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11.3 Piaget's Theory of Cognitive Development 191

I cover my eyes, can you still see me?). Complex concrete operations, such as putting items
in height order, are easily achievable. Classification becomes more sophisticated, yet the
mind still cannot master purely logical operations based on abstract logical representations
of the observational world.

• Formal: Abstract deductive reasoning, the process of forming, then testing hypotheses, and
systematically reevaluating and refining solutions, develops at this stage, as does the ability
to reason about purely abstract concepts without reference to concrete physical objects.
This is adult human-level intelligence. Note that the capability for formal operations is
intrinsic in the PLN component of CogPrime, but in-principle capability is not the same as
pragmatic, grounded, controllable capability.

Very early on, Vygotsky [Vyg86] disagreed with Piaget's explanation of his stages as inherent
and developed by the child's own activities, and Piaget's prescription of good parenting as
not interfering with a child's unfettered exploration of the world. Some modern theorists have
critiqued Piaget's stages as being insufficiently socially grounded, and these criticisms trace back
to Vygotsky's focus on the social foundations of intelligence, on the fact that children function
in a world surrounded by adults who provide a cultural context, offering ongoing assistance,
critique, and ultimately validation of the child's developmental activities.

Vygotsky also was an early critic of the idea that cognitive development is continuous,
and continues beyond Piaget's formal stage. Gagne [RBW92] also believes in continuity, and
that learning of prerequisite skills made the learning of subsequent skills easier and faster
without regard to Piagetan stage formalisms. Subsequent researchers have argued that Pi-
aget has merely constructed ad hoc descriptions of the sequential development of behaviour
[Gib78, Bro84, CP05]. We agree that learning is a continuous process, and our notion of stages
is more statistically constructed than rigidly quantized.

Critique of Piaget's notion of transitional "half stages" is also relevant to a more compre-
hensive hierarchical view of development. Some have proposed that Piaget's half stages are
actually stages [Bro84]. As Commons and Pekker [CP05] point out: "the definition of a stage
that was being used by Piaget was based on analyzing behaviors and attempting to impose
different structures on them. There is no underlying logical or mathematical definition to help
in this process ..." Their Hierarchical Complexity development model uses task achievement
rather than ad hoc stage definition as the basis for constructing relationships between phases
of developmental ability – an approach which we find useful, though our approach is different
in that we define stages in terms of specific underlying cognitive mechanisms.

Another critique of Piaget is that one individual's performance is often at different ability
stages depending on the specific task (for example [GE86]). Piaget responded to early critiques
along these lines by calling the phenomenon "horizontal décalage," but neither he nor his suc-
cessors [Fis80, Cas85] have modified his theory to explain (rather than merely describe) it.
Similarly to Thelen and Smith [TS94], we observe that the abilities encapsulated in the defini-
tion of a certain stage emerge gradually during the previous stage – so that the onset of a given
stage represents the mastery of a cognitive skill that was previously present only in certain
contexts.

Piaget also had difficulty accepting the idea of a preheuristic stage, early in the infantile
period, in which simple trial-and-error learning occurs without significant heuristic guidance
[Bic88], a stage which we suspect exists and allows formulation of heuristics by aggregation of
learning from preheuristic pattern mining. Coupled with his belief that a mind's innate abilities
at birth are extremely limited, there is a troublingly unexplained transition from inability to
ability in his model.

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