RICHARD DALLAWAY, DPhil thesis, February 1994

A connectionist view of arithmetic skills

(and other information)
The nutshell guide to
the thesis
"I have no doubt that when we do mental arithmetic we are doing something well, but it is not arithmetic" (Marr 1982, p.348)

CHILDREN HAVE ACQUIRED a host of impressive skills by the time they are taught formal arithmetic: they have learned a language and can navigate in a hostile environment. In contrast, the "simple" tasks of arithmetic require at least a further five or six years of schooling. Once the skills are learned there are many opportunities for error. Adults, for example, make plenty of mistakes recalling multiplication facts -- especially on the "tricky" problems, such as 8x4 or 9x8. Arithmetic, it seems, is not an easy skill to come by.

So what is the "something" that we do well when we solve arithmetic problems? The view taken here is that the things we do well are those tasks suited to a certain kind of computation -- namely connectionism. Difficult tasks, such as arithmetic, need to be turned into pattern matching problems. That is, "we succeed in solving logical problems not so much through the use of logic, but by making the problems we wish to solve conform to problems we are good at solving" (Rumelhart, Smolensky, McClelland & Hinton 1986, p. 44). Exactly how this is done for arithmetic is the topic of this thesis.

Two elementary arithmetic skills are considered: adult memory for multiplication facts and children's errors in long (multicolumn) multiplication.