Recent research in problem solving has shifted its focus to actual classroom implementation and what is really going on during problem solving when it is used regularly in classroom. This book seeks to stay on top of that trend by approaching diverse aspects of current problem solving research, covering three broad themes. Firstly, it explores the…

The contributions of the Persian scholar Karaji are described. Perhaps his greatest contributions were his arithmetization of algebra and the geometrical representation of algebraic operations. (MNS)

Geometric interpretations of algebraic statements are proposed, both to clarify the meaning of algebra and to encourage students to look for spatial or diagrammatic models. Binomial grids for several equations are illustrated. (MNS)

A possible method of derivation of prescriptions for solving problems, found in Babylonian cuneiform texts, is presented. It is a kind of "geometric algebra" based mainly on one figure and the manipulation of or within various areas and segments. (MNS)

How to determine when there is a unique solution when two sides and an angle of a triangle are known, using simple algebra and the law of cosines, is described. (MNS)

How tedious algebraic manipulations for simplifying general quadratic equations can be supplemented with simple geometric procedures is discussed. These procedures help students determine the type of conic and its axes and allow a graph to be sketched quickly. (MNS)

A paper staircase drawn on a grid forms the basis for a discovery lesson in algebra. (DT)

Discussed are two approaches to determining which regular polygons, either inscribed within or circumscribed about the unit circle, exhibit rational area or rational perimeter. One approach involves applications of abstract theory from a typical modern algebra course, whereas the other approach employs material from a traditional…

Algebraic and transcendental curves are discussed, with attention focused on computing the area of some special regions bounded by the curves. (MNS)

Illustrated is how students can be directed to move from a geometric to an algebraic mode of thought and back again in linear algebra, thereby deepening their understanding. Rotations and line reflections in the plane are used as the example. (MNS)

Korean secondary school students preparing for college learn about a simple algebraic formula for area bounded by a parabola and line. The approach does not seem well-known among American students. It is noted that, while the formula derivations rely on integration, algebra students could use the formulas without proofs. (MP)

Demonstrates that the equation of a circle (x-h)2 + (y-k)2 = r2 with center (h; k) and radius r reduces to a quadratic equation x2-2xh + (h2 + k2 -r2) = O at the intersection with the x-axis. Illustrates how to determine the center of a circle as well as a point on a circle. (Author/ASK)

The author observes that the predictor functions determined geometrically by Ranucci (EJ 090 184) can be algebraically derived, and that capable students might enjoy verifying them. Algebraic formulations and derivations are provided. (SD)

Provided is an analysis, using concepts from geometry, algebra, and trigonometry, to explain the apparent loss of area in the rug-cutting puzzle. (MNS)

One mathematician's view of real problems is given. Described (with some humor) are magic retail numbers, the highway bypass, the elliptical table, and the senior-class schedule. (MNS)