That the principal axis theorem does not extend to any finite field is demonstrated. Presented are four examples that illustrate the difficulty in extending the principal axis theorem to fields other than the field of real numbers. Included are a theorem and proof that uses only a simple counting argument. (KR)

Described is a decomposition theory from which the Cayley-Hamilton theorem, the diagonalizability of complex square matrices, and functional calculus can be developed. The theory and its applications are based on elementary polynomial algebra. (KR)

Given are instructions for teaching spreadsheet use for problem solving in mathematics. Examples of relevant problems from algebra through calculus are provided. The advantages of using computer spreadsheets in this application are stressed. (CW)

Presented is a lesson in which the patterns that occur within and between sequences of polygonal numbers present an opportunity for students to analyze, represent, and generalize relationships. Materials, objectives, levels, and directions for this activity are discussed. Worksheets to accompany the activities are provided. (CW)

A simple algebraic model relating the mean achievement in a class to the students' ability range is presented. Agreement between the model and data on the relationship between mean achievement and class size is described. Data concerning opinions of 15 primary school through university teachers quantitatively support the model. (TJH)

Identified are variables affecting performance of algebraic word problems for eighth graders. An appropriate readability level would seem necessary but not sufficient for student success. The structural variables used appeared to be good predictors of both the correct strategies and solving time. (Author/YP)

Discusses using the computer to promote versatile learning of higher order concepts in algebra and calculus. Generic organizers, generic difficulties, and differences between mathematical and cognitive approaches are considered. (YP)

Computer software with graphic representations of functions can teach algebraic concepts, or can blur or obscure concepts of great importance. Making good use and bad use of visual imagery are each discussed. Then a general theory of interpretation of graphs is proposed, and a list of examples is presented. (MNS)

The author defines and discusses the cognitive theory of constructivism as it relates to teaching mathematics. It is suggested that the philosophical and theoretical view of knowledge and learning embodied in constructivism offers hope that educational processes will be discovered enabling students to acquire deep understanding rather than…

Deficits common among secondary students with learning disabilities or mathematics deficiencies are considered, along with a strategy to teach upper level mathematics, such as algebra or calculus. The strategy involves use of a mentor to help students to comprehend mathematics vocabulary, develop their own problem-solving strategy, and create a…

Describes an experimental course in which the programming language ISETL served as a tool for exploring concepts in abstract algebra, and identifies instructional strategies that facilitate the successful use of ISETL. Offers informal observations regarding the effectiveness of this approach. (13 references) (Author/MKR)

Presented is a series of examples that illustrate a method of solving equations developed by Leonhard Euler based on an unsubstantiated assumption. The method integrates aspects of recursion relations and sequences of converging ratios and can be extended to polynomial equation with infinite exponents. (MDH)

Discussed is a method of approximating the roots of a quadratic that allows the discovery of relationships between parabolas and circles and between the use of geometry and algebra. Included are the procedure and justification of the method. (KR)

Presents the use of spreadsheets as an alternative method for precalculus students to solve maximum or minimum problems involving surface area and volume. Concludes that students with less technical backgrounds can solve problems normally requiring calculus and suggests sources for additional problems. (MDH)

A study analyzed data from the International Association for the Evaluation of Educational Achievement survey of 13 year olds from 20 countries to examine gender differences in mathematics achievement. Multivariate analysis indicated differences varied from country to country and a significant difference in favor of boys in geometry achievement.…