Tandem Conceptual Progression (TCP): Pilot case study of a cognitively diverse student's use of adjacent problem-solving and answer checking schemes

Children experiencing difficulties learning mathematics often have a long-embedded coping mechanism of looking to others as authorities for the correctness of their solutions. In this pilot case study, we demonstrate ways in which promoting their checking of their own answers can empower their development. Specifically, we examine answer-checking schemes that a cognitively diverse 6th grader with difficulties learning mathematics used when solving additive tasks. We draw on constructivist scheme theory as a framework to analyze data from a year-long teaching experiment, demonstrating a rather rapid progress in his problem-solving schemes, from counting-all to break-apart-make-ten. Along with this rapid conceptual progress, we found that his answer-checking schemes developed in tandem with the problem-solving schemes, typically being one cognitive step behind the latter, that is, advancing from no answer-checking to a numerical count-on scheme. During problem-solving, he may have used schemes at either a participatory or anticipatory stage, whereas for answer-checking he mostly used schemes at the anticipatory stage. We discuss theoretical and practical implications of these novel findings about numerical progress in a cognitively diverse student.