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Error Correction: The Specialization Theory 239

Table 7.3. Declarative knowledge in the HS model of children’s counting skill.

Label Content
A One-One Mapping Constraint: If a number N is associated with object O i , O i
ought to be the only object that N is associated with.
B Cardinality Constraint: If object O i is the last object to be associated with some
number N, and N has been designated as the answer to count({O}), then
every object in {O} ought to be associated with some number.
C Regular Traversal Constraint: If each number from 1 to N has been associated
with some object, and M is the current number, then M ought to be the
successor to N.
D Order Imposition Constraint: If O is chosen as the current object, O should not
already be associated with a number.
E Coordination Constraint: If O is the current object and O is associated with
number N, then N ought to be the current number.

Based on Ohlsson and Rees, 1991a, Table 4.

We then gave HS repeated practice on counting sets with two to five
objects. In the beginning, the model made the same types of errors that chil-
dren do: skipping numbers, skipping objects, counting one object more than
once and issuing a seemingly arbitrary number as the answer. However, the
constraints that encoded the counting principles enabled the model to detect
its errors, and the specialization mechanism gradually restricted the condi-
tions of its rules. Even when practice was limited to small sets, the model, like
children, learned a general strategy for how to count sets of any size. If the
initial rules and the constraints are stated in general terms so that they apply
to any type of object, then the model learns a counting procedure that applies
to any type of object.
Consider a particular run with the counting model, summarized in
Table 7.4. The model acquired the correct counting procedure in 22 trials, per-
47
forming correctly on trial 23. The path illustrates several properties of the model’s
behavior. The third column shows the constraint that was violated, that is, which
type of error was committed, on each trial. Some constraints were violated sev-
eral times, like D (trials 2, 3 7, 9, 10, 14, 15, 17, 18, 19 and 20), while others were
violated less often, like C (trials 4, 6 and 21) and A (trials 1, 5 and 16). Sometimes
constraints were violated again and again on adjacent trials, like D on trials 17
through 20, while others were violated on trials separated by other trials on
which they were not, like E on trials 13 and 22. The model did not work on one
type of error until it was completely eradicated, and then turn its attention to the
next error type. Instead, it responded to errors as they occurred in the course of