A Microgenetic Study of the Conceptual Development of Inversion on Multiplication/Division Inversion Problems
Bibliographic record
Abstract
The purpose of this study was to conduct a microgenetic study of the development of the concept of inversion as it applies to multiplication and division inversion problems. The study was modelled on Siegler and Stern’s (1998) study in which Grade 2 participants solved addition and subtraction inversion problems (a + b- b) for 6 weekly sessions. In session 7, modified inversion problems (b + a- b) as well as lure problems (b- a + b) were also included. In the current study, Grade 6 participants solved multiplication and division inversion problems (d x e ÷ e) during 6 weekly sessions. Previous research has shown that this latter type of inversion problems is more difficult than the former type. The present results indicate that there are differences in how frequently participants discover and apply the inversion concept compared to Siegler and Stern’s (1998) work. The findings add to the recent body of knowledge indicating that the concept of inversion as it applies to multiplication and division is significantly more difficult than it is for addition and subtraction.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 source (direct Gemma or distilled Codex), 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".