This study sought to compare the effectiveness of teaching with manipulatives versus algorithms and procedures. Assessment data were collected before instruction, after instruction with algorithms and procedures, and after instruction with modeling and manipulatives. Data were collected from 113 students from the 5th, 6th, and 7th grades within…

Describes a lesson on long division using chip trading which follows that algorithm for long division. (MKR)

Discusses the use of manipulative materials to model operations and algorithms. Indicates that they clarify the several interpretations of each operation, establish a basis for correct mathematical language, and show why algorithms work. (JN)

A strategy to make the transition from manipulative materials to a written algorithm for division is outlined in dialogue form. (MNS)

Renaming fractions with the dot method is described with illustrations. It can be used to introduce renaming at the manipulative level in a meaningful way prior to moving to a more abstract level where prime factorization will be involved. (MNS)

The purpose of this study was to compare the repeated subtraction method of long division of a partitioning distributive method, both taught by recording manipulation of popsicle sticks. Two instructional units were prepared. Twelve fourth grade classes were used, with six being taught one approach and six using the other. Tests consisting of…

The use of developmental algorithms to develop skills is described. Children make the transition from using concrete materials to using a standard algorithm. (MNS)

Describes a research project designed to monitor the transition from work based on concrete materials to the more formalized aspect of mathematics found in secondary schools. The topic of subtraction was chosen by three teachers who were involved in the investigation. (PK)

A method is presented for helping children make a smooth transition from using manipulative materials to using symbols only in solving multidigit addition problems. (MP)

The ideas and techniques involved in learning about fractions were investigated with students in grades 1-12, in the first 3 years of colleges, in community college mathematics courses, and in graduate school. Also included were some high school mathematics teachers, some mathematicians, and some retired persons. Part I provides the rationale and…

Discusses using what students already know about taking away objects when teaching subtraction and gives six lessons to develop language for discussing and recording subtraction situations that give meaning to the subtraction algorithm. (MKR)

The instruction of finger counting and finger calculation, also known as Chisanbop, is promoted as a natural method of introducing and teaching the basic processes of addition, subtraction, multiplication and division to children, particularly to those who are mentally and physically handicapped. The sequential process for teaching finger…

The concept of multiplication is described and illustrated using several different representational systems. A conceptual approach to teaching mathematics is compared with the procedural approach commonly found in the school curriculum. Four different methods of representing the multiplication process with numbers larger than ten are presented:…

Traditional methods of teaching addition include algorithms that involve right-to-left procedures. This article describes efficient procedures for left-to-right addition and subtraction involving computation and computational estimation that reflect children's natural behaviors observed during activities with unifix cubes. (MDH)