A microgenetic study of the multiplication and division inversion concept.
Bibliographic record
Abstract
This microgenetic study investigated the discovery and development of the multiplication and division concept of inversion. Little is known about multiplicative concepts relative to additive concepts, including the inversion concept. Grade 6 participants (mean age = 11 years 6 months) solved multiplication and division inversion problems (e.g., d x e/e) for several weeks. In the final week they solved inversion, modified inversion (e.g., e x d/e), and lure problems (e.g., d/e x d) to investigate transfer of knowledge. Despite years of formal arithmetic instruction and repeated exposure to inversion problems, over a third of the participants failed to discover the inversion-based shortcut whereas another third used the shortcut almost exclusively. Almost all participants had difficulty appropriately generalising the inversion concept. Current theories of mathematical understanding may need to be modified to include the developmental complexities of multiplicative concepts.
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How this classification was reachedexpand
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 | 0.000 | 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 | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".