Development of Mathematical Concepts of Two-Dimensional Space in Grid Environments: An Exploratory Study

• Published in: Cognition and instruction 2003-01, Vol.21 (3), p.285-324
• Main Authors: Sarama, Julie, Clements, Douglas H., Swaminathan, Sudha, McMillen, Sue, González Gómez, Rosa M.
• Format: Article
• Language: English
• Subjects: Cognitive Development; Computers in education; Concept Formation; Conceptualization; Coordinate systems; Coordinates; Curricula; Elementary Education; Elementary School Mathematics; Elementary School Students; Field Tests; Geometric lines; Geometry; Grade 4; Grids; Knowledge Level; Learning Strategies; Mathematical Concepts; Mathematics Curriculum; Mathematics education; Posttests; Pretests; Rectangles; Scaffolding (Teaching Technique); Spatial Ability; Spatial Tasks
• ISSN: 0737-0008; 1532-690X
• DOI: 10.1207/S1532690XCI2103_03

We investigated the development of two-dimensional space concepts within a mathematics unit on grids, coordinates, and rectangles, part of a large-scale curriculum development project funded by the NSF. Data from case studies, interviews, paper-and-pencil tests, and whole-class observations were collected during field tests with 4 fourth-grade classrooms. Students had to overcome substantial hurdles in learning to spatially structure two-dimensional grids, including interpreting the grid's components as line segments rather than regions; appreciating the precision of location the lines required, rather than treating them as fuzzy boundaries or indicators of intervals; and learning to trace vertical or horizontal lines that were not axes. We found no evidence for a relationship between students' strategies for naming and locating coordinates and their other knowledge of grid and coordinate systems; instead, such knowledge was related to students' levels of competence in three aspects of more complex coordinate situations: number sense, spatial-geometric relationships, and the ability to discriminate and integrate the two numbers constituting a coordinate pair and the two axes constituting a coordinate plane. On this basis we describe elements of students' mental structuring of grid and coordinate systems as two-dimensional spaces, demarcated and measured with conceptual rulers. Relationships between students' conceptualizations and various representations, including real-world analogies, were complex, with the real-world contexts serving an important scaffolding role for most students at the early phases of learning, but impeding further mathematical abstraction in some students at later phases. Computer representations were significant in aiding learning, especially mathematical generalization and abstraction, in both grid and coordinate systems and in the learning of concepts of rectangles. Teacher and computer scaffolding also played a critical role in helping students integrate information from within these areas and broadly between their spatial and numeric schemes.