Fang Yanping

Students’ limited view of mathematics and their strong orientation on numbers have been major hindrances in developing students’ capacity in problem solving. Our study aims to understand students’ challenges in reasoning quantitatively and gain insights on how they overcome these challenges. To this end, seven pairs of Grade 5 students and eight pairs Grade 7 were observed and videotaped when they solved one routine task and one nonroutine task. Their processes in solving these two tasks were analyzed using the method of conceptual analysis (Glasersfeld, 1995). Findings suggest that there were two major areas of difficulties that they experienced in reasoning quantitatively. Some of the major challenges behind those difficulties were the lack of quantitative understanding in whole numbers and fractions, the stronger focus on the magnitudes of quantities than the quantities themselves, and the stronger reliance on visually-perceived relationships in the model representation of situation than the quantitative relationships. A case of a Grade 7 pair is presented to discuss a context of pair collaboration that fosters the overcoming challenges in reasoning quantitatively that suggests a practical implication of using collaborative learning to support students’ development of quantitative reasoning capacity.

Video cases have been shown as effective tools to support teacher learning and teacher development. This study aims to design web-based video cases to support teachers in mathematical problem solving. Two video cases were built by drawing on data from a three-year funded project in Singapore. The first case is built on two Grade 5 teachers’ different pedagogy in conducting problem solving activities while the second case draws on different challenges that two pairs of Grade 5 students experienced in the process of solving problems. The framing includes case commentary, guiding questions, and discussion forum to help users reflect on their practices and initiate peer-sharing and -learning among users. The long-term goal of the design is to build a virtual community of practice.

This study aims to understand students‟ mathematical learning in a collaborative setting from real cases of Singaporean students‟ problem-solving pair work. Videotaped observations, with a think-aloud instruction, were conducted on 7 Fifth Grade and 8 Seventh Grade pairs of Singaporean students. The data were analysed using compareand-contrast method (Glaser & Strauss, 1967) across all student pairs to generate cases of pair interactions that fostered or hindered students in overcoming the challenges that they encountered. Findings reveal at least four aspects of the pair interactions that either fostered or hindered these processes. Important implications were drawn to support teachers in developing insights about the enabling and disabling factors for effective collaborative work in their classrooms. This paper presents the detailed analysis of all these cases.

"The eventual goal of the project is to broaden the access of teachers to online collegial interactive platforms and opportunity to analyze classroom discourse, teacher decision making, tools of teaching and student learning in regular lessons. ... To achieve the goal, the project focused on addressing the following research questions: 1) What constitutes and how to develop a rich and educative video case on problem solving (bridging algebra through the model method) facilitated by interactive digital media and web technology? 2) What constitutes and how to design an effective online platform for teacher learning anchored the video cases of teaching surrounding the bridging issues in Singapore? 3) What have we achieved in piloting the design in one local school to build an online PLC?"-- [p. 4].

A project on lesson study carried out in 2006 and 2007 aimed to find out how it can be used to bring about enhanced teacher learning and instructional improvements. This two-year intensive collaboration between researchers and a local primary school involved planning, teaching and observing research lessons together, as well as studying the processes of school-based curriculum development and teacher learning. The challenges of implementing lesson study in the Singapore context and its evolving local adaptations were carefully documented and analysed. We found that with strong school support, lesson study could bring about continual instructional improvement and a sense of community among the teachers, even though its time-consuming nature may pose numerous challenges for the teachers and schools.

This paper presents findings from observations of the case teacher’s unit of lessons that informed the development of the video cases and teacher participation in the online PLCs. A simplified version of Nassaji & Wells’ (2000) IRF-discourse structure and the Mathematics Curriculum Framework (CPDD, 2005) reveals that the bridging from model method to algebra was done through ‘discussion of problem sums’ by purposefully and carefully building upon students’ work using model method. The lessons followed predominant IRF structure where teachers’ initiated discourse plays key role in developing students’ processes, metacognition, and attitudes towards building connection between the two methods. Such classroom discourse patterns informed the development of viewing questions that scaffold teachers in viewing the video cases.