Generalizing patterns is an important feature of algebraic reasoning that is accessible to students across grade-levels because it connects their numerical reasoning to algebraic reasoning. In this article, the authors describe how teachers can use the game Crack the Code to introduce generalizing to their students or can extend students'…

The authors who are mathematics teacher educators, have found that classifying, composing, and transforming shapes (in particular, rotations and reflections) are areas of difficulty for adults as well as for children. However, these are also some of the most important geometric ideas. They are fundamental topics in the K-8 Geometry and Measurement…

This chapter introduces three applications of Bayesian inference to common and novel issues in second language research. After a review of the critiques of conventional hypothesis testing, our focus centers on ways Bayesian inference can be used for dealing with missing data, for testing theory-driven substantive hypotheses without a default null…

There is a long-standing debate about the principles constraining the combinatorial properties of suffixes. Hay 2002 and Hay & Plag 2004 proposed a model in which suffixes can be ordered along a hierarchy of processing complexity. We show that this model generalizes to a larger set of suffixes, and we provide independent evidence supporting the…

Purpose: To introduce a Clinical Forum focused on the Complexity Account of Treatment Efficacy (C. K. Thompson, L. P. Shapiro, S. Kiran, & J. Sobecks, 2003), a counterintuitive but effective approach for treating language disorders. This approach espouses training "complex" structures to promote generalized improvement of simpler, linguistically…