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

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)

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)

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)

Elementary School Teachers were instructed on finding number patterns in a figure in one of three groups: individual discovery, group discovery, and expository. They were tested for recall, retention, and transfer. The expository groups did better on recall, but group discovery was more effective for retention and recall. (SD)

This 19-page workbook presents the elementary student with topological concepts through sequences of pictures with related questions. Generally the questions ask, "How are the pictures different?" and "How are they the same?" Several topological concepts are presented in this manner: connectivity, number of holes, closed and open curves, networks,…

When a simple game was presented as a challenge to a primary-school girl, she solved the problem, and went on to generalize her solution. (SD)

Mathematical concepts occur spontaneously from many topics and can be developed in the framework of cross-curricular schoolwork. Consequently, students can gain knowledge of where mathematics arises and insight about its purposes. (Included are activities dealing with the reflectional and rotational symmetries of automobile hubcaps evolved from…

Geometric concepts and theorems can be discovered by high school students using the materials described. Among the topics explored are Platonic solids and Euler's formula. (SD)

Details of an investigation into polygonal numbers are given. From a study of the tabled series, patterns are discovered both within and between the various polygonal series. From a study of these results and the corresponding figures, formulas are developed. (LS)