Certain properties of lenses provide a physical model of the mathematical concepts of multiplication of integral numbers and of similarity transformations in geometry. Further, they can provide a realistic concrete representation for rules governing multiplication of signed numbers. Suggestions for problems and classroom demonstrations involving…

The instruction of probability and statistics is promoted as an area for exploration in elementary school. Activities are suggested that are designed to provide pupils with opportunities to learn desired concepts through informal, exploratory exercises. A total of eight different activities are suggested, with four each under probability and…

A mathematical system based on six functions is illustrated. (MP)

Some easy-to-state number properties are suggested as the subjects of proofs that are readily understood and interesting to pupils. (MP)

The following ideas are presented: plans for constructing a calculator bin rack that provides a place for a school to store and charge calculators; a lesson in geometry based on a news article about salt containers; and a very simple approach to the concept of infinite geometric series. (MP)

The use of graph paper to outline arrays of multidigit multiplication problems is promoted. (MP)

Every student at every level is seen to learn mathematics individually and personally, putting an individual stamp on each incoming perception. (MP)

How the use of calculators can illuminate mathematics and improve the level of problem-solving discussion in classes is presented. (MP)

The creation of "false laws" of mathematics by many pupils as they attempt to rephrase new ideas and make them understandable and useful is discussed. The reasons for these mistakes and some ways teachers can help students adopt correct rules are reviewed. (MP)

The concepts of groupings, their characteristics, and possible developments in a context of concept formation processes are illustrated. (MP)

Research on the conceptual notions of fractions held by students at various levels of schooling is discussed. A method of approach to instructing students on fractions that seems to hold considerable promise is also covered in detail. (MP)

Students recognized as less successful individuals in mathematics are tested for their understanding of fractions. The data reveals that most were only able to apply remembered rules to problems without actually knowing if the rule worked for the given situation. (MP)

Results of the second mathematics assessment of the National Assessment of Educational Progress (NAEP) indicate how 9-year-olds handled decimals prior to much formal instruction, and give insight into areas of difficulty for 13-year-olds who have received instruction. (MP) Aspect of National Assessment (NAEP) dealt with in this document: Results…

Seven activities designed to lead students to the idea that the ratio of the perimeter of a square and its diagonal is a fixed number for all squares are presented. The activities are groundwork for preparing students for the ratio between the circumference and diameter of a circle. (MP)

A number of general ideas intended to be helpful in analyzing differences in concept images among individuals are formulated. These ideas are applied to the specific concepts of continuity and limits, as taught in the secondary school and university. (MP)