THE ANALYSIS OF ACTIVITY THAT GIFTED STUDENTS
CONSTRUCT DEFINITION OF REGULAR POLYHEDRA1
KyungHwa Lee*, EunSung Ko* and SangHun Song**
*Korea National University of Education /
**Gyeongin National University of Education
This study was conducted with the focus on the process of constructing a definition and
produced definitions rather than gifted students’ conceptions of a mathematical
definition. Accordingly, instead of a mathematical subject that students would come
into contact with as part of the curriculum or in their ordinary lives, this study used
regular polyhedron as its subject matter which students are not familiar with even if
they may have encountered it in their ordinary lives. In this study, students were asked
to make platonic polyhedra, observe them and then construct a definition of regular
polyhedron based on their observations. We sought to gain various suggestions
through the analysis of the observations and definition laid down by the students and
through the characteristics shown by the students in the process of defining the
concept.
INTRODUCTION
There have been many different opinions regarding the definition of giftedness and
gifted children and many scholars have had different perspectives about them, as the
criteria for giftedness and gifted children have differed with the changes in the times,
cultural and social values (Song, 1998). For the purpose of this study, gifted children
are restrictively defined as children who was selected as a gifted children by experts of
institute for science gifted education supported by the government.
Up until now, studies on characteristics on the way of thinking of mathematically
gifted students have focused on generalization, abstraction, justification, reasoning
ability, etc. that are at play during the process of problem-solving and proving
(Krutetskii, 1976; Lee, 2005; Sriraman, 2003; 2004).
Definition accounts for the important part of mathematics and mathematics education
(Harel, Selden, & Selden, 2006; Ouvrier-Buffet, 2002; Shir & Zaslavsky, 2001) and
the significant roles that definition plays in grasping mathematical concepts, solving
problems and proving have been emphasized by numerous researchers (Shir &
Zaslavsky, 2002; Skemp, 1971; Vinner, 1991). In mathematical learning, construction
of definition rather than the provision of constructed definition is regarded as
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. 3, pp. 153-160. Seoul: PME.
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important as problem solving, guessing, generalization, and proving (De Villiers,
1998; Mariotti & Fischbein, 1997; Ouvrier-Buffet, 2004; 2006).
This study seeks to deal with another aspect of mathematically gifted students by
analysing the characteristics displayed by students in the process of making five types
of regular polyhedra, observing them and constructing the definition of a regular
polyhedron. This study is also intended to give some suggestions as to the selection of
mathematically gifted students and the curriculum for their education.
BACKGROUND
Kang & Cho (2002) have identified 5 definition-methods that are used in the geometry
of school mathematics - synonymous, denotative, implicative, constructive, analytic.
And they categorized them into practical and scientific methods. The former 3
definition-methods are classified as practical whereas the latter 2 definition-methods
are classified as scientific. In making definitions, practical methods select directly
perceived attributes and directly useful characteristics while scientific methods select
‘causality’, ‘generation’ or ‘relationship,’ which show how things are mutually
dependent on one another and how they interact mutually. Accordingly, the latter
methods enable us to identify connectivity between the discrete pieces of information.
In this study, we used Kang & Cho (2002)’s study to examine whether students
depended on directly perceived attributes or took notice of the relationship between the
components that make up a regular polyhedron.
Fischbein (1987) claimed that examples play a core role in intellectual activities and
emphasized the importance of denotative method. According to him, ‘paradigmatic
model’ is basically an example but is beyond a mere example. An example of a concept
refers to the object that carries all the attributes of the concept. Example as
paradigmatic model not only carries all the attributes of a concept but also plays a core
role in intellectual activities. Skemp (1971) attached a particular significance to
conceptual learning as it lays foundation for higher level of mathematical principles
and problem solving. According to him, the best way to administer conceptual learning
in mathematics is through inductive reasoning whereby proper examples related to the
concept at hand are presented to help students identify commonalities of the examples
and construct the concept from the commonalities. This study is based on Fischbein
(1987) and Skemp (1971) and determines that platonic polyhedra give a proper
situation for constructing the definition of a regular polyhedron.
In this study, students participating in this experiment were asked to make platonic
polyhedra using materials (Znodome system), observe them, and construct a definition
of each regular polyhedron based on their observations. That is, they were asked
construct a definition through inductive reasoning using examples.
Abstracting is one of the most important things and generalizing and synthesizing form
a prerequisite basis to abstracting. Generalization is to derive or induce from
particulars, to expand familiar processes, and abstracting is constructive process
building mental structures from properties of and relationships between mathematical
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objects (Dreyfus, 1991). According to Dreyfus (1991), this process depends on the
isolation of proper properties and relationships. Such constructive mental activity on
the part of a student depends on the student’s attention being focused on those
structures which are to form part of the abstract concept and drawn away from those
which are not relevant in the intended context. Synthesizing means to combine or
compose parts in such a way that they form a whole, an entity. Unrelated facts
hopefully merge into a single picture, within which they are all composed and
interrelated. This process of merging into a single picture is a synthesis.
METHODOLOGY
Participants
Participants in this study are 21 intellectually gifted elementary school students in the
5th grade (11 years old) - 14 boys and 7 girls - who are being instructed under the
program of institute for science gifted education attached to a National University and
supported by Korean government. But they had no This institute selects gifted students
through 3-stage steps : (1) Recommendation by a principal, (2) Testing of students by
experts on high level of mathematics problem solving, and (3) Testing of students by
experts on abilities on a solve problems requiring ingenuity. The students in this
institute educated 60 hours in science and 42-hours in mathematics for one year. Math
programs, which are 3-hour long each, deal with various fields such as algebra,
geometry, probability, etc. with the focus on improving students’ abilities on
problem-solving, reasoning, and justification. We have confirmed that these students
did not experience any class previously on how to construct definition on a certain
concept through the regular curriculum of education either at their schools or at the
institute.
Activities
The teaching experiment designed for this study was part of the regular curriculum of
this institute and was administered to the students for three hours on end after dividing
the students into 3 groups. The experiment consisted of 4 steps and the details on these
steps are as follows.
Step 1: Making regular polyhedra (in group). For starters, pictures of platonic
polyhedra were presented to the students. Each group was asked to make the regular
polyhedra using the materials based on these pictures.
Step 2: Observing regular polyhedra (in individual). Students were asked to
observe the regular polyhedra that their group had made and record their observations
(characteristics, attributes, etc.) about each type of regular polyhedron in the activity
sheet. They were asked to record as many observations as possible and a mutual
discussion within each group was permitted.
Step 3: Defining regular polyhedra (in individual) Students were asked to construct
a definition and record it on the activity sheet based on the observations they have
made in step 2.
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Step 4: Further develop their definition (in group) Students of each group engaged
in a group discussion to refine the definition of regular polyhedron constructed by
individual members of each group in step 3.
RESULTS
In this study, analysis was conducted with the focus on step 2 and 3. The main data to
analysis was the activity sheet that students prepared and one-on-one interviews were
conducted in case the need arose to clarify certain terms and expressions used by
students. The discussions in this study were not classified and organized by a
conventional framework but, rather, are the results that were derived through an
inductive method based on the responses of the students (Denzin & Lincoln, 1994;
Goetz & LeCompte, 1984).
According to analysis results, students were categorized into three groups based on the
relationship between the responses of students in step 2 of observing five platonic
polyhedra and recording their observations and the responses of students in step 3 of
defining a regular polyhedron based on these observations. Part of the responses shown
by students of each group is as follows:
Group
(sample
students)
The critical components of the figures by students
in step 2
Regular
4-hedron
Regular
6-hedron
Regular
8-hedron
Regular
12-hedron
Regular
20-hedron
The definitions by
students in step 3
The shape of face. The number of vertex and edge. A regular polyhedron
G S1
r
o
u
p
1
S2
has the same area of
faces and the same
length of sides. The
angles
formed
by
adjacent edges are the
same.
The length and number of edge. The angle A regular polyhedron
formed by adjacent edges. The number of has sides of the same
vertex. The shape and number of face.
length, angles of the
same size and faces of
the same area.
G
r
o
S3
u
p
2
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The length and number of edge. The number of A regular polyhedron
vertex and face. V+F-E=2.
is a solid that has edges
of the same length and
faces of the same area.
V-E+F=2. The numbers
of edges, vertices and
faces are all even.
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Lee, Ko & Song
G
r
o
S4
u
p
3
The
number of
face and
diagonal.
The
number
and shape
of
face.
The
number of
vertex.
Making
method.
The
number
and shape
of
face.
Looks like
a top.
The
number
and shape
of
face.
The
number of
vertex.
The
number of
vertex to
meet on
each
vertex.
Making
method. It
has many
edges and
skewed
parts.
It should have faces.
The
numbers
of
vertices are four, eight
and twelve. That is, the
numbers
are
multiplied by 2. The
numbers of the edges
are multiples of six. It
should consist of plane
figures.
Table 1: The critical components of the figures & definitions by students
[On group 1] Students in Group 1 constructed a definition that is logically congruent
with the mathematical definition. Not only they took notice of the critical components
but also the components observed were consistent in observing the characteristics of
each regular polyhedron in step 2. For examples, student 1 recorded their observations
with a consistent focus on the number of edges and vertices, the shape and number of
faces that make up the regular polyhedra while student 2 recorded his observations
with a focus on the size of angles formed by two adjacent edges, area of faces, the
number of vertices and edges, and the number and shape of faces.
Definition by student 1 satisfies the mathematical definition of a regular polyhedron,
“A regular polyhedron is composed of congruent regular polygons”. The statement that
“All the faces are the same area while all the edges are the same length” means that all
the plane figures that make up a regular polyhedron are congruent. An addition of the
statement that “the angles of the adjacent edges are the same” indicates that all the
plane figures are congruent regular polygons. Though this definition an enumeration of
the factual observations that represent the characteristics of regular polyhedra, this
satisfies the mathematical definition of a regular polyhedron. That is, they tried to
present sufficient conditions that are needed to make a mathematical definition. The
definitions of student 2 also satisfy the mathematical definition of a regular
polyhedron.
It was confirmed, through student 1 and 2 that students in group 1 grasped the
relationship between the components that make up a regular polyhedron in observing
five types of regular polyhedra, and they perceived regular polyhedra’ structures so
that recognized the fundamental attributes of regular polyhedron. They recognized the
critical components that make up a regular polyhedron and defined with the
relationship between the components. In addition, in defining a regular polyhedron,
student 1 recognized the fact- angles formed by two adjacent edges are the same size that were not expressed in step 2, and used the fact to define a regular polyhedron. It
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shows that the student recognized the relationship between the components that make
up a regular polyhedron and the fundamental attributes.
[On group 2] Students in Group 2 constructed a imperfect definition. They took notice
of the critical components and the components observed were consistent in observing
the characteristics of each regular polyhedron as the students from Group 1 did. Foe
example, student 3 observed the regular polyhedra with a consistent focus on lengths
of sides, number of vertices, number of faces and edges, the relationship (vertices +
faces –edges = 2) between the numbers of vertices, faces and edges. Even though the
definition of student 3 used the results of observations appropriately, it simply
enumerates observations in a superficial fashion while failing to identify the
relationship between the components that make up a regular polyhedron. Thus, this
definition includes part of Catalan polyhedron in addition to a regular polyhedron.
[On group 3] Students in Group 3 had difficulty defining a regular polyhedron.
Student 4 was not able to put a consistent focus when observing platonic polyhedra in
step 2. That is, he was not taking a systematic approach. Though student 4 observes
regular polyhedra with a focus on number and shapes of faces, number of vertices,
number of edges that meet on a vertex, number of diagonals, how to make regular
polyhedra using materials and overall shape, etc., the student’s focus changes
depending on the type of the regular polyhedron - number of faces and diagonals in the
case of a regular tetrahedron and number & shape of faces and number of vertices in
the case of a regular hexahedron. Accordingly, this student presents the number of
components instead of defining a regular polyhedron by identifying the relationship
between the components that make up a regular polyhedron.
CONCLUSION
According to the studies of Shir & Zaslavsky (2002) and Zaslavsky & Shir (2005),
students showed a tendency not to adopt the definition that uses something other than
critical components that make up the figure (faces, edges and vertices in the case of a
regular polyhedron). However, students of Group 1 voluntarily used things other than
critical components of a regular polyhedron to construct definition. It shows that
students recognized the relationship between the components that make up a regular
polyhedron and the fundamental attributes. Students of Group 1 could not only
generalize and synthesize the facts that were observed through platonic polyhedra but
also abstract a regular polyhedron by grasping the relationship between the
components that make up a regular polyhedron and capturing the critical
characteristics of a regular polyhedron. Thus they were able to construct a definition
that is logically congruent with the mathematical definition.
Students from Group 2 succeeded in generalizing the facts that were observed through
platonic polyhedra but they failed to synthesize the facts that were observed and
abstract a regular polyhedron by capturing the critical characteristics of a regular
polyhedron so that they were not able to construct a complete definition. Kang & Cho
(2002) argued that for a learner to be able to define things through examples,
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attentiveness and comprehensive faculty are prerequisites and Dreyfus (1991) said that
abstracting is possible when the student’s attention is focused on those structures
which are to form part of the abstract concept, and drawn away from those which are
not relevant in the intended context. In constructing definition of a regular polyhedron,
students of Group 2 seems to be lacking in understanding needed to recognize the
relationship between the components and in attentiveness needed to distinguish
between different types of polyhedra.
According to the study of Mariotti & Fischbein (1997), students who tend to view
geometrical figure as visual gestalts have a tendency to rely on unnecessary
characteristics while overlooking decisive characteristics of the figure. Students from
Group 3 had difficulty defining definition of a regular polyhedron as they failed to
generalize and overlooked important characteristics by perceiving the solids as visual
gestalts only.
It was confirmed, through students in Group 1, defining a mathematical concept was a
useful activity through which the abilities of generalization, synthesizing and
abstraction that are characteristics of gifted students as confirmed by various studies
(Krutetskii, 1976; Sriraman, 2003) could be verified and participating students could
exercise the abilities. It was confirmed, through students from Group 2, that in order to
define a mathematical concept, the ability to recognize the relationship between the
components that make up the concept and to capture fundamental characteristics are
needed in addition to the abilities of generalization. With this experiment, it was
confirmed that the activity of defining concepts could be used for selecting gifted
students and developing programs for gifted students. It was also found out that the
ability to recognize the relationship between the components that make up the concept
and to capture fundamental characteristics should be taken into consideration in
addition to the abilities of problem-solving, generalization, and justification. It was
confirmed, through students in Group 2 and 3, defining a mathematical concept only
with examples are difficult even to gifted students. It is important to give counter
examples in considering the level of learners.
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