In his remarks on “Some Issues in the Psychology of Mathematics Instruction,” Gagné (1983) has stimulated a renewed discussion of the cognitive phenomena involved when children learn mathematics and of the implications of cognitive learning theory for mathematics instruction. Gagné gives a three-phase performance model along with the following core message: Students should understand how to mathematize a concrete situation that is described verbally and how to validate a solution once it is obtained, but they need *not* understand how the solution is derived. Instead, the skills involved in the computation phase should be made automatic for the sake of optimal overall performance, and lots of practice should be devoted to automatizing such skills. Taken out of their context and inserted into the current climate in mathematics education, Gagné's statements are likely to be misunderstood, giving the wrong impression to teachers and perhaps causing researchers to reject his ideas.

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### Ipke Wachsmuth

### Merlyn J. Behr, Ipke Wachsmuth, Thomas R. Post and Richard Lesh

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.

### Thomas R. Post, Ipke Wachsmuth, Richard Lesh and Merlyn J. Behr

Fourth-grade students' understanding of the order and equivalence of rational numbers was investigated in an 18-week teaching experiment. Data from observations of, and interviews with, two children were employed to identify patterns over time in the strategies used in performing tasks. Three related characteristics of thinking are hypothesized to be related to the successful performance of tasks on order and equivalence: (a) thought flexibility in coordinating between-mode translations, (b) thought flexibility for within-mode transformations, and (c) reasoning that becomes increasingly independent of specific concrete embodiments.

### Merlyn J. Behr, Ipke Wachsmuth and Thomas R. Post

This report from the Rational Number Project concerns the development of a quantitative concept of rational number in fourth and fifth graders. In a timed task, children were required to select digits to form two rational numbers whose sum was as close to 1 as possible. Two versions of the task yielded three measures of the skill. The cognitive mechanisms used by high performers in individual interviews were characterized by a flexible and spontaneous application of concepts of rational number order and fraction equivalence and by the use of a reference point. Low performers tended either not to use such cognitive mechanisms or to apply concepts in a constrained or inaccurate manner.

### George W. Bright, Merlyn J. Behr, Thomas R. Post and Ipke Wachsmuth

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.