Attentional Anchor


If you were to teach someone to stand with good posture, you might say something like, “Straighten your back, raise you chest a bit, drop and soften your shoulders, tuck your chin slightly, and don’t forget to relax!” A slight exaggeration, perhaps, but nonetheless an illustration of how even seemingly simple physical tasks can be composed of myriad distinct parts. An alternative instruction for good posture might be, “Stand as if a string were supporting you from the top of your head.” (Try it). All the pieces of the previous directive are subsumed within this new one, and somehow everything falls into place; one focus organizes many actions. This is the crux of attentional anchors (Hutto & Sánchez-García, 2015; and see Abrahamson, Sánchez-García, & Smyth, 2016, on metaphors as productive constraints on action).

Just as there was no literal string at the top of your head, attentional anchors need not be physically present in the environment (Abrahamson & Sánchez-García, 2016). Rather, these anchors arise from a need to perform some action and coordinate the various subparts of that action (Hutto, Kirchhoff, & Abrahamson, 2015). Yet these anchors are also tied to the perceptual environment; after all, performing an action depends on the environment in which it is performed.

Consider some examples from our work with the Mathematics Imager Trainer for Proportion. When moving their hands up and down in the air or on a touch screen, some students speak of manipulating an invisible diagonal line extending from one hand to the other or a rectangle configured by their hands and locations on the screen. Such shapes do not actually appear on the screen but are instead “highlighted” as would-be objects, imagined percepts that the students engage and sustain as a means to organize their action; instead of focusing on moving their left and right hands, students focus on manipulating one thing – the attentional anchor. What was a two-handed (bimanual) coordination task has thus been reduced in complexity to the manipulation of a single object.

We believe that research on attentional anchors in situated mathematics tasks could enhance the kinds of environments and interactions students experience as part of their mathematics learning. More broadly, we see attentional anchors as key theoretical constructs for developing the ecological dynamics approach to mathematics-education research in dialogue with both constructivist and sociocultural perspectives.

The emergence of an attentional anchor in the goal-oriented sensorimotor field.

In the figure, a young student with a history of difficulty in mathematics is solving a bimanual coordination problem. The left hand moves a cursor up and down, while the right hand moves another cursor right and left. The task objective of this orthogonal manipulation activity is to make the screen green. The screen will be green only when the right hand is doubly away from the origin (i.e. the bottom-left corner of the Cartesian space) as the left hand (a 1:2 ratio). The student does not know this initially. The orange dot shows the researchers where the student is gazing, but the student cannot see the dot in real time. The efficacy and consistency of the student’s actions suggest that he has invented a strategy whereby he attends to non-salient features of the visual field so as to enhance the effect of his physical actions. In turn, new phenomenal categories may become accessible to the student’s consciousness and discourse, leading to the articulation of new mathematical insight. We call this privileged visualization of the environment an attentional anchor; we theorize how attentional anchors relate to constructivism and sociocultural theory; and we design activities for students to learn mathematical concepts through moving in new ways and inventing action strategies focused on new objects. We conceptualized this approach to the study and support of learning as the ecological dynamics of mathematics education. This approach draws on the kinesiological science of motor-action learning and control. In particular, we are curious about the onset of non-equilibrium phase transitions in bimanual coordination afforded by the spontaneous invention of an attentional anchor. We wonder about the relations between apperceptions of external objects (or quasi-external imaginary constellations of stimuli, i.e., the attentional anchors) and the phase transition into new attractors facilitating effective interaction. Note how the dot rises diagonally along the trajectory of a linear function y = 1/2 x. What if the teacher could see these data in real time? What if the child could see them?

The attentional anchor (AA) is an enactive artifice, a subjective phenomenological construction productively constraining an agent’s dynamical sensorimotor interaction with the environment and potentially with other agents. In performing simple tasks, the AA may be degenerate, such as focusing on an actual feature in the environment that we are about to manipulate, for example a coffee cup. However, in performing complex tasks, an AA may emerge as a coordination structure, an action-bound perceptual gestalt wedged in the agent-environment system. The AA is thus a goal-oriented, sensorimotor, coordinative objectification of latent features at the agent–environment interface, and this constellation may include actual as well as imagined features. While they can be initially tacit, under appropriate conditions, contexts, and framings these constellations may rise to conscious reflection and disciplinary activity, entified as bonafide (immaterial) objects bearing properties and relations that can be monitored, described, inscribed, elaborated, measured, quantified, represented, schematized, modeled, calculated, predicted, etc., in turn becoming elements in yet more complex structures.


Duijzer, A. C. G., Shayan, S., Van der Schaaf, M. F., Bakker, A., & Abrahamson, D. (2017).  Touchscreen tablets: Coordinating action and perception for mathematical cognitionFrontiers in Psychology, 8(144).

ABSTRACT: Proportional reasoning is important and yet difficult for many students, who often use additive strategies, where multiplicative strategies are better suited. In our research we explore the potential of an interactive touchscreen tablet application to promote proportional reasoning by creating conditions that encourage transitions to multiplicative strategies. The design of this application (Mathematical Imagery Trainer) was inspired by arguments from embodied-cognition theory that mathematical understanding is grounded in sensorimotor schemes. This study draws on a corpus of previously treated data of 9-11 year-old students, who participated individually in semi-structured clinical interviews, in which they solved a manipulation task that required moving two vertical bars at a constant ratio of heights (1:2). Qualitative analyses revealed the frequent emergence of visual attention to the screen location halfway along the bar that was twice as high as the short bar. The hypothesis arose that students used so-called “attentional anchors” (AAs)—psychological constructions of new perceptual structures in the environment that people invent spontaneously as their heuristic means of guiding effective manual actions for managing an otherwise overwhelming task, in this case keeping vertical bars at the same proportion while moving them. We assumed that students’ AAs on the mathematically relevant points were crucial in progressing from additive to multiplicative strategies. Here we seek farther to promote this line of research by reanalyzing data from 38 students (aged 9-11). We ask: (1) What quantitative evidence is there for the emergence of attentional anchors?; and (2) How does the transition from additive to multiplicative reasoning take place when solving embodied proportions tasks in interaction with the touchscreen tablet app? We found that: (a) AAs appeared for all students; (b) the AA-types were few across the students; (c) the AAs were mathematically relevant (top of the bars and halfway along the tall bar); (d) interacting with the tablet was crucial for the AAs’ emergence; and (e) the vast majority of students progressed from additive to multiplicative strategies (as corroborated with oral utterance). We conclude that touchscreen applications have the potential to create interaction conditions for coordinating action and perception for mathematical cognition.

Abrahamson, D., & Bakker, A. (2016). Making sense of movement in embodied design for mathematics learning. In N. Newcombe & S. Weisberg (Eds), Embodied cognition and STEM learning [Special issue]. The Psychonomic Society—Cognitive Research: Principles and Implications (CRPI), 1(1), Article #33. 

  • Abstract: Embodiment perspectives from the cognitive sciences offer a rethinking of the role of sensorimotor activity in human learning, knowing, and reasoning. Educational researchers have been evaluating whether and how these perspectives might inform the theory and practice of STEM instruction. Some of these researchers have created technological systems, where students solve sensorimotor interaction problems as cognitive entry into curricular content. However, the field has yet to agree on a conceptually coherent and empirically validated design framework, inspired by embodiment perspectives, for developing these instructional resources. A stumbling block toward such consensus, we propose, is an implicit disagreement among educational researchers on the relation between physical movement and conceptual learning. This hypothesized disagreement could explain the contrasting choices we witness among current designs for learning with respect to instructional methodology for cultivating new physical actions: Whereas some researchers use an approach of direct instruction, such as explicit teaching of gestures, others use an indirect approach, where students must discover effective movements to solve a task. Prior to comparing these approaches, it may help first to clarify key constructs. In this theoretical essay we draw on embodiment and systems literature as well as findings from our design research so as to offer the following taxonomy that may facilitate discourse about movement in STEM learning: (a) distal movement is the technologically extended effect of physical movement on the environment; (b) proximal movement is the physical movements themselves; and (c) sensorimotor schemes are the routinized patterns of cognitive activity that becomes enacted through proximal movement by orienting on so-called attentional anchors. Attentional anchors are goal-oriented phenomenological objects or enactive perceptions (“sensori-”) that organize proximal movement to effect distal movement (“-motor”). All three facets of movement must be considered in analyzing embodied learning processes. We demonstrate that indirect movement instruction enables students to develop new sensorimotor schemes including attentional anchors as idiosyncratic solutions to physical interaction problems. These schemes are by necessity grounded in students’ own agentive relation to the world while also grounding target content, such as mathematical notions.
  • Significance: Engineering developments in computational technology have created unprecedented opportunities for industry to build and disseminate mathematics-education applications (“apps”). Thousands of these applications are now literally at the fingertips of any child who can access a tablet, smartphone, or personal computer with responsive touchscreen. Educational researchers could contribute to the quality of these ubiquitous consumer products by offering design frameworks informed by theories of learning. However, existing frameworks are derived from interaction theories drawing on epistemological assumptions that are no longer tenable, given the embodiment turn in the cognitive sciences. A proposed systemic reconceptualization of mathematical objects as grounded in sensorimotor schemes for material interaction offers educational designers heuristics for creating activities in which students learn by discovering motion patterns.

Abrahamson, D. (2016). The ecological dynamics of mathematics education: The emergence of proportional reasoning in fields of promoted action (Invited keynote in W. van Dooren & G. Wiiliams [Co-Chairs], Topic Study Group 27: Learning and cognition in mathematics). In G. Kaiser (Ed.), Proceedings of the 13th International Congress on Mathematical Education. Hamburg: University of Hamburg.

ABSTRACT: Originating in the fields of kinesiology and sports science, the theory of ecological dynamics draws on ecological psychology and dynamical-systems theory to explain processes of motor-action learning in sociocultural contexts. From this systemic view, learning is the achievement of new task-oriented dynamical stability between an agent and its environment, and teaching is the sociocultural orchestration of constraints that steer the agent toward developing new motor-action coordination. Ecological dynamics could therefore frame empirical investigations into students’ multimodal behaviors as they engage in solving physical-interaction problems designed to ground mathematical concepts. I summarize analyses of integrated clinical, action-logging, and eye-tracking data from a design-research project centered on the Mathematical Imagery Trainer. I propose that recent technological innovations for fostering as well as analyzing multimodal action may revive Piagetian scholarship and design.

Shayan, S., Abrahamson, D., Bakker, A., Duijzer, A. C. G., & Van der Schaaf, M. F. (in press). Eye-tracking the emergence of attentional anchors in a mathematics learning tablet activity. In A. W. Christopher, F. J. Sansosti, & B. J. Morris (Eds.), Eye-tracking technology applications in educational research. Hershey, PA: IGI Global.

FIRST PARAGRAPHS: Eye-tracking is a technique for collecting data in the context of conducting empirical studies of human perception, cognition, and behavior by determining aspects of participants’ sensory perception in the visual modality—where they are looking. In turn, locations of visual perception can be used to infer foci and patterns of gaze as these are relevant to making sense of human cognition and behavior. Over the years, eye-tracking hardware has advanced to the point that the instruments are now mobile, so that gaze data can be collected not only in research laboratories but also in the field, such as in investigating the perceptual behavior of supermarket consumers.

Although eye-tracking technology has been used quite widelyin cognitive psychology for many decades now, it has only quite recently been employed in the context of conducting educational research. In particular, eye tracking has been used to study how students solve problems in the domains of physics and mathematics (De Corte, Verschaffel, & Pauwels, 1990; Hegarty & Just, 1993; Hegarty, Mayer, & Green, 1992; Landy, Jones & Goldstone, 2008; Suppes, 1990; Suppes, Cohen, Laddaga, Anliker, & Floyd, 1983; Verschaffel, De Corte, & Pauwels, 1992), reading and comprehension (Paulson & Henry, 2002; Rayner, Chace, Slattery & Ashby, 2006), and multimedia learning and interaction (van Gog & Jarodzka, 2013; van Gog & Scheiter, 2010 ; Scheiter & van Gog, 2009). The particular contribution of eye-tracking to educational research has been by combining it with data of students’ physical movements and verbal utterances. Knowing where and possibly what students are looking at as they interact in a designed environment has enhanced micro-level analyses of learning to a level that had not been possible without this multi-modal approach.

Abrahamson, D., & Sánchez-García, R. (2016). Learning is moving in new ways: The ecological dynamics of mathematics educationJournal of the Learning Sciences25(2), 203-239.

ABSTRACT: Whereas emerging technologies, such as touchscreen tablets, are bringing sensorimotor interaction back into mathematics learning activities, existing educational theory is not geared to inform or analyze passages from action to concept. We present case studies of tutor–student behaviors in an embodied-interaction learning environment, the Mathematical Imagery Trainer. Drawing on ecological dynamics—a blend of dynamical-systems theory and ecological psychology—we explain and demonstrate that: (a) students develop sensorimotor schemes as solutions to interaction problems; (b) each scheme is oriented on an attentional anchor—a real or imagined object, area, or other aspect or behavior of the perceptual manifold that emerges to facilitate motor-action coordination; and (c) when symbolic artifacts are introduced into the arena, they may both mediate new affordances for students’ motor-action control and shift their discourse into explicit mathematical re-visualization of the environment. Symbolic artifacts are ontological hybrids evolving from things you act with to things you think with. Students engaged in embodied-interaction learning activities are first attracted to symbolic artifacts as prehensible environmental features optimizing their grip on the world, yet in the course of enacting the improved control routines, the artifacts become frames of reference for establishing and articulating quantitative systems known as mathematical reasoning. [Shorter version appears in PME-NA 2015 ; See also ICLS 2016 on metaphors as constraints on action.]

Abrahamson, D., Shayan, S., Bakker, A., & Van der Schaaf, M. F. (2016). Eye-tracking Piaget: Capturing the emergence of attentional anchors in the coordination of proportional motor action. Human Development, 58(4-5), 218-244. 

ABSTRACT: The combination of two methodological resources—natural-user interfaces (NUI) and multimodal learning analytics (MMLA)—is creating opportunities for educational researchers to empirically evaluate theoretical models accounting for the emergence of concepts from situated sensorimotor activity. 76 participants (9-12 yo) solved tablet-based presymbolic manipulation tasks designed to foster grounded meanings for the mathematical concept of proportional equivalence. Data gathered in task-based semi-structured clinical interviews included action logging, eye-gaze tracking, and videography. Analysis of these data indicates that successful task performance coincided with spontaneous emergence of stable dynamical gaze-path patterns soon followed by multimodal articulation of strategy. Significantly, gaze patterns included unmanipulated, non-salient screen locations. We present cumulative evidence that these gaze patterns served as ‘attentional anchors’ mediating participants’ problem solving. By way of contextualizing our claim, we also present case studies from the various experimental conditions. We interpret the findings as enabling us to revisit, support, refine, and perhaps elaborate on seminal claims from Piaget’s theory of genetic epistemology and in particular his insistence on the role of situated motor-action coordination in the process of reflective abstraction. (Shorter version appears in ICLS 2016)

Abrahamson, D., Sánchez-García, R., & Smyth, C. (2016). Metaphors are projected constraints on action: An ecological dynamics view on learning across the disciplines. In C.-K. Looi, J. L. Polman, U. Cress, & P. Reimann (Eds.), “Transforming learning, empowering learners,” Proceedings of the International Conference of the Learning Sciences (ICLS 2016) (Vol. 1, “Full Papers”, pp. 314-321). Singapore: International Society of the Learning Sciences. [Selected as Best Paper ICLS 2016]

ABSTRACT:  Learning scientists have been considering the validity and relevance of arguments coming from philosophy and cognitive science for the embodied, enactive, embedded, and extended nature of individual learning, reasoning, and practice in sociocultural ecologies. Specifically, some design-based researchers of STEM cognition and instruction have been evaluating activities for grounding subject content knowledge in interactive sensorimotor problem solving. In so doing, we submit, the field stands greatly to avail of theoretical models and pedagogical methodologies from disciplines oriented explicitly on understanding, fostering, and remediating motor action. This conceptual paper considers potential values of ecological dynamics, a perspective originating in kinesiology, as an explanatory resource for tackling enduring Learning Sciences research problems. We support our position via an ecological-dynamics reexamination of the function of metaphor in the instruction of sports skills, somatic awareness, and mathematics. We propose a view of metaphors as productive constraints reconfiguring the dynamic system of learner, teacher, and environment.

Hutto, D. D., Kirchhoff, M. D., & Abrahamson, D. (2015). The enactive roots of STEM: Rethinking educational design in mathematics. In P. Chandler & A. Tricot (Eds.), Human movement, physical and mental health, and learning [Special issue]. Educational Psychology Review, 27(3), 371-389.

ABSTRACT: New and radically reformative thinking about the enactive and embodied basis of cognition holds out the promise of moving forward age-old debates about whether we learn and how we learn. The radical enactive, embodied view of cognition (REC) poses a direct, and unmitigated, challenge to the trademark assumptions of traditional cognitivist theories of mind— those that characterize cognition as always and everywhere grounded in the manipulation of contentful representations of some kind. REC has had some success in understanding how sports skills and expertise are acquired. But, REC approaches appear to encounter a natural obstacle when it comes to understanding skill acquisition in knowledge-rich, conceptually based domains like the hard sciences and mathematics. This paper offers a proof of concept that REC’s reach can be usefully extended into the domain of science, technology, engineering, and mathematics (STEM) learning, especially when it comes to understanding the deep roots of such learning. In making this case, this paper has five main parts. The section “Ancient Intellectualism and the REC Challenge” briefly introduces REC and situates it with respect to rival views about the cognitive basis of learning. The “Learning REConceived: from Sports to STEM?” section outlines the substantive contribution REC makes to understanding skill acquisition in the domain of sports and identifies reasons for doubting that it will be possible to apply the same approach to knowledge-rich STEM domains. The “Mathematics as Embodied Practice” section gives the general layout for how to understand mathematics as an embodied practice. The section “The Importance of Attentional Anchors” introduces the concept “attentional anchor” and establishes why attentional anchors are important to educational design in STEM domains like mathematics. Finally, drawing on some exciting new empirical studies, the section “Seeing Attentional Anchors” demonstrates how REC can contribute to understanding the roots of STEM learning and inform its learning design, focusing on the case of mathematics.

Hutto, D. D., & Sánchez-García, R. (2015). Choking RECtified: Embodied expertise beyond DreyfusPhenomenology and the Cognitive Sciences, 14(2), 309-331.

ABSTRACT: On a Dreyfusian account performers choke when they reflect upon and interfere with established routines of purely embodied expertise. This basic explanation of choking remains popular even today and apparently enjoys empirical support. Its driving insight can be understood through the lens of diverse philosophical visions of the embodied basis of expertise. These range from accounts of embodied cognition that are ultra conservative with respect to representational theories of cognition to those that are more radically embodied. This paper provides an account of the acquisition of embodied expertise, and explanation of the choking effect, from the most radically enactive, embodied perspective, spelling out some of its practical implications and addressing some possible philosophical challenges. Specifically, we propose: (i) an explanation of how skills can be acquired on the basis of ecological dynamics; and (ii) a non-linear pedagogy that takes into account how contentful representations might scaffold skill acquisition from a radically enactive perspective.