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Making Sense of Instructional Devices: The Emergence of Transparency in Mathematical Activity
*Luciano Meira* This article examines the mathematical sense-making of children as they use physical devices to learn about linear functions. The study consisted of videotaped problem-solving sessions in which pairs of 8th graders worked on linear function tasks using a winch apparatus, a device with springs, and a computerized input-output machine. The following questions are addressed: How do children make sense of physical devices designed by experts to foster mathematical learning? How does the use of such devices enable learners to access selected aspects of a mathematical domain? The concept of transparency is suggested as an index of access to knowledge and activities rather than as an inherent feature of objects. The analysis shows that transparency is a process mediated by unfolding activities and users' participation in ongoing sociocultural practices. |

An Investigation of African American Students' Mathematical Problem Solving
*Carol E. Malloy, M. Gail Jones* In this study we examined the problem-solving characteristics, strategy selection and use, and verification actions of 24 African American 8th-grade students. Students participated in individual, talk-aloud problem-solving sessions and were interviewed about their problem solutions and attitudes about learning mathematics. Students displayed approaches attributed to African American learners in the literature, regularly using holistic rather than analytic reasoning; their display of confidence and high self-esteem did not appear to be related to success. Students' problem-solving actions matched previously reported characteristics of good mathematical problem solvers: successful use of strategies, flexibility in approach, use of verification actions, and ability to deal with irrelevant detail. Success was highly correlated with strategy selection and use and moderately correlated with verification actions. |

Preschoolers' Counting and Sharing-** FREE PREVIEW!**
*Kristine L. Pepper, Robert P. Hunting* Both counting and sharing require action on discrete elements, entailing the logic of one-to-one correspondence. How counting and sharing relate to one another was the focus of an experiment conducted to examine strategies preschool children used to subdivide items. We designed tasks in which applications of counting skills, of visual cues such as subitizing, and of informal measurement skills were made more difficult. Children exhibited alternative strategies, suggesting use of a recipient as a mental cycle marker and an adjacent recipient strategy, with pauses between allocations suggesting a re-presentation of lots corresponding to the number of recipients. Results supported the view that dealing competence does not relate directly to counting skill. |

Prospective Teachers' Use of Computing Tools to Develop and Validate Functions as Mathematical Models
*Rose Mary Zbiek* This study explored the strategies used by 13 prospective secondary school mathematics teachers to develop and validate functions as mathematical models of real-world situations. The students, enrolled in an elective mathematics course, had continuous access to curve fitters, graphing utilities, and other computing tools. The modeling approaches fell under 4 general categories of technology use, distinguished by the extent and nature of curve-fitter use and the relative dominance of mathematics versus reality affecting the development and evaluation of models. Data suggested that strategy choice was influenced by task characteristics and interactions with other student modelers. A grounded hypothesis on strategy selection and use was formulated. |

Kindergarten Students' Organization of Counting in Joint Counting Tasks and the Emergence of Cooperation
*Heide G. Wiegel* The purpose of the study was to investigate and document possibilities for and manifestations of collaborative work with pairs of kindergarten students while they worked on tasks designed to promote early number development. Ten students, paired to be compatible with respect to their development of counting, were taught weekly for a period of 4 months. The students were addressed as pairs and provided with only one set of counting materials. The students generated 4 strategies to organize their counting: counting parts side by side, counting all at the same time, taking turns, and counting cooperatively. Cooperative counting was defined as a counting episode in which the counting acts of both students merged into a single activity with the partners working toward a common goal. Three themes emerged from the analysis of the cooperative solutions: (a) the relation between cooperation and the specific requirements of a counting task, (b) the relation between the ability to work cooperatively and the students' development of counting, and (c) the students' need to complement a cooperative venture with a solution of their own. |

Relationship Between Computational Performance and Number Sense Among Sixth- and Eighth-Grade Students in Taiwan
*Robert E. Reys, Der-Ching Yang* This research provides information on the number sense of Taiwanese students in Grades 6 and 8. Data were collected with separate tests on written computation and number sense. Seventeen students were interviewed to learn more about their knowledge of number sense. Taiwanese students' overall performance on number sense was lower than their performance on written computation. Student performance on questions requiring written computation was significantly better than on similar questions relying on number sense. There was little evidence that identifiable components of number sense, such as use of benchmarks, were naturally used by Taiwanese students in their decision making. This research supports the need to look beyond correct answers when computational test results are reported. |