The embodied, proceptual, and formal worlds in the context of functions

Abstract

In this study we use Tall et al.’s (2000) theory on mathematical concept development, which describes three worlds of operations, the embodied, the proceptual and the formal (Tall, 2004; Tall, 2003; Watson, Spirou, & Tall, 2002). The purpose of the study is threefold: first, to identify mathematical tasks in the context of function that reflect the three worlds of operations; second, to investigate whether students’ thinking corresponds to the embodied, the proceptual, and the formal modes of thinking; and third, to reveal the structure of and relationships among the three worlds of operations as these unfold through students’ responses. The study was conducted with first‐year university students. The results suggested that mathematical tasks can be categorized on the basis of Tall et al.’s (2000) theory and indicated that students exhibit different kinds of thinking, which reflect to a large extent the three worlds of operations. Three classes of students were identified in terms of the difficulty level of the tasks: (a) the proceptual, (b) the proceptual‐emhodied, and (c) the formal. According to Tail's theory, embodied, proceptual, and formal thinking develop in sequence in an individual's life. This study indicates that freshmen university students, who have had mathematics as a major in higher secondary school, are only able to deal with the embodied tasks once they had been successful with the proceptual ones. The leap to formal thinking could only be achieved when proceptual manipulations were enhanced with competence in embodied tasks.