Proportional reasoning: Reducing the interference of natural numbers through an intervention based on the problem-solving framework of executive functions
Bibliographic record
Abstract
According to the problem-solving framework of executive functions, the first step is to construct a problem space, i.e., the representation of the problem and its possible solutions. We explored how different problem spaces affect students' proportional reasoning. Proportional reasoning is important in school and in everyday life. It involves the comparison of ratios and is known to be difficult. Previous studies have shown that difficulties in proportional reasoning may stem from the interference of the automatic comparison of the salient natural numbers that comprise the ratios. We designed two equivalent comparison of ratios tests that were visually very similar, the Drops test and the Juice test. In the Drops test, tenth graders were asked to compare the intensity of color of mixtures of red and white paint drops. In the Juice test, they were asked to compare the amount of juice each child receives when equally dividing the contents of cups of juice among children in each group. The Juice test was aimed at presenting the task in a mode leading to a problem space that directs students to calculate "rate per unit," thereby reducing the interference of the automatic comparison of the salient natural numbers. The findings indicated that success in the Juice test was higher than in the Drops test. Moreover, success in the Drops test was higher when performed after the Juice test. The current study suggests using modes of presentation that lead to problem spaces that direct students to use appropriate solution strategies, hence aiding them in overcoming difficulties. Using modes or orders of presentation could serve as important tools for educators in science and mathematics and could lead to higher academic achievements among their students.
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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 | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.002 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".