In this study, we analyzed students' understanding of a complex calculus graphing problem. Students were asked to sketch the graph of a function, given its analytic properties (1st and 2nd derivatives, limits, and continuity) on specific intervals of the domain. The triad of schema development in the context of APOS theory was utilized to study students' responses. Two dimensions of understanding emerged, 1 involving properties and the other involving intervals. A student's coordination of the 2 dimensions is referred to as that student's overall calculus graphing schema. Additionally, a number of conceptual problems were consistently demonstrated by students throughout the study, and these difficulties are discussed in some detail.

This study examines the extent to which the New York City Teaching Fellows (NYCTF) has delivered on its promise of improving mathematics teacher diversity, preparedness, effectiveness, and retention in hard-to-staff city schools. As a program theory evaluation study, it articulates the theory of action for selective alternative route programs and uses this to evaluate NYCTF’s program for secondary mathematics. The analysis draws on longitudinal data from 620 secondary mathematics teachers who began NYCTF in the prior decade. While the results point to potential improvements, it provides evidence that selective programs like NYCTF serve to maintain important gaps in teacher quality that they were designed to address.

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