Changing Local Practice for Good: Walking Scale Geometry as Designed Disruptions for Productive Hybridity

This dissertation presents the first iteration of a design study that investigates an instructional setting that I call Walking Scale Geometry (WSG). WSG tasks are geometry problems that are meant to be solved at very large scale, outdoors, using everyday materials, by groups of students. This setting disrupts typical classroom mathematics in four ways: a) problems are solved outdoors, in large open spaces, rather than in classrooms at desks; b) classroom representational and conceptual tools like paper and pencil, rulers, protractors, and hands are replaced with everyday materials like ropes, lawn flags, and students whole bodies; c) students see geometric figures from intrinsic perspectives, rather than extrinsic, birds-eye- views; d) the division of labor is such that problems cannot be solved individually. WSG is designed to constitute a mediating setting that supports the recruitment of resources for learning and doing mathematics, and access to participation (opportunities to learn) not typically available in classroom instruction. Students shared experiences in the WSG setting can be leveraged in classroom instruction to support productive hybridity, improving geometry learning for all students. I compare implementation of WSG tasks in two different instructional contexts: an urban seventh grade mathematics class and a summer enrichment course for high achieving rising ninth and tenth graders. Data include audio, video, and photographic records of design and instruction, field notes, student work artifacts, and interviews. Micro-analysis focused on students joint, whole-bodied accomplishment of WSG tasks and the resources that students recruited for participation and problem-solving. I demonstrate that aspects of the multi-party, whole-bodied activity, the use of everyday materials in inventing new representational practices, and the interconnectedness of the inscriptional system support access to participation with a variety of forms of mathematical engagement, and make available novel resources for problem-solving. I conclude with comments regarding the next iteration of WSG task design, classroom instruction that can legerage students experiences in the WSG setting, and thoughts about what a design for ensemble mathematics might entail.