Understanding Mathematics through Resolution of Paradoxes
Bibliographic record
Abstract
Brain-challenging puzzles have attracted people for a very long time. Paradoxes constitute a special type of puzzle aimed to reveal and emphasize an inconsistency or contradiction resulting from some mental experiments in mathematics. Their resolution teaches us to stay alert and be aware of possible flaws of various kinds. Many paradoxes, such as those of Zeno and Russell, greatly influenced the shape of mathematics as we know it today. That suggests a possibility to incorporate the study of paradoxes in standard mathematical courses. But how productive may it be? At which stage of their study will students benefit from being exposed to paradoxes? How one can practically do it in the classroom? This paper is an attempt to address some aspects of these important questions. We discuss the nature and role of paradoxes in the process of understanding, along with potential problems and advantages of their use in study. We give several examples of mathematical paradoxes in both the historical and the classroom context. A short survey results outline an idea of the audience reaction and suggests further directions for research. We conclude that the pedagogical payoff of the use of paradoxes in the classroom is currently underestimated and a comprehensive and consistent study of the impact of paradoxes on learners will allow us to develop a teaching strategy which takes an advantage of the natural curiosity of the mind towards puzzles.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.008 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| 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".