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Number & Operations for Teachers Copyright David & Cynthia Thomas, 2009 |
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Measurement Model A measurement
model for addition and subtraction of whole numbers may be based on the
number line. The measurement model for
addition and subtraction represents whole numbers as arrows of different
lengths. When two whole numbers are
added, their arrows point in the same direction (i.e., to the right). When one number is subtracted from another,
their arrows point in opposite directions.
Figure 2.4 shows concept models for the operations 2 + 3 = 5 and 5 -3 =
2. Note that the first term in either
addition or subtraction is represented as an arrow that begins at the origin
and points in the positive direction.
The second term is represented as another arrow arranged head-to-tail
with the first.
Figure
2.4: Measurement models for addition and subtraction The
measurement model is particularly useful when considering questions of the
following sort: “Does the indicated
operation 2 – 5 have an answer that is a whole number?” Using the measurement model, it is clear
that, if any answer exists, it must lie on the number line to the left of
zero. Since no whole number exists to
the left of zero, the answer must belong to a different set. This insight leads naturally to the need
for the set of integers. By contrast,
using the set model, trying to “take away” 5 disks from 2 disks generally
leads to the observation “You can’t do that.”
This observation is valid as far as it goes but does little to
motivate the development of signed numbers.
In general, it
is better to make use of both set models and measurement models on a regular
basis, rather than one or the other.
Doing so facilitates the development of flexible mathematical thinking
with regard to representations and communication. These are dispositions that should be
developed early and supported throughout a child’s mathematics education.
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