In this paper, we describe our extension of the action, process, object, schema (APOS) theory to capture the complex interplay of mathematics and physics. We do this in the context of the 2D heat equation. We describe a hypothetical learning trajectory of the 2D heat equation, a preliminary genetic decomposition that stresses the conceptual…

Adaptive teachers use student data to guide instruction. Learn about using an anecdotal record form to support adaptive teaching. In this article, the authors describe how teachers can use learning trajectories to make adaptive decisions to meet the needs of their students. They provide an example using the U .S. Math Recovery Council's learning…

The burgeoning field of third language (L3) acquisition has increasingly focused on intermediate stages of language development, aiming to establish the groundwork for comprehensive models of L3 learning that encompass the entire developmental sequence. This article underscores the importance of a robust epistemological foundation, advocating for…

This brief provides findings from the full report, "Adapting to Changes in California's English Learner Population," which examines how the size and composition of the English Learner (EL) population has changed and whether services, program offerings, and learning trajectories have evolved in response to these changes. Key findings…