Some Assumptions and Facts about Arithmetic Facts

Abstract In the present review we will deal with the following questions: What are arithmetic facts? How are they related to other cognitive functions? What are characteristic features of fact retrieval performance in healthy adult subjects? Which typical patterns of breakdown can be observed in acalculia? How do current models account for the representation of and access to arithmetic facts? Key words: simple calculation, mental arithmetic, fact retrieval Simple arithmetic problems, such as 3+6 or 5x4, are frequently encountered and routinely solved in every-day life. Those problems whose solution does not require further computational processes or strategies but can directly be retrieved from long-term memory are commonly referred to as arithmetic facts. Usually, problems with one-digit operands (2+4; 3x5; 6-3) are subsumed under this label, though a precise definition is rarely given. In the following, we will describe dissociations and associations of arithmetic fact retrieval and related cognitive functions, reviewing neuropsychological evidence as well as findings from experimental and developmental psychology. In this way we will try to characterize in more detail what arithmetic facts actually are. Finally, some approaches will be presented to model the acquisition, representation, and retrieval of arithmetic facts. Concepts, procedures, and facts Three main types of arithmetic knowledge can be distinguished: concepts, procedures, and facts (Delazer, 2003). Conceptual knowledge provides understanding of arithmetic operations and principles. This type of knowledge is the prerequisite to make inferences and to relate different information involved in arithmetic. Conceptual knowledge is flexible, can be adapted and applied to new tasks and thus provides adaptive expertise (Hatano, 1988). Procedural knowledge guides the execution of algorithms. It can only be applied in familiar contexts. Accordingly, it can be characterised as routine expertise (Hatano, 1988). Arithmetic facts, finally, can be conceptualised as stored in and directly retrieved from (declarative) long-term memory (e.g., Ashcraft, 1987; Campbell, 1995; Dehaene & Cohen, 1995; Rickard & Bourne, 1996; Siegler, 1988; for a different view see Baroody, 1994). During development, arithmetic facts evolve from conceptual and procedural knowledge. For instance, it has been demonstrated that the pattern of response latencies and errors in the retrieval of simple multiplication facts of older children and adults reflects the number and type of errors made in the backup strategies used by children at an earlier developmental stage solving the same problems (Lemaire & Siegler, 1995; Siegler, 1988). Neuropsychological studies support the assumption that arithmetic facts are stored separately from other numerical skills. In a seminal paper Warrington (1982) reported a patient who was no longer able to retrieve simple arithmetic problems from memory, but was able to give the approximate result of arithmetic problems, both in simple and more complex calculation, to estimate visually presented quantities of dots, to give adequate numerical cognitive estimates, to judge the relative size of a number and to give accurate definitions of arithmetic operations. Several later case reports confirmed the separate storage and selective vulnerability of arithmetic facts knowledge. However, other studies questioned the assumption that healthy, educated adults retrieve the solutions for all simple arithmetic problems from memory (i.e., as facts) (Lefevre, Bisanz, Daley, Buffone, Greenham, & Sadesky, 1996; Lefevre, Sadesky, & Bisanz, 1996; for methodological considerations see Kirk & Ashcraft, 2001). Lefevre and colleagues found that in single digit addition, non-retrieval strategies (e.g., counting) accounted for 29% of all trials and for about half the problems with sums above 10. In simple multiplication, up to 19% of all trials were solved by non-retrieval strategies (e. …