Abstract
In the past decade, research on learning trajectories has progressed from an agenda for research on student learning to include an agenda for research on teaching. In this paper, we use a design experiment to examine a professional development program created to support teacher learning of one learning trajectory. Using a re-interpretation of Ball and colleagues’ mathematics knowledge for teaching framework, we conducted a retrospective analysis to examine how a purposefully selected sample of three teachers used their knowledge to participate in discussions of the professional learning tasks designed to support teacher learning of the trajectory. Findings indicate that professional learning tasks focusing on pedagogical content knowledge present in learning trajectories also allow for teacher learning of subject matter knowledge; this learning is mediated by teachers’ prior mathematics knowledge for teaching. We conclude with considerations for researchers and professional developers seeking to support teachers in learning about learning trajectories.
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Notes
See Confrey (2012) for a complete description of the EPLT.
This was original identified and named by Pothier as “composition of factors” but because Confrey et al. (2009) use it to introduce multiplication rather than to build it in light of knowledge of factors, it was renamed to be derived from knowledge of splitting.
This emergent relationship has since been renamed as the Property of Equality for Equipartitioning (PEEQ) in light of the fact that if two or more equal-sized shapes are split into the same number of parts, those parts will be equal. (J. Confrey personal communication).
One notes that in this case, the four triangles have equivalent areas but only two pairs of triangles are congruent. The EPLT uses this task to have students reformulate their earlier belief that equal-sized parts must be both the same size and the same shape (Confrey 2012).
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This report is based upon work supported by the National Science Foundation under Grant Number DRL-1008364. Any opinions, findings, and conclusions or recommendations expressed in this report are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Appendix
Appendix
Day | Goal | Engagement | Exploration | Formalization | Application |
|---|---|---|---|---|---|
1 | Three equipartitioning criteria | What do students know about equipartitioning based on their work? | Investigation of selected work samples | Group discussion to articulate the 3 equipartitioning criteria from the EPLT | Identify the criteria within the student work samples |
2 | LT levels for sharing collections | What strategies do students use when sharing collections? | Viewing of clinical interviews | Group discussion of strategies from EPLT | Analyze curriculum to identify instances of equipartitioning |
2/3 | LT levels for factor-based change and reallocation (Sequence I) | How do students use compensation when sharing collections? | Viewing of clinical interviews | Group discussion, of qualitative compensation, factor-based change, and reallocation from the EPLT | Anticipate student responses to assessment tasks targeted at these EPLT levels |
3 | LT level for composition of splits (Sequence II) | What emergent properties are unique to sharing wholes and how do task parameters interact students’ approaches? | Paper folding task | Group discussion of task parameters from the EPLT | Order EPLT assessment tasks in increasing difficult and analyze student responses to the items |
3 | LT levels for justification and transitivity (Sequence III) | How do students justify fair shares? | Viewing of clinical interviews and justifying the equivalence of non-congruent shares | Group discussion of justifications and emergent relationships from the EPLT | Analyze classroom video of an equipartitioning lesson |
4/5 | LT Level for sharing multiple wholes (Sequence IV) | What strategies and naming practices do students use when sharing multiple wholes? | Conducting and participating in clinical interviews | Group discussion of strategies from EPLT | Analyze student responses to assessment tasks targeted at these LT levels |
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Wilson, P.H., Sztajn, P., Edgington, C. et al. Teachers’ use of their mathematical knowledge for teaching in learning a mathematics learning trajectory. J Math Teacher Educ 17, 149–175 (2014). https://doi.org/10.1007/s10857-013-9256-1
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DOI: https://doi.org/10.1007/s10857-013-9256-1