2013 Conference Proceedings - University of Nevada, Las Vegas

One camera was fixed at the front to catch the general context of the whole teaching process.Another camera was fixed at the back of the class to catch the activities near the blackboard. Thethird and fourth were carried by the two researchers to capture classroom discussion, whichprimary focal point was the teacher. Documents from Mr. Wu’s former presentations and severalvideo clips of his previous teaching were also used. Data were discussed among the researchers.Case study methods were used for data analysis (Huberman & Miles, 1994).ResultsFindings indicates a recursive inquiry-based instructional design that emphasized theprocesses of conjecturing, testing, revising, and proof; evolving discourse with increasingmathematical understanding and reasoning skills was also documented. In particular, we tracedthe discourse throughout the lesson to understand the advancement of students’ mathematicalthinking and the strategies employed by the expert teacher in designing and developing rich anddynamic discourse.The TaskIn the lesson, the main learning task was to investigate under what conditions twoquadrilaterals are congruent. The cognitive demand of the problem was “doing mathematics”,which involved higher-level mathematical thinking (Stein et al., 2000). The task requiresstudents to make conjectures and prove the conjecture. It allows multiple entrances to tackle theproblem and embodies the essential skills of reasoning and different ways of proof. Mr. Wuindicated in the debriefing session that the objective of the lesson was to help students learnmathematical inquiry and to think mathematically. The designing of the task was aligned withthe higher goal.The Unfolding DiscourseThe lesson began with an introduction activity to mathematical inquiry in which the teacherlet students explore the ratio between the length and the width of a common sheet of size A4paper. Mr. Wu initiated the conversation as below.Wu: When I am looking at this piece of paper, there are a lot of questions I want to askyou. My first question is: have you ever paid attention to the length and width of it? What is theratio between the length and width approximately? What do you think about it?One student immediately said 3:2. Mr. Wu continually asked for other students’ opinions.When seeing no more different responses, Mr. Wu posed an argument, “Why not 5:3?” He thenProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 215