Folding back and growing mathematical understanding: a longitudinal study of learning
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Purpose The purpose of this paper is to summarize some of the key findings and approaches used in documenting the authors’ longitudinal studies of mathematical learning and understanding. In particular, it focuses on “folding back,” a theoretical construct originally developed by Susan Pirie and Tom Kieren, to show how, over the last two decades, the authors have taken up, built-upon, and elaborated this construct in relation to Pirie and Kieren’s wider theorizing and in relation to classroom practice. Design/methodology/approach The paper documents the various methodologies and methods the authors have used to elaborate theory and contribute to extending teaching practice in a number of related research studies. Findings This paper describes the role of folding back in the growth of students’ mathematical understanding, initially at the level of the individual, more recently at that of the collective – and currently with a specific consideration of the role of the teacher. It notes that the longitudinal nature of the work has allowed it to respond to shifting perspectives in the field of mathematics education and to become a more nuanced and powerful analytic and teaching tool. Originality/value The paper discusses the significance of a longitudinal, shared program of research, deeply rooted in mathematics classrooms, that builds theory systematically and over an extended period of time.
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.
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 | 0.002 | 0.003 |
| 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 | 0.000 |
| 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 it