Measurement instruction has at least three distinct agendas. The first and most important is to help children understand what measurement is and how to go about doing it. This includes an understanding of the attribute that is being measured, how units are used in measuring, and an understanding of how measuring tools work. A second agenda is very much related to the first. It can be termed measurement sense; a familiarity of the most commonly used standard units, an ability to estimate with commonly used measures, and some flexibility with the use of related units. The third agenda is different from the first two: the development and use of a few standard formulas. Generally in the K-8 curriculum, these are area and volume formulas for common shapes.
A General Measurement Scheme
Although it may be over simplified, I have tried to present a common scheme or a general plan of instruction that can guide the development of most instructional units on measurement. This scheme is outlined in Table 20.1, p. 376 and in an abbreviated form on Transparency T-111. With an understanding of this scheme, teachers should be able to take stock of their students' current understanding relative to the objectives at hand and plan a developmentally appropriate unit of instruction. The idea is relatively simple. Students need to have clearly in mind what attribute they are measuring, how units are used to measure that attribute (including what units for that attribute are), and, when appropriate, an understanding of traditional tools for measuring. Corresponding to these three agendas are three types of activities: students can make comparisons, use actual units as directly as possible, and construct measuring tools using actual units of measurement to do so.
Within the activities for this scheme there is considerable flexibility and need for decision-making. When, why, and for how long should informal units be used? How quickly do you progress? What is the role of estimation and how can that be built into the program? My goal when teaching this chapter is to help teachers appreciate this general framework and gain some perspective concerning these decisions. Helping teachers see this general plan is an efficient method of getting at all measurement in school. You will never be able to discuss the measurement of every attribute that is measured in the elementary school.
Though estimation is discussed explicitly in at least two different sections of the chapter (an outlining of the value of estimation and a description of specific estimation techniques), estimation should never be a separate issue in instruction. Transparency T-112 is a listing of the reasons and values for including estimation in measurement. Be sure that you include estimation in some form in nearly every activity you do with teachers and encourage them to do likewise with students. Estimation in measurement is probably the most important thing that we can do to enhance a student's measurement sense. Estimation of a measure is a highly reflective activity that avoids the procedural aspects of using rulers and scales and getting answers. It helps develop benchmarks for units and enhances familiarity with important units. It develops flexible thought patterns for each measurement area.
Most of us have a great deal more skill with length estimation, of both short and long distances, than we do with weight or capacity estimates. I suspect that we simply have done more length estimation in our everyday lives. The effect is that we have a much better measurement sense for length than most any other area. This fact seems like a strong argument for estimation.
I do not spend a lot of time on this topic other than to downplay it. One of the mistakes that was made in attempting to convert the US to the metric system was to try to get people to learn it in its entirety instead of developing some initial familiarity with the most commonly used units and let the rest come as required. Therefore, my message to teachers is to focus on the most important units and let students experience them directly, develop personal reference points or benchmarks for them, and do lots of estimation using them. Some unit conversions or inter-relationships among units may be important even in everyday life (It's nice to know 12 inches make a foot and 3 of those make a yard.) but a focus on conversions quickly distracts from every other agenda related to measurement.
Time, Clock Reading, and Elapsed Time
The short section on measuring time can probably be read by your students without class discussion. There is a brief discussion of duration and actually measuring short intervals of time in the same manner as with other attributes. Clock reading is not really measurement but the skill of reading a dial with two hands. The text suggests an alternative to the traditional sequence of teaching time first to the hour, then the half hour, and so on. Instead, approximate clock reading for all times on the clock can be developed from the outset with a focus on the hour hand. The minute hand comes last as a refinement of the hour hand to tell more precisely how much after the hour it may be. The approach is not new or novel. At the same time (no pun intended) it is not yet in standard textbooks for children. However, it does work and offers teachers a conceptual approach. Finally, you will find a discussion of elapsed time in which a suggestion for use of an empty number line is made to keep track of intervals and to make the passage of noontime more obvious.
One message that I want to get across is that children should never use formulas without participating in the development of those formulas. For the purposes of school mathematics, every formula that needs to be known can at least be understood if not actually discovered. To have students use a formula that is mysterious is antithetical to the spirit of this book. Furthermore, plugging numbers mindlessly into formulas is not really measurement but usually only third-grade arithmetic. Perhaps teachers should stop and think about that for a bit.
A second idea concerning formulas is that they do represent an opportunity to engage children in some nice mathematics. This is a great place to help students see that when mathematical ideas are c related one to another, they are relatively easy to understand. In this section you will find a coordinated development of area and volume formulas that I do not believe is presented in other texts. Formulas for parallelograms (beginning with rectangles), triangles, trapezoids, circles, cylinders (including the special case of prisms), and cones (including the special case of pyramids) are all related to a single concept of the product of base times height. This sequence of formula development is a nice example of how conceptual big ideas connect mathematics.