2013 Conference Proceedings - University of Nevada, Las Vegas

on students’ explanation, constructing and justification and indicates a spiraling developmentaround the main task. The conversation system determined the directions of discourse rather thanan authority (Davis & Sumara, 2006; Sfard, 2008). The discourse became evolved and dynamicrather than convergent and linearly patterned. In the main session, students learned not only howto do mathematical inquiry, but also strategies, such as how to categorize situations and constructcounter-examples. The lesson closed with the teacher and students applying and reflecting onthose methods of inquiry.Overall, the discourse demonstrated a recursive process of developing students’mathematical thinking. At each stage, both the teacher and students made conjectures, revised,and proved the conjectures, leading to a higher level of mathematical thought. Figure 2 belowcaptures the iterative and inquiry-based structure across the introduction, main session andclosing. In the development of the discourse, Mr. Wu sometimes posed himself as someone whoturned on students for solutions as he asked “what do we do?”, “what do you think?”. He showednot only his respect but also his passion to students’ solutions. Moreover, Mr. Wu was sensitiveto student ideas and also flexible to adjust his teaching based on the dynamics of classroominteractions. When students’ answers were different from his expectation, instead of guidingstudents to what he was expecting, he made modifications of his instruction. His adjustment ofhis instruction was a comprehensive consideration of emerging ideas, students’ understanding,and learning objectives of the lesson. He was also able to help students develop their thinkingwhile also lead the discussion back to his original plan in a later time. In the process, studentideas were a “generator of meaning” (Peressini & Knuth, 1998). Both the teacher and studentswere equal participants of the inquiry process. Overall, the features demonstrated in Mr. Wu’slesson for developing the discourse include 1) a democratic environment to engage students inthe investigation of the problem; 2) passion and appreciation for students’ ideas; 3) questionsrequiring explanation and justification; 4) arguments stimulating sense-making; 5) flexibleinstruction guided by a comprehensive consideration of emerging ideas, students’ understanding,and lesson objectives; 6)recursive conversation built on the co-evolvement of the teacher andstudents.Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 217