Nonlinear Learning Trajectories From Natural to Decimal Numbers in Zambia Based on Process-Object Dualism
Bibliographic record
Abstract
Understanding the transition from natural to decimal numbers is a central challenge in mathematics education, especially in developing countries. The present study investigates how Zambian primary students conceptualize decimal numbers and which cognitive difficulties they encounter, using Sfard’s process–object dualism as a theoretical framework. To capture children’s conception, I focused on the sub-item of conception, knowledge. Data were collected over two years from 204 sixth- and seventh-grade students through assessments and interviews. To capture students’ cognitive trajectories, a seven-stage learning trajectory was developed. From prior knowledge, the students moved from operational knowledge of natural numbers and decimals to pseudo-structural and structural knowledge. The results revealed that many students demonstrated computational fluency without a deep conceptual grasp and that pseudo-structural knowledge persisted. Typical misconceptions included overgeneralizing natural number rules (e.g., 0.8 ÷ 2 = 4) and misinterpreting place value. Importantly, the results showed that learning does not proceed linearly from one stage to the next. Rather, students often moved back and forth between stages, revisiting earlier forms of reasoning even after demonstrating higher-level understanding. This non-linear and dynamic nature of learning suggests that operational and structural knowledge interact in complex ways. The study proposes a classification system to diagnose students’ conceptual stages and provide targeted instructional strategies. By extending process–object dualism to primary-level learning and emphasizing the recursive, interactive nature of concept development, this research offers new insights into improving mathematics instruction and assessment in low-resource educational settings.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 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".