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.
