Measurement comprises a set of understandings and skills that form a foundation for essential learning, especially in science and mathematics. These understandings and skills routinely come into play in everyday life and as preparation for further investigation of other topics—fractions, proportion, area and volume, and density to name a few. The importance of measurement is underscored by the array of math science standards for which it forms the basic underpinning. This is discussed at length as both narrative and cited examples in national and state mathematics and science education standards documents.

Measurement Understandings, Skills, and Conceptual Growth

Most of what we know regarding measurement—especially linear measurement, the most basic building block of conceptual understanding of measurement—is based either directly or indirectly on studies by Piaget and his colleagues. As of publication of NCTM's Principles and Standards for School Mathematics (NCTM, 2000), Piaget's definition of measurement as the synthesis of subdivision (i.e., understanding that space or an object's length can be "partitioned") and change of position (i.e., being able to partition a unit from an object and iterate that unit without gaps or overlaps) (Piaget & Inhelder, 1956; Piaget et al., 1960) remains intact. The term "synthesis" has special meaning, as it refers to the ability to coordinate understanding and application of the two and to furthermore do so with insight—not randomly, or by trial and error, or even as an act of accurate measurement when done only by formalized steps devoid of meaning and interpretation. The importance of this notion is further supported by the Principles and Standards for School Mathematics. The concepts of inclusion and conservation of length are implicit in this view of measurement—they form as well the most sound developmental continua of which we are aware, as both have been found to grow as a child learns to measure (Inhelder, Sinclair, & Bovet, 1974). The question of the influence of one on the other, and by default the order in which each is realized, is a matter of debate. Of significance, however, is that the close correlation of the concepts of inclusion and conservation of length with the act of measuring has resulted in our ability to categorize children's development of measurement concepts into several stages (Piaget et al., 1960; Carpenter, 1976; Copeland, 1974; and Sinclair, 1970). The following illustrates the distinctions between stages based on their progress in their understanding and ability to appropriately apply these concepts.

Carpenter, T. (1976). Analysis and synthesis on existing research on measurement. In R. Lesh & D. Bradford (Eds.), Number and measurement. [Papers from a research workshop]. (ERIC Document Reproduction Service No. ED 120027)

Copeland, R. (1974). How children learn mathematics: Teaching implications of Piaget's research (2nd ed.). New York: Macmillan.

Inhelder, B., Sinclair, H., & Bovet, M. (1974). Learning and the development of cognition. Cambridge, MA: Harvard University Press.

Piaget, J., & Inhelder, B. (1956). The child's conception of space. London: Routledge & Kegan Paul.

Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child's conception of geometry. New York: Basic Books.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Sinclair, H. (1970). Number and Measurement. In M. Rosskopf, L. Steffe, & S. Taback (Eds.), Piagetian cognitive-development research and mathematical education (pp. 135-149). Washington, DC: National Council of Teachers of Mathematics.