STANDARD 9 - MEASUREMENT
Measurement helps describe our world using numbers. We use numbers to describe simple things like length, weight, and temperature, but also complex things such as pressure, speed, and brightness. An understanding of how we attach numbers to those phenomena, familiarity with common measurement units like inches, liters, and miles per hour, and a practical knowledge of measurement tools and techniques are critical for students' understanding of the world around them.
Meaning and Importance
Measurement is important because it helps us to quantify the world around us. Although it is perfectly natural to think about length, area, volume, temperature, and weight as attributes of objects that we measure, a little reflection will produce many other measurable quantities: speed, loudness, pressure, and brightness, to name just a few. An understanding of the processes of measurement, the concept of a unit, and a familiarity with the tools and common units of measurement, are all critical for students to develop an understanding of the world around them.
This standard is also, in many ways, the prototypical "integrated" standard because of its strong and essential ties to almost every one of the other content standards. Measurement is an ideal context for dealing with numbers and with numerical operations of all sorts and at all levels. Fractions and decimals appear very naturally in real-world measurement settings. Metric measures provide perhaps the most useful real-world model of a base-ten numeration system we can offer to children. Similarly geometry and measurement are almost impossible to consider separately. For instance, treatments of area and perimeter are called "measurement" topics in some curricula and "geometry" topics in others because they are, quite simply, measurements of geometric figures. Another of the content standards which is inextricably linked to measurement is estimation. Estimation of measures should be a focus of any work that students do with measurement. Indeed, the very concept that any continuous measurement is inexact - that it is at best an "estimate" - is a concept that must be developed throughout the grades.
Think about how many different content standards are incorporated into one simple measurement experience for middle school students: the measurement of a variety of circular objects in an attempt to explore the relationship between the diameter and circumference of a circle. Clearly involved are the measurement and geometry of the situation itself, but also evident are opportunities to deal with patterns in
the search for regularity of the relationship, estimation in the context of error in the measurements, and number sense and numerical operations in the meaning of the ratio that ultimately emerges.
K-12 Development and Emphases
Throughout their study and use of measurement, students should be confronted explicitly with the important concept of a measurement unit. Its understanding demands the active involvement of the learner; it is simply not possible to learn about measurement units without measuring things. The process of measurement can be thought of as matching or lining up a given unit, as many times as possible, with the object being measured. For instance, in its easiest form, think about lining up a series of popsicle sticks, end to end, to see how many it takes to cover the width of the teacher's desk; or consider how many pennies it takes to balance the weight of a small box of crayons on a pan balance. At a slightly more sophisticated level, multiple units and more standard units might be used to add precision to the answers. The desk might be measured with as many orange Cuisenaire Rods as will fit completely and then with as many centimeter cubes as will fit in the space remaining; the crayons, with as many ten-gram weights as can be used and then one-gram weights to get a better estimate of its weight. These types of activities - this active iteration of units - make the act of measurement and the relative sizes of units significantly more meaningful to children than simply reading a number from a measurement instrument like a meter stick or a postal scale. Of course, as the measures themselves become the focus of study, rather than the act of measurement or the use of actual physical units, students should become knowledgeable in the use of a variety of instruments and processes to quickly and accurately determine measurements.
Much research has dealt with the development of children's understanding of measurement concepts, and the general agreement in the findings points to a need for coherent sequencing of curriculum. Young children start by learning to identify the attributes of objects that are measurable and then progress to direct comparisons of those attributes among a collection of objects. They would suggest, for instance, that this stick is longer than that one or that the apple is heavier than the orange. Once direct comparisons can consistently be made, informal, non-standard units like pennies or "my foot" can be used to quantify how heavy or how long an object is. Following some experiences which illustrate the necessity of being able to replicate the measurements regardless of the measurer or the size of the measurer's foot, these non-standard units quickly give way to standard, well-defined units like inches and grams.
Older students should continue to develop their notions of measurement by delving more deeply into the process itself and by measuring more complex things. Dealing with various measurement instruments, they should consider questions concerning the inexact nature of their measures, and to adjust for, or account for, the inherent measurement error in their answers. Issues of the degree of precision should become more important in their activities and discussion. They need to appreciate that no matter how accurately they measure, more precision is always possible with smaller units and better instrumentation. Decisions about what level of precision is necessary for a given task should be discussed and made before the task is begun, and revised as the task unfolds.
Older students should also begin to develop procedures and formulas for determining the measures of attributes like area and volume that are not easily measured directly, and to develop indirect measurement techniques such as the use of similar triangles to determine the height of a flagpole. Their universe of measurable attributes should expand to include measures of a whole variety of physical phenomena (sound, light, pressure) and a consideration of rates as measures (pulse, speed, radioactivity).
Connections are another strong focus of students' work with measurement. The growth of technology in schools opens up a wide range of new possibilities for students of all ages. Inexpensive instruments attached to appropriately programmed graphing calculators and computers are capable of making and recording measurements of temperature, distance, sound and light intensity, and many other physical phenomena. The calculators and computers may then be used to graph those measurements with respect to time or any other measure, to present them in tabular form, or to manipulate them in other ways. These opportunities for scientific data collection and analysis using this technology are unlike any that have been available to mathematics and science teachers in the past and hold great promise for real-life investigations and for the integration of these two disciplines.
In summary, measurement offers us the challenge to actively and physically involve students in their learning as well as the opportunity to tie together seemingly diverse components of their mathematics curriculum like fractions and geometry. It is also one of the major vehicles by which we can bring the real worlds of other disciplines such as the natural and social sciences, health, and physical education into the mathematics classroom.
Note: Although each content standard is discussed in a separate chapter, it is not the intention that each be treated separately in the classroom. Indeed, as noted in the Introduction to this Framework, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences.
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