Fourth-grade students' understanding of the order and equivalence of rational numbers was investigated in 11 interviews with each of 12 children during an 18-week teaching experiment. Six children were instructed individually and as a group at each of two sites. The instruction relied heavily on the use of manipulative aids. Children's explanations of their responses to interview tasks were used to identify strategies for comparing fraction pairs of three types: same numerators, same denominators, and different numerators and denominators. After extensive instruction, most children were successful but some continued to demonstrate inadequate understanding. Previous knowledge relating to whole numbers sometimes interfered with learning about rational numbers.
This study investigated the ways students represented fractions on number lines and the effects of instruction on those representations. Two clinical teaching experiments and one large-group teaching experiment were conducted with fourth and fifth graders (N = 5, 8, and 30) The instruction primarily concerned representing fractions and ordering fractions on number lines. Tests and videotaped interviews indicated that unpartitioning, in particular, is difficult for students, although the instruction seemed to help. Associating symbols with representations also seems difficult and may depend on an understanding of the unpartitioning process.
This study investigated the effects of two context variables, ratio type and problem setting, on the performance of seventh-grade students on a qualitative and numerical proportional reasoning test. Six forms of the qualitative and numerical-tests were designed, each form using a single context (one of two settings for each of three ratio types). Different ratio types appear to have a stronger impact on the difficulty of the qualitative and numerical proportional reasoning problems than small differences in problem setting. However, the familiarity of problem setting did show an increasingly large effect on qualitative reasoning as the difficulty of ratio type increased. We also investigated the nature of the relationships between rational number skills, qualitative reasoning about ratios, and numerical proportional reasoning. Qualitative reasoning appears to be sufficient, but not necessary for numerical proportional reasoning. The evidence for the requisite nature of rational number skills for proportional reasoning was equivocal. The implications of these findings for science education are discussed.
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