Equivalence in symbolic and nonsymbolic contexts: Benefits of solving problems with manipulatives.
Bibliographic record
Abstract
Children's failure on equivalence problems (e.g., 5 + 4 = 7 + _) is believed to be the result of misunderstanding the equal sign and has been tested using symbolic problems (including = ). For Study I (N = 48), we designed a nonsymbolic method for presenting equivalence problems to· determine whether Grade 2 children's difficulty is due to the presence of symbols or to a more fundamental misunderstanding of equivalence. Children's superior performance on nonsymbolic versus symbolic problems suggests that children fail to map their understanding of equivalence onto problems presented with the symbols of arithmetic. For Study 2 (N = 32), we implemented a within-subject design to assess whether experience with nonsymbolic problems would facilitate performance on symbolic problems. This hypothesis was confirmed. Exposure to nonsymbolic problems may have enabled children to map their successful concepts and strategies to symbolic equivalence problems
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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 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".