**FEATURES** |

Developing Children's Understanding of the Rational Numbers: A New Model and an Experimental Curriculum
*Joan Moss, Robbie Case* A new curriculum to introduce rational numbers was devised, using developmental theory as a guide. The 1st topic in the curriculum was percent in a linear-measurement context, in which halving as a computational strategy was emphasized. Two-place decimals were introduced next, followed by 3- and 1-place decimals. Fractional notation was introduced last, as an alternative form for representing decimals. Sixteen 4th-grade students received the experimental curriculum. Thirteen carefully matched control students received a traditional curriculum. After instruction, students in the treatment group showed a deeper understanding of rational numbers than those in the control group, showed less reliance on whole number strategies when solving novel problems, and made more frequent reference to proportional concepts in justifying their answers. No differences were found in conventional computation between the 2 groups. |

Advancing Children's Mathematical Thinking in Everyday Mathematics Classrooms
*Judith L. Fraivillig, Lauren A. Murphy, Karen C. Fuson* In this article we present and describe a pedagogical framework that supports children's development of conceptual understanding of mathematics. The framework for Advancing Children's Thinking (ACT) was synthesized from an in-depth analysis of observed and reported data from 1 skillful 1st-grade teacher using the Everyday Mathematics (EM) curriculum. The ACT framework comprises 3 components: Eliciting Children's Solution Methods, Supporting Children's Conceptual Understanding, and Extending Children's Mathematical Thinking. The framework guided a cross-teacher analysis over 5 additional EM 1st-grade teachers. This comparison indicated that teachers often supported children's mathematical thinking but less often elicited or extended children's thinking. The ACT framework can contribute to educational research, teacher education, and the design of mathematics curricula. |

Creating a Context for Argument in Mathematics ClassYoung Children's Concepts of Shape
*Terry Wood* Evidence shows that class discussion is important in students' development of mathematical conceptions. Theoretically, the process of contradiction and resolution is central to the transformation of thought. This article is a report of an 18-month investigation of a teacher's actions during class discussions in a 2nd-grade classroom in which students' disagreement was resolved by argumentation. Although the teacher valued children's reports of their reasoning, the context of argument in discussion was characterized by the high priority she afforded their roles as critical listeners. Her sensitivity in communicating her expectations for students' participation was evident during both discussion and disagreement. Moreover, the teacher participated with the students to create patterns of interaction and discourse that enabled children to shift their cognitive attention from making social sense to making sense of their mathematical experiences. |

Consistency and Representations: The Case of Actual Infinity
*Pessia Tsamir, Dina Tirosh* In this article we demonstrate how research-based knowledge about students' incompatible solutions to various representations of the same problem could be used to raise their awareness of inconsistencies in their reasoning. In the first part of the article we report that students' decisions as to whether 2 given infinite sets have the same number of elements depend on the specific representation of the infinite sets in the problem. We used these findings to construct an infinite-set activity with the aim of encouraging students to reflect on their own thinking about infinity. The findings indicate that taking part in this activity led a number of the participating students to realize that producing 2 contradictory reactions to the same mathematical problem is problematic; yet, few chose to avoid these contradictions by using 1-to-1 correspondence as a criterion for comparing infinite sets. |

The Effects of a Graphing-Approach Intermediate Algebra Curriculum on Students' Understanding of Function
*Jeannie C. Hollar, Karen Norwood* In this study, we extended O'Callaghan's computer-intensive algebra study by using his component competencies and the process-object framework to investigate the effects of a graphing-approach curriculum employing the TI-82 graphing calculator. We found that students in the graphing-approach classes demonstrated significantly better understanding of functions on all 4 subcomponents of O'Callaghan's Function Test, including the reification component, than did students in the traditional-approach classes. Additionally, no significant differences were found between the graphing-approach and traditional classes either on a final examination of traditional algebra skills or on an assessment of mathematics attitude. |