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…

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)

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 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)

Described is a way to use knots to relate a three-dimensional object to a two-dimensional representation of the object. The results are used to produce an algorithm or rule to explain a general case. Included are examples, diagrams, procedures, and explanations. (KR)

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)

Describes teaching strategies in which children are encouraged to develop computational procedures that are meaningful to them. Authors state that classroom observation reveals most primary students to be capable of developing their own accurate solution procedures for multi-digit addition and subtraction as well as for simple multiplication and…

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)