Page 266 - Deep Learning

Error Correction: The Specialization Theory 249

It required two training problems to learn augmenting procedurally, five
to learn augmenting conceptually. It required four training problems to master
regrouping in both the procedural and the conceptual conditions. The number
of tutoring messages required for correct performance varied between 20 and
31 in the four conditions. Given that children do worksheets with hundreds
of problems, these numbers seem low, but the model was learning in a one-
on-one tutoring scenario, it learned from tutors who caught every error and,
being a machine, suffered no distractions, lack of motivation or working mem-
ory limitations. The number of learning events (rule revisions) required for
mastery varied between 16 and 32. Interestingly, the regrouping method is easy
to learn when there are no blocking zeroes, that is, zeroes that force the learner
to borrow from the next column. Mastery required only 16 learning events.
However, the model required 32 learning events to learn the conceptual ver-
sion of the regrouping procedure for problems with blocking zeroes. For both
the conceptual and procedural versions, the regrouping method, but not the
augmenting method, was strongly affected by the presence of blocking zeroes.
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How realistic are these numbers? Gaea Leinhard has shown that mastery
of subtraction might require six classroom lessons. Six lessons represent 4.5
hours of instruction. The HS model required 32 learning events in the concep-
tual regrouping condition, the approach used in most American classrooms.
Thirty-two events per 4.5 hours comes to one learning event per eight minutes.
Empirical estimates of the rate of learning are hard to come by but this learn-
ing rate is of the right order of magnitude.
The tutoring simulation illustrates the idea that learning from instruc-
tion is embedded within unsupervised learning. The very same computational
mechanism that was invented to learn from error in the presence of declar-
ative principles such as the counting principles or the laws of chemistry can
also learn when the constraints arrive in a succession of tutoring messages. No
additional cognitive machinery was required for HS to be able to learn from
tutoring instead of from prior declarative knowledge.

Constraint-Based Tutoring
The question arises whether it is possible to use the constraint-based theory
to design instructional software systems that tutor human students in the
same way that we tutored the HS model. Instructional software systems that
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attempt to mimic human tutors are called intelligent tutoring systems. The key
features of this class of instructional systems are that the target skill is repre-
sented explicitly, the student’s incomplete and possibly erroneous knowledge