A string number-line lesson sequence to promote students' relative thinking and understanding of scale, key elements of proportional reasoning
Bibliographic record
Abstract
Proportional reasoning involves relative thinking: the ability to think about multiple quantities simultaneously, and in relative terms as opposed to absolute terms (Ontario Ministry of Education, 2012). Concepts related to relative and absolute thinking and the ability to think relatively and multiplicatively are essential to understanding proportionality (Hilton, Hilton, Dole, & Goos, 2015). It takes time for students to develop proportional reasoning and without targeted teaching, many students fail to develop the important skills and conceptual understanding that underpins it (Bangert- Drowns, Hurley, & Wilkinson, 2004; Kastberg, D'Ambrosio, & Lynch-Davis, 2012; Lamon, 2012). Proportional reasoning is essential for students to succeed in many mathematical areas, including ratio and proportion, measurement and unit conversions, geometry, and probability. It is also necessary in other subjects, such as geography and science (Akatugba & Wallace, 2009). In fact, scale and proportion have been identified as crosscutting concepts fundamental to understanding and reasoning in science (National Research Council, 2012).
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".