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Highlights: [•] We investigate pre-service secondary teachers’ covariational reasoning. [•] We characterize their covariational reasoning in multiple coordinate systems. [•] Different images of change play a role in their activity. [•] Covariational reasoning is important for representational activity. [•] Polar coordinates offer a context for promoting covariational reasoning. [Copyright &y& Elsevier]

• 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]