Three examples are presented which involve constructions employing a compass, a centimeter ruler, and a calculator. Employing mathematical tools to support understanding and discovery of mathematical principles is viewed as being clearly a desire of mathematics educators, but it is felt such tools are not frequently used to their fullest. (MP)

Most algebra textbooks use a trial-and-error method, but such a method is not necessary. A procedure is described which has been tested in basic and intermediate algebra classes for four semesters, with significant improvement found in pupil ability to handle trinomial factoring. (MP)

It is seen as important that mathematics teachers encourage open, imaginative investigations that give students an opportunity to probe, find patterns and relations, and derive new patterns and relations. The article focuses on multiplication tables and classes, including questions for further exploration. (MP)

Many children do not appear to see any need to worry about the conventions for ordering operations or using brackets in algebra problems, due to the "intuitive" mode of approach they use in mathematics. The need to teach pupils the necessity of such conventions is discussed. (MP)

Evaluated the possibility that three-year-olds solve small-number conservation problems by an empirical procedure whereby the sets are quantified each time presented. Children chose the more numerous of two arrays, one containing two elements and the other three elements. Results disconfirmed claims that three-year-olds can conserve small numbers.…

A process of teaching fractions is detailed that respects pupil ideas and concept images and builds on these in a way that helps children to develop personal algorithms. This approach is an attempt to get away from perceived one-sided views found in most textbooks. (MP)

Three experiments were conducted to examine children's use of multiplying and proportionality rules in judgments of area. In the first two experiments, 7- through 8-year-olds were asked to judge the area of rectangles. In the third, 8- through 11-year-olds were tested on ratio of a rectangle compared to a horizontal line. Results indicated…

This study took place among undergraduates enrolled in first-year university courses and graduates enrolled in teacher-training courses. The test used is presented in full. Results indicated that misconceptions received in secondary-level courses tend to remain and factors relating to presenting and learning mathematics dominate all other…

Activities that can be used for maintenance of computational skills and can lead to the discovery of a pattern verifiable by an algebraic procedure are noted. The language of algebra is used in nonroutine settings, with several challenges for further exploration. (MP)

Ways of teaching students to read scales are presented as process instructions that are probably overlooked or taken for granted by most instructors. Scales on such devices as thermometers, rulers, spring scales, speedometers, and thirty-meter tape are discussed. (MP)

A program designed to plot three-dimensional graphs is described and presented. The program allows the user to select maximum and minimum ranges for X, Y, and Z, and can enlarge a graph on the screen or look at a portion away from the origin. It is in Applesoft BASIC. (MP)

Ways to make a discussion of the commutative properties of addition and multiplication more interesting to students are reviewed. An alternative promoted is a transposition-of-digits property, seen as useful in a number of instructional situations. The proposed investigation could be structured around calculator use. (MP)

Materials designed to promote student investigation of factoring quadratic polynomials over the integers through using a physical model are presented. Pupils are expected to discover regularities that will permit them to factor such polynomials at the symbolic level. Worksheet masters designed for classroom use are provided. (MP)

Examples from arithmetic, algebra, and calculus show that analogy and metaphor are as central to the expression of mathematical meaning as they are to expression of meaning in natural language. (MP)

A discussion on the general equation of a line shows how students can be lead to discover mathematical properties, find how one discovery leads to another, see how different branches of mathematics can lead to a solution, and be provided with a starting point for studying special mathematical topics. (MP)