Analysis of Mathematical Thinking through Interactions with a Diagram in Mathematics Education: The Meaning of a Diagram Obtained from the Networking between Semiotics and Embodiment

Abstract

　　 A diagram is a kind of an inscription.  In mathematics classroom, students think with different diagrams mathematically, so a diagram is considered a special type of an inscription, that is, it has an ability to enhance students’ mathematical thinking.  In this research, we focus on the meaning and function of a diagram, and the goal of this article is to analyze students’ mathematical thinking with a diagram from the several theoretical perspectives - semiotics and embodiment.  We assume that a diagram has a dual, complementary nature - cultural and cognitive - and it is the reason why we have chosen these theories.  By standing on these theoretical perspectives, we can really understand actual students’ mathematical thinking.

　　 The results obtained from discussions in this article demonstrate that a diagram has two important rules for construction and usage.  From the semiotic perspective, these are special characters to identify an inscription as a diagram; on the contrary, from the embodiment theory, these are based on or emerged from bodily actions as a certain typical pattern.

　　 Plus, we compared these perspectives by using the networking methodology.  Comparing with each other in the term of a basic principle, research question, methodology and result, we could describe some characters of mathematical thinking with a diagram such as:

● Mathematical thinking is a process to construct and use a diagram, while it is a process of diagramming.

● Mathematical thinking is directed by mathematical signs, while it is embodied.