Interacting neighbors: A connectionist model of retrieval in single-digit multiplication

For most adults, retrieval is the most common way to solve a single-digit multiplication problem (Campbell & Xue, 2001). Many theories have been proposed to describe the underlying mechanism of arithmetical fact retrieval. Testing their validity hinges on evaluating how well they account for the basic findings in mental arithmetic. The most important findings are the problem size effect (small multiplication problems are easier than larger ones; cf. 3 X 2 and 7 X 8), the five effect (problems with 5 are easier than can be accounted for by their size), and the tie effect (problems with identical operands are easier than other problems; cf. 8 X 8 and 8 X 7). We show that all existing theories have difficulties in accounting for one or more of these phenomena. A new theory is presented that avoids these difficulties. The basic assumption is that candidate answers to a particular problem are in cooperative/competitive interactions and these interactions favor small, five, and tie problems. The theory is implemented as a connectionist model, and simulation data are described that are in good accord with empirical data.