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Marta Kobiela, Kara J. Jackson, Annie Savard and Emily Shahan

All too often, definitions of mathematical ideas and objects are presented as facts to memorize in mathematics classrooms (de Villiers 1998; Keiser 2000). This is unfortunate, as it means that students are not provided opportunities to engage in a form of reasoning that is arguably at the heart of mathematics—definitional reasoning. Making sense of, constructing, and using definitions to determine what counts as an object (e.g., an odd number, a triangle) develop students' ability to communicate about mathematical ideas and their conceptual understanding of properties and relations (Keiser 2000).

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Kara Jackson, Anne Garrison, Jonee Wilson, Lynsey Gibbons and Emily Shahan

This article specifies how the setup, or introduction, of cognitively demanding tasks is a crucial phase of middle-grades mathematics instruction. We report on an empirical study of 165 middle-grades mathematics teachers' instruction that focused on how they introduced tasks and the relationship between how they introduced tasks and the nature of students' opportunities to learn mathematics in the concluding whole-class discussion. Findings suggest that in lessons in which (a) the setup supported students to develop common language to describe contextual features and mathematical relationships specific to the task and (b) the cognitive demand of the task was maintained in the setup, concluding whole-class discussions were characterized by higher quality opportunities to learn.

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Jonee Wilson, Mahtab Nazemi, Kara Jackson and Anne Garrison Wilhelm

This article outlines several forms of instructional practice that distinguished middle-grades mathematics classrooms that were organized around conceptually oriented activity and marked by African American students' success on state assessments. We identified these forms of practice based on a comparative analysis of teaching in (a) classrooms in which there was evidence of conceptually oriented instruction and in which African American students performed better than predicted by their previous state assessment scores and (b) classrooms in which there was evidence of conceptually oriented instruction but in which African American students did not perform better than predicted on previous state assessment scores. The resulting forms of practice can inform professional learning for preservice and in-service teachers.

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Kara J. Jackson, Emily C. Shahan, Lynsey K. Gibbons and Paul A. Cobb

Consider four important elements of setting up challenging mathematics problems to support all students' learning.