A design of useful implementation principles for the development, diffusion, and appropriation of knowledge in mathematics classrooms

Much of the reform in mathematics education advocates collaborative learning and approaches that require students to explain their mathematical ideas. However, in many mathematics classrooms, students are still introduced to new ideas through the teacher\u27s presentation of them to the entire class. Likewise, students are often introduced to new tools such as computer programs and graphing devices through some form of direct presentation. On the contrary, this study investigates classrooms where students are introduced to new tools, tool-related practices, concepts, facts, and problem solving strategies through the spread of student-initiated ideas throughout the classroom. The identification of factors that influence the development, diffusion, and exchange of new knowledge among middle school students who work collaboratively on thought revealing mathematical activities (Lesh, Hoover, Hole, Kelly, & Post, 2000) forms the basis of this study. The product of this classroom design study (Collins, 1992) is a set of multi-revised principles for teachers for diffusing innovative mathematical knowledge in classrooms. These implementation principles are intended to guide teachers in modifying the classroom environment to promote the sharing, spread, and exchange of mathematical ideas, facts, concepts, problem solving strategies, and tool usages. The principles were designed using diffusion theory (Rogers, 1995) and communities of practice (Wenger, 1998) as theoretical frameworks. The effectiveness of the principles was tested using the premise of a new type of design research that is modeled after design research in applied fields such as engineering. The principles underwent four testing iterations, where they were revised and tested until a satisfactory set was constructed based on consensus of the design team