Quantifying Path Length: Fourth-Grade Children's Developing Abstractions for Linear Measurement

Abstract

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.

Curriculum Interventions for Area Measurement Instruction to Enhance Children’s Conceptual Understanding

“…According to Barrett and Clements (2003), mathematical tasks involving the following components are important to support children's strategies for abstracting length measurement: recognition of integral relationships among unit of length, sides of polygons, and perimeter measurement; and stimulation of the use of partitive, iterative, and counting strategies. Moyer (2001) proposed that conceptual understanding of the relationship between the perimeter and area of a rectangle depended on the ability to distinguish the attributes and explain the processes of the two measurements.…”

“…Moyer (2001) proposed that conceptual understanding of the relationship between the perimeter and area of a rectangle depended on the ability to distinguish the attributes and explain the processes of the two measurements. Barrett and Clements (2003) suggested that knowledge of 2-D geometry improves children's conceptualization of perimeter. We thus hypothesized that an instructional intervention integrating spatial geometry with area measurement in which children are guided to pay attention to the relationship between number and space (e.g.…”

Children’s Conceptions of Area Measurement and Their Strategies for Solving Area Measurement Problems

“…Furthermore, Barrett and Clements (2003) and Malloy (1999) argued that a better concept of area and perimeter grounded on acquisition of geometric concepts. Thus, it is argued that children's application of area formulas and their performance in identifying geometric shapes and in modifying their erroneous solutions of area measurement are related to the quality of understanding of area measurement.…”

Evaluating and Improving a Learning Trajectory for Linear Measurement in Elementary Grades 2 and 3: A Longitudinal Study

“…We describe this as Consistent Length Measurer (CLM) in Table 1. As children reflect on their own operations with collections of units to measure using ever-smaller units they make goal-directed decisions about ways to curtail their motion along objects (Barrett & Clements, 2003). Children gradually expect a collection of units to be conserved as they develop what Steffe (1991) called a conceptual ruler-the ability to mentally iterate units that are not perceptually available along the object-as early as grade 3.…”

“…Children gradually expect a collection of units to be conserved as they develop what Steffe (1991) called a conceptual ruler-the ability to mentally iterate units that are not perceptually available along the object-as early as grade 3. As children in grades 4 and 5 reflect on their activities with path length problems, they integrate motions, drawings, and verbal narratives to act on an internal image of a nested sequence of units (Barrett & Clements, 2003;Bragg & Outhred, 2004;Chiu, 1996;Fischbein, 1993;Outhred & Mitchelmore, 2004). Thus, more sophisticated levels may exist beyond Conceptual Ruler measurer (CR).…”