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.
Background: Fractions continue to pose a critical challenge for students and their teachers alike. Mathematics education research indicates that the challenge with fractions may stem from the limitations of part-whole concepts of fractions, which is the central focus of fractions curriculum and instruction in the USA. Students' development of more sophisticated concepts of fractions, beyond the part-whole concept, lays the groundwork for the later study of important mathematical topics, such as algebra, ratios, and proportions, which are foundational understandings for most STEM-related fields. In particular, the Common Core State Standards for Mathematics call for students to develop measurement concepts of fractions. In order to support such concepts, it is important to understand the underlying mental actions that undergird them so that teachers can design appropriate instructional opportunities. In this study, we propose a learning progression for the measurement concept of fractions-one that focuses on students' mental actions and informs instructional design. Results: A hierarchy of fraction schemes is charted outlining a progression from part-whole concepts to measurement concepts of fractions: (a) part-whole scheme (PWS), (b) measurement scheme for unit fractions (MSUF), (c) measurement scheme for proper fractions (MSPF), and (d) generalized measurement scheme for fractions (GMSF). These schemes describe concepts with explicit attention to the mental actions that undergird them. A synthesis of previous studies provides empirical evidence to support this learning progression. Conclusions: Evidence from the synthesis of a series of research studies suggests that children's measurement concept of fractions develops through several distinct developmental stages characterized by the construction of distinct schemes. The mental actions associated with these schemes provide a guide for teachers to design instructional opportunities for children to advance their construction of a measurement concept of fractions. Specifically, the collection of quantitative studies suggest that students need opportunities to engage in activities that support two kinds of coordinations-the coordination of partitioning and iterating, and the coordination of three levels of units inherent in fractions. Instructional implications are discussed with example tasks and activities designed to provoke these coordinations.
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