The influence of the place-value structure of the Arabic number ...

In the holistic model, two input arrays, each reflecting a holistic representation of the
numbers ranging from 11 to 99, are implemented. These input arrays are connected to two
output nodes, whereof the left one represents the “first number larger“ decision and the right
one the “second number larger” decision.
Training of connection weights
The initial connection weights were pseudo-random values ranging from 0 to 1 taken
from a uniform distribution. Basically, the same learning algorithm was used for all three
models. Nevertheless, as the fully decomposed and the hybrid model involved a hidden layer
learning followed a back-propagation approach in these two models. For the holistic model
which did not incorporate a hidden layer the learning algorithm used reflected delta rule
guided learning, respectively. Please note that back-propagation is a generalization of the
delta rule approach to networks involving at least one hidden layer (cf. Rumelhart, Hinton, &
Williams, 1986). Based on this and the fact that the same mathematical algorithm was used
for the training of connection weights in all three models we are confident that differences in
the learning algorithm did not determine the current results. Finally, the learning constant η
was arbitrarily set to 0.1.
The training phase comprised 50.000 for all models. In each training circle two
randomly chosen numbers between 11 and 99 (excluding multiples of 10) were presented to
the network. The frequency of occurrence of each number during the training cycles was
determined by its frequency of occurrence in daily life as assessed by taking the entries
referenced in Google for the respective numbers as an estimate of their every-day occurrence
(see Appendix A; Verguts & Fias, 2008 for a similar approach).
The procedure of the training phase will be described for the case of the decomposed
model: At first, two numbers between 11 and 99 were randomly chosen with the exception of
tie numbers (e.g., 44) and multiples of 10 as empirical studies showed anomalous processing
248