Method of defining in a mathematical inquiry:

Focusing on normative aspects

Abstract

The purpose of this study has been to clarify a method of defining in a mathematical inquiry. Since the 1990s, the number of publications on defining has been increasing in mathematics education research. Although they have clarified different descriptive aspects of studentsʼ defining through empirical investigation, they have not subjected the key term “defining” to theoretical investigation. Therefore, there has been a need to examine the normative aspects of defining through a conceptual analysis of defining within the context of defining special quadrilaterals. In this study, mathematical inquiry is positioned as the context and defining is the main activity. On the one hand, based on the humanistic inquiry approach (Borasi, 1992), this study defined a mathematical inquiry as continuous activities of problem solving to reduce uncertainty, conflict, and doubt that include organizing the familiar and creating the unfamiliar. On the other hand, based on previous studies of definition and defining, defining is taken to mean continuous activities of constructing and revising definitions that include investigating examples and properties of an object; refining a definition according to the requirements of mathematical definition; and achieving a purpose while interacting with others. By referring to these concepts, the current study identifies a method of defining in a mathematical inquiry that comprises five aspects, including selecting the definition or naming an object by its purpose, constructing a hierarchy of definitions, pursuing an exact definition, pursuing minimal definition, and confirming the consistency of definitions. Specifying the object to define is seen as the first step, while the remaining four serve to then hone the definition into one that is more mathematically based. The current study exemplifies the possibility of applying the method specifically to defining special quadrilaterals. We start from Problem 1, which asks imaginary students to position kites and isosceles trapezoids within a Venn diagram of squares, rhombuses, rectangles, and parallelograms. This example shows how the definition of kites and that of isosceles trapezoids can be constructed and then sophisticated using the proposed method of defining in a mathematical inquiry. Problem 2 then requires imaginary students to find quadrilaterals that fit into a blank Hasse diagram of special quadrilaterals involving trapezoids, cyclic quadrilaterals, and tangential quadrilaterals. This example also shows that it is possible to extend the definition to include ellipse quadrilaterals, in which the sum of two adjacent sides is equal to the sum of the remaining two adjacent sides. The significance of this study is to clarify the normative aspects of defining that involve how to identify the object to define and then how to refine the definitions to become more mathematical. The act of defining suggests the dynamic interplay between object examples or properties requiring definition (object-level) and the requirements of mathematical definition (meta-level). Therefore, the method of defining within the context of a mathematical inquiry can be the foundation of empirical studies related to definition and defining when designing tasks and planning interventions.