How Language Influences Thought: The Case of Multiplying Fractions
Bibliographic record
Abstract
This study addresses the question of whether and how language influences mathematical cognition. Canadian students in Grades 4–7 (N = 348) were asked fraction multiplication questions in one of the two ways. In one condition, students were asked a series of standard multiplication questions (e.g. What’s ½ × ½?). In the other condition, children were asked the same multiplication questions but the word “of” replaced the “×” symbol (e.g. “What’s ½ of ½?”). We hypothesized that the “of” condition would lead to better performance due to its familiar use to signify a proportion of a quantity or amount, encouraging children to interpret fraction multiplication as a “fraction of a fraction.” Contrary to our predictions, children in the “×” group answered more questions correctly and made fewer errors than children in the “of” group. While the word “of” prompted more diagrammatic reasoning, the “×” symbol prompted higher use of purely symbolic reasoning. Thus, diagrammatic reasoning did not help make fraction multiplication more intuitive and accurate. These findings suggest that changing the question format alone is insufficient to improve the comprehension of fraction multiplication. Given the poor performance of both groups, more research is needed on effective ways to improve the teaching and learning of fraction multiplication.
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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 | 0.001 |
| 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 | 0.000 | 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".