Students' problem solving and understanding in learning mathematics through conceptually- and procedurally-focused instruction : a situated discourse approach
Bibliographic record
Abstract
The nature and effect of two approaches to instruction in mathematics were studied from a situated cognition perspective, which emphasizes the pivotal role of discourse in knowledge comprehension. Typically, students must construct conceptual knowledge representations that integrate different types of information from diverse resources. Currently, little is known about how students' cognitive processes are influenced by distinct instructional approaches. The study focused on two problem-based methods: conceptually-focused instruction (CFI) and procedurally-focused instruction (PR). Both methods implemented common principles based on cognitive theories. Moreover, different characteristics were identified in the cognitive theories having different research traditions as their source. These differences provided insights for devising the two instructional approaches. A one-on-one tutoring setting was implemented for investigation. Of the two individual students in each matched pair, one was randomly assigned to CFI, and the other to PFI. There were a total of twelve one-on-one tutoring sessions, with six grade nine students in each condition. The researcher, an experienced high school mathematics teacher and counselor for the French Montreal School Board, was also the tutor in both conditions. The same two problems on linear functions, with gradual increase in difficulty, were solved in each method. Points of disjunction of approaches were achieved through variation in the type and degree of explanation (modeling strategy in Problem 1) and of scaffolding (coaching strategy in Problem 2). The results indicated that the different instructional conditions were implemented with consistency. Within the problem-based instruction framework and for the same problem-solving activity, it was possible for the tutor to maintain the focus on concepts vs. procedures. Although both approaches were equally beneficial to students, they affected the students' cognitive activity differently. Students' talk was more procedural in both approaches, but their speech was more balanced between concepts and procedures in CFI than in PH, indicating that students' cognitive processes were transferred differently from the modeling to the coaching discourse. These findings have implications for policymakers in education and for Quebec Ministry of Education standards. Future research is recommended to extend such findings to other populations of students and to additional types of teaching approaches.
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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 | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".