A Fusion of the School Arithmetic and the School Algebra

Abstract

In this paper we present our views on algebra as a school subject. To say pre-cisely what is the content of this subject, the language of abstract algebra has to be used. Accordingly, this content consists of the establishing and application of the properties of operations in the system of natural numbers which are then carried over to the extended systems. In the last of these extensions—the set of rational numbers, these properties can be taken to be axioms of ordered field and the main facts that are deduced from them. Of course, in the classroom practice, this content is didactically transformed and shaped to serve the learner and is aimed at the development of the skill of transforming numerical and literal expressions. On the other hand, the school arithmetic, when structured properly, consists of the gradual building of the number blocks up to 10, to 20, to 100 and to 1000, each of them having its own package of didactical tasks that are supposed to be accomplished. We take that both school arithmetic and school algebra have for their intuitive ground, the phenomenology which consists of sets at the sensory level and their configurations in the form of additive and multiplicative schemes. And while arithmetic deals with specific numbers denoted by their decimal notations, algebra deals with species of numbers (variables) denoted by letters. As for the properties of operations, in arithmetic they are related to specific schemes whose extent is determined by specific numbers, while in algebra they are related to the species of such schemes where letters replace specific numbers. The way we view school arithmetic and school algebra inspires us to suggest a fusion of these two subjects. In this paper we sketch how this fusion should be carried out, avoiding any possible abrupt semantic jumps from specific to general cases. The ultimate aim of teaching and learning both, arithmetic and algebra is the building of the systems of natural numbers, integers and rational numbers and as this aim is attained in the last classes of elementary school, in this paper we confine our considerations to the primary level. At this level we suggest and sketch: Elaboration of arithmetic in the form of gradual building of number blocks, based on the permanent meaning of ad-dition and multiplication. Derivation of properties of operations and their pro-cedural expression from the very beginning (within blocks of numbers up to 20 and 100). Teaching situations which are selected to help and encourage devel-opment of the idea of variable (letters in the role of the unknown, evaluation of literal expressions, etc.). Derivation of properties of operations in symbolic form, based directly upon the experience of species of schemes.