Problem solving activities of two middle school students with distinct levels of units coordination

The aim of this study is to investigate relationships between students' arithmetical knowledge and their proportional reasoning. Two of seven students for whom we conducted clinical interviews were selected as participants in the study. An analysis of their solutions to four different types of multiplicative problems (equal sharing, reversible multiplicative relationship, fraction composition, and proportional relationship) was conducted. Based on the analysis, we found that the student who coordinated two three-level units structures prior to activity in the first three problem types could also solve the proportion problem using the units coordination. In contrast, the student who coordinated two three-level units structures only in activity in the first three problem types could not solve the proportion problem. Given the importance of units coordinating operations in solving diverse problems, implications for further research on students' construction of proportional reasoning are discussed.