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Appendix B
Concepts of Measurement
At least eight concepts form the foundation of children’s understanding
of length measurement. These concepts include understanding of the attri-
bute, conservation, transitivity, equal partitioning, iteration of a standard
unit, accumulation of distance, origin, and relation to number.
Understanding of the attribute of length includes understanding that
lengths span fixed distances (“Euclidean” rather than “topological” concep-
tions in the Piagetian formulation).
Conservation of length includes understanding that lengths span fixed
distances and the understanding that as an object is moved, its length does
not change. For example, if children are shown two equal length rods
aligned, they usually agree that they are the same length. If one is moved
to project beyond the other, children 4½ to 6 years often state that the
projecting rod is longer (at either end; some maintain, “both are longer”;
the literature is replete with different interpretations of these data, but
certainly children’s notion of “length” is not mathematically accurate). At
5 to 7 years, many children hesitate or vacillate; beyond that, they quickly
answer correctly. Conservation of length develops as the child learns to
measure (Inhelder, Sinclair, and Bovet, 1974).
Transitivity is the understanding that if the length of object X is equal
to (or greater/less than) the length of object Y and object Y is the same
length as (or greater/less than) object Z, then object X is the same length
as (or greater/less than) object Z. A child with this understanding can use
an object as a referent by which to compare the heights or lengths of other
objects.
Equal partitioning is the mental activity of slicing up an object into the
359

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360 MATHEMATICS LEARNING IN EARLY CHILDHOOD
same-sized units. This idea is not obvious to children. It involves mentally
seeing the object as something that can be partitioned (or cut up) before
even physically measuring. Asking children what the hash marks on a ruler
mean can reveal how they understand partitioning of length (Clements and
Barrett, 1996; Lehrer, 2003). Some children, for instance, may understand
“five” as a hash mark, not as a space that is cut into five equal-sized units.
As children come to understand that units can also be partitioned, they
come to grips with the idea that length is continuous (e.g., any unit can
itself be further partitioned).
Units and unit iteration. Unit iteration requires the ability to think
of the length of a small unit, such as a block as part of the length of the
object being measured, and to place the smaller block repeatedly along the
length of the larger object (Kamii and Clark, 1997; Steffe, 1991), tiling the
length without gaps or overlaps, and counting these iterations. Such tiling,
or space filling, is implied by partitioning, but that is not well established
for young children, who also must see the need for equal partitioning and
thus the use of identical units.
Accumulation of distance and additivity. Accumulation of distance is the
understanding that as one iterates a unit along the length of an object and
count the iteration, the number words signify the space covered by all units
counted up to that point (Petitto, 1990). Piaget, Inhelder, and Szeminska
(1960) characterized children’s measuring activity as an accumulation of
distance when the result of iterating forms nesting relationships to each
other. That is, the space covered by three units is nested in or contained in
the space covered by four units. Additivity is the related notion that length
can be decomposed and composed, so that the total distance between two
points is equivalent to the sum of the distances of any arbitrary set of seg-
ments that subdivide the line segment connecting those points. This is, of
course, closely related to the same concepts in composition in arithmetic,
with the added complexities of the continuous nature of measurement.
Origin is the notion that any point on a ratio scale can be used as the
origin. Young children often begin a measurement with “1” instead of zero.
Because measures of Euclidean space are invariant under translation (the
distance between 45 and 50 is the same as that between 100 and 105), any
point can serve as the origin.
Relation between number and measurement. Children must reorganize
their understanding of the items they are counting to measure continuous
units. They make measurement judgments based on counting ideas, often
based on experiences counting discrete objects. For example, Inhelder,
Sinclair, and Bovet (1974) showed children two rows of matches, in which
the rows were the same length but each row was comprised of a different
number of matches as shown in Figure B-1. Although, from the adult per-
spective, the lengths of the rows are the same, many children argued that

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APPENDIX B 361
FIGURE B-1 Relationship between number and measurement.
the row with 6 matches was longer because it had more matches. Thus,
in measurement, there are situations that differ from the discrete cardinal
situations. For example, when measuring withB-1
Figure a ruler, the order-irrelevance
principle does not apply and every element (e.g., each unit on a ruler)
R01420
should not necessarily be counted (Fuson and Hall, 1982).
bitmapped fixed image
Concepts of Area Measurement
Understanding of area measurement involves learning and coordinat-
ing many ideas (Clements and Stephan, 2004). Most of these ideas, such as
transitivity, the relation between number and measurement, and unit itera-
tion, operate in area measurement in a manner similar to length measure-
ment. Two additional foundational concepts will be briefly described.
Understanding of the attribute of area involves giving a quantitative
meaning to the amount of bounded two-dimensional surfaces.
Equal partitioning is the mental act of cutting two-dimensional space
into parts, with equal partitioning requiring parts of equal area (usually
congruent).
Spatial structuring. Children need to structure an array to understand
area as truly two-dimensional. Spatial structuring is the mental operation
of constructing an organization or form for an object or set of objects in
space, a form of abstraction, the process of selecting, coordinating, unify-
ing, and registering in memory a set of mental objects and actions. Based
on Piaget and Inhelder’s (1967) original formulation of coordinating dimen-
sions, spatial structuring takes previously abstracted items as content and
integrates them to form new structures. It creates stable patterns of mental
actions that an individual uses to link sensory experiences, rather than the
sensory input of the experiences themselves. Such spatial structuring pre-
cedes meaningful mathematical use of the structures, such as determining
area or volume (Battista and Clements, 1996; Battista et al., 1998; Outhred
and Mitchelmore, 1992). That is, children can be taught to multiply linear
dimensions, but conceptual development demands this build on multiplica-
tive thinking, which can develop first based on, for example, their thinking
about a number of square units in a row times the number of rows (Nunes,
Light, and Mason, 1993; note that children were less successful using rulers
than square tiles).

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362 MATHEMATICS LEARNING IN EARLY CHILDHOOD
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