A growing body of research implicates students' ability to coordinate multiple levels of numerical units as an important aspect of their mathematical development. In this paper, we consider relationships between the ways students coordinate units with whole numbers (their multiplicative concepts) and the ways students coordinate units with fractions (their fractions schemes). Interviews with 50 sixth-grade students suggest commonality in the number of levels of units-within-units structures that students construct for the two contexts, consistent with Steffe's (2001) reorganization hypothesis. The results suggest that fractions may be a fertile domain for elementary and middle grades students to continue to develop whole number understandings.
In considering mathematical development across multiple domains, researchers have implicated the critical role of an individual's ability to produce and coordinate units. Here, we describe a theoretically grounded instructional approach for promoting growth in units coordination. Our approach is informed by neuroscience, as well as existing research on units coordination. We present promising results from implementing our approach in a ten-week teaching experiment with a sixth-grade student named Cody. We demonstrate how, over the course of the 14 teaching sessions, Cody progressed from activity involving the coordination of two levels of units, to activity involving the coordination of three levels of units.
We describe the process of adjusting the balance between computerbased learning and peer interaction in a college algebra course. In our first experimental class, students used the adaptive-learning program ALEKS within an emporium-style format. Comparing student performance in the emporium format class with that in a traditional lecture format class, we found an improvement in procedural skills, but a weakness in the students' conceptual understanding of mathematical ideas. Consequently, we shifted to a blended format, cutting back on the number of ALEKS (procedural) topics and integrating activities that fostered student discourse about mathematics concepts. In our third iteration using ALEKS, we made use of ALEKS-generated data to design peer-to-peer activities that matched student progress.
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