Analytic Justifications and Empirical Predictions for the Model of  Hierarchical Complexity

Michael Lamport Commons

Harvard Medical School

The Model of Hierarchical Complexity is part of a theory that simplifies and generalizes from Jean Piagetís stage theory.  The model describes discrete orders of hierarchical task complexity, suggesting that these orders underlie stage-like performance.  Furthermore, it suggests that the analytical properties of hierarchical orders may be defined by the following three axioms.  A higher order task-required action must:

  1. be definedd in terms of the lower-stage actions; and

  2. coordinate the lower-stage actions in a

  3. nonarbitrary way.

These axioms have to be satisfied in order to define a task sequence and the corresponding stage-of- performance sequence.  It does not posit detailed empirical forms of stages, or the empirical processes that cause stage change.  Nor does the existence of  stages or of stage sequences shown through task analysis depend on any particular psychological assumptions about stage or stage change.

This theory of hierarchical complexity mathematically shows that:

  1. The orders of hierarchical complexity are scaled by the natural numbers

  2. Orders of hierarchical complexity are therefore an interval scale.

A number of testable implications following from 1 and 2:

  1. There are gaps in task difficulty (and resultant performance) due to the fact that there are no non natural number orders of complexity

  2. Groups of tasks within a sequence at different orders of hierarchical complexity should cluster in well-defined and equally spaced groups in the appropriate analysis.

  3. Stages of performance should be equally spaced because orders of hierarchical complexity of the tasks are equally spaced.

  4. All stage transitions are therefore equally difficult, from # 5

  5. Quantal nature of task hierarchy means there can be no intermediate performances.  A task either meets conditions (1), (2), and (3) or does not.

  6. There cannot be any other stages other than the 14 we have proposed except for ones beyond 14.  The 14 stages have been proposed by a larger number of other researchers (e.g. Case, 1978, 1985; Fischer, 1980; Pascual_Leone, 1970, 1976).  Three of them are somewhat controversial, the 13th (paradigmatic) and 14th (crossparadigmatic), and the 5th (sentential).

Empirical findings.

  1. Six stages are found from the beginning of schooling to adulthood.  Earlier stages also posited to exist have been extensively studied by others.  Some later stages have only been studied using historical examples.

  2. No stage performance has been found to be acquired out of sequence and no stages skipped.

  3. No mixing of stage scores for items takes place.  A Saltus model ( Wilson, 1989) shows that there is no continuity between the stage items.

  4. Gaps in scaled difficulty of items shown using a Rasch analysis with a Saltus model have been shown to exist between at least some of the stages and  with some tasks 

  5. These gaps are relatively equal in size, suggesting that the task demands of transitioning from one stage to another are similar.

  6. People generally perform in a consistent manner across items from the same task sequence of the same hierarchical complexity.