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This article explores three attributes of teachers’ understanding of fraction magnitude: the accuracy and reasonableness of teachers’ estimations in response to fraction arithmetic tasks as well as the alignment of the estimation strategies they used with the concept of fraction magnitude. The data were collected from a national sample of mathematics teachers in Grades 3–7 in which fraction concepts were taught (N = 603). The results indicated the teachers’ estimations were only partially accurate and reasonable, particularly when fraction division was involved. Furthermore, teachers’ credentials and the grade level at which they taught mathematics were significantly related to teachers’ understanding of fraction magnitude. 

This article explores three attributes of teachers’ understanding of fraction magnitude: the accuracy and reasonableness of teachers’ estimations in response to fraction arithmetic tasks as well as the alignment of the estimation strategies they used with the concept of fraction magnitude. The data were collected from a national sample of mathematics teachers in Grades 3–7 in which fraction concepts were taught (N = 603). The results indicated the teachers’ estimations were only partially accurate and reasonable, particularly when fraction division was involved. Furthermore, teachers’ credentials and the grade level at which they taught mathematics were significantly related to teachers’ understanding of fraction magnitude. 

In this paper, we focus on the design of assessments of mathematics teachers’ knowledge by emphasising the importance of identifying the purpose for the assessment, defining the specific construct to be measured, and considering the affordances of particular psychometric models on the development of assessments as well as how they are able to communicate learning or understanding. We add to the literature by providing illustrations of the interactions among these critical considerations in determining what inferences can be drawn from an assessment. We illustrate how the considerations shape assessments by discussing both existing and ongoing research projects. We feature discussion of two projects on which the authors of this paper are collaborating to demonstrate the affordances of attending to all three considerations in designing assessments of mathematics teachers’ knowledge to provide readers with opportunities to see those considerations in use. 

The development of a model that explains how teachers learn from teaching is critical for informing the design of quality professional development, which in turn can support teachers’ effectiveness and student learning. This article reports the authors’ effort to develop a model that brings together critical findings from existing research to unpack when and under what conditions teachers learn from teaching. Grounded in evidence drawn from research relating to teachers’ learning and practice, the authors build a rationale for the Learning from Teaching (LFT) model, introduce each component of the model and propose two conditions that increase the likelihood of teachers’ learning from their own teaching. 

In this study, we explored teachers’ attention to and flexibility with referent units as well as how teachers’ understanding of referent units is related to their performance on other fraction concepts and their professional background. By using data collected from 246 U.S. mathematics teachers in Grades 3–7 where fractions are taught, we found that teachers’ attention to and flexibility with referent units were moderately related. Whereas some teachers’ professional background variables could explain their flexibility with referent units, none of the variables was linked to their attention to referent units. Furthermore, both teachers’ attention to and flexibility with referent units seemed to be associated with their performance on other fraction concepts.