#### The Child's Ideas for *3. Investigating Standard Measures*

# The Challenges in Learning about Weight Measurement

Ultimately, learning to measure weight (and, later, volume) opens the door for discovering powerful regularities in the world.

Learning to measure weight draws on, and brings together, children's prior understandings of both weight and numbers. In the process, children's understanding of both can be transformed.

For example, the weight of an object goes from being *felt weight* (that is, an imprecise and subjective “quantity” that can be only crudely assessed) to a precisely measured and objective quantity such as 16 grams. Further, although children can begin to measure weight drawing on only their knowledge of counting numbers (i.e., the numbers 1, 2, 3, 4 … in the count list) and the operations of addition and subtraction, precise measurement ultimately calls for them to deal with fractional units and the operations of division. Thus, learning to measure is an important context for extending and changing children's concept of numbers.

Learning to measure weight involves much more than learning a simple set of measurement procedures—it also involves developing a deeper “theory of measure.” Because many of the assumptions underlying weight measurement are not obvious to children, these assumptions need to be explicitly discussed. The use of external visual representations—such as a weight line—are critical to making these assumptions public and open to discussion. The weight line integrates many elements of a theory of measure and allows students to use the line to make novel inferences.

For example, weight measurement involves identifying a unit of weight—such as one gram—and then determining how many grams it takes to balance with the object on a scale. In this way, the weight of an object is analyzed as the sum of the number of weight units; in contrast, with felt weight, there are no component units—weight is just assessed holistically. The weight line clearly represents this aspect of weight measurement by marking off “equal size” intervals that correspond to different numbers of units, and placing objects along the weight line.

Weight measurement also involves subdividing units into fractional units. As children begin to measure the weight of objects, they find that many objects have a weight somewhere “between” two weights (e.g., 10 and 11 grams). Asking children to place objects on a continuous weight line not only supports inferences about the existence of weights between two units, but also representing those weights and ordering their magnitudes visually. This in turn supports learning about fractional units, and treating them as magnitudes that can be ordered.

Weight measurement starts at 0 with increasing values along a common (continuous) dimension. In this way, measures are unlike counts, which start at 1, and everyday perceptual experience where the starting point is not clearly defined. Indeed, many small objects feel like they weigh “nothing at all,” and children use the word “nothing” quite loosely (rather than the mathematical sense of 0). In everyday language, children also contrast “heavy” and “light,” which may be initially conceived as opposite qualities, rather than the *same quality* (just in different amounts). Further, as children begin to map weight to number, many think that light objects may have negative weights, and confuse fractional values with negative numbers. Again, the weight line provides an anchoring context for discussing and sorting out these important issues. Where will the weight of successive halves of pieces of clay fall on the line? Will you ever get to 0?

Learning about measurement also involves developing an understanding of “measurement error”—the difference between the true value of a quantity and the value obtained from an actual measurement—as well as understanding that measuring instruments vary in sensitivity and that what measurement units are appropriate may vary for different problems. Thus, children need to be mindful about measuring—always actively thinking about what the sources of error may be, what level of precision is important for a certain kind of problem, and how to make measurements more accurately. For example, children may put a tiny piece of clay on the balance, which doesn't make the balance move. Does that mean that the piece doesn't weigh anything at all, or that the scale isn't sensitive enough to detect its weight? Students might get different values when they measure the weight of the same object at different times. Does that mean the weight of the object has really changed, or does it stem from some measurement error? If the latter, what are the sources of measurement error? Are there small random fluctuations, or some other more systematic variation, such as failure to properly calibrate the balance or putting one's thumb on one side? Without an understanding of measurement error, students may mistakenly “trust” a measurement that is obviously wrong (because it is an outlier from others). Alternatively, they might regard every difference in readings as real, instead of noticing the differences are small and in no particular direction, clustering around a central value that is likely to be the true value.

Ultimately, learning to measure weight (and, later, volume) opens the door for discovering powerful regularities in the world. Without measurement, how could children tell that the weight of an ice cube equals the weight of the water produced when it melted, or that the weight of an object made of a given material is directly proportional to its volume and that a given material can be characterized by a (constant) density?

—Carol L. Smith