KNOWING AND USING MATHEMATICS IN TEACHING: CONCEPTUAL AND EPISTEMOLOGICAL CLARIFICATIONS
Bibliographic record
Abstract
The body of research on teachers' mathematical knowledge for teaching has been growing in importance in recent years in the international research community (Adler & Davis, 2006; Ball & Bass, 2003; Davis & Simmt, 2006; Margoli nas, Coulanges & Bessot, 2005). These studies, grounded in empirical data and theoretical reflections about teachers' mathematical knowledge, offer new ways of thinking about teachers' mathematical education. They challenge the math ematics teacher education structures prevalent in most universities, where an emphasis is placed on university aca demic mathematical training something that has been argued to be quite disconnected from the mathematical prac tices teachers enact in their classrooms (see, e.g., Moreira & David, 2008; Proulx & Bednarz, 2008). That said, as francophones who work in the field of didac tique des mathematiques in Quebec (Canada) [2], we admit to having been surprised by the excitement that the body of research on teachers' mathematical knowledge, pioneered by Ball and Bass, has recently provoked in the scientific community. We were under the impression that this discourse (and the examples given to explicate this new field) had been present for a number of years in francophone communities, at least in Quebec, around the various developments enacted by mathematics teacher educators in pre-service teacher edu cation (see, e.g., Bednarz, 2001; Bednarz, Gattuso & Mary, 1995; Bednarz & Proulx, 2005; Janvier, 1996; Janvier & Hosson, 1999). However, this brought us to realize that these interventions and developments, and their underpinning principles, had not been theorized as well and as recent work in the domain of teachers' mathematical knowledge. [3] We thus perceive an opportunity to explain our conceptu alizations about knowing and using mathematics in teaching. We hope to contribute to the current reflections on teachers' mathematical knowledge for teaching and stim ulate discussions around this significant component of teachers' knowledge. We ground our discourse in mathematics teachers' actual practices, using vignettes taken from a collaborative study with a teacher. Our work has been refined through collabo rative studies with teachers (see, e.g., Bednarz, 2004,2009), which enabled us to better understand the knowledge enacted by teachers in mathematics teaching/learning situa tions. This theorization is also inspired from a posteriori reflections on the interventions developed in our secondary level mathematics teacher education program, established in the 1970s at the Universite du Quebec a Montreal (UQAM). Our discussion combines these two complementary axes: (1) theorizations about knowledge teachers enact in their mathe matics teaching practices and (2) illustrations, drawn from our program, of efforts to develop this professional knowledge.
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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 | 0.001 |
| 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 | 0.000 |
| 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 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".