An exploration of pre-service elementary teachers' mathematical knowledge for teaching
Bibliographic record
Abstract
AbstractMathematical knowledge for teaching, or MKT, is a critically important knowledge of mathematics unique to teachers and teaching. One aspect of MKT, specialized content knowledge (SCK), involves the ability to interpret nonstandard student solutions, represent relevant mathematical content non-symbolically, and explain standard math algorithms (Ball, Thames, & Phelps, 2008). A number of measures are currently available to those seeking to assess teachers' SCK, and still, this knowledge domain has "yet to be fully mapped" (Hill, 2010, p. 537). Nowhere is this lack of mapping more apparent than among pre-service teachers, who, due to a lack of teaching experience, are likely to exhibit SCK that is markedly different from that displayed by their in-service counterparts (Hill, 2010). Teacher practices and dispositions, while not a part of existing frameworks for SCK, are likely to play a key role in shaping the nature of this unique mathematical knowledge. This study sought to examine the nature of 11 pre-service teachers' specialized content knowledge, practices, and dispositions through the use of hour-long structured interviews (Ginsburg, Jacobs, & Lopez, 1998). In each interview, study participants were asked to interpret non-standard student solutions to two math problems, one involving a comparison of fractions and another involving multi-digit multiplication. Data gathered in this study indicate that pre-service teachers, unlike experienced teaching professionals, require greater support in two key aspects of SCK: generating non-symbolic representations and interpreting non-standard student solutions (Hill, 2007; Hill, 2010). Additionally, it would appear as though pre-service teachers would benefit from efforts to foster a flexible disposition, as such flexibility appears to augment one's specialized content knowledge (Hill. Dean & Goffney, 2007). The results of this study will inform teacher educators, who must make difficult choices when deciding how to design courses so as to make efficient use of what little time they are given to prepare pre-service teachers (Ball, Sleep, Boerst, & Bass, 2009; Hill, 2010; Kajander, 2010).Keywords: mathematical knowledge for teaching, specialized content knowledge, disposition, pre-service teacher
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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 | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.003 | 0.002 |
| Scholarly communication | 0.005 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.013 | 0.001 |
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".