Conjecture and proof, as opposed to numerical calculation, are presented as being the primary focus for teaching mathematics. (MP)

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

Ways of finding the sums of some interesting sequences of numbers are discussed. Finding the patterns creates a challenge, but the patterns are not too difficult for average pupils to discover. Mathematical induction can then be used to prove the formulas. (LS)

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)

Activities suitable for use with primary school students are described. The author offers advice concerning expectations of the students, use of the vocabulary of probability, integration of activities with other classroom work, and related topics. (SD)

Playing with number patterns can lead students to discover some divisibility tests. (JT)

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)

Having students find and cut out of graph paper the 12 pentominoes can provide an enjoyable exercise in perception, discovery, and double checking. The pentominoes can then be used to illustrate informally several geometric concepts. (SD)

In three experiments, three normal preschoolers were presented with sorting tasks ordinarily insoluble for four-year-olds. Found that, although the subjects had difficulty correctly sorting the pictures when told what criterion to sort by, they could provide correct answers when asked what they were looking for. This self-instructional effect was…

Mathematical investigations pursued by two gifted students are presented. One is written by a boy age 9 with his follow-up at 11; he studied patterns to derive the factors for the difference of two squares. Another boy of 11 discovered formulas for slopes of tangent lines and areas under curves. (LS)

An activity project is described which encourages students to question and experiment. Appendices provide examples of student results. (MNS)