Empowering students with specific learning disabilities: Jim's concept of unit fraction

摘要

Cognitive differences have historically led to deficit assumptions concerning the mathematical experiences that children with learning disabilities (LD) can access. We argue that the problem can be located not within children but instead as a mismatch between features of instruction and children's unique learning abilities. In this paper, we investigate how one elementary school child, Jim, with specific visual motor integration differences constructed a unit fraction concept. Through ongoing and retrospective analysis of data drawn from seven sessions, we uncovered three key features of Jim's development that become connected and coordinated over time: (a) Use of midpoints and "sameness" to construct unit fractions as one-level structures, (b) Connecting constructed unit fractions with number relationships, and (c) Coordinating unit fractions across multiple wholes through iteration as two-level structures. Contributions of this work include insights into Jim's unique abilities alongside instructional design components helpful for intervention planning: (a) Utilizing wide, accessible tasks that encourage multiple means of contextualizing, representing, and expressing knowledge, (b) Carefully considering constraints of representational materials to support operations and connections, and (c) Bridging existing knowledge to new activity. Results support a continued exploration and gathering of evidence toward a new framework for interventions called Small Environments that begins in the teacher's responsiveness to children's complex mathematical reasoning and specific learning abilities.