Cognitive Foundations for a Multiplicative Structures Curriculum

中年级的乘法结构课程，包括乘法、除法、比率、比例推理等，被普遍认为是课程难度大、成绩差的根源。这些研究人员将进行为期一年的密集教学实验，涉及8个班级、5个实验者和3个对照，旨在建立一个强大的数据库，并为新的、更有效的中高年级乘法结构课程奠定理论基础。四个六年级实验班和一个能力水平低于平均水平的八年级班将使用专门设计的软件和材料进行教学，这些班级的学生一再未能学习主题。控制将使用传统的文本材料进行教学。一年中积累的概念变化的全班书面测量将与对每一班实验者和对照的具有代表性的子样本的为期一年的临床研究相叠加。这项研究将利用一系列精心设计的软件环境，包括由ETC开发的“从具体到抽象的软件坡道”。此渐变的较低级别在基于对象的计算环境中具有具体的激活乘法、除法和比率，而较高级别包括链接到数据表和坐标图的动态变化的代数表示的比例。在整个过程中，该软件采用了积极链接的数学概念和程序的多种表示法，利用了学校负担得起的计算机新提供的表示能力。这项研究还将比较由同一教师授课的3个中等能力水平的班级，其中一个班级以教师为中心，仅使用一台计算机，第三个班级作为对照。其目的是确定这些类型的学习环境的表征能力是否以及如何在计算机稀缺的环境中得到有效利用，而交互能力需要计算机密集的实验室环境，因此需要更昂贵的实验室环境。哈佛大学和韦斯顿中学的教育技术中心共同承担了这个项目的费用。

The multiplicative structures curriculum in the middle grades, covering multiplication, division, rates, ratios, and proportional reasoning, is widely seen as the locus of serious curricular difficulty and underachievement. These investigators will conduct an intensive set of year-long teaching experiments involving 8 classes, 5 experimentals and 3 controls,intended to build a robust database and a theoretical foundation for a new, more effective multiplicative structures curriculum for the mid-upper elementary grades. Four experimental 6th grade classes and 1 below average ability level 8th grade class whose students have repeatedly failed to learn the topics will be taught using specially designed software and materials. Controls will be taught using conventional text materials. Whole-class written measures of conceptual change accumulated through the year will be overlaid with a year-long clinical study of a representative subsample of both experimentals and controls in each class. This research will utilize an elaborated sequence of software environments comprising a "concrete-to-abstract software ramp" developed at ETC. Lower levels of this ramp feature concretely enactive multiplication, division and ratios in object-based calculation environments, whereas upper levels include dynamically varying algebraically represented proportions linked to tables of data and coordinate graphs. Throughout, the software employs actively linked multiple representations of mathematical concepts and procedures, exploiting the representational power newly available in school affordable computers. The study will also compare 3 mid ability level classes taught by the same teacher, one of which uses it in the teacher-centered, one-computer mode only, and a third acts as control. The intent is to determine if and how the representational power of these types of learning environments can be used effectively in computer-scarce contexts apart from its interactive power, which requires computer-dense, hence more expensive, lab contexts. The Education Technology Center at Harvard and Weston Middle School are sharing the costs of conducting this project.