“It won’t work every time”: Prospective elementary teachers’ counterexamples for students’ false arguments about fractions
Bibliographic record
Abstract
In today’s elementary mathematics classroom, students are urged to construct arguments. If this is to enhance students’ learning, teachers must be able to identify and refute students’ false arguments. This requires substantial knowledge, yet little research has examined the nature of this knowledge with prospective elementary teachers. We asked 17 prospective teachers to assess the validity of students’ arguments regarding the comparison of fractions and to refute those that were false using counterexamples. Teachers did well with the mathematical aspects of this task, successfully identifying false arguments and refuting them with correct counterexamples. The pedagogical aspects of the task were more challenging, as only one counterexample explained why an argument was false and counterexamples were hampered at times by distractors. We propose that teacher educators emphasize pedagogical considerations in preparing prospective elementary teachers for such work. However, which considerations to emphasize requires additional research examining elementary students’ reactions to counterexamples. • Prospective elementary teachers assessed students’ arguments about fractions. • Teachers successfully identified and refuted false arguments with counterexamples. • Few counterexamples explained why an argument was false. • At times, counterexamples were hampered by potential distractors or “noise”. • Which pedagogical considerations matter most for elementary students is unknown.
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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 | 0.012 | 0.052 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.004 | 0.006 |
| Scholarly communication | 0.007 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.003 | 0.005 |
| 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".