The Concept of Measurement

What does it mean to measure something? According to the National Council of Teachers of Mathematics (2000), "Measurement is the assignment of a numerical value to an attribute of an object, such as the length of a pencil. At more-sophisticated levels, measurement involves assigning a number to a characteristic of a situation, as is done by the consumer price index." An early understanding of measurement begins when children simply compare one object to another. Which object is longer? Which one is shorter? At the other extreme, researchers struggle to find ways to quantify their most elusive variables. The example of the consumer price index illustrates that abstract variables are, in fact, human constructions. A major part of scientific and social progress is the invention of new tools to measure newly constructed variables.

To be able to assign a numerical value to an attribute of an object, we must first be able to identify the attribute, and then we must have some kind of unit against which to compare that attribute. Most often we need a measurement tool that supplies us with our units. If our units are smaller than the attribute in question, then our measurement is in terms of numbers (quantities) of those units. On the other hand, if our units are larger than the attribute in question, then our measurement is in terms of parts (partitions) of the unit. Most often measurement includes a quantity of whole units along with a part of a unit. The fractional part of a unit determines the precision with which we measure. Greater precision results from smaller partitions.

The units that we use to measure are most often standard units, which means that they are universally available and are the same size for all who use them. Sometimes we measure using nonstandard units, which means that we are using units that we have invented and that are unknown outside our local context. Either standard or nonstandard units may be used in the classroom, depending on the teachers' immediate objectives.

Try This!

Without using any standard measurement tool, find a way to measure the length of and width of the room in which you spend most of your time. As you undertake this task, keep track of all the decisions you make along the way. Take some time to compare your list with your colleagues' lists. Does the number of decisions involved in this relatively simple task surprise you?

To illustrate one family's unusual use of nonstandard measurement units, the story of Elizabeth is interesting. When Elizabeth was born, her mother, an engineer, noted Elizabeth's length of 19 inches. She then made a Plexiglas measuring stick of that length and labeled it "One Elizabeth." As Elizabeth grew, on each of her birthdays her mom measured her in Elizabeth-units. By the time Elizabeth entered first grade she had more than doubled her original height. She looked forward to reaching the height of "three Elizabeths." When Elizabeth was in third grade her mom marked off the Plexiglas measuring stick into fractional Elizabeths. She began with quarters. Within a relatively short period of time, with Elizabeth's help, she divided these into eighths and eventually sixteenths. Now Elizabeth could keep track of her growth more precisely, because the sixteenths-of-an-Elizabeth were smaller milestones. Eventually Elizabeth became curious about how many feet and inches she was tall, and the family worked out a separate measuring system using feet and inches. In middle school, although Elizabeth no longer kept track of her height in Elizabeth-units, she worked out a conversion formula that allowed her to convert feet and inches to Elizabeth-units or Elizabeth-units to feet and inches. When Elizabeth's teacher heard this story she asked each of her students to go home and try to find out what their birth-length was. This teacher then replicated Elizabeth's experience as a classroom activity. She took it one step further and had each student do the same thing with their birth weight. Her question became, "How many 'me'-units do I weigh now?"

When the topic of measurement is approached using nonstandard units in this way, learners can develop a deep and meaningful understanding of measurement. It is particularly powerful to use nonstandard units that have special personal meaning, such as the Elizabeth-units. The process of construction of idiosyncratic, nonstandard units for measurement is yet one more case in which the powerful idea of composition is used. The units are composed by our abstract mental activity. Once we understand that measurement is simply a matter of comparing the object being measured with some composed unit, then we can advance in our use of various kinds of units. Standard units can be shown to be special units that are widely available and useful for communication among widely disparate parties. All measurement, whether measurement of length, weight, volume, time, temperature, area, or any other attribute, has these common characteristics.

Choosing the Best Measurement Unit

Try This!

Find out what your birth-length was, and use it to figure out how many "you"-units, including fractional "you"-units, you have become as an adult!

One school exercise that seems to receive more attention than it should is an exercise in which children must choose a correct (or best) measurement unit to use for a specific measurement task. For example, there may be an illustration of a swimming pool, and children are asked whether to measure its capacity using a gallon jug, a quart bottle, or a teaspoon. Surprisingly, for some children exercises like these are quite difficult. Each measurement tool would work, since each choice is a way to measure volume. The "correct" answer, the gallon, is considered correct because we typically hear about swimming pool capacities in terms of the number of gallons of water they contain. Furthermore, using the gallon jug would give us the answer most quickly and efficiently. However, due to their lack of experience with this sort of context, many children do not see that the unit of a gallon has any particular superiority over either of the other choices. Children find exercises of this type very difficult, while adults find the answers to this type of question obvious. This disparity should tell us that the skill involved here is a skill that is best learned through the accumulated experience of daily life.



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