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Abstract BackgroundWhat and how teachers learn through teaching without external guidance has long been of interest to researchers. Yet limited research has been conducted to investigate how learning through teaching occurs. The microgenetic approach (Siegler and Crowley, American Psychologist 46:606–620, 1991) has been useful in identifying the process of student learning. Using this approach, we investigated the development of teacher knowledge through teaching as well as which factors hinder or promote such development. ResultsOur findings suggest that teachers developed various components of teacher knowledge through teaching without external professional guidance. Further, we found that the extent to which teachers gained content-free or content-specific knowledge through teaching depended on their robust understanding of the concept being taught (i.e., content knowledge), the cognitive demand of the tasks used in teaching, and the lesson structure chosen (i.e., student centered vs. teacher centered). ConclusionsIn this study, we explored teacher learning through teaching and identified the sources leading to such learning. Our findings underscore the importance of teachers’ robust understanding of the content being taught, the tasks used in teaching, and a lesson structure that promotes teachers’ learning through teaching on their own. 

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.