CALCULABILITY OF WRITTEN INSTRUMENTS FOR PROBLEM SOLVING
Bibliographic record
Abstract
In order to study spontaneous phenomena of semiotic instrument genesis in mathematical problem solving, we have designed an experimental situation in which students from 6 th grade to 11 th grade had to solve a series of problems commonly identified as "algebra problems with two unknowns". We examine how the students' writings allow to achieve the various cognitive functions involved in problem solving (i.e. their "calculability"), how the writings evolve into an autonomous status from the activity which produced them, as well as how they develop thanks to the repetition of isomorphic problems. This paper demonstrates that a significant minority of students makes use or invents written instruments. It also illustrates two categories of genesis phenomena : the adaptation of a curricular instrument to the problem constraints and the composition of a new instrument from previous isolated instruments in a "bricolage" perspective. Instrumental evolution from one problem to the next is rather rare and when it does occur it is mostly directed towards extending the use of previous instruments. In particular, the transformation of the pattern ax + by = d from a proof into a calculable representation, seems to be a privileged enough transition between arithmetic and algebra to be further studied.
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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 | 0.003 | 0.028 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| 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".