Starting to add worse: effects of learning to multiply on children's addition.

A major stumbling block in acquiring a new skill can be integrating it with old but related knowledge. Learning multiplication is a case in point, because it involves integrating new relations with previously acquired arithmetic knowledge (in particular, addition). Two studies explored developmental changes in the relations between single-digit addition and multiplication. In the first study, third-graders, fifth-graders, and adults performed simple addition or multiplication in mixed- and blocked-operations formats. Substantial interfering effects from related knowledge were found at all age levels, but were more pronounced for younger subjects. Thus in the early stages of learning multiplication, one consequence of learning a new operation is interference in performance of an earlier, related, but less recently studied skill. Consideration of error patterns supported the view that the problem of integrating operations is a prominent one even in the early stages of mastering multiplication. Patterns of errors were generally consistent across all age groups, and all groups were much more likely to give a correct multiplication response to an addition problem than the reverse. A second, longitudinal study confirmed this finding, showing evidence for impaired performance of addition over time within individual children (second-, third-, and fourth-graders) tested on simple addition and multiplication over a 5-month period. Analysis of reaction times for addition indicated that second-graders in advanced math classes and third-graders in regular math classes tended to slow down over the year in responses to addition problems. Fourth-graders, on the other hand, tended to increase their speed of addition over the course of the year. Multiplication showed a different pattern during this period, with no evidence for slowing among children who were able to perform this task. Disruption of previously learned knowledge in the course of acquiring new skills provides evidence that new knowledge and old knowledge are being integrated. This kind of non-monotonic development may provide an empirical method for determining the functional limits of a domain of knowledge.