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cut this cube with two vertical planes parallel to
the surface of the screen, then with two vertical
planes perpendicular to it, finally with two hori-
zontal planes. How many small cubes do you have
now (nine of them can be seen on each side of the
big cube)? Please tell:
and Metzler in 1971 (Thomas, 2007). The test
comprised twenty problems, and each test item
consisted of a criterion figure and four alterna-
tive figures, two of which were the same as the
criterion and two incorrect. The two correct al-
ternatives were rotated versions of the criterion
figure. For example, the stimulus figure pairs used
by Shepard and Metzler contained (Nigel J. T.
Thomas, Stanford Encyclopedia of Philosophy):
How many cubes have three blue sides,
How many cubes have two blue sides,
How many have one blue side, and
How many cubes are all red?
• Identical objects differing by a rotation in
the plane of the page
• Identical objects differing by a rotation in
depth
• Mirror-image objects (also rotated
Answer: there are 8 cubes with 3 blue sides, 12
cubes with 2 blue sides, 6 cubes with 1 blue side,
0 cubes with no blue side, and 1 small cube inside
a big one would be all red.
in
depth)
Doing simple mental operations like this one,
without sketching it first, is considered improving
one's spatial abilities.
CONCLUSION
Both communication with others and exchange
of knowledge, insight, and information can be
done in non-verbal way. Our communication with
the world and people goes through our senses in
many ways, not only by sight, hearing, touch and
haptic experience, smell, or taste. Several senses
that gather information from the external and in-
ternal receptors take part in our defense systems.
Explorations about the sense of numbers take a
form of a discussion about nonverbal counting
and searching whether there is thought, especially
math thought, occurring without language. We are
able to capture light and then recognize, name, and
also categorize it with our senses. We are still not
sure which senses we had already lost, and which
ones we ignore or are not aware of, while they may
be possessed by some other living creatures. We
observe, research, and record nature in terms of
making inventions, and also by creating art; we
apply our senses for that. Issues discussed in this
chapter are discussed in further chapters with
diverse frames of reference, from physiological,
physical, technological, to aesthetical.
Measurement of Spatial Abilities
Piaget and Inhelder used models and an interview
method. Paper and pencil tests are often used to
determine spatial visualization and spatial orienta-
tion. Typically, spatial visualization abilities are
measured with cognitive tests; mental rotation
can be accomplished using spatial visualization.
Cognitive tests used to measure spatial abilities
are, among others: the Spatial Visualization Abili-
ties Test (SVAT) that is suitable for 4th graders;
the adult level Thurston's Paper Folding Test,
VZ-2 (Paper Folding, with a sequence of folds
and a set of holes punched), and VZ-3 (Surface
Development, with a flat shape with numbered
sides and a three-dimensional shape with lettered
sides) - results were better at the formal opera-
tional level); a Form Board Test (with a shape
and a set of smaller shapes); Baltista, Talsm, and
Wheatly Test; Purdue Spatial Visualization Test
on Rotation; Vandenburg/Kuse Mental Rotation
Test (a test of 3-D spatial visualization based on
stimuli constructed from figures used by Shepard
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