This article describes how children build increasingly abstract knowledge of linear measurement, emphasizing ways they relate space and number. Assessments indicate children struggle to understand measurement, especially concepts related to complex paths as in perimeter tasks. This article draws on developmental accounts of children's knowledge of measurement to describe the coordination of cognitive processes as a progression through increasingly abstract layers of strategy (Clements, 2003;Lehrer, 2003) within a constructivist perspective (Steffe & Cobb, 1988;Steffe & Thompson, 2000). This article reports 4 case studies from a 6-month teaching experiment with Grade 4 students to examine ways of promoting their strategy development. Attending to children's understanding of length along straight objects alone proved inadequate in predicting and validating models of children's developing strategies and concepts for perimeter. Setting perimeter tasks that emphasized integral relations among unit of length, sides of polygons and perimeter, and prompting children to integrate partitive, iterative, and counting schemes appeared to promote increasingly abstract length measurement strategies. The researchers came to distinguish between a naïve use of units to find length (Level 2a) and a more abstract strategy for unitizing and iterating (Level 2b), extending an earlier framework (Clements, Battista, Sarama, Swaminathan, & McMillen, 1997). Finally, implications for instruction are drawn from the development of tasks and questioning sequences.We are interested in describing ways children build increasingly abstract knowledge of linear measurement. More broadly, we seek to understand how children learn to relate space and number through school-based, increasingly formal mathematical systems for measurement. This article describes children's learning as they engage in path length and perimeter tasks. As background, we begin by looking at assessments of performance and teaching and move into research on the learning of linear measurement.

We examined children's development of strategic and conceptual knowledge for linear measurement. We conducted teaching experiments with eight students in grades 2 and 3, based on our hypothetical learning trajectory for length to check its coherence and to strengthen the domain-specific model for learning and teaching. We checked the hierarchical structure of the trajectory by generating formative instructional task loops with each student and examining the consistency between our predictions and students' ways of reasoning. We found that attending to intervals as countable units was not an adequate instructional support for progress into the Consistent Length Measurer level; rather, students must integrate spaces, hash marks, and number labels on rulers all at once. The findings have implications for teaching measure-related topics, delineating a typical developmental transition from inconsistent to consistent counting strategies for length measuring. We present the revised trajectory and recommend steps to extend and validate the trajectory. Measurement of geometric space is an important aspect of the broad curriculum of mathematics in elementary education, yet research on teaching and learning measurement is less robust and less extensive than research on other domains such as number. Further, the research on learning measurement has not been thoroughly integrated with instructional design and pedagogical concerns. The goal of this study is to implement and improve a learning trajectory in the domain of measurement. We are particularly interested in the potential integration of developmental progressions that sequence increasingly sophisticated types of thinking in a domain with pedagogical progressions that list appropriate types of tasks and instructional activities suited to specific levels along that developmental progression. By integrating learning progressions and pedagogical progressions within the context of theoretical accounts of children's cognitive operations on mental images and concepts, we expect to forge more practical and productive learning trajectories. Such trajectories provide important infrastructure for improving assessment tools, professional development models, and curriculum design.

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Measurement is a critical component of mathematics education, but research on the learning and teaching of measurement is limited, especially compared to topics such as number and operations. To contribute to the establishment of a research base for instruction in measurement, we evaluated and refined a previously developed learning trajectory in early length measurement, focusing on the developmental progressions that provide cognitive

We examined ways of improving students' unit concepts across spatial measurement situations. We report data from our teaching experiment during a six-semester longitudinal study from grade 2 through grade 5. Data include instructional task sequences designed to help children (a) integrate multiple representations of unit, (b) coordinate and group units into higher-order units, and (c) recognize the arbitrary nature of unit in comparison contexts and student's responses to tasks. Our results suggest reflection on multiplicative relations among quantities prompted a more fully-developed unit concept. This research extends prior work addressing the growth of unit concepts in the contexts of length, area, and volume by demonstrating the viability of level-specific instructional actions as a means for promoting an informal theory of measurement.