All students will develop their understanding of measurement and systems of measurement through experiences which enable them to use a variety of techniques, tools, and units of measurement to describe and analyze quantifiable phenomena. |
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 numerical operations of all sorts and at all levels. Fractions and decimals especially appear very naturally in real-world measurement settings. In fact, metric measures provide perhaps the most useful real-world model of a base-ten numeration system we can offer to children. Geometry and measurement are similarly almost impossible to think about separately. Very similar treatments of area and perimeter, for instance, are called "measurement" topics in some curricula and "geometry" topics in others because they are, quite simply, measurements of geometric figures. Yet another of the content standards which is inextricably linked to measurement is estimation. Estimation of measures should be one of the focuses of any work that children do with measurement. Indeed, the very concept that any measurement is inexact -- 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 grades 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 operations in the meaning of the ratio that ultimately presents itself.
Much research has been done into the development of children's understanding of measurement concepts and the general agreement in the findings leads to a coherent sequence in 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 pointing out 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 grams and inches.
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 be asked to confront 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.
Students should also begin to develop procedures and formulas for determining the measures of attributes like area and volume that are not easily directly measured, and also to develop indirect measurement techniques such as the use of similar triangles to determine the height of a flagpole. Their universe of measureable attributes expands to include measures of a whole variety of physical phenomena (sound, light, pressure) and a consideration of rates as measures (pulse, speed, radioactivity).
The growth of technology in the classroom also opens up a wide range of new possibilities for students of all ages. Inexpensive instruments that attach to 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, when programmed with simple software, are then capable of graphing those measurements over time, presenting them in tabular form, or manipulating them in other ways. These opportunities for scientific data collection and analysis are unlike any that have been available to math and science teachers in the past and hold great promise for some true integration of the two disciplines.
IN SUMMARY, measurement offers us the challenge to actively and physically involve children in their learning as well as the opportunity to tie together seemingly diverse components of their mathematics curriculum like fractions and geometry. It also serves as one of the major vehicles by which we can bring the real worlds of the natural and social sciences, health, and physical education into the mathematics classroom.