6.1 Memory for arithmetic facts
Chapter 2 contains a review of the literature and previous models of adult memory for multiplication facts. It was noted that RTs tend to be lower for smaller problems than larger problems, although there are exceptions to this rule. Most errors are operand errors---answers that are correct for a problem that shares an operand with the presented problem. With the exception of McCloskey & Lindermann's MATHNET model, previous models lack details about learning, response mechanisms or the spread of activation.
The cascade model presented in chapter 3 was trained on the multiplication facts and captures the main aspects of the phenomena. Recall from the network is based on a build up of activation in the hidden and output units. The RT is measured as the number of processing cycles required before a product unit exceeds some randomly selected threshold. When the threshold is low, incorrect products can be selected as the answer. These errors are mostly operand errors.
Earlier experiments used different assumptions about representation of operands and the frequency with which problems occurred. These experiments, together with an analysis of the networks, suggest that the following factors contribute towards the problem-size effect and error distribution: variations in input representation, especially the relative "sharpness" of the encoding; how frequently each problem occurs in the training set; and the nature of the arithmetic facts themselves. It also was noted that coarse encoding is equivalent to training on false associations. Some models assume that false associations are learned, and some do not. This thesis indicates that the question of interest is not whether or not false associations are formed, but by which method they are formed.
Preliminary simulations were presented of network damage and recall of zero and ones problems. Finally, possible accounts of verification and priming were discussed.
The various network models show some degree of consensus regarding the phenomena associated with recall of arithmetic facts. For example, the problem is considered as the spread of activation between operand units and answer units. However, there are many interesting differences between the models, including: input and output representation, intermediate representations, activation rules, training assumptions. The importance of these differences remains to be explored. For example, one of the assumptions in the cascade model is that the presence of a tie problem is made explicit in the input to the network. For adults tie problems are solved quickly, but for the network, this is only achieved by the use of a tie flag. At the moment it appears that tie problems are difficult for network models to account for without a measure equivalent to the tie flag.
6.2 Multicolumn arithmetic
Chapter 4 described children's errors in multicolumn multiplication. Previous accounts of buggy behaviour were considered---especially VanLehn's Sierra model. Sierra is an extension to repair theory and includes an inductive learning mechanism. VanLehn's model predicts that when children reach impasses, general purpose repairs are made to the local problem solver. The errors that are observed depend on what kind of impasse occurred and which repair was carried out.
A number of observations were made of why connectionism can contribute to this domain. It was noted that the notion of a impasse does not directly apply to connectionist networks: given an input, the network will produce an output. Networks may be able to automatically repair undefined situations because of such properties as similarity-based processing and automatic generalization.
Using some of the assumptions of VanLehn, and taking ideas from Suppes et al.'s model of eye-movement, a connectionist model of multicolumn multiplication was built (chapter 5). To study bugs, rather than slips, the network was trained to activate procedures to carry out the details of multiplication, such as adding, multiplying, and keeping track of registers. The recurrent network was trained on problems of ever-increasing difficulty, from 1+1 to 12x99. During training, the network was tested on unseen problems from the curriculum, and errors occurred at this point.
The errors made by the system do not match the empirical observations very well, although there are difficulties in comparing the errors to children's errors. The set of output operations, although sufficient for solving multiplication problems, requires further work to capture children's errors in detail. An analysis of the errors made by the network shows some interesting results. The system is behaving as a graded state machine: it has many of the properties of finite state machines, but does not "get stuck" when encountering novel inputs. Errors were characterized as perturbations to the desired trajectory, rather than perturbations to a rule set. The errors are a result of unlearned state transitions, and the details of a particular error depends on its similarity to previous experienced problems.
The state of the network was visualized by plotting the principal components of the hidden unit activations. Although it is not obvious that this reduction in dimensions (from 35 hidden units to 2 axis) will provide any interesting information about the system, in practice it does. The mistakes made by the system are capture errors: the system is temporarily attracted into a region of state space which represents an arithmetic subroutine. This is clearly visible with the PCA trajectory diagrams.
The representations learned by the system have a great deal of structure. The model suggests that this may be exploited to account for errors without reference to explicit impasses or repairs. It was noted that the output layer of the network exhibits an increase in residual error at moments that correspond to impasses. If the processing details of the network were changed, this increase in residual error could be observed as an increase in RT. This raises the question of whether impasses are important moments for the learner, or simply a by-product of the processing mechanisms. The model requires more work before this suggestion can be more thoroughly explored and tested.
This is, of course, just one of many possible connectionist views of impasses. From the point of view of Soar, for example, Rosenbloom (1989) suggests that connectionist impasses may occur when a number of output units are above threshold---meaning that there is no uniquely specified course of action to take.
6.3 Future work
Specific future work, in the short- and mid-term, was outlined at the end of chapters 3 and 5. More general comments are made here.
Our understanding of the representation of number and of the training environment is poor. In both models the training environment---frequency or order of problems---is important. Empirical evidence needs to be accumulated to understand what problems children actually encounter.
Experiments in part I showed that changes in the "sharpness" of operand representation changed the results of the simulation. In part II, emphasis was placed on arithmetic perceptual skills, and in particular on eye-movements. Without an understanding of these details it will be difficult to build an appropriate operation set for the multicolumn model. It seems that more study is needed of preschool number abilities and foundational skills, such as number comparison and counting. The representation of number assumed and developed by the recall network should be applied to these other number skills. In this way it may be possible to determine the validity of the various representations, and evaluate the plausibility of a product level of representation and a tens and units level.