Addressing the struggle to link form and understanding in fractions instruction
Bibliographic record
Abstract
BACKGROUND: Although making explicit links between procedures and concepts during instruction in mathematics is important, it is still unclear the precise moments during instruction when such links are best made. AIMS: The objective was to test the effectiveness of a 3-week classroom intervention on the fractions knowledge of grade 5/6 students. The instruction was based on a theory that specifies three sites during the learning process where concepts and symbols can be connected (Hiebert, 1984): symbol interpretation, procedural execution, and solution evaluation. Sample. Seventy students from one grade 5/6 split and two grade 6 classrooms in two public elementary schools participated. METHOD: The students were randomly assigned to treatment and control. The treatment (Sites group) received instruction that incorporated specific connections between fractions concepts and procedures at each of the three sites specified by the Sites theory. Before and after the intervention, the students' knowledge of concepts and procedures was assessed, and a random subsample of 30 students from both conditions were individually interviewed to measure their ability to make specific connections between concepts and symbols at each of the three sites. RESULTS: While all students gained procedural skill (p < .001), the students in the Sites condition acquired significantly more knowledge of concepts than the control group (p < .01) and were also better able to connect fractions symbols to conceptual referents (p < .025). CONCLUSIONS: The current study contributes to the literature because it describes when it might be important to link concepts and procedures during fractions instruction.
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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 | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| 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".