Involvement of Algebraic-Geometrical Duality in Shaping Fraction’s Meaning and Calculation Strategies with Fractions

Abstract

Many mathematical concepts and processes, besides the algebraic form of their presentation, can be modeled as well geometrically through diagrams and graphics. Both these aspects of concepts demonstration (algebraic and geometrical aspect) are present on mathematical textbooks of pre-university education. In this paper we consider algebraic and geometrical aspect on 6 th grade math textbooks and in particular algebraic-geometrical duality, aiming that the fraction concept and the fraction calculation strategy to be assimilated better by the students. A study was made with 78 students to understand their abilities to express using algebraic symbols and to introduce mathematical situations with geometrical models for "Fractions" chapter. After the analysis of calculative strategies applied by students, in the article it is suggested that algebraic-geometrical duality must be included in teaching based on a complete framework. This will enable students to fully realize the deep understanding of concepts and the calculative strategies they are using.

Preservice Elementary Teachers’ Mathematical Knowledge on Fractions as Operator in Word Problems

“…Among the different mathematical contents, one interesting element is the study of fractions due to, among other reasons, their importance in the learning of several science areas. Students possess the knowledge that allows them to represent fractions, establish equivalences and operate with them, but they do not identify the meaning of the mathematical procedures involved, a situation that is fundamental for a deep and lasting learning of fractions [59,60]. Furthermore, the research on preservice teachers' conceptual knowledge of fraction showed that from the five sub-constructs considered, the participants were most successful with part-whole, presenting little knowledge of the rest [37].…”

Reflective Abstraction of Junior High School Students in Reconstructing The Factorization Concept

“…There are three forms of abstraction which are; the empirical abstraction where the focus is on the objects properties and knowledge on the object derives from his own properties, the pseudo-empirical abstraction where the focus is on actions which leaves out the properties that the action of the subjects have introduced into objects and reflective abstraction where further constructions can then be accomplished by its existing structures used to construct new structures [4].…”