Thesis available

[Richard Dallaway]

FTP-host: ftp.cogs.susx.ac.uk
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The following thesis is available via anonymous ftp.

DYNAMICS OF ARITHMETIC: A CONNECTIONIST VIEW OF ARITHMETIC SKILLS

	Richard Dallaway 
	email: richardd at cogs.susx.ac.uk
	
	Cognitive Science Research Paper CSRP-306
	School of Cognitive & Computing Sciences
	University of Sussex, Brighton, UK

SUMMARY: Connectionist models of adult memory for multiplication
facts and children's multicolumn multiplication errors. Full abstract
at then end of this message.

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155 pages. 552567 bytes compressed, 1922143 bytes uncompressed

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	School of Cognitive & Computing Sciences
	University of Sussex
	Falmer, Brighton, UK.

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ABSTRACT:
Arithmetic takes time.  Children need five or six years to master the
one hundred multiplication facts (0x0 to 9x9), and it takes adults
approximately one second to recall an answer to a problem like 7x8.
Multicolumn arithmetic (e.g., 45x67) requires a sequence of actions,
and children produce a host of systematic mistakes when solving such
problems. This thesis models the time course and mistakes of adults
and children solving arithmetic problems.  Two models are presented,
both of which are built from connectionist components.

First, a model of memory for multiplication facts is described. A
system is built to capture the response time and slips of adults
recalling two digit multiplication facts. The phenomenon is thought of
as spreading activation between problem nodes (such as 7 and 8) and
product nodes (56).  The model is a multilayer perceptron trained with
backpropagation, and McClelland's (1988) cascade equations are used to
simulate the spread of activation.  The resulting reaction times and
errors are comparable to those reported for adults.  An analysis of
the system, together with variations in the experiments, suggest that
problem frequency and the "coarseness" of the input encoding have a
strong effect on the phenomena.  Preliminary results from damaging the
network are compared to the arithmetic abilities of brain-damaged
subjects.

The second model is of children's errors in multicolumn
multiplication.  Here the aim is not to produce a detailed fit to the
empirical observations of errors, but to demonstrate how a
connectionist system can model the behaviour, and what advantages this
brings. Previous production system models are based on an
impasse-repair process: when an child encounters a problem an impasse
is said to have occurred, which is then repaired with general-purpose
heuristics. The style of the connectionist model moves away from
this. A simple recurrent network is trained with backpropagation
through time to activate procedures which manipulate a multiplication
problem.  Training progresses through a curriculum of problems, and
the system is tested on unseen problems.  Errors can occur during
testing, and these are compared to children's errors.  The system is
analysed in terms of hidden unit activation trajectories, and the
errors are characterized as "capture errors".  That is, during
processing the system may be attracted into a region of state space
that produces an incorrect response but corresponds to a similar
arithmetic subprocedure.  The result is a graded state machine---a
system with some of the properties of finite state machines, but with
the additional flexibility of connectionist networks. The analysis
shows that connectionist representations can be structured in ways
that are useful for modelling procedural skills such as arithmetic. It
is suggested that one of the strengths of the model is its emphasis on
development, rather than on "snap-shot" accounts. Notions such as
"impasse" and "repair" are discussed from a connectionist
perspective.
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