Measurement Approach to Teaching Fractions: A Design Experiment in a Pre-service Course for Elementary Teachers
Bibliographic record
Abstract
In this paper, we present a design experiment in a "Teaching Mathematics"course for prospective elementary teachers where we sought to develop a measurement approach to fractions. We focus on the conceptualization of the mathematical content of the approach. We attribute our progress in the conceptualization to our efforts to overcome the challenges we had to face in bringing our students - prospective teachers - to thinking about fractions in a theoretical way. We describe some of these challenges in the paper. The approach was inspired by an approach under the same name, but addressed to children, developed by the psychologist V.V. Davydov and described in the paper by Davydov and Tsvetkovich (1991). Davydov's measurement approach proposes that, in order to develop a concept of fraction with sources in reality, children's attention should be directed to multiplicative relationships between quantities defined in terms of concrete units (such as kilos, inches or cups) rather than to indeterminate objects such as pizzas or cakes. Our original contribution is a systemic study of these relationships and operations on them, as a theoretical system, using definitions derived from measurement situations, mathematical reasoning based on these definitions and generalizations of observed patterns. It is intended to support a gradual process of abstraction of the notion of fraction as an abstract number that represents a measure of the relationship between two quantities. Furthermore, our proposal for conceptualizing fractions in this way is addressed to teachers, not to children. By its focus on relational reasoning about quantities and gradual construction of a theoretical system, the approach both requires and is expected to foster the development of quantitative reasoning and theoretical thinking.
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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 | 0.017 | 0.039 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.004 | 0.004 |
| Insufficient payload (model declined to judge) | 0.007 | 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".