Levels 1 to 3

The exemplars in this progression focus on the key mathematical concept of measurement. The study of measurement is important because of the frequency with which we make measurements in everyday life. Studying measurement also provides opportunities for students to learn about and apply other mathematical concepts, such as number operations and geometric ideas.

Measuring is fundamentally about making comparisons, beginning with directly comparing one object with another (in this case, a table with a doorway) and leading to comparing the attributes of a given object with a standard unit.

Use direct comparison

At this stage in the measuring progression, students are able to compare two objects directly by placing one next to the other. This method of comparison is valid but limiting because the students can compare the two objects only if one of them can be moved.

Use indirect comparison

At this stage in the measuring progression, students are able to compare two objects indirectly by using a third object (in this case, a piece of string or an arm span). The third object allows the students to state which of the two objects being compared is wider or longer but does not allow them to quantify the "size" of the difference.

Use repeated non-standard units of measurement

At this stage in the measuring progression, students are able to make repeated use of a non-standard unit to compare two objects. Non-standard units introduce most of the principles associated with measurement, namely, that:

• measurements are expressed by counting the total number of units used
• the unit must not change during the measurement activity
• the units of measure used are not absolute but are chosen for their appropriateness.

Use standard units of measurement

At this stage in the measuring progression, students are able to compare two objects directly by accurately using standard units of length. Students at this stage are familiar with the size of each unit, the correct language for naming each unit, and the conventions and appropriate symbols for writing each unit.

Use reasoned measurement

Students are now able to understand and use all the learning steps in the progression and to select the appropriate method of measurement for a given situation. This may not be the most sophisticated method that the students understand.

Background to the task
The measuring progression is illustrated in this set of exemplars by the specific example of a comparison of lengths. The teacher asks the students, "Will the table fit through the door?" The teacher tries to elicit each student's most sophisticated thinking about measurement.

Although this problem may seem trivial, its very simplicity gives the students freedom to explore and demonstrate a range of approaches to finding solutions. The exact dimensions of the table and the doorway are unimportant, but should be similar enough for the students to need some form of measurement to be sure that the table will fit through the doorway.

This problem challenges the students to compare two lengths (the width of the table and the width of the doorway) but could easily be altered to a comparison of masses, volumes, or areas.

Parallel tasks
Other possible tasks to explore in this progression include:

• Will this letter fit through the slot of the letterbox?
• Will the water from this container fit into this [other] container?
• Will three rolls of wallpaper cover this section of wall?
• Will 30 5-cent coins weigh the same amount as this blackboard duster?
• Will 20 A4 sheets of paper cover this door?

References

Department of Education (1985–1989). Beginning School Mathematics (BSM): Cycles 1-8. Wellington: School Publications.

Ministry of Education (1992). Beginning School Mathematics: Cycles 9-11. Wellington: Learning Media: Ministry of Education.

Ministry of Education (1993). Beginning School Mathematics: Cycle 12. Wellington: Learning Media, Ministry of Education.

Ministry of Education (2001). Figure It Out, Levels 3-4. Wellington: Learning Media.

Ministry of Education (1992). Mathematics in the New Zealand Curriculum. Wellington: Learning Media.

Ministry of Education (1996). Te Whāriki: He Whāriki Mātauranga mō ngā Mokopuna o Aotearoa/Early Childhood Curriculum. Wellington: Learning Media.