In this chapter, we use an analytic framework called Mathematical Discourse in Instruction (MDI) to investigate and describe the opportunities for learning to solve quadratic inequalities in Mr T.s lesson event. MDI brings to the fore two commonplaces of mathematics teaching: exemplification and explanatory talk that together can, and in our view should, enable access to scientific concepts (Vygotsky, 1978). Our analysis shows potential across examples for relating equations and inequalities, and for connecting different representations of the solution to a particular quadratic inequality, but this was thwarted by the accompanying explanatory talk, by how Mr T. legitimated the solution strategy he offered in the lesson, and in how he named what he talked about. Consequences follow for learning and, we will argue, deliberate work across research and practice.

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