Why bother with informal units?
When informal or non-standard units such as hand spans, paperclips or popsticks are used to measure a length, the units have to be either aligned along the length, or one unit has to be repeated and the endpoint of each length marked in some way. However, when formal units are used to measure length, the measurement can usually be read from a scale on a ruler or tape, which shows units of a particular size. If students are not shown the relationship between the informal and formal measurement procedures, they may not understand the principle underlying the use of a ruler. Similarly, measuring areas and volumes with informal units assists students to understand the calculation formulae when these are taught, providing the principles underlying the informal and formal processes are understood.
Students cannot successfully estimate, compare and measure if they have inadequate understandings of what they are measuring and the structure of units employed in measuring. We want students to understand what it is that they are measuring, to choose and use appropriate units to compare quantities and, to use measuring instruments in ways that assist in using the structure of units to determine calculated quantities.
Students can develop an initial appreciation of the attribute (length, area, volume) being compared through an explicit focus on the attribute when directly comparing quantities. They also need to know that the quantity is unchanged if it is rearranged (conservation). Students learn to recreate a copy of the object using multiple units of the same size when measuring, before developing the capacity to create tools, such as rulers, through iterating a single unit to form a representation of quantified multiple units.
A key understanding of the measurement process is the repetition (or iteration) of units. Unit iteration involves knowledge of repeatedly placing identical tightly packing units so that there are no overlaps or gaps. For example, accurately aligning units along a length, constructing an array of units to measure the area of a rectangle, or packing a container to determine its volume.
The length of the line is five paper clips
Six units in each row and three rows result in an area of 6 x 3 =18
Another key understanding is that the units must be equal in size. In the first year of a three-year longitudinal study of children initially in years 1 to 3, Lehrer, Jenkins and Osana (1998) found that over 80% of the children saw no problem with mixing two different-length paper clips. Over time, however, 80% of children in grades 4 and 5 said that the units needed to be the same. Unless students understand length measurement they will not have the basis for developing area and volume concepts.
As well as recognising the need for units to be equal in size, students need to recognise the relationship between number and units of measure. When a length is measured in two ways, using different but equal sized units (as below), some students consider that the greater number of units represents a greater length (Inhelder, Sinclair, & Bovet, 1974).
Equal length shown with different numbers of matchsticks.
Individual units can be combined into composite units, such as a row of tiles or a layer of blocks. Experience with composite units will help students to learn that fewer large units are needed to measure a quantity, but smaller units give more precision. Composite units can be used in calculations and linked to early multiplication, e.g., 4 rows of 3 squares. Accounting for parts of units will extend students’ knowledge of fractions of continuous quantities.