A square lattice is used to generate classroom activities, such as counting the number of squares in a lattice of a given size. (MP)

Described are results obtained by students investigating double-diagrams, including discoveries of properties and accompanying proofs. (MP)

The use of dice and dominoes in the classroom is discussed as leading from counting activities to pattern recognition and discovery learning. (MP)

A problem concerning the generation of an array of numbers is posed. The author examines the use of this problem in several secondary-school and college classes. (SD)

Details of pupil exploration as to the largest number of sides that a polygon could have on a geoboard are presented. The problem is not seen as open-ended, but many different avenues of pursuit stem from it. (MP)

Rule learning in mathematics among elementary school children is described. Samples of dialogue between experimenters and children are presented, and the concept of rule learning is described in relation to two subgroups, gifted children and "regular" ones. (MP)

An example of a lesson involving calculators that focused on calculator use is given. An examination of the traditional algorithm by the students led to student-directed investigations. (MP)

The number of triangles having a fixed length of longest side is investigated. (SD)

Two students (aged 10) and a teacher described their exploratory approach to problems involved in sliding blocks games. (SD)

A series of simple games can be analyzed inductively to determine strategies. Students explore possibilities, hypothesize solutions, prove their solutions correct, and then generalize to more complex games. (SD)

Aspects of the instruction of trigonometry in secondary school mathematics are reviewed. Portions of this document cover basic introductions, a student-developed theorem, the cosine rule, inverse functions, and a sample outdoor activity. (MP)

In workshops for teachers the author provides each participant with five sticks and challenges them to create a variety of configurations. In the process the teachers hypothesize and test a number of geometric notions. (SD)

Allowing students to use a polystyrene cutter to fashion their own three-dimensional models is suggested as a means of allowing individuals to experience problems and develop ideas related to solid geometry. A list of ideas that can lead to mathematical discovery is provided. (MP)

The author describes a dialogue with his three-year-old daughter about numbers. He discusses aspects of mathematics education and how his views about how students acquire mathematical knowledge have changed. (MP)