Inventing (in) early geometry, or How creativity inheres in the doing of mathematics
Bibliographic record
Abstract
Inventing is fundamental to mathematical activity, should one be a professional mathematician or a primary school student. Research on mathematical creativity generally is organized along three axes according to its focus on the final product, the overall process, or the individual person. Through these conceptualizations, however, research rarely considers how mathematical actions themselves are fundamentally creative. In an action-oriented perspective, every single act is recognized as creative, whereas discovery and invention emerge as the result of the incoming of the unexpected qua unexpected—which can take place at any moment in the most mundane, everyday action. In this article, we conceptualize mathematical actions as inherently creative of the activity within which professional mathematicians and primary school students experience (some) mathematics for a first time. To make our case, we develop the microanalysis of an exemplary episode of third-grade geometry (age 8-9 years) in which two children and an adult work with a tangram set. Our analysis characterizes inventing (in) geometry as a serendipitous, open-ended experience of working with traces in the receiving and the offering of something novel. In concluding, we propose considering that inventing in early geometry is also inventing geometry itself: an inventing-in-the-act which also result in being invented as a (professional or school) geometer
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.014 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".