This paper presents a conceptual analysis for students’ images of graphs and their extension to graphs of two-variable functions. We use the conceptual analysis, based on quantitative and covariational reasoning, to construct a hypothetical learning trajectory (HLT) for how students might generalize their understanding of graphs of one-variable functions to graphs of two-variable functions. To evaluate the viability of this learning trajectory, we use data from two teaching experiments based on tasks intended to support development of the schemes in the HLT. We focus on the schemes that two students developed in these teaching experiments and discuss their relationship to the original HLT. We close by considering the role of covariational reasoning in generalization, consider other ways in which students might come to conceptualize graphs of two-variable functions, and discuss implications for instruction.

The purpose of this article is to describe the development of elementary school teachers’ mathematical knowledge for teaching as they participated in a Modeling Instruction environment that placed heavy emphasis on improving their subject-matter knowledge as a basis for affecting the development of their pedagogical content knowledge. We investigate the development of the teachers’ content knowledge and pedagogical content knowledge by considering the results of our iterative revisions with supporting documentation of the insights we made as we refined the course to explore teachers’ knowledge. We conclude that Modeling Instruction helped the teachers conceive of mathematics as a tool to explain scientific phenomena and provided the teachers with opportunities to reflect upon the process of learning mathematics, which were both foundational to the development of their subject matter knowledge and their pedagogical content knowledge.