Conditions under which group work leads to learning have been studied in collaborative settings. Little is known, however, about whether and how the interplay between collaboration and cooperation impinges on group learning. In this paper, we study this interplay in the context of mathematical problem-solving. We focus on how training students to learn together influences this interplay, and on the relations of this interplay with mathematical problem-solving. Five groups of Grade 8 students participated in a course aimed at fostering learning to solve mathematical problems in small groups. Before and after the course, they solved a mathematical problem. An increase in the ratio of cooperation episodes out of total group work time was observed, as well as advancements in mathematical problem-solving. In addition, we found a mid-high correlation between instances of cooperation and mathematical activity: up to a certain threshold, cooperating more in a group yielded an increase in the individual generation of mathematical claims and arguments. We identified the critical role of coordination: for group learning to be productive, students should continuously negotiate and adjust their goals through communication before or while they cooperate on different tasks. We conclude that teachers aiming at fostering group work should encourage the diversification of modes of group work, for the advancement of mathematical problem-solving or of any case in which individual settings are too challenging.

中文翻译：

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分手但团结在一起：解决数学问题中的协作与合作

在协作环境中研究了小组工作导致学习的条件。然而，关于协作与合作之间的相互作用是否以及如何影响小组学习的了解很少。在本文中，我们将在数学问题解决的背景下研究这种相互作用。我们专注于培训学生如何共同学习如何影响这种相互作用，以及这种相互作用与数学问题解决的关系。五组八年级学生参加了旨在促进学习以小组形式解决数学问题的课程。在课程前后，他们解决了一个数学问题。观察到合作情节占小组总工作时间比例的增加，以及数学问题解决的进展。此外，我们发现了合作实例与数学活动之间的中高相关性：达到某个阈值时，在小组中进行更多的合作会增加数学主张和论点的个体生成。我们确定了协调的关键作用：为使小组学习富有成效，学生应在开展不同任务之前或之时通过沟通不断地协商和调整目标。我们得出的结论是，旨在促进小组工作的教师应鼓励小组工作模式的多样化，以促进数学问题解决或在任何情况下个人设置过于挑战的情况。在小组中更多地合作产生了更多的数学主张和论据。我们确定了协调的关键作用：为使小组学习富有成效，学生应在开展不同任务之前或之时通过沟通不断地协商和调整目标。我们得出的结论是，旨在促进小组工作的教师应鼓励小组工作模式的多样化，以促进数学问题解决或在任何情况下个人设置过于挑战的情况。在小组中更多地合作产生了更多的数学主张和论据。我们确定了协调的关键作用：为使小组学习富有成效，学生应在开展不同任务之前或之时通过沟通不断地协商和调整目标。我们得出的结论是，旨在促进小组工作的教师应鼓励小组工作模式的多样化，以促进数学问题解决或在任何情况下个人设置过于挑战的情况。