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. 2014 Jul 30;8:551. doi: 10.3389/fnhum.2014.00551

Table 1.

Summary of Piagetian and dynamical systems concepts for a theory of equilibration.

Piagetian concept DS definition Notation Example
SM coordination scheme Class of SM coordinations, defined e.g., by region in SM space, task constraints, etc. A, B, C… The class of movements and sensations that belong to the subject’s experience of pushing objects toward the ground; absorbing impacts with the hands etc.
Environmental response structure Those environmental variables most directly affecting the sensory variables in A, B, C. i.e., the projection of the whole dynamic system, when engaged in SM coordinations A, B, C, onto relevant environmental variables. A′, B′, C′… Sound of the ball hitting the floor; height of the ball above ground; force exerted by the ball on the hand
SM coordination Instance of SM coordination belonging to class A, B…, i.e., a trajectory in SM space that belongs to the respective SM coordination class. a(t), b(t),… A particular instance of pushing the ball towards the ground
Environmental response Instance of environmental response of class A, B… a′(t), b′(t), … The sound of the impact for this particular bounce
SM coordination and environmental response tuple Simultaneous occurrence of SM coordination a(t) ∈ A and corresponding environmental trajectory a′(t) ∈ A′ in the coupled system. <a, a′>
The set of all tuples <a, a′>. A × A′
Sensorimotor organization or sequence scheme Sensorimotor strategy. A sequence of SM coordination classes (and their corresponding environmental projections). O: A × A′ → B × B′ → … → A × A′ Ball bouncing sequence of coordinations that includes pushing the object towards the ground, hearing the impact, waiting for its return, preparing muscles for contact, absorbing the impact and pushing it back.
Assimilation of A′ by A in O (1) Stability condition: all a′(t) ∈ A′ are environmental responses corresponding to SM coordinations a(t) ∈ A. Or simply: all a(t) are true SM coordinations.
(2) Transition condition: all a(t) ∈ A are special SM coordinations, namely reliable transients leading to the next scheme in O (e.g., B).
Continuous, stable ball bouncing despite small variations in motor pattern or wind speeds
Accommodation of Y′ into O by A Parametric changes that re-establish a closed set of schemes O or O1 such that Y′ becomes the environmental projection of the whole system when engaged in A, which is a scheme belonging to organization O. This can involve modifying the previous A or creating a new scheme A1 and integrating it into O. Learning to bounce a ball on a slope
Lacuna: perturbation of SM scheme due to a “gap” in understanding Violation of the transition condition. Something is manifestly “unknown” about the world, since the presumed “right” handling of the situation (A × A′) does not lead to the next stage in the cycle (B × B′). Bouncing a ball on a slope for the first time. Ball does not return to the same position.
Obstacle: perturbation of SM scheme due to contradictions and disturbances. Violation of the stability condition. Something in the sensorimotor coordination has failed where in the past it used to work. Attempting to bounce a new ball that is significantly heavier than the one that had been accommodated. Bouncing demands more strength.
Equilibration: ongoing adaptive process involving assimilation and accommodation that stabilizes the totality of SM schemes against perturbations by an ever changing environment (lacunae, obstacles) and internal tensions. A potentially never ending series of parametric changes of the totality of SM organization, aimed at maximizing the stability of each organization against violations of the transition and stability conditions resulting from environmental perturbations or internal tensions. The process of learning to bounce the ball under a variety of conditions (size and weight of the ball, slope and friction of the floor, etc.).