A series of nine activities that illustrate prime factorization and other elementary number theory ideas in concrete, representative, and symbolic form are described. (MNS)

Proportion problems were given to pupils in grades 3-5. Success rates were high, with elementary strategies based on addition often used. Familiarity with the use of ratios in the context, the presence of a mixture, and the use of manipulatives were variables that influenced performance. (Author/MNS)

Four types of mathematics centers are described: listening, game, skill, and manipulative. Illustrations are included. (MNS)

Describes materials mastery and symbolic development as two discrete but related processes that preschool children learn. Uses the preschool setting to identify the developmental stages of construction play. (AS)

Described are activities for thinking through patterns using toothpicks, paper strips, wooden cubes, and people patterns; thinking with attribute items such as geocards, namestrips, and treasures; and thinking with geometric materials such as geoboards and dot paper, cubes, and simple puzzle pieces. (MNS)

How children can be guided to see and feel the power of thinking with and about mathematical symbols is discussed. A strategy to help them bridge the gap between manipulative models and symbols is detailed. (MNS)

The use of a variety of models to develop number concepts is advocated. Four models are discussed, with illustrations: the cardinal number of a set, Cuisenaire rods, the number line, and the Papy Minicomputer. (MNS)

Using manipulative materials and pictures is considered valuable, but the need to use models appropriate for individual children is stressed. Suggestions for using well-structured physical space are given. (MNS)

Investigated what effects two instructional strategies (related to student use of manipulative materials) would have on second-grade students' (N=96) development of logical structures. Also examined the effects of sex on logical development. Results indicate that supplying manipulative materials alone is not adequate in promoting development of…

The relationship in the Pythagorean Theorem is introduced to low-ability seventh graders using Cuisenaire rods. The sequence of instruction is described. (MNS)

Discusses how teachers can help students make the transition from using manipulative materials to abstract mathematical symbols. Includes: (1) a definition of the transition state; (2) ways to bridge the gap (by asking questions and by using activities); and (3) a specific example (teaching place value). (JN)

The preparation of prospective teachers to use manipulatives in the classroom involved three dimensions. Each of these dimensions is discussed within the context of the elementary and early childhood teacher education program at Pennsylvania State University. (JN)

Findings from a study with first-grade children on addition and subtraction help make a case for a conceptual rather than a mechanized approach to the operations. How base ten blocks were used in the program is illustrated. (MNS)

The game of Fractional Dominoes is illustrated. (DT)

Designs on a circular geoboard lead to an investigation of factors, primes, relative primes, and modular arithmetics. (DT)