Compelling research evidence shows benefits for student learning from explaining one's ideas and engaging with the ideas of others. However, whether certain patterns of group interaction may engender this productive student participation is unknown. Using data from two third grade mathematics classrooms, and over the course of six days during a five-month span, we investigated how students interacted with each other to solve problems when the teacher was not driving the interaction. We identified multiple profiles of group interaction that yielded highly-detailed participation for some or all students in the group. These profiles varied in terms of whether students interacted in an ongoing, sustained manner or interacted periodically but not continually, whether one or multiple students initiated problem-solving strategies, and whether group members worked jointly or largely separately on their strategies. No single profile of group interaction was either necessary or sufficient to lead to highly-detailed participation for all students in the group.