Levels of participatory conception of fractional quantity along a purposefully sequenced series of equal sharing tasks: Stu's trajectory

• We present key understandings of fractional quantity, evidenced through problem solving strategies, observable operations, and naming/quantification of one fifth grader with learning disabilities.
• Data analysis uncovered four distinct points of understanding within the child's understanding that is reflective of a conceptual trajectory.
• Results of this work suggests that, when intervention is employed that engages children with LD in solving problems within their mathematical activity and responds to their ideas and ways of thinking, children with LD can and do advance their conceptions. Further, instruction for any child, including those labeled LD, should focus on what the child is able to understand, learn, and ultimately do mindfully.

Current intervention research in special education focuses on children's responsiveness to teacher modeled strategies and not conceptual development within children's thinking. As a result, there is a need for research that provides a characterization of key understandings (KUs) of fractional quantity evidenced by children with learning disabilities (LD) and how growth of conceptual knowledge may occur within these children's mathematical activity. This case study extends current literature by presenting KUs of fractional quantity, evidenced through problem solving strategies, observable operations, and naming/quantification of one fifth grader with LD before, during, and after seven instructional sessions situated in equal sharing. The researchers utilized a characterization of evolving fraction conceptions developed from research of children without disabilities that was ultimately productive in facilitating conceptual advances of the child with LD. We hypothesize that the trajectory of the child's conceptions is a case of something more general. Pending future research, the trajectory may be a useful tool to practitioners wishing to plan thoughtful, conceptually-based fraction instruction that is responsive to all children's evolving conceptions of fractions as quantities built through their own mathematical activity.