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Anyone involved in mathematics education will at some point encounter a classroom situation in which a student's statement provides an opportunity for the teacher to enhance students' mathematical understanding. This paper aims to characterize such classroom situations, known as critical events, as noticed by pre-service mathematics teachers during classroom observations and teaching. Analysis of 80 critical events reported by 20 pre-service mathematics teachers during a field-based teaching preparation program yielded a model that enables teacher-educators and researchers to characterize critical events while reflecting on the wide variations within these events. The model comprises three axes: the participants in the event, the focus of the event, and the dimensions in learning and teaching mathematics (i.e., cognitive, affective and social). The paper discusses the theoretical and empirical construction of the model and demonstrates its potential contribution using examples taken from our research. • Pre-service teachers identified critical events during classroom observations. • A model for characterizing critical events was constructed theoretically and empirically. • The model affords eliciting the multidimensional nature and complexity of critical events. • The model's potential contribution is presented using examples from our research. [ABSTRACT FROM AUTHOR]

• Mathematicians may value pedagogical content knowledge but lack confidence in teaching for its development. • Mathematicians and mathematics educators may value pedagogical content knowledge for similar reasons. • Expectancy-value theory may explain a lack of focus on developing teachers' pedagogical content knowledge. Recent studies suggest that change is needed in undergraduate mathematics capstone courses for prospective secondary teachers. One promising but infrequently used strategy for improvement is to incorporate tasks that explicitly focus on pedagogical content knowledge (PCK). Expectancy-value theory provides an account for why instruction of these courses does not more regularly employ this strategy. To make this argument, this paper uses an interview-based study of 9 mathematicians that investigated the process of prioritizing tasks and goals for these courses. As the study found, these mathematicians valued developing teachers' PCK. However, they were unconfident of their ability to teach with tasks and goals focused on developing teachers' PCK relative to more purely mathematical tasks and goals. The central implication is that interventions in mathematicians' teaching must take into account the possibility that it may be just as important to improve confidence and resources as it is to change values. [ABSTRACT FROM AUTHOR]

• We examine the relationship between teacher meanings and conventions. • Teachers often do not dissociate conventions from logico-mathematical operations. • We provide implications for teaching in relation to conventions of graphing. In this paper, we use relevant literature and data to motivate a more detailed look into relationships between what we perceive to be conventions common to United States (U.S.) school mathematics and individuals' meanings for graphs and related topics. Specifically, we draw on data from pre-service (PST) and in-service (IST) teachers to characterize such relationships. We use PSTs' responses during clinical interviews to illustrate three themes: (a) some PSTs' responses implied practices we perceive to be conventions of U.S. school mathematics were instead inherent aspects of PSTs' meanings; (b) some PSTs' responses implied they understood certain practices in U.S. school mathematics as customary choices not necessary to represent particular mathematical ideas; and (c) some PSTs' responses exhibited what we or they perceived to be contradictory actions and claims. We then compare our PST findings to data collected with ISTs. [ABSTRACT FROM AUTHOR]