This study is an investigation of the factors that influence the effectiveness of collaboration on open-ended mathematical tasks. Students in Grades 3, 6, and 9 worked in groups of 3 on 2 chance and data tasks-1 related to fair dice and the other related to associations among variables presented on data cards. The groups' outcomes and the types of collaboration observed are investigated in relation to issues raised in the literature. Various phenomena are identified that influence cognitive "lifting,""hovering," and "falling," that is, improvement, no change, and reduction in levels of functioning, respectively. These phenomena include cognitive factors, social or interpersonal factors, and external factors.
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