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决定学生是否能够理解重要的数学原理和过程。我们过去的研究表明，教师使用各种形式的视觉脚手架来连接不同的数学表示。这个项目的目的是了解教师视觉脚手架的变化是如何影响学生的学习的。我们特别关注教师在教学情节中产生的非语言支持，这些情节将数学信息的相关表征联系起来。特别是，我们研究了那些在物理环境或熟悉的行为、经历或表征中为思想奠定基础的非语言支持。该研究有三个目的：(1)调查如果教师用手势（而不是单独使用言语）来建立要联系的思想，是否会促进学生的学习；(2)研究某些类型的非语言支持是否对学习特别有益（具体而言，冗余手势与互补手势，指向手势与表征手势）；(3)检查手势是否提供了一种“特殊”的方式来视觉支撑思想，从某种意义上说，它们比其他非手势的视觉支撑方法更有效。我们将在中学生学习线性方程的实验中解决这些问题。实验将包括视频课程，这些视频课程将改变教师的手势或用于强调相关表征（手势或数字图标）方面的媒介。我们将在课前和课后评估学生对线性方程的概念性和程序性知识，以便我们可以评估教师视觉脚手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，准备将这一调查方式扩展到学习统计学的大学生。这项初步研究将调查教师如何在置信区间教学中使用语音和手势将表征联系起来。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。通过实验操作数学概念之间的关系传达方式，并探索学习的后果，我们将对获得数学理解所涉及的认知过程有更深入的了解。这项工作将为教师如何有效地使用视觉脚手架的建议提供经验基础。

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.