MATHEMATICALLY GIFTED STUDENTS’ SPATIAL
VISUALIZATION ABILITY OF SOLID FIGURES1
HyunAh Ryu*, YeongOk Chong** and SangHun Song**
*Konkuk University / **Gyeongin National University of Education
This research, with 7 mathematically gifted students as subjects, looked into their
spatial visualization ability of solid figures by suggesting geometric tasks that require
distinction of the constituents of a solid figure. As a result of analysis, which was
made based on McGee (1979)’s spatial visualization ability, the ability to imagine the
rotation of a represented object, to visualize the configuration, to transform a
represented object into other shape, and to manipulate an object in the mind were
found in some of the mathematically gifted subjects, which are similar to the spatial
visualization ability theorized by McGee. On the other hand, it was found that some
students had difficulty in imagining a 3-dimensional object in space from its 2dimensional representation in a plane.
INTRODUCTION
According to Freudenthal (1973), geometry is grasping space in which the child lives,
breathes and moves and that they have to know, explore and conquer in order to live,
breathe and move better in it. With regard to spatial ability, many researches have
been made, including those by Thurstone (1983), French (1975), McGee (1979),
Lohman (1979), Bishop (1980), Del Grande (1987), etc. Soviet mathematicians, in
the past, emphasized spatial thinking in geometry, particularly the spatial
visualization ability, arguing that “visualizations are used as a basis for assimilating
abstract geometric knowledge and individual concepts (Yakimanskaya, 1971,
p.145).” Presmeg (1986), also, made researches into visualization in mathematics,
which includes the visualization ability of mathematically gifted students.
Nevertheless, specific researches into the spatial visualization ability of
mathematically gifted elementary school students are insufficient.
Therefore, the purpose of this research is to analyse the spatial visualization ability of
mathematically gifted elementary school students using tasks that require them to
distinguish relevant constituents of a three-dimensional object from its two dimensional representation by mentally manipulating or rotating it.
SPATIAL VISUALIZATION
Gutiérrez (1996) regarded visualization, visual imagery, spatial thinking defined by
Yakimanskaya, Dreyfus and Presmeg as equivalents and defined “visualization” in
mathematics, either mental or physical, as a kind of reasoning activity based on the
1
This work was supported by Korea Research Foundation Grant funded by Korea
Government(MOEHRD, Basic Research Promotion Fund) (KRF-2005-079-BS0123)
2007. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of
the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 137-144. Seoul: PME.
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use of visual or spatial elements. According to the analysis of Gutiérrez (1996), the
“spatial thinking” that Yakimanskaya (1991) mentioned possible to create spatial
images is a form of mental activity that can manipulate them in the course of solving
various practical and theoretical problems. Spatial image, here, is created from
perceptive cognition of spatial relations, which can be expressed in diverse graphic
forms including diagrams, pictures, drawings, outline, etc. Therefore, in spatial
visualization, the interaction between creation of spatial images and external
representation is important.
On the other hand according to Lohman (1979), spatial ability can be defined as the
ability to create, maintain, and manipulate abstract spatial images (Clements, 1981, p.
35). McGee (1979) divided the elements that compose spatial ability into spatial
visualization and spatial orientation; and Lohman (1979) divided them into spatial
relation, spatial visualization and spatial orientation (Clements, 1983). Later, Linn
and Peterson (1985) divided spatial sense into the spatial perception, spatial rotation
and spatial visualization. As have been mentioned, elements that compose spatial
ability differ by researcher, but that spatial visualization functions as an important
factor in spatial ability is agreed by them all.
McGee’s spatial visualization ability refers to the ability to manipulate, rotate, change
the position in mind of an object depicted as a picture, in other words the ability,
using mental image, to rotate, arrange, or manipulate a depicted object. Lohman’s
spatial visualization ability means the ability of arranging the pieces of an object to
complete paper folding or overall shape. And, that of by Linn and Peterson means the
ability to make given spatial information visible and draw it in one’s mind.
The meaning of spatial visualization can also be interpreted differently in accordance
with the viewpoint of each researcher; and, the same applies to sub-abilities that
compose thereof. Of all the different classifications, McGee classified spatial
visualization abilities as follows (Gutiérrez, 1996):
-Ability to visualize a configuration in which there is movement among its parts
-Ability to comprehend imaginary movements in three dimensions, and the manipulate
objects in the imagination
-Ability to imagine the rotation of a depicted object, the (un)folding of a solid, and the
relative changes of position of objects in space
-Ability to manipulate or transform the image of a spatial pattern into other arrangement
The task suggested in this research is possible to be solved by mentally manipulating,
rotating or changing the direction of depicted objects; and is deemed to require the
spatial visualization ability McGee suggested.
RESEARCH METHODS
Tasks
The geometry task used in this research –the one suggested in the doctoral thesis of
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Raquel(2001) with a view to explore the role of Geometer’s Sketch Pad(GSP) in
improving students’ geometric thinking and spatial ability– is the problem that
compares the side lengths and the angle sizes in each picture that depicts a regular
icosahedron without the dotted line (Figure 1). It is descriptive of a regular
icosahedron in the plane.
Problem 1
problem 2
problem 3
Compare two real lengths indicated
bold and explain the reason.
Compare two real angles indicated
bold and explain the reason
E
A
C
problem 4
B
A
A
C
C
C
B
B
D
D
D
F
D
E
F
E
G
Figure 1: The task for the spatial visualization
The identification of a plane in a 3D/2D representation is a very important problem
which also concerns the first steps in the geometrical representation of space
(Rommevaux, 1997). In other words, the first thing required in distinguishing the
relevant parts of a solid figure in spatial geometry is to distinguish the different plane
parts. This task is judged to be useful to observe the characteristics of spatial
visualization that requires such ‘change of dimensions’ (Duval, 1998).
Participants
There are 6 students (aged 11~12; T, U, V, W, X, Y) in the 6th grade and 1 student
(aged 13; Z) in the 7th grade receiving gifted education in the institute attached AUniversity supported by Korean government. They belong to the upper 0.1% group in
their respective school years.
Procedures
In preliminary experiment, we met 7 students (1 mathematically gifted 6th grade, 1
mathematically gifted 7th grade and 5 ordinary 7th grade) with a view to design the
method of analysing the understanding of the task, approach to the given task and
visualization ability needed to solve the problem.
Data collection and analysis for this research were done from Nov. to Dec. of 2006.
The participants were asked to solve the task for 60 minutes, individually, without
using a ruler or a compass. After that, interviews on the problem-solving process
were conducted and videotaped; and the activity sheets of students were collected.
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We analysed the data based on McGee’s spatial visualization ability and found a
relation from Duval and Del Grand theories.
RESULTS AND DISCUSSION
Spatial visualization ability displayed in problem solving
The spatial visualization abilities mainly found in the students’ problem solving
process are the ability to imagine the rotation of a depicted object, to visualize its
configuration, to transform it into a different form and to manipulate it in ones
imagination.
Ability to visualize configuration
The visualization ability that was found most in the problem solving process of this
research were ability to visualize a configuration in which there is movement among
its parts explained by Mc Gee. This is the ability to clearly see a partial configuration
out of an overall configuration that is useful in solving the problem. An example is
that from the figure of problem 3, all the students visualized each regular pentagon
that includes angle ABC and angle DEF; and another example is that Student Y,
while solving problem 1, clearly visualized spatial figure from a plane figure. He
explained this as follows:
Student Y: When you see a regular icosahedron, there are
vertexes, one at the top and the other at the bottom. And
there are two pentagonal pyramids of which, the vertex is
the former and the latter, respectively. And if they are
linked when they do not meet, that makes the longest line
segment.
A
C
B
D
Interviewer: Did you know it from the beginning?
Student Y: Yes. In the figure there are two of them here
(Figure 2a) and here (Figure 2b). And there is no shared part
between them, and accordingly CD becomes the diameter
and the longest line.
Fig. 2a
A
C
B
Interviewer: What does it mean that they do not have a shared
part? Do you mean the two pentagons do not meet?
Student Y: No. What I mean is that when viewed threedimensionally, the two pentagonal pyramids share no part.
And AB is not the diameter.
D
Fig. 2b
Interviewer: Then what line is it?
Student Y: It is just a line. Since the two pentagonal pyramids, of which the vertex is A
and B, respectively, have a common part, AB is not a diameter.
According to Duval (1999), the operative apprehension is carried out by transforming
a visual operation in looking a figure. Here, various operations caused figural
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changes that can play a heuristic function and provide the
insight necessary to solve a problem. It can be said that this
visualization ability also visualizes spatial figures, which in
turn play a heuristic function that is necessary for problem
solving in spatial geometry, and provides insight necessary
for problem solving. In this viewpoint, the ability to visualize
a configuration of spatial figure depicted in plane played a
heuristic function that is necessary for insight for problem
solving
Fig. 3
On the other while, Student Z, in problem 4, could distinguish a regular pentagon
from the figure in the question that includes angle DEG as shown in Figure 3. This
means he perceived a figure in a difficult and complex background where the two
lines overlap one another and dots were hard to be distinguished from line, which can
be classified as Figure-ground perception ability suggested by Frostig and Horne of
the 7 sub-categories of space perception theorized by Del Grande (1987).
Ability to manipulate an object in imagination
The students were able to mentally arrange or manipulate a 3-dimensional object
which is depicted in 2-dimensional plane. This is included in the ability to
comprehend imaginary movements in three dimensions, and the manipulate objects in
the imagination mentioned by Mc Gee. An example of this is the case where Student
Z, in problem 2, cut off each pentagonal pyramid of which the vertex is E and F,
respectively, made a solid figure of which the base plane is
a regular pentagon and the side faces are regular triangles A
standing straight and headlong alternately, assumed the
C
B
height of the pentagonal pyramids to be a and that of the
G
F
rest prism to be b and explained EF equals a+2b and CD
D
E
also equals a+2b. In another case, Student U, while solving
problem 1, explained the section that includes CD has
hexagonal shape by manipulating the object in mind and
Fig. 4
marked the vertexes of the hexagon A, C, F, G, D, E as
shown in Figure 4.
Ability to imagine the rotation of a depicted object
The students were able to mentally rotate a 3-dimensional figure depicted in 2dimensional plane and change the positions of its constituents. For an instance,
Student V said if the figure depicted in problem 2 is revolved, the positions of CD
and EF look changed and in the end the lengths of the two lines are the same. This is
included in the ability to imagine the rotation of a depicted object, the (un)folding of
a solid, and the relative changes of position of objects in space classified by Mc Gee.
Ability to transform a depicted object into a different form
The students were able to change the form of a depicted object by mentally cutting it
or adding to it. For instance, Student Z changed the figure in problem 2 by cutting off
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pentagonal pyramids of which the vertexes are E, F, respectively into a solid figure of
which the base plane is a regular pentagon and side faces are regular triangles
standing straight and headlong alternately. After that, he imagined the length of EF
by separating the length in the two pentagonal pyramids from that in the newly
formed solid figure. This is included in the ability to manipulate or transform the
image of a spatial pattern into other arrangement mentioned by Mc Gee.
Errors that occur in the spatial visualization process
Dependence on visual facts
Despite a figure given in a problem depicts a solid figure, one fails to imagine it as a
spatial object and depends on the visual facts of the plane
figure.
A
E
For example, while solving problem 2, Student X thought
C
H
FE>CD since FE=AB and AB>CD; while Student W thought
EF is longer than CD by GF and HE since CD=GH (Figure 5).
G
D
This is the phenomenon found among students who are F
B
accustomed to pictures that express a three-dimensional object
in two dimensions using dotted lines: they cannot see the
Fig. 5
object in perspective when all the lines are solid lines as in the
given questions.
Confusion in distinction of edges
Though a number of students knew that all the facets of a regular icosahedron in the
task of this research are regular triangles and the lengths of all edges are the same,
they were confused about which lines in the depicted figure becomes edges of the
polyhedron and misunderstood that all the lines marked in the question are edges.
For example, Student X, while solving problem 1, said
AB>CD, citing the reason that the lengths of CD and CB
are the same because both of them are sides of a regular
triangle. Student W, in problem 4, said angle DEG=60°,
citing the reason that DGEH in Figure 6 is a regular
tetrahedron where all the lengths of sides are the same. In
the case of Student W, though he created a spatial image by
visualizing partial configuration, he got confused a little in
distinguishing edges.
B
A
C
H
E
D
G
Fig. 6
Difficulty in imagining the section of a solid figure
It can be said that the main idea required to solve the task of this research is how to
distinguish, from a solid figure represented in a plane, relevant line segments and
planes. Actually, several difficulties were found in imagining spatial planes from a
picture in a plane. A number of students marked a part that cannot be a section of a
solid figure and argued that it was a plane.
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B
B
A
C
B
A
C
A
C
D
D
E
E
D
E
G
G
Figure 7a
Figure 7b
Figure 7c
For example, Student U, while solving problem 1, with a view to compare ED and
CD the length of which is the same as that of AB, marked the plane that includes the
two line segments as shown in Figure 7a. In problem 4, he also marked the plane that
includes the angle DEG as shown in Figure 7b. And Student T, in problem 4, marked
a section inside a regular icosahedron as a plane that includes the angle DEG as
shown in Figure 7c.
CONCLUSION
This research, with 7 mathematically gifted students as subjects, looked into how
they mentally manipulate or rotate a solid figure represented in a plane and
distinguish relevant constituents – their spatial visualization ability.
Though 2 out of the 7 subjects displayed characteristic spatial visualization ability
carrying out all the tasks in this research, most of the other 5 students had some
difficulty in mentally manipulating an object depicted in a plane as a spatial object.
The spatial visualization abilities mainly found in the students’ problem-solving
process are the ability to mentally rotate a 3-dimensional solid figure depicted in 2dimensional representation and thus change the positions of its constituents, to
transform a depicted object into a different form by mentally cutting it or adding to it,
to see a partial configuration of the whole that is useful to solve the problem, and to
mentally arrange or manipulate a 3-dimensional object depicted in 2-dimensions.
These abilities are similar to that of McGee (1979). Of these abilities, all the students
displayed the ability to visualize partial configuration that is useful for solving the
problem with easier pictures that have no overlapping lines or dots; however, only
one Student V is played the ability to discover a partial configuration with complex
pictures that have overlapping lines or dots. This can be classified as Figure-ground
perception suggested by Frostig and Horne of the 7 sub-categories of space
perception as theorized by Del Grande (1987).
On the other hand, with compared to the ordinary students, it was found that some
students who display excellent characteristics in algebra or other fields of geometry
had, to some extent, difficulty in the spatial visualization process. In the case where
one depends upon the visual facts as represented in a plane picture, he get confused in
distinguishing the edges of a spatial object from the depicted picture and has
difficulty in distinguishing planes in 3-dimensional object from its 2-dimensional
representation.
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