Theoretical analysis of whole number-based calculation strategies and digit-based algorithms for multi-digit multiplication and division reveals that strategy use includes two kinds of reasoning: reasoning about the relations between numbers and reasoning about the relations between operations. In contrast, algorithms aim to reduce the necessary…

The Common Core State Standards (2010) involve the demonstration of conceptual knowledge of numbers and operations. For students who struggle with mathematics and have not responded to instruction, it is important that interventions emphasize this understanding. In order to address conceptual understanding of numbers and operations in meeting the…

Research addressing the effectiveness of feedback for learning has focused on many dimensions of feedback, including the timing (Kulik & Kulik, 1988), type (Mory, 2004), and amount of available information (Dempsey et. al, 1993). Much of the feedback research in education has tacitly assumed that the available information is perceived, and any…