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Emergent graphical shape thinking (EGST) involves interpreting or constructing a graph as dynamically generated, which is useful across science, technology, engineering, and mathematics fields. Although evidence suggests that students as young as middle school can engage in EGST with support, other research indicates most college students and U.S. teachers do not spontaneously engage in such reasoning when potentially productive. We describe a local instruction theory (LIT) to support middle school students developing EGST as part of their graphing meanings. We then present a case study to show how two students engaged with a task sequence designed with the LIT in mind to develop meanings for EGST. This article illustrates general principles researchers and educators could use to promote students’ graphing meanings.

We investigated how the time elementary preservice teachers (PSTs) spent studying certain mathematics topics during teacher education coursework was related to performance on teaching-related tasks administered after graduation. In two studies, participants completed tasks assessing their specialized content knowledge (SCK) for teaching 12 mathematical topics addressed to varying degrees in the preparation program. We found that instructional time was positively associated with SCK demonstrated both immediately postgraduation and 2 years later. Several possible confounding factors were assessed; one, entering PSTs’ average SCK for topics, appeared to influence the relationship. Accounting for professional learning postgraduation, such as attending professional development, did not change the underlying relationship. Considering these findings, we identify policy implications for the mathematics curriculum of PST education.