The present study examined how comparing and explaining both correct and incorrect solutions by sharing group results with the whole class after group discussion in a collaborative arithmetic class may facilitate individuals’ understanding. Second graders (N=32) solved arithmetic problems about the sum of fractions in 3 stages: pre-test, during the lesson (collaborative learning), and post-test. The results indicated that, (a) through the collaborative learning, children’s individual understanding on fractions was facilitated, (b) in the collaborative learning, harmonization of children’s own ideas on fractions was facilitated, and (c) regarding the sum of fractions, individuals’ understanding was facilitated by harmonizing their own ideas. These results suggest that, in the group discussion, though the explanation of the children may not be sufficiently elaborated, the change of the collective explanation directly affects the change of the explanation of the individual, while on the other hand, in the whole-class exchanges though the teacher’s support encourages elaboration of the explanation of the children, the influence of elaboration of the collective explanation on the individuals’ explanation is likely to differ depending on their existing knowledge.
The present study examined the promotion of association of knowledge by collectively examining relevance of the solutions in the collaborative process in the mathematics class of junior high school. Eighth graders (3 classes; total N=88) solved arithmetic problems about mathematical consideration of events using linear functions in 3 stages: pre-test, during the lesson (3 conditions: collaborative process to examine relevance of the solutions, collaborative process to investigate the validity of the solutions and teacher’s explanations of relevance of the solutions), and post-test. The results indicated that, (a) in both of the two collaborative processes, the students who assumed the linear function change were encouraged to relate and interpret the rate of change and the inclination of the graph according to the event, and, thus, grasp clearly the basis for capturing changes in events by linear functions, and (b) in examining relevance, a change was observed for students who were originally unable to interpret the rate of change or the inclination of the graph and who became able to do it.