How Does Teachers' Visual Scaffolding Support Students' Mathematics Learning

数学推理需要理解数学信息的不同表示之间的联系。数学表示在课堂上的联系方式可能会决定学生是否理解重要的数学原理和程序。我们过去的研究表明，教师使用各种形式的视觉支架连接不同的数学表征。本研究的目的是了解教师视觉支架的变化如何影响学生的学习。我们的具体重点是非语言的支持，教师产生的教学情节，链接相关的数学信息表示。特别是，我们研究那些非语言的支持，服务于地面的想法在物理环境或熟悉的行动，经验或表示。该研究有三个目标：（1）研究教师以手势作为联结概念的基础，是否有助学生学习（与单独使用语言相反）;（2）检查某些类型的非语言支持是否对学习特别有益（具体地，冗余手势与互补手势，以及指向手势与代表性手势）;以及（3）检查手势是否提供了一种“特殊”的方式来视觉支架的想法，在这个意义上，他们更有效地这样做比其他，非手势的视觉支架方法。我们将在中学生学习线性方程的实验中解决这些目标。实验将涉及视频课程，改变教师的手势或用于突出链接表示（手势或数字图标）方面的媒体。我们将在课前和课后评估学生对线性方程的概念和程序知识，以便我们可以评估教师视觉支架的变化如何影响学生的学习。我们还将进行一项试点研究，为我们将这一调查范围扩大到学习统计的大学生做好准备。本研究将探讨教师如何在教学中使用言语和手势来联系表征的置信区间。这项工作将有助于我们从具身认知的角度对学习和教学的科学理解。通过实验操纵数学思想之间的关系传达的方式，并探索学习的后果，我们将获得更深层次的理解认知过程中涉及获得数学理解。这项工作将提供一个经验的基础上，建议教师如何有效地使用视觉支架。

Mathematical reasoning requires understanding connections between different representations of mathematical information. The way mathematical representations are linked in the classroom may determine whether students come to understand important mathematical principles and procedures. Our past research showed that teachers use various forms of visual scaffolding to link different mathematical representations. The purpose of this project is to understand how variations in teachers' visual scaffolding affect students' learning. Our specific focus is on the nonverbal supports that teachers produce in instructional episodes that link related representations of mathematical information. In particular, we examine those nonverbal supports that serve to ground ideas in the physical environment or in familiar actions, experiences or representations. The research has three aims: (1) to investigate whether students' learning is facilitated if teachers ground the to-be-linked ideas with hand gestures (as opposed to using speech alone); (2) to examine whether certain types of nonverbal supports are especially beneficial for learning (specifically, redundant vs. complementary gestures, and pointing vs. representational gestures); and (3) to examine whether gestures offer a "special" way to visually scaffold ideas, in the sense that they are more effective at doing so than other, non-gestural methods of visual scaffolding. We will address these aims in experiments with middle school students learning about linear equations. The experiments will involve video lessons that vary the teachers' gestures or the medium used to highlight aspects of the linked representations (hand gestures or digital icons). We will assess students' conceptual and procedural knowledge of linear equations before and after the lessons, so that we can evaluate how variations in teachers' visual scaffolding affect students' learning. We will also conduct a pilot study to prepare us to extend this line of inquiry to college students learning about statistics. This pilot study will investigate how teachers link representations using speech and gesture in instruction about confidence intervals. This work will contribute to our scientific understanding of learning and instruction from an embodied cognition perspective. By experimentally manipulating the ways in which relations between mathematical ideas are conveyed, and exploring the consequences for learning, we will gain a deeper understanding of the cognitive processes involved in acquiring mathematical understanding. This work will provide an empirical basis for recommendations about how teachers can use visual scaffolding effectively.