**FEATURES** |

Young Children's Perceptions of Mathematics in Problem-Solving Environments
*Megan Loef Franke, Deborah A. Carey* Thirty-six first graders from 2 different school systems participated in individual interviews to determine the children's stated perceptions regarding what it means to engage in mathematics and the rationale and conditions under which they held such perceptions. These children were in classrooms that reflected the spirit of the current reform movement in mathematics education. Generally, the children perceived of mathematics as a problem-solving endeavor in which many different strategies are considered viable and communicating mathematical thinking is an integral part of the task. The children recognized and accepted a variety of solution strategies, with many of the children valuing all solutions equally and assuming a shared responsibility with the teacher for their mathematics learning. Children had varying perceptions of what it meant to succeed in mathematics, but success was not determined by speed and accuracy. As we begin to understand children's perceptions of mathematics as they participate in reform-minded mathematics classrooms, we become aware of issues concerning the impact of the children's perceptions on both the development of their future perceptions and their mathematics learning. |

Conceptual Units Analysis of Preservice Elementary School Teachers' Strategies on a Rational-Number-as-Operator Task
*Merlyn J. Behr, Helen A. Khoury, Guershon Harel, Thomas Post, Richard Lesh* This study explores preservice teachers' understanding of the operator construct of rational number. Three related problems, given in 1-on-1 clinical interviews, consisted of finding 3/4 of a pile of 8 bundles of 4 counting sticks. Problem conditions were suggestive of showing 3/4 of the number of bundles (duplicator/partition-reducer [DPR] subconstruct) and 3/4 of the size of each bundle (stretcher/shrinker [SS] subconstruct). This study provides confirming instances that students use these 2 rational number operator subconstructs. The SS strategies are identified when the rational number, as an operator, is distributed over a uniting operation. With these SS strategies, rational number is conceptualized as a rate. However, the SS strategies were used less often than the DPR strategies. Detailed cognitive models of these strategies in terms of the underlying conceptual units, their structures, and their modifications, were produced, and a "mathematics of quantity" notational system was used as an analytical tool to describe and model the embedded abstractions. |

Students' Development of Length Concepts in a Logo-Based Unit on Geometric Paths
*Douglas H. Clements, Michael T. Battista, Julie Sarama, Sudha Swaminathan, Sue McMillen* We investigated the development of linear measure concepts within an instructional unit on paths and lengths of paths, part of a large-scale curriculum development project funded by the National Science Foundation (NSF). We also studied the role of noncomputer and computer interactions in that development. Data from paper-and-pencil assessments, interviews, and case studies were collected within the context of a pilot test of this unit with 4 third graders and field tests with 2 third-grade classrooms. Three levels of strategies for solving length problems were observed: (a) apply general strategies such as visual guessing of measures and naive guessing of numbers or arithmetic operations; (b) draw hatch marks, dots, or line segments to partition lengths to serve as perceptible units to quantify the length; (c) no physical partitioning--use an abstract unit of length, a "conceptual ruler," to project onto unsegmented objects. Those students who had connected numeric and spatial representations evinced different and more powerful problem-solving strategies in geometric situations than those who had forged fewer such connections. |

The Evolution With Age of Probabilistic, Intuitively Based Misconceptions
*Efraim Fischbein, Ditza Schnarch* The purpose of this
research was to investigate the evolution, with age, of probabilistic,
intuitively based misconceptions. We hypothesized, on the basis of previous
research with infinity concepts, that these misconceptions would stabilize
during the emergence of the formal operation period. The responses to
probability problems of students in Grades 5, 7, 9, and 11 and of prospective
teachers indicated, contrary to our hypothesis, that some misconceptions grew
stronger with age, whereas others grew weaker. Only one misconception
investigated was stable across ages. An attempt was made to find a
theoretical explanation for this rather strange and complex situation. |