Comparing and Transforming: An Application of Piaget’s Morphisms Theory to the Development of Class Inclusion and Arithmetic Problem Solving
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Bibliographic record
Abstract
In his final works, Piaget suggested formalizing the developmental processes operating in different fields in terms of the mathematical theory of morphisms and categories. This new approach is used to account for the stages of development in two distinct fields: the understanding of the concept of inclusion observed via quantification tasks and the resolution of complex additive problems in arithmetic. We shall demonstrate that the distinction between intramorphic, intermorphic and transmorphic levels proposed by Piaget accounts for the stages of development in these two fields as well as for numerous empirical facts which have been considered incompatible with the earlier structuralist approach. As Piaget suggested, the main difficulty which children encounter in trying to resolve inclusion and additive problems is indeed the transition from state-oriented to transformation-oriented reasoning.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it