ABSTRACT The goal of this design-based research study was the creation and evaluation of a mini-unit intended to foster perceptually grounded understandings of the concept of slope in middle-school students. Central to this unit was an innovative device designed to create a productive pedagogical space between student intuition for steepness and formal definition of slope. Nine students from a US middle school engaged with the design in one-on-one video-recorded interviews. Analysis of the data indicates that a majority of students appropriated the mathematics of slope to support their intuitive judgments regarding steepness. Comparing the cases of three students suggests conceptual learning milestones. Overall, the findings show that students' successful coordination of steepness and slope demand that their analytical and intuitive constructions be surfaced and integrated multiple times throughout the learning experience. To support students in connecting the mathematics of slope to their perceptions of steepness, educators can plan for students' multiple ways of viewing situations involving slope.

中文翻译：

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中学生坡度认知中坡度数学基础的具体化设计

摘要 这项基于设计的研究的目标是创建和评估一个迷你单元，旨在培养中学生对坡度概念的感性理解。该单元的核心是一个创新设备，旨在在学生对陡度的直觉和坡度的正式定义之间创造一个富有成效的教学空间。来自美国一所中学的九名学生在一对一的视频采访中参与了设计。数据分析表明，大多数学生利用斜率数学来支持他们对陡度的直觉判断。比较三个学生的案例表明了概念性学习的里程碑。总体而言，调查结果表明，学生的 陡峭和坡度的成功协调要求他们的分析和直觉结构在整个学习过程中多次浮出水面和整合。为了支持学生将坡度的数学与他们对陡度的感知联系起来，教育工作者可以为学生规划涉及坡度的多种情况。