Spatial Structuring as a Coordination of Mental Models and Schemes for Measuring Perimeter and Path Length:
Second-Grade Children's Accounts of Linear Quantity
Jeffrey E. Barrett
Illinois State University
Children’s ways of measuring length and perimeter for polygons are often thought to depend on their experiences with the appropriate tools, and their ability to estimate length. My interest is to identify obstacles and supportive structures from a psycho-cognitive viewpoint. This report emphasizes the need for coordination of collections of unitary line segments beyond the 1D world of linearity, emphasizing the 2D aspect of most perimeter tasks. Part of a study of 4 second-grade children’s understanding and practices for measuring length and perimeter, this report examines the role of spatial structuring as Reba worked to coordinate units of length along sides of polygons, and around their perimeter.
The cognitive constructions students make in measuring perimeter for polygons, or path length in a plane have been described by researchers in terms of one-dimensional, one-directional constructs (Cannon, 1992; Clements, 1997; Hiebert, 1981). In contrast, other researchers have attended to spatial structuring as a theoretical construct allowing analysis of students' work in more than one dimension, emphasizing the importance of coordination of units without strict boundaries on the properties of dimensionality ( Battista & Clements, 1996; Battista, Clements, Arnoff, Battista, & Van Auken Borrow, 1998). In this report, I employ a theory of spatial structuring as a way of describing children's evolving strategies for measuring paths and finding perimeter for polygons like triangles and rectangles during task-based interviews. This report focuses on children’s ways of coordinating length units to relate 1D quantity along a linear object to its 2D context as a polygon in a plane. By accounting for children's strategy development, this report examines the psychological aspects of learning to measure length, informing task development (Tzur, 1999).
The Principles and Standards for School Mathematics (NCTM, 2000) calls for increased attention to measurement concepts and practices by including measurement as one of the five content standards in the K-12 curriculum. If teachers can learn what kinds of trajectories, and what kind of cognitive stumbling blocks to expect as children learn to measure length and perimeter, then they may develop instructional practices that promote increasingly abstract levels of understanding for length measurement. Much of the current research on length identifies the problem of point counting as an imagistic stumbling block, but fails to explain the cognitive aspects of this struggle. Point counters report the number of subdivision marks rather than the number of intervals.
Specifically, this study will describe children's ways of organizing collections of line segments as iterative sequences of length units for enumerating perimeter as a way of explaining the phenomenon of point counting for children measuring perimeter. The most critical mental process for organizing collections of unitary segments along a path is spatial structuring. Constructivist research on children's learning of geometry emphasizes spatial structuring as a fundamental mental process whereby children come to connect and organize collections of objects taken to be related in a whole; by synthesizing collections of related parts, one may establish an ordered sequence to be used for storing and retrieving images of that whole. For example, Battista and Clements (1996) explained children's ways of understanding and enumerating 3-D cube arrays by describing children’s use of terms like 'layers' or 'rows', or 'stacks' to identify relevant spatial structures. They emphasized the connections children used as they composed parts of cube arrays into a whole, attending to their capacity for coordinating various sub-collections, or their failures to integrate several viewpoints as they were counting up the unitary cubes. These researchers argued that such structuring implies a complex interaction between numerical and spatial structuring (1996, p 291). The present report extends this perspective to examine student's ways of relating units of length through iterative sequences in one and two dimensions.
This study examines patterns and reasoning habits exhibited by second-grade students as they attempted to find several cases of triangles or rectangles having a perimeter of exactly 24. I examined the generality of the four-part account of length knowledge and strategies supported by earlier research with fourth-graders (Barrett, 1998). I examined whether this would provide adequate discrimination among strategies exhibited by second-grade level students. There are four observed length strategies:
§ Level 1-Partially structured efforts to assign number for length,
§ Level 2a-Enumerating single rows by making hash marks,
§ level 2b-Enumerating sequences of rows by making hash marks,
§ level 3-Establishing quantity by applying imagistic nested units of units to coordinate enumeration for collections of parts of paths (Barrett, 1998; Clements, 1997)
I interviewed ten children individually in videotaped, transcribed sessions of approximately 40 minutes duration. I followed Goldin's (1999) method for task-based structured interviewing, using a task from previous studies with modifications to emphasize point counting as a potential cognitive obstacle for enumerating perimeter (Cannon, 1992).
Figure 1. Side View of a Plastic Straw, as Used in Interviews.
As an example of the data suggesting the importance of spatial structuring for measuring perimeter, consider Reba, a student who was moving from a strategy described as Level 1:Partially-Structured Collections of Segments into a more abstracted strategy described as Level 2a: Enumerating Single Rows of Countable Items Along Segments. Reba described length along a decreasing sequence of straw sections (I asked her to show me first a length of three, then two, one, and finally zero units) by explaining that the countable items along the straw must begin and end exactly, “in the middle” of the notches. Her way of structuring the observations of the straw eventually lead her to describe a segment of length one as having a length of one and a half. Reba tried to be consistent in identifying countable items by calling a unit the distance between one middle and the next:
I: Why don’t you show me 3 pieces on your straw?
REBA: [bends the straw at the 3rd notch]
I: Okay, now show me 2 pieces.
REBA: [bends the straw at the 2nd notch]
I: Okay, that’s 2 pieces, now show me one.
REBA: [bends the straw in between the 1st and 2nd notch]
I: Now show me none.
REBA: [looks up and puts the straw down, indicating none.]
….
I: So how about if you were going to show me 2 [again]?
REBA: I would probably...fold it in the middle, [takes the straw and folds it in between the second and third notch] like this.
Reba did not impose a consistent part-whole structuring to partition the straw, even when a consistent partition was visible with the notches. Unlike students who simply counted notches (common point counting), Reba counted the center points between each pair of notches. Nonetheless, Reba tried to enumerate length by counting collections of specific points along the object (center points between notches), and thereby failed to isolate unit objects with length. Point counting appears to be a mismatch of dimensions, a counting of points that do not have dimensionality to measure a linear path with dimension one. It is interesting that Reba struggled to enumerate perimeter around this straw based on straw units, but she was later able to impose inches as interiorized units. This suggests children's strategies for imposing units may be interiorized for a specialized context, especially those based on standard units, yet still lack generality.
Like Reba, other students exhibiting strategy Level 2a often failed to make use of rows of segments as a composite unit. Many of the second-grade children in this study did not represent parts of collections of length units in relation to the other parts at the internal, representational level of figural-imagistic processing (Steffe & Cobb, 1988). They were not yet able to visually iterate a single unit, or disembed the unitary structure that is implicit in the notched intervals along the straw. For example, for Reba the zero point was not at the position along the straw that she expected. Thus, she struggled to find a way of identifying exactly one unit. This corroborates the claim that without the spatial structuring process of iterating a unitary piece along the straw, one is left to guess at the meaning of the sub-dividing hash marks, the notches along the straw. Students ways of structuring sequences of unit-length parts to coordinate along various sides of polygons affects their ability to enumerate length along the entire perimeter (cf: Battista et al., 1998). Further research is needed to explain students' ways of appropriating verbal narratives, their own drawings, and diagrams in measures of perimeter and path length.
References
Barrett, J. E. (1998). Representing, connecting and restructuring knowledge: The growth of children's understanding of length in two-dimensional space. Unpublished Dissertation, Graduate School of Education, State University of New York at Buffalo.
Battista, M. T., & Clements, D. H. (1996). Students' understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27, 258-292.
Battista, M. T., Clements, D. H., Arnoff, J., Battista, K., & Van Auken Borrow, C. (1998). Students' spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29(5), 503-532.
Cannon, P. L. (1992). Middle grade students’ representations of linear units. In W. Geeslin & K. Graham (Eds.), Proceedings of the Sixteenth PME Conference (Vol. I) (pp. 105-112). Durham, NH: Program Committee of the 16th PME Conference.
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Goldin, G. A. (1999). A Scientific Perspective on Structured, Task-Based Interviews in Mathematics Education Research. In R. Lesh & E. Kelly (Eds.), Handbook of research design in mathematics and science education . Mahway, New Jersey: Lawrence Erlbaum Associates.
Hiebert, J. (1981). Cognitive development and learning linear measurement. Journal for Research in Mathematics Education, 12(3), 197-211.
National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics (Draft). Reston, VA: NCTM.
Steffe, L. P., & Cobb, P. (1988). Construction of Arithmetical Meanings and Strategies. New York: Springer-Verlag.
Tzur, R. (1999). An integrated study of children's construction of improper fractions and the teacher's role in promoting that learning. Journal for Research in Mathematics Education, 30(4), 390-416.