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Topic: The Mind and Numbers
Replies: 3   Last Post: Feb 6, 1998 2:04 PM

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Kim or Melodie Mackey

Posts: 272
Registered: 12/6/04
The Mind and Numbers
Posted: Feb 4, 1998 2:23 PM
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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]

regards, Kim Mackey

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