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Appendix B Concepts of Measurement At least eight concepts form the foundation of children’s understanding of length measurement. These concepts include understanding of the attri- bute, conservation, transitivity, equal partitioning, iteration of a standard unit, accumulation of distance, origin, and relation to number. Understanding of the attribute of length includes understanding that lengths span fixed distances (“Euclidean” rather than “topological” concep- tions in the Piagetian formulation). Conservation of length includes understanding that lengths span fixed distances and the understanding that as an object is moved, its length does not change. For example, if children are shown two equal length rods aligned, they usually agree that they are the same length. If one is moved to project beyond the other, children 4½ to 6 years often state that the projecting rod is longer (at either end; some maintain, “both are longer”; the literature is replete with different interpretations of these data, but certainly children’s notion of “length” is not mathematically accurate). At 5 to 7 years, many children hesitate or vacillate; beyond that, they quickly answer correctly. Conservation of length develops as the child learns to measure (Inhelder, Sinclair, and Bovet, 1974). Transitivity is the understanding that if the length of object X is equal to (or greater/less than) the length of object Y and object Y is the same length as (or greater/less than) object Z, then object X is the same length as (or greater/less than) object Z. A child with this understanding can use an object as a referent by which to compare the heights or lengths of other objects. Equal partitioning is the mental activity of slicing up an object into the 359

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360 MATHEMATICS LEARNING IN EARLY CHILDHOOD same-sized units. This idea is not obvious to children. It involves mentally seeing the object as something that can be partitioned (or cut up) before even physically measuring. Asking children what the hash marks on a ruler mean can reveal how they understand partitioning of length (Clements and Barrett, 1996; Lehrer, 2003). Some children, for instance, may understand “five” as a hash mark, not as a space that is cut into five equal-sized units. As children come to understand that units can also be partitioned, they come to grips with the idea that length is continuous (e.g., any unit can itself be further partitioned). Units and unit iteration. Unit iteration requires the ability to think of the length of a small unit, such as a block as part of the length of the object being measured, and to place the smaller block repeatedly along the length of the larger object (Kamii and Clark, 1997; Steffe, 1991), tiling the length without gaps or overlaps, and counting these iterations. Such tiling, or space filling, is implied by partitioning, but that is not well established for young children, who also must see the need for equal partitioning and thus the use of identical units. Accumulation of distance and additivity. Accumulation of distance is the understanding that as one iterates a unit along the length of an object and count the iteration, the number words signify the space covered by all units counted up to that point (Petitto, 1990). Piaget, Inhelder, and ­ Szeminska (1960) characterized children’s measuring activity as an accumulation of distance when the result of iterating forms nesting relationships to each other. That is, the space covered by three units is nested in or contained in the space covered by four units. Additivity is the related notion that length can be decomposed and composed, so that the total distance between two points is equivalent to the sum of the distances of any arbitrary set of seg- ments that subdivide the line segment connecting those points. This is, of course, closely related to the same concepts in composition in arithmetic, with the added complexities of the continuous nature of measurement. Origin is the notion that any point on a ratio scale can be used as the origin. Young children often begin a measurement with “1” instead of zero. Because measures of Euclidean space are invariant under translation (the distance between 45 and 50 is the same as that between 100 and 105), any point can serve as the origin. Relation between number and measurement. Children must reorganize their understanding of the items they are counting to measure continuous units. They make measurement judgments based on counting ideas, often based on experiences counting discrete objects. For example, Inhelder, Sinclair, and Bovet (1974) showed children two rows of matches, in which the rows were the same length but each row was comprised of a different number of matches as shown in Figure B-1. Although, from the adult per- spective, the lengths of the rows are the same, many children argued that

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APPENDIX B 361 FIGURE B-1  Relationship between number and measurement. the row with 6 matches was longer because it had more matches. Thus, in measurement, there are situations that differ from the discrete cardinal situations. For example, when measuring withB-1 Figure a ruler, the order-irrelevance principle does not apply and every element (e.g., each unit on a ruler) R01420 should not necessarily be counted (Fuson and Hall, 1982). bitmapped fixed image Concepts of Area Measurement Understanding of area measurement involves learning and coordinat- ing many ideas (Clements and Stephan, 2004). Most of these ideas, such as transitivity, the relation between number and measurement, and unit itera- tion, operate in area measurement in a manner similar to length measure- ment. Two additional foundational concepts will be briefly described. Understanding of the attribute of area involves giving a quantitative meaning to the amount of bounded two-dimensional surfaces. Equal partitioning is the mental act of cutting two-dimensional space into parts, with equal partitioning requiring parts of equal area (usually congruent). Spatial structuring. Children need to structure an array to understand area as truly two-dimensional. Spatial structuring is the mental operation of constructing an organization or form for an object or set of objects in space, a form of abstraction, the process of selecting, coordinating, unify- ing, and registering in memory a set of mental objects and actions. Based on Piaget and Inhelder’s (1967) original formulation of coordinating dimen- sions, spatial structuring takes previously abstracted items as content and integrates them to form new structures. It creates stable patterns of mental actions that an individual uses to link sensory experiences, rather than the sensory input of the experiences themselves. Such spatial structuring pre- cedes meaningful mathematical use of the structures, such as determining area or volume (Battista and Clements, 1996; Battista et al., 1998; Outhred and Mitchelmore, 1992). That is, children can be taught to multiply linear dimensions, but conceptual development demands this build on multiplica- tive thinking, which can develop first based on, for example, their thinking about a number of square units in a row times the number of rows (Nunes, Light, and Mason, 1993; note that children were less successful using rulers than square tiles).

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362 MATHEMATICS LEARNING IN EARLY CHILDHOOD REFERENCES Battista, M.T., and Clements, D.H. (1996). Students’ understanding of three-dimensional rect- angular arrays of cubes. Journal for Research in Mathematics Education, 27, 258-292. Battista, M.T., Clements, D.H., Arnoff, J., Battista, K., and Borrow, C.V.A. (1998). Students’ spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Educa- tion, 29, 503‑532. Clements, D.H., and Barrett, J. (1996). Representing, connecting and restructuring knowledge: A micro-genetic analysis of a child’s learning in an open-ended task involving perimeter, paths and polygons. In E. Jakubowski, D. Watkins, and H. Biske (Eds.), Proceedings of the 18th Annual Meeting of the North America Chapter of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 211-216). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Clements, D.H., and Stephan, M. (2004). Measurement in preK-2 mathematics. In D.H. C ­ lements, J. Sarama, and A.-M. DiBiase (Eds.), Engaging Young Children in Mathemat- ics: Standards for Early Childhood Mathematics Education (pp. 299-317). Mahwah, NJ: Erlbaum. Fuson, K.C., and Hall, J.W. (1982). The acquisition of early number word meanings: A con- ceptual analysis and review. In H.P. Ginsburg (Ed.), Children’s Mathematical Thinking (pp. 49-107). New York: Academic Press. Inhelder, B., Sinclair, H., and Bovet, M. (1974). Learning and the Development of Cognition. Cambridge, MA: Harvard University Press. Kamii, C., and Clark, F.B. (1997). Measurement of length: The need for a better approach to teaching. School Science and Mathematics, 97, 116-121. Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W.G. M ­ artin, and D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics (pp. 179-192). Reston, VA: National Council of Teachers of Mathematics. Nunes, T., Light, P., and Mason, J.H. (1993). Tools for thought: The measurement of length and area. Learning and Instruction, 3, 39-54. Outhred, L.N., and Mitchelmore, M.C. (1992). Representation of area: A pictorial perspec- tive. In W. Geeslin and K. Graham (Eds.), Proceedings of the Sixteenth Psychology in Mathematics Education Conference (vol. II, pp. 194-201). Durham, NH: Program Com- mittee of the Sixteenth Psychology in Mathematics Education Conference. Petitto, A.L. (1990). Development of number line and measurement concepts. Cognition and Instruction, 7, 55-78. Piaget, J., and Inhelder, B. (1967). The Child’s Conception of Space. (F.J. Langdon and J.L. Lunzer, Trans.). New York: W.W. Norton. Piaget, J., Inhelder, B., and Szeminska, A. (1960). The Child’s Conception of Geometry. Lon- don, England: Routledge and Kegan Paul. Steffe, L.P. (1991). Operations that generate quantity. Learning and Individual Differences, 3, 61-82.