Proportional-Integral-Derivative (PID) controllers are essential in ensuring the stability and efficiency of numerous scientific, industrial, and medical processes. However, teaching the principles of PID control can be challenging, especially when the introduction focuses on the underlying mathematical framework. To address this, we developed the…

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)

Indicates that there has been a lot of work done and that a great deal needs to be done in the future to explore the world of children's early number. Discusses the counting, the use of algorithm, practical mathematics, the use of manipulatives, individual differences and pedagogical concerns, and classroom applications. Contains 18 references.…

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)

Twelve fourth-grade classes in two schools were taught division using either the subtractive approach to the Greenwood algorithm or a partitioning approach to the standard algorithm. Analysis of class means on immediate, retention, and delayed retention tests indicated that use of the standard algorithm was more effective. (JP)

Patterns for a set of Napiers Bones and for a denominate number slide rule are provided, with instructions on how to use each of the calculating instruments. (DT)

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

Described are two methods of calculating developed by the Dutch WISKOBAS Project: addition and subtraction with a loop abacus and multiplication using a grid model. (MP)

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)

This article discusses the use of concrete materials as conceptual models which serve as references for later learning of algorithms. (DT)

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…