component, e.g., a new bimanual motion coordination centered on the attentional anchor). In particular, attentional anchors, such as a linear interval between two points, are what we have come to call goal-oriented sensorimotor objectifications. Students construct attentional anchors as their spontaneous solution to motor-control interaction problems. These attentional anchors constitute new proto-mathematical objects amenable to reflecting, modeling, articulating, and expressing in formal symbolic notation. The studies also demonstrate that, whereas light-handed guidance is sufficient for the emergence of attentional anchors, mathematical re-description of these interaction solutions requires more heavy-handed intervention.

In summary, proximal movements may result from a variety of sensorimotor orientations toward the task space – one cannot judge a sensorimotor scheme by its proximal movement cover. In order to understand how students learn mathematics through participating in interaction tasks, it is not enough to describe their distal and proximal movements. We need to dig deeper to find out how they are orienting toward the situation. In particular, more nuanced observation and measurement is necessary for looking under the distal and proximal covers so as to implicate underlying sensorimotor schemes. Educational investigations of embodied learning, we propose, must be geared to theorize, measure, and analyze sensorimotor orientations, as these may constitute the psychological source of proto-mathematical objects. Understanding students’ sensorimotor orientations with the help of attentional anchors should help us better theorize mathematical learning and, in turn, better design mathematics learning systems. For this to occur, educational researchers need to adopt a taxonomy of movement and develop research methods for investigating mathematical moving.