Mathematical Tasks as Boundary Objects in Design-Based Implementation Research

Abstract

This paper describes collaboration involving school district curriculum supervisors, mathematics teachers, university researchers, and web engineers engaged in design-based implementation research of the adaptation of an Algebra 1 curriculum to meet the demands of the Common Core State Standards for Mathematics. In this effort, mathematical tasks operated as boundary object, acting to organize cooperative work despite lack of consensus regarding their meaning or purpose. Exposing and understanding this lack of consensus during joint work at community boundaries provided opportunities for learning in the form of changed practice and activity. Evidence for learning through coordination is presented, such as the communication of curricular vision and the routinization of task rating process. Where lack of consensus led to potential conflict, as in the case of task adaptation, evidence for learning in the form of new practices was less evident. Major Issue The adoption of the Common Core State Standards for Mathematics (CCSSM) has created need for many teachers and school districts to consider changing their curricular materials. For the 44 U.S. States that have adopted these standards, they generally represent more focused and demanding target than prior standards (Porter, McMaken, Hwang, & Yang, 2011). While some schools will purchase new curriculum materials that better align to the CCSSM, many others will attempt to modify and improve existing materials. This kind of adaptation can require considerable effort and expertise, and implementation strategies for the CCSSM are not yet well-developed (Cobb & Jackson, 2011). Collaboration with researchers can be productive option for not only developing quality curriculum, but for better understanding the processes by which standards-based curriculum is designed and implemented. A useful unit of mathematics curriculum is the task , which is sometimes defined as a classroom activity, the purpose of which is to focus students' attention on particular mathematical idea (Stein, Grover, & Henningsen, 1996, p. 460). As intended by the CCSSM, high-quality mathematical tasks are likely to be inquiry-based activities that engage learners in multiple Standards for Mathematical Practice (NGA Center/CCSSO, 2010). While adapting curricula to include such tasks may be goal shared by teachers, district curriculum supervisors, researchers, and other reform agents, the meaning and significance of mathematical task may vary depending on one's position and perspective. In this way, mathematical tasks are potential boundary object (Star & Griesemer, 1989; Star, 2010). In this paper, we ask: How do mathematical tasks operate as boundary objects, and what do mechanisms for learning related to boundary objects and boundary crossing imply for collaborative efforts to adapt and implement new curricula? Contextualization and Significance Our research is an example of design-based implementation research (DBIR; Penuel, Fishman, Cheng, & Sabelli, 2011), an approach that expands classroom-based design research to consider perspectives of other stakeholders in an educational system. DBIR leverages iterative and collaborative development of processes and products to enhance system's capacity to sustain and scale change related to persistent problems of practice. Participants in this DBIR project represented four communities: university researchers, curriculum supervisors from an urban school district, high school algebra teachers from the district, and an engineering team coordinating with the researchers to develop web-based catalog of curricular resources. The district supervisors selected teachers with the goal of representing varying levels of teaching experience and expertise with curriculum development. Approximately ten teachers participated at any one time on Teacher Design Team (TDT), with some leaving and others being added as the project changed phases or as individual circumstances demanded. This project is significant not only in its purpose to support curricular change in response to the CCSSM standards, but in its DBIR approach and researcher access to both the district's teachers and their supervisors. The focus of this paper is the initial phase of the project, period from July 2012 through the first TDT meeting in December of 2012, full-day workshop involving all four communities. During this phase there were weekly meetings of the research team as well as weekly phone conferences between the researchers and the district supervisors. One member of the engineering team attended these weekly meetings in significant liaison role.