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Review of Literature This report was produced under contract to the Ministry of Education, Contract No. 323-1642 by Andrew Tagg with the help of Derek Holton and Gill Thomas


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Measurement

Theory


Lehrer (2003) describes the conceptual development of measurement as “change in a network or web of ideas about measurement”. He lists eight components that provide a basis for this network (p. 181):


  1. Unit-attribute relations: What units can/should be used?

  2. Iteration: Subdivision into congruent parts or repetition of a unit.

  3. Tiling: Gaps must not be left between the units.

  4. Identical units: If units are identical a count represents the measure.

  5. Standardisation: Using a standard unit makes communication of measures easier (possible).

  6. Proportionality: Different units can be used to measure and can be compared.

  7. Additivity: A line segment can be divided into several smaller line segments whose sum will equal the original length.

  8. Origin: When using a scale to measure it is important to identify the zero point, although this can be any point on the scale.

These eight components are seen as important in the development of understanding of all measurement attributes (length/area/volume/mass/angle/time). The development of a conceptual grasp of measures of the different attributes is neither simultaneous nor sequential in a linear way, but understanding of the eight components can be extended from one attribute to another. For example, a student who understands that when measuring a length using hand spans it is important not to leave gaps, is more likely to understand that measuring an area using tiles must also leave no gaps.


Lehrer et al. (2003) describe a similar set of ‘central concepts’, with two significant differences: the addition of the concept of precision – that all measurement is inherently approximate and the choice of units (based on the context) determines the level of precision; and the omission of standardization from the list.

Curriculum/Implementation

National Numeracy Strategy


The National Numeracy Strategy (DFEE, 1999) describes objectives for students from Reception to Year 6. Within the measures, shape and space strand, the objectives related to measurement can be broadly grouped into four categories: vocabulary and units; estimating and measuring; calculating perimeters and areas; and time. The objectives within each of these categories describe a clear progression of complexity from simplest to most complex. For example, the progression of objectives related to vocabulary and units is:
Reception: Use language such as more or less, longer or shorter, heavier or lighter…

Year 1: Understand and use the vocabulary related to length, mass and capacity.

Year 2: Use and begin to read the vocabulary related to length, mass and capacity.

Year 3: Read and begin to write the vocabulary related to length, mass and capacity. Begin to use decimal notation for metres and centimetres.

Year 4: Use, read and write standard metric and imperial units. Know and use the relationships between familiar units of length, mass and capacity. Begin to convert between smaller and larger units.

Year 5: Use, read and write standard metric units. Convert larger to smaller units. Know imperial units.

Year 6: Use, read and write standard metric units. Convert larger to smaller units. Know imperial units and their rough equivalents.

NCTM Principles and Standards


Principles and Standards for School Mathematics (NCTM, 2000) describes objectives for students from pre-kindergarten to grade 12. Within the measurement strand, the objectives are grouped into two categories:
Understand measurable attributes of objects and the units, systems, and processes of measurement; and

Apply appropriate techniques, tools, and formulas to determine measurements.


The objectives within each of these categories describe a clear progression of complexity from simplest to most complex.

Exemplars


The New Zealand Curriculum Exemplars (Ministry of Education, 2003) describe a progression in development of understanding of the concept of measurement as applied to length.
Level 1a: Direct comparison.

Level 1b: Indirect comparison.

Level 1c: Non-standard units.

Level 2: Standard units.

Level 3: Use Reasoned measurement.
While this sequence of stages is considered to be the same for length, area, volume and mass, the levels (of the New Zealand curriculum) are given for length comparisons, as students are likely to develop this understanding before the more complex attributes of area, volume and mass.

asTTle Curriculum Map


The development of the asTTle assessment tool included work carried out on the ‘mapping’ on Mathematics in the New Zealand Curriculum. The achievement objectives were grouped into categories and ordered by level, with several additional objectives added to fill perceived gaps in the curriculum. An attempt was made to identify progressions through the levels of MiNZC. The subcategories within measurement identified in the asTTle map were:

Count Me Into Measurement


The core of the Count Me Into Measurement program is the Learning Framework in Measurement, which aims to describe the stages students progress through in developing an understanding of measurement. The Learning Framework describes three key stages:
Identification of the attribute (direct comparison/partitioning/conservation);

Informal measurement (counting units/relating number of units to quantity/comparison of measurements); and

Unit structure (replicating a single unit/relating size of units to number required)

(Outhred et al., 2003, p. 85).


Students are perceived as passing through the same three stages in their understanding of each of length, area, volume/capacity and mass, though not at the same time, as increasing the number of dimensions measured leads to increasing complexity of concept.

TIMSS


The TIMSS (Trends in Mathematics and Science Study) 2003 framework is intended to describe “important content for students to have learned in mathematics and science” (Mullis et al., 2003, p. i). Within the Measurement strand the objectives are grouped into two categories:


  • Attributes and units

  • Tools, techniques, and formulas
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