In Errors in Children's Subtraction in Cognitive Science 1981, Young and O'Shea set up a PS model of subtraction. They say that in standard educational theory, faulty arithmetic is due to a failure to remember ``number facts'', aggravated by bad setting-out, fatigue, carelessness, poor concentration, etc. So standard theory has concentrated on models for recall of number facts.
But the children's errors may be due to failures in execution. We should regard the child as faithfully following wrong algorithms, not wrongly executing the correct one. Of course, this is a hypothesis; but it's worth taking seriously, because if true, it gives us more control over teaching.
Let's see whether a production system model could show how children might acquire wrong algorithms rather than correct ones. We'll express these as rules, of course. Young and O'Shea devised a set of PS rules which modelled correct subtraction, and then demonstrated that by slight changes to some of the rules, the model could generate most of the observed errors. This model accounts for more than 2/3 of such errors (from an analysis of 1500 sums by ten-year olds).
Some of these rules are faulty because of wrong actions; others, because of wrong conditions. For example,
50 46 -- 16 --may have been caused by a rule whose conditions don't discriminate between having the zero under a digit, or above.
What are the implications for teaching? Conventionally, a new skill or aspect of skill is viewed as a new component or procedure. So teachers ``consolidate'' it by repeated practice or ``consolidation training''. For example, language textbooks that introduce past tenses and then set translations involving all past tenses and no present tenses. But the rule model shows we must also train the learner in discriminating when to apply rules - to get the conditions right. Such training may actually be hindered by consolidation. In the language example, old present tense rules may be lost or changed in favour of past tense ones.
For a survey of this and other work on PS models of arithmetic skill, see the introduction to Mind Bugs: The Origins of Procedural Misconceptions by Kurt VanLehn (MIT 1990; PSY BF:V 032). Note that VanLehn treats PS models as descriptions in the same way that differential equations describe circuits or grammars describe linguistic behaviour (page 3).
See Rules of the Mind by John Anderson (LEA 1993; PSY BH:A 547) for a recent work by somebody who still believes in the psychological reality of production rules, and who contrasts his views with those of some connectionists. His arguments for rules are on page 10.