Gestalt psychologists proposed two distinct learning mechanisms. Associative learning occurs gradually through the repeated co-occurrence of external stimuli or memories. Insight learning occurs suddenly when people discover new relationships within their prior knowledge as a result of reasoning or problem solving processes that re-organize or…

People spontaneously discover new representations during problem solving. Discovery of a mathematical representation is of special interest, because it shows that the underlying structure of the problem has been extracted. In the current study, participants solved gear-system problems as part of a game. Although none of the participants initially…

This study investigated the emergence of combinatorial competence in early adolescence and the effectiveness of a programmed discovery training procedure. Significant increases in combinatorial skill with age were shown; it was found that the expression of this skill was significantly facilitated if problems involved concrete material of low…

Describes three activities that the teacher can employ to help students develop thinking skills through mathematics instruction: (1) memorization using the technique of chunking; (2) higher order thinking with magic squares; and (3) predicting games. Identifies eight facets of the teacher's role in promoting thinking skills. (MDH)

Preservice elementary teachers worked cooperatively to explore the relationship between the measure of the central angle and the number of sides of regular polygons using the reflection of a line in a hinged mirror. The experience involved conjecture and experimentation, communication, and building connections between science, geometry, and…