In this study, I examined the environment within a mathematics classroom and observed studentsʼ creation of new mathematical knowledge and the support provided by reciprocal acts in the creation of this knowledge. Furthermore, I developed a descriptive framework that captures the childrenʼs negotiations in the mathematics classroom. In addition, I set the subject of this study to clarify, using examples, the aspects of negotiations that are characterized by the mathematical knowledge required to reach an agreement. As supporting evidence for the subject of this research, I planned and implemented an experimental class that followed a single 'line graph' based on the perspectives of agreement, consultation, and composition of Grade 4 elementary school children, with constructivism agreement as its theoretical background, and considered the following two points. First, by determining who the negotiator was in this study (the subject of the negotiation), the effectiveness of the speech act theory was elucidated by understanding the negotiation. In addition, a descriptive framework was developed to capture the verbal intentions of a child during the negotiations. Second, based on the descriptive traits of the framework that was derived from the conclusions of the first experiment, an attempt was made to clarify, using examples, the negotiations that were characterized to achieve an agreement, as well as the mathematical knowledge that aims to form an agreement. Based on the configuration rules evident during a conversation according to Searle, the results of this study established the following descriptive frameworks in order to capture the verbal intentions seen during a negotiation; 'Assert' , 'Request' , 'Question' , 'Advise' , 'Agree' and 'Permit'. In addition, when a classroom analysis was conducted based on this descriptive framework, for the 'Assertion' statement that left a sense of vagueness in terms mathematical reason, a 'Request' that questioned this basis and reason was submitted. However, for 'Assertions' based on mathematically sound observations, such as numerical values, a 'Question' rather than a 'Request' was presented to verify the authenticity of the statement. Moreover, there were some children who seemed to agree to this assertion. Furthermore, there were some children who corrected their own knowledge during the negotiations.

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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.

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