An investigation of students' metaphors, metonymies and prototypes in algebraic reasoning and symbol manipulation

The primary purpose of this study was to identify the metaphors, metonymies and prototypes used by students to express their understanding of algebraic reasoning and symbol manipulation tasks. A qualitative study was conducted in a teacher's eighth and ninth grade classes. Classroom observations, didactic exchanges and students' writings were collected as data for the development of an understanding of students' algebraic symbol manipulation. Clinical interviews were conducted with students as a means to explore their algebraic reasoning. During the interviews students were provided number sequences and asked to provide a pattern, then a generalization and then a formalization for the number sequence.;The results indicate students' use of elaborate visual metaphors in their interpretation of algebraic symbols. Students applied orientational, quantification, operational, comparative and container metaphors as well as metaphors of exclusion. Students then metonymically interpreted what the operational symbols stood for in the algebraic expression. The order in which the students applied the operations to the algebraic expressions depended on students' operational prototypes. Furthermore, students applied metaphors of arithmetic to their understanding of algebraic reasoning tasks. Students primarily perceived each term in the number sequence as a collection of objects and the sequence as a journey. This resulted in students exploring the number sequences at various levels of sophistication. These ranged from simple recursive strategies to complex forms of generalization and consequent formalization of their patterns.