Changing Conceptions of Mathematics
Bibliographic record
Abstract
This descriptive case study explores how a conceptual understanding of fractions develops in pre-service elementary teachers enrolled in a reform-based, remedial mathematics skills course set at a middle school level, during their final year of an education program in a small Canadian university. I compared the learning trajectory of these adults to that of children and then developed a landscape of (re)learning fraction concepts within a constructivist-oriented environment. Fourteen prospective teachers completed pre/posttest content exams and interviews that followed-up each of these test instruments, as well as two paired problem-solving interviews that used a modified think aloud protocol, and open ended questionnaires. All interviews were videotaped using two cameras, capturing both the participant(s) and a zoomed in view of their hands and written work. Data collection took place at four points in the school year between September and March, providing snapshots of the developmental progression of fraction understanding in the pre-service teachers over time. Analysis occurred at multiple layers: first with the individual pre-service teachers, then with three clusters of participants grouped according to their mathematical abilities, and finally with the group as a whole. The theoretical framework utilized a constructivist
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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 | 0.009 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.007 | 0.022 |
| Scholarly communication | 0.006 | 0.006 |
| Open science | 0.001 | 0.005 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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".