One of the most fundamental strategies in mathematics instruction is practice problems because teachers know that practicing a skill improves performance. At the same time, teachers also know that just because students can correctly answer practice problems does not mean they fully understand the concept or how to apply a formula--especially not…

When students encounter a set of concepts (or terms or principles) that are similar in some way, they often confuse one with another. For instance, they might mistake one word for another word with a similar spelling (e.g., allusion instead of illusion) or choose the wrong strategy for a mathematics problem because it resembles a different kind of…

Most mathematics assignments consist of a group of problems requiring the same strategy. For example, a lesson on the quadratic formula is typically followed by a block of problems requiring students to use the quadratic formula, which means that students know the appropriate strategy before they read each problem. In an alternative approach,…

Two experiments examined conditional discrimination in 4- to 6-year-olds. Children learned to choose one of two objects (e.g., circle) when the background was, say, red and to choose the other object (e.g., triangle) when the background was, say, blue. Awareness was assessed and interpreted as a marker of relational processing. In Experiment 1,…

In this article, I discuss the concepts of "private stimuli," "covert responses," and "private events," emphasizing three aspects: the conditions under which private stimuli may acquire discriminative functions to verbal responses, the conditions of unobservability of covert responses, and the complexity of events or phenomena described as…