The Mediation of Arithmetic Problem-solving by Third-graders: the Procedure of “Assembling Sets”

The purpose of our study was to reveal students’ typical difficulties when performing trivial transformations of several quantities, which are to be coordinated. We have designed two tasks that required to build up the "sets", and to keep their ratio. 97 third-graders from three secondary Moscow schools were recruited for this study. The participants solved the first task individually in the written form, and afterwards there were 25 couples randomly selected to solve the similar task jointly in an oral interview with the experimenter. The analysis of the results revealed the weakness and confusion of most of the surveyed students in solving such problems. The in-depth consideration of the written works and interviews allowed us to characterize the model means, used by students, as formal or meaningful. Among them, only the “portion-by-portion” measurement of two independent quantities, performed through drawing or using the counting material, provided by the experimenter, turned out to be effective. The study showed that the source of difficulties in solving problems related to “assembling sets” is the lack of adequate model mediation, and confirmed the relevance of considering the "assembling sets problem" in the general line of development of the number concept in primary mathematics education within the framework of V.V. Davydov's theory of learning activity.