Bibliographic record
Abstract
In this research, we analysed two didactic situations in order to study the functioning of arithmetic knowledge in the context of the transition from arithmetic to algebra, considering intellectual validation as a fundamental issue in this change of practice. These situations were tested in four classes, two classes in Argentina (level corresponding to the 1st year of secondary education in Quebec) and two classes in Quebec (2nd year of secondary education). The aim of our study was to highlight some of the limitations of arithmetic practices and the associated validation practices, when it comes to resolving a contradiction and validating the general nature of an algebraic formula. We were also interested in analysing the development of situations in the context of classes characterised by different didactic contracts: efforts to adapt the pupils' knowledge in order to provoke a change in practice (change of contract); the teacher's work within the constraints imposed by the situations.The design of the situations was based on the theory of situations and on several studies in the didactics of algebra. The problematic of our research led us to a more detailed analysis of certain aspects of these theorisations. Taking into account the need to develop a practice of intellectual validation, as a problematic in itself and as a possible source for giving meaning to algebraic practices, led to a closer examination of the notion of adidacticity and validation situations, in the theory of didactic situations developed by Brousseau (1998).Our attempt to set up a validation practice enabled us to observe a different way in which arithmetical knowledge functioned, by making it possible to give meaning to the pupils' entry into algebraic practices, and then to develop a particular language. At the same time, the development of this language has encouraged new objects to enter the classroom - 'numerical expressions', which were previously a means of calculation, are now becoming objects of reflection. But recognition of the explanatory and operative nature of writing requires a different use of the properties of operations, which are the fruit of social agreements and cultural conventions specific to a new practice. Our research has enabled us to characterise this change in practice and contract, and to realise the complexity of the teacher's role in this context.The development of social agreements and cultural conventions specific to a new practice involves the production of situational knowledge. Our research has characterised the nature of this production and the nature of the knowledge involved, by identifying different levels of knowledge in play. We have analysed this knowledge and the conditions of its production in the light of existing theorising; finally, we have formulated questions that have not yet been addressed in research into the didactics of mathematics.
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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 | 0.011 | 0.031 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.005 | 0.014 |
| Scholarly communication | 0.008 | 0.005 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.003 | 0.004 |
| Insufficient payload (model declined to judge) | 0.009 | 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".