The central concern of the present study is the psychological processes involved in the learning of mathematics by children who, in Piaget's sense, are in the stage of "concrete operations" and are not, presumably, yet able to deal readily with formal propositions. Better to understand how mathematics learning of a highly symbolized type might occur, we worked with a small number of children, observing them in minute detail to determine the steps involved in grasping mathematical ideas. Such an approach is, in our opinion, most pressingly needed at this stage of development of new mathematical instruction. It is closely akin to the detailed study of the naturalist and clinician. Perhaps such study can serve to aid more large-scale psychometric testing or, indeed, to elucidate the nature of instruction. It would be disingenuous to say that we (or any naturalist, for that matter) worked without due regard to some theory. Our theoretical predilections were, we should say, far clearer when we finished than when we started. They will also be plain to the reader as our account progresses. The observations to be reported were made on four eight-year-old children, two boys and two girls, who were given an hour of daily instruction in mathematics four times a week for six weeks. The children were in