As a theory of skill acquisition, the instance theory of automatization posits that, after a period of training, algorithm-based performance is replaced by retrieval-based performance. This theory has been tested using alphabet-arithmetic verification tasks (e.g., is A + 4 = E?), in which the equations are necessarily solved by counting at the…

An onto-semiotic analysis of the mathematical connections established by one in-service mathematics teachers and university students when solving a problem about launching a projectile using the derivative was carried out. Theoretically, this research was based on the articulation between the Extended Theory of Mathematical Connections and the…

This study explored the potential of a rich assessment task to reveal students' multiplicative thinking in respect to a hypothetical learning trajectory. Thirty pairs of students in grades 5 and 6 attempted the task. Twenty-two pairs applied multiplicative structure to find the number of items in arrays. However counting and computational errors…

Units of measure are critical in many scientific fields. While instructors often note that students struggle with units, little research has been conducted about the nature and extent of these difficulties or why they exist. We investigated calculus students' unit use in area and volume computations. Seventy-three percent of students gave…

This paper reports use of sample stimulus control shaping procedures to teach arbitrary matching-to-sample to 2 capuchin monkeys ("Cebus apella"). The procedures started with identity matching-to-sample. During shaping, stimulus features of the sample were altered gradually, rendering samples and comparisons increasingly physically dissimilar. The…

This study investigated the development of the many-to-one counting strategy in 4- year-old children. In the first experiment, 52 children participated. Their development with respect to two kinds of tasks, a hidden-items task and a needed-items task, was studied over four sessions. Children (n = 28) who accurately used the many-to-one strategy in…