The article discusses the ways that less successful mathematics students used graphing software with capabilities similar to a basic graphing calculator to solve algebra problems in context. The study is based on interviewing students who learned algebra for 3 years in an environment where software tools were always present. We found differences between the work of these less successful students and the traditional problem-solving patterns of less successful students. These less successful students used the graphing software to obtain a broader view, to confirm conjectures, and to complete difficult operations. However, they delayed using symbolic formalism, and most of their solution attempts focused on numeric and graphic representations. Their process of reaching a solution was found to be relatively long, and the graphing software tool was often not used at all because it did not support symbolic formulation and manipulations.
One major part of the effort to reform secondary school mathematics is the project of changing the goal of studying school algebra from the mastery of symbolic manipulations to the ability to reason algebraically. Another major component of these reform efforts is the creation of opportunities for students to communicate within and about mathematics. The ability to generalize, especially when the generalization requires a major breakthrough in habits of mind, is one indication of algebraic reasoning. In this article, I describe generalization activity as an opportunity to learn about seventh graders' understanding of functions. A group of students who had studied functions modeled a multivariable situation. Through individual and group work, they designed, described, and discussed various representations for functions of 2 variables. Their modeling efforts allowed them to analyze their understanding of representations of quantities, relationships among quantities, and relationships among the representations of quantities in both single- and multivariable functions.
Michal Yerushalmy and Shoshana Gilead
What is it that a rich society owes to its citizens? Not, I think, warehousing under the pretext that they are in “ninth-grade algebra or learning Macbeth,” but something far deeper, more valuable, more personal, more meaningful. When a student is bored, I do not think it is necessarily a failure of the student, nor of the teacher, but more often a failure of the curriculum. (Davis 1994, 318)
Michal Yerushalmy and Richard A. Houde
Traditionally, the teaching of high school geometry has emphasized the principles of deductive systems. This approach often forces students to learn how to manipulate mathematical systems while it denies them an equal opportunity to create geometry. Geometry teachers have always faced the dilemma of having to instil in their students an appreciation of deductive mathematical systems while at the same time offering them an opportunity to create mathematics. This article describes our approach in dealing with this dilemma.