The Spontaneous Emergence of Elementary Number-Theoretic Concepts and Techniques in Interaction with Computing Technology
Bibliographic record
Abstract
We present in this paper five case studies of instrumental genesis of number-theoretic concepts and techniques involving multiples, divisors, and numerical decomposition. The pupils, who were 12 to 15 years old, used the multi-line screen display of the graphing calculator to explore numerical tasks related to the “Five steps to zero” problem. The paper begins with a mathematical task analysis of various techniques that could be used to determine divisibility by 9. Findings show that the use of the technological tool permits students of this age to spontaneously develop variants of these techniques within tasks that simultaneously afford the co-emergence of related conceptual notions. A body of current research in computer-supported mathematics learning environments is oriented toward the theoretical development and empirical study of the interactions among the following concepts: artefact, instrument, technique, task, and theory (e.g., Artigue, 2001; Guin & Trouche, 1999; Guzman & Kieran 2002; Lagrange, 2000; Mariotti, 2002; Trouche, 2000; Varillon & Rabardel, 1995). In particular, these studies discuss the role played by the instrument in the process of instrumental genesis within the context of specific tasks and with pupils of different ages and technological experience. In a paper presented last year at PME (Guzman & Kieran, 2002), we reported, for example, on the perceptions expressed by case-study students of the role played by the calculating tool in the evolution of their numerical conceptual schemas. The present paper, which is quite different, analyses the nature of the mathematical techniques that emerged spontaneously in representative participants of the study. By means of two case studies from each of Secondary 1 and 2, and one from Secondary 3, we illustrate the number-based interactions between students and their calculators and show how, within a given set of tasks, the students developed numerical techniques that were afforded by the technology-supported environment in which their numerical explorations occurred. The emergence of these techniques provides further evidence of the intertwining nature of the development of procedural and conceptual learning in mathematics. THE RESEARCH STUDY
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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 | 0.004 | 0.015 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.003 | 0.010 |
| Scholarly communication | 0.004 | 0.003 |
| Open science | 0.001 | 0.006 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".