Algebra slaves and agents in a Logo-based mathematics curriculum

Abstract

For many students the ideas and methods of algebra appear obscure and mysterious, their sense and purpose unclear, and their applicability to anything genuinely real or interesting very remote. Students often fail to acquire an understanding of the key concepts, despite their inherent simplicity. Even when they gain the notion of variables, expressions and equations, students often lack the strategic knowledge required to motivate and direct the global planning and detailed execution of an attack on a problem. These conceptual and strategic difficulties are compounded by the needs for precise performance of the arithmetic and symbolic operations required in manipulating expressions. Extended operations like subtracting an expression from both sides of an equation or expanding a product of three terms, are very difficult for beginning students. Their buggy performance in carrying out the detailed manipulative work greatly confounds and frustrates their acquisition and assimilation of the most important and central ideas. In an effort to confront these difficulties and show how they can be overcome, we are developing a Logo-based introductory algebra course for sixth graders. Our approach has three major components: work on Logo programming projects in algebraically rich contexts whose content is meaning ful and compelling to students, the use of algebra microworlds with concrete iconic representations of formal objects and operations, and the introduction of the algebra workbench, an expert instructional system to aid students in performing extended algebraic operations. The algebra workbench will employ a set of powerful symbolic manipulation tools for performing the standard manipulations of high school algebra. It will have two main modes of use: demonstration mode, which uses an expert tutor program to solve algebra problems incrementally, explaining its strategy and its step by step operations in straightforward terms along the way; and practice mode, in which the student tries to solve a problem with the assistance of the tutor, which performs the operations requested by the student at each step and which can be called at any point to advise the student of the correctness of a step, to perform or explain any step, to evaluate the student's solution, or to perform a problem that she poses. These powerful aids make it possible to effectively separate out the difficulties in performing the formal and manipulative aspects of algebra work from those encountered in learning the central conceptual and strategic content. Distinctly different kinds of instructional tools and activities-Logo programming, expert tutors, or algebra microworlds-can thus be brought to bear where each is most appropriate and effective.