The harmonic mean is defined, and examples of problems viewed suitable for classroom exploration are presented. Such exercises provide many opportunities for students to engage in interesting nontrivial problem-solving activities and practice on many algebraic skills. Class discussion of the arithmetic, geometric, and harmonic means is suggested.…

The trick focuses on a theorem that the sum of the digits of the difference between any natural number and the sum of its digits is divisible by nine. Two conditions of using the trick are noted. The reason that the theorem works is established through a proof. (MP)

A scheme for diagnostic teaching is discussed, and aspects of the approach are illustrated from three teaching experiments. Instruction that aims towards free and relaxed discussion of single points of difficulty or conflicts and towards pupils seeking to learn how a particular system works is promoted. (MP)

A program written in PASCAL designed to find the number of binary trees possible for a given number of nodes is presented. The problem was found to be highly motivating and exciting for the group of introductory computer science students with whom it was used. (MP)

Utilizes 42 second-grade rural students to determine if using multiembodiments rather than a single embodiment of concepts related to three-digit numbers would result in greater understanding of selected place value concepts. Reveals no significant differences on the group main effect or group-by-time interactions. (AH)

The following five steps are designed to help teach a directed mathematics lesson: (1) introducing, (2) presenting, (3) analyzing, (4) practicing, and (5) applying. It is noted that teaching a directed lesson means to help students master specific mathematical skills, as well as to formulate concepts and understand principles. (MP)

Problems designed to show the meaningful use of logarithms in the age of calculators are presented. The emphasis is placed on viewing a logarithm as an inverse operation to raising to a power. (MP)

A science fiction approach is used to explore some unusual properties associated with the concept of infinity. (MP)

This article, written by a high school junior, shows that there can never be more than two isosceles triangles having the same perimeter and area. (MK)

The way that fractions are taught to fourth-grade pupils at the Village Community School in New York City is detailed. The approach is thought to make the often frightening symbols of fractions into something that even beginning students find meaningful. Meaning rather than computational technique is emphasized. (MP)

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)