The Function of Conceptual Understanding in the Learning of Arithmetic Procedures

School children learn arithmetic procedures by rote rather than by constructing them on the basis of their understanding of numbers. Role learning produces lack of flexibility, nonsensical errors, and other difficulties in learning. Mathematics educators have proposed that, if arithmetic procedures were constructed on the basis of conceptual understanding of arithmetic principles, then procedure acquisition would not suffer from these difficulties. However, little effort has been invested in clarifying this hypothesis or in proving its viability. We propose a theory of conceptual understanding and its role in the learning and execution of arithmetic procedures. Our hypothesis is that conceptual understanding constrains problem states and, thereby, enables the learner to monitor his or her own performance and to detect and correct his or her errors. A novel knowledge representation, the state constraint, captures this view of principled knowledge. We propose that learning occurs when state constraints are violated. A particular learning mechanism that can correct procedural rules on the basis of the information contained in constraint violations has been developed. We have implemented our theory in the Heuristic Searcher (HS), a computer model that learns arithmetic procedures. We have simulated (a) the discovery of a correct and general counting procedure in the absence of instruction, feedback, or solved examples; (b) flexible adaptation of an already learned counting procedure in response to changed task demands; and (c) self-correction of errors in multicolumn subtraction. The state constraint theory provides a new interpretation of the role of conceptual understanding in arithmetic learning, generates testable predictions about human behavior, deals successfully with theoretical issues that cause difficulties for other theories of learning, and fares well on evaluation criteria such as generality and parsimony.