The continuing posts on numbers and multiplication finally forced me to dig out the big guns moldering in my bookshelfs. (heh, heh, you guys are in for it now!)
The author of the following excerpt is Stanislas Dehaene who has a master's in mathematics and a doctorate in something nueropsycholgical. He works at the Laboratoire de Sciences Cognitives et Psycholinguistique in Paris, France. He has written the book "Number Sense" which is excellent and also on my bookshelf waiting for another time and topic. The title of the article is "Towards an Anatomical and Functional Model of Number Processing" and appeared in 1995 in the Journal "Mathematical Cognition".
"At the simplest level, studies of normal subjects have repeatedly suggested that the most simple arithmetic facts such as 2 x 3 = 6 are stored and retrieved from memory (for review see e.g. Ashcraft, 1992). Addition and multiplication facts can be activated automatically even when they are irrelevant to the task (e.g. LeFevre, Bisanz, & Mrkonjic, 1988). Bilingual subjects, long after they have moved to a different country and shifted to a second language, often continue to activate arithmetic facts in the language in which they originally acquired them (Shannon, 1984). Such evidence is consistent with our hypothesis that rote arithmetic facts belong to the general class of rote verbal memories (Dehaene, 1992). Our model supposes that no semantic knowledge is required in order to retrieve the most simple of arithmetic facts. Rather, they can be recovered mechanically, without regard to the quantities involved.
At a second level of complexity, however, other arithmetic problems require a _semantic elaboration_ before rote memory can be accessed. Even for simple single-digit additions and multiplications, normal subjects do not possess a complete and error-free memory (e.g. Campbell & Graham, 1985). Hence, when faced with an unknown or irretrievable fact, subjects may resort to strategies other than memory retrieval (e.g. Siegler, 1988). For instance addition problems may be decomposed into simpler memorised facts (e.g. 9+7 = 9+1+7-1 = 10+6 = 16). Such recoding strategies obviously require a good understanding of the quantities involved in the original problem (e.g. noticing that 9 is close to 10). Accordingly, we subsume them under the heading of "semantic elaboration." Semantic elaboration may be needed both before and after memory retrieval, in order to check the plausibility of a retrieved answer. In particular, accessing a representation of magnitudes may permit the filtering out of the grossly false results such as operand confusions (e.g. 5 + 6 = 30; Winkelman & Schmidt, 1974). Thus, we postulate that success in arithmetic fact retrieval can be boosted considerably by accessing a semantic representation of the magnitude of the operands and of the tentative result.
At a third level of complexity, arithmetic problems often involve working memory. Working memory is obviously required when the operands are presented auditorily and must be remembered throughout the calculation. It is also needed for problems that require the temporary storage of intermediate results, for instance during carry or borrow operations. Research with normal subjects has demonstrated the involvement of a short-term verbal store, called the articulatory loop, in complex calculation (e.g. Hitch, 1978; Logie et al., 1994). A visuo-spatial store may also be used to maintain on-line the spatial layout and digits of an ongoing multidigit calculation.
Finally, at a fourth level of complexity, the most complex arithmetic problems require sequential planning and control processes. Multi-digit operations involve the resolution, in a strict order, of many single-digit problems. The selection and execution of each elementary operation must be controlled and possibly corrected. This also calls into play visuo-spatial resources, due to the spatial organisation of calculation algorithms. With considerable practice, parts of these algorithms may become routinised. For most of us, however, complex operations, especially subtraction and division, involve sequential trial and error processes (van Lehn, 1990). In yet other problems, such as word problems, the solution strategy is often not known or not transparent in the formulation, and a novel algorithm must be planned."[p.103-105]