Algorithmic and investigative approaches to mathematics are compared and discussed. Their mutually contradictory spirit is explored. Examples of the application of each method to a mathematics problem are presented. (CW)

Proposed is a way for teachers to distinguish between rich, challenging material that encourages mathematical thinking and material that is unsuitable. Included are multistep problems that encourage a broader understanding of mathematics. (KR)

Described are four classroom activities that may be interspersed into a curriculum or offered as homework. Systematic procedures, alternative strategies, and trial and error are emphasized in these counting exercises. (KR)