Considerations ConcerningBalacheff’s 1988 Taxonomy ofMathematical Proofs
Bibliographic record
Abstract
Current school curriculum documents require that justification and proof become a significant part of the mathematics classroom culture. In order to determine how well secondary level student teachers can complete a valid mathematical proof the researcher administered, the same mathematical task of Balacheff (1988) to a group of student teachers who were at the last semester of their teacher education program. The student teachers’ written responses were then classified using Balacheff’s Taxonomy of Proofs (BToMP). To assist in classifying student teachers’ work, the researcher generated examples corresponding to Balacheff’s taxonomy of proof. The purpose of this article is to confront the results of Balacheff and also to determine the various levels of proficiency with which the student teachers approached the task on the basis of BToMP. Along with the analysis of the results, the difficulties that the researcher encountered in categorizing student teachers’ written work according to BToMP, for the same task he administered in his study is also discussed in this article This study raises questions concerning the applicability of BToMP, especially with advanced level students who have preconceived ideas about what would constitute a “preferred” approach to the proving task. It also suggests a need for further research into the thought processes and cognitive skills that are necessary, no matter what one’s age, in solving mathematical proof tasks.
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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 | 0.033 | 0.063 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.007 | 0.006 |
| Science and technology studies | 0.006 | 0.020 |
| Scholarly communication | 0.007 | 0.016 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.004 | 0.009 |
| 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".