The meaning of isometries as function of a set of points
and the process of understanding of geometric
transformation
Xhevdet Thaqi, Joaquim Gimenez, Ekrem Aljimi
To cite this version:
Xhevdet Thaqi, Joaquim Gimenez, Ekrem Aljimi. The meaning of isometries as function of a set
of points and the process of understanding of geometric transformation. Konrad Krainer; Naďa
Vondrová. CERME 9 - Ninth Congress of the European Society for Research in Mathematics
Education, Feb 2015, Prague, Czech Republic. pp.591-597, Proceedings of the Ninth Congress
of the European Society for Research in Mathematics Education. <hal-01287028>
HAL Id: hal-01287028
https://hal.archives-ouvertes.fr/hal-01287028
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The meaning of isometries as function
of a set of points and the process of
understanding of geometric transformation
Xhevdet Thaqi1, Joaquim Gimenez2 and Ekrem Aljimi1
1
University “Kadri Zeka” Gjilan, Gjilan, Republic of Kosovo, [email protected], [email protected]
2
Unversity of Barcelona, Barcelona, Spain, [email protected]
In this paper, we try to show that in the process of understanding of isometric transformations, the meaning of isometric transformations is characterized as a
function of whole figure to whole figure, as a function
of the parts of the figure to the correspondent parts of
the figure, and as a function of the set of points of figure
to set of points of the same or other figures. This perception of isometric transformation has been observed in
an experimental study which enabled us to define and
understand different levels of difficulties in recognition
of isometric transformation, from which the details are
presented in this paper.
Keyword: Geometric transformation, function, teaching
geometry, concept images.
INTRODUCTION
Considering low level of geometric reasoning of future teachers observed in the experiences and showed
by different investigations (Thaqi, 2009), our interest
is the foundations of the professional development of
the prospective teacher of mathematical education.
For that reason we have to design, plan and implement a practice on learning to teach the mathematics;
to analyze elements of the constructions of personal meanings of future teachers about mathematics
and, to recognize the difficulties of the students to
understand, relate and organize mathematical contents, terms and properties associated to that content.
During last years with my colleges we tried to contribute in this process focused concretely in geometrical transformations (Thaqi, Gimenez, & Rosich, 2011;
Thaqi & Gimenez, 2012). In this paper we research
nature and causes of difficulties in teaching/learning
geometrical transformation and the relation of such
CERME9 (2015) – TWG04
difficulties with concept images constructed about
geometrical transformation.
Some investigations have highlighted the reasons
and advantages that provide the study of geometrical transformations (Jackson, 1975; Küchemann,
1980; Jaime, 1993; Harper, 2003; Jagoda & Swoboda,
2011; Thaqi, 2009). A main reason to study the transformation is curricular, since “The transformations
are applications of the geometrical functions, and
this treatment is fundamental for all the mathematics” (Jackson, 1975, p. 554). Other reason is that the
transformations provide geometrical dynamic task.
Despite these reasons and advantages that the transformations teaching offers, generally we know that
the students show a low level of learning about the
transformations (Thaqi, Gimenez, & Rosich, 2011) and
we highlight that a program for the education of the
teachers must integrate the same objectives that the
scholar geometries classes have; the need of additional formation in formal geometry and empathize the
conceptual understanding, starting from the analysis of the geometrical environment with conceptual
explorations. The study of how prospective teachers
build a meaning of the concept of isometrics transformation in process of understanding of geometrical
properties and their relation with difficulties is one
of the aims of this research.
THEORETICAL FRAMEWORK
Sundry authors have distinguished the everyday
concepts (known as spontaneous) and so did the
scientific (Piaget, 1970; Vygotsky, 1987). Fischbein
(1993) considered that there have been seen three
types of conceptual constructions in the investigation: inductive, deductive and inventive building of
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The meaning of isometries as function of a set of points… (Xhevdet Thaqi, Joaquim Gimenez and Ekrem Aljimi)
concept. Later on, the same author tells us that the
concepts are the results of accumulated social experiences (Fischbein, 1993). We consider that there has
to be built the conceptual meaning in interaction with
contexts grounded in the experiences, to later build
images and abstractions. Some authors like Sowder
(1996) proposed that what characterizes a concept is
to state an idea that is given like an answer to non
similar stimulation (to varnish the understood like
examples). In the opinion of Fischbein (1993) what
characterizes the concept is the fact to state a idea,
general representation of a class that is based in common characteristics.
The conceptual construction rests on a set of processes of construction, visualization, exploration of properties, elaboration of explanations and classifications,
within others. Among more and better experiences
we have, the conceptual image gets closer to the concept because, like Vinner and Hershkowitz states:
to acquire a concept means, to acquire a mechanism of construction and identification through
what will be possible to identify and build all the
examples of the concept, the same way as that is
conceived from the mathematical community
[Cited for Jaime & Gutierrez, 1995].
As it concerns to the various investigations about
the subject (Jaime & Gutiérrez, 1995; Pearman, 1990)
generally they put the manifest that between the
Piagetian concept of conservation of the length and
the invariance there is a tight relation that they can
be saved (Jaime & Gutiérrez, 1995).
The constructs “concept image” and “concept definition” (Tall & Vinner, 1981) will be useful to us also to
describe the status of the knowledge of the individual
fellow with a relation to a mathematical concept. It
is meant to the mental entities that are introduced
to distinguish the mathematical concepts formally
defined and the cognitive concepts through which
they are conceived. With the expression:
concept image describes that the cognitive
structure is totally associated to a concept, that
includes the mental images and associated processes and properties (Tall & Vinner, 1981, p. 152).
and the examples that has been seen or used, both in
a scholar and extra-scholar context have a basic role.
Frequently the examples are few and the students
convert them in prototypes. Jagoda & Swoboda (2011)
study the process of construction of the concept in
rotation spotlighting that “recognition of a specific
figure to figure position is only a static image of this
relationship, not connected with the movement of one
object onto the other”. In fact they confirm that the
idea of geometrical transformation is necessary to
conceive the specific movement that is transforming
the initial figure into the final one, through which it
is important that such conception stems from mental reflection on the phenomenon of movement. Our
position is based on the meaning of the concept of
geometrical transformation as a function of (whole)
figure to (whole) figure that is the basic level of knowledge about geometrical transformations. It has also
been shown by several authors that pre-service elementary teachers have difficulties in determining: (1)
the correct attributes of transformation and motion
to move an object from one point to another; (2) the
results of transformations involving multiple combinations of figures; (3) the use of transformations as
mathematically-general operations which require the
specification of inputs, but as particular actions, each
with given prototypical parameters (Harper 2003).
A recent study concerning prospective teachers‘
knowledge of rigid transformations (Yanik & Flores
2009) revealed that scholars: (1) started by referring
to transformations as undefined motions of a single
object which is equivalent with Static Arrangement
Figure To Figure (Jagoda & Swoboda, 2011), followed
by (2) using transformations as defined motions of a
single object, and (3) the understanding of transformations as defined motions of all points on the plane
which is equivalent with transformation as function of
set of points to set of points. In our research we will try
to explain that precisely these three ways of change
of concept images of geometrical transformations
enable us to explain and understand the difficulties
of the understanding of the concept of geometrical
transformation. A head we will show the relationship
between the process of construction of the concept of
geometrical transformation and the function of the
set of the points of figure to the set of points of the
(same or) another figure.
Jaime & Gutiérrez (1995) state that in the formation of
images of a concept that a person has, the experience
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The meaning of isometries as function of a set of points… (Xhevdet Thaqi, Joaquim Gimenez and Ekrem Aljimi)
materials, of these context is the same as showed in
the deep study (Thaqi, 2009), not detailed in this paper.
METHODOLOGY
The methodology was adapted to several techniques
that allowed to approach the construction of the goals
of the investigation and allowed to rate it as a theoretical formal study within the interpretative focus. In
this way we have elaborated a design in which was
combined the own techniques of the studies with case
studies. Considering our own experience in the mathematical education of the teachers, we considered
adults who acquire a professional scientific knowledge, and our work that has transformed a vision of
the processes in the creation of a meaning, we identify that the investigation, action and formation are
the three sides of a same methodological triangle. So,
the investigation comprises theoretical component
with epistemological and cultural character, also an
experimental type component. These two parts of
the investigation are intimately related giving the
product - analysis of the speech and analysis of the
personal constructions, to be able to form the conclusions of the investigation.
Participants of the study were 18 students of Faculty
of Education in University of Gjilan – Kosovo. This
is because two authors of this study have worked in
regular lectures with these students in the program of
preparing prospective teachers. Participants where
purposefully chosen, and voluntary participated for
this study. They are 20–22 years old students – from
different, rural and urban places, from both sexes and
with different studies done in prior studies. A previous curricular-cultural analysis based on textbooks,
official curricular proposals and teachers’ training
During the development of the practical sessions
dedicated learning to teach geometrical transformations, there has been prepared for every student, work
sheets, to be able to have their productions and later
analyze them. The quantity of the practical sessions
and their length is what we present in the following
table (see the main ideas in Table 1).
The results of a final semi structured questionnaire,
was the basic data considered in this paper. Data were
collected from descriptive notes, reflective notes, interviews and video records. The collected data was
analysed using method described by Strauss & Corbin
(1998) along with analytic induction. Such a questionnaire is the last step for a more wide developmental
study in which group of students have the same training about learning to teach geometrical transformations (Thaqi, 2009).
The activities have been realized in usual classrooms
of the Faculty of Education. In the group there were 18
students of 3rd grade of the study program of primary
education. Some other questions were added to identify reasoning and specific cultural elements about
geometric transformations, ideas about teaching
and learning, and about their thinking about future
classrooms in teaching geometrical transformations.
The students come to the final test after having taken
training course (Table 1) about teaching geometric
transformations in the school, during spring semester
2013 course. The students were given the question-
Aspect of meaning of geometric transformation
Identified Activities
Isometrics and the everyday
life (SI)
Presentation: An experience about isometric transformation - (SIP)
1.2. The activities about the isometric transformations (SIA)
1.3. Didactical activity: Presentation - video of teaching symmetry (SID).
Learning the usage and the
value of the sources to teach
the transformations (SR)
2.1.Presentation: Scientific article as resource for teacher education (SRP)
Projections and shadow (SP)
3.1. Recognition of work about shadow in primary school (SPP)
2.2. Activities about didactical sources and geometrical transformations (SRA)
3.2. Properties of shadow (SPA)
Reasoning, arguing and justification of geometric transformations (SA)
4.1. Presentation of topics (SAP)
4.2. Activities about reasoning, proof and justification of geometr. transformations.
(SAA)
Table 1: Sets of questions related to mathematical ideas about transformations
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The meaning of isometries as function of a set of points… (Xhevdet Thaqi, Joaquim Gimenez and Ekrem Aljimi)
naire the last day of the course, and all students have
responded to the questionnaire. The issue in focus is
the identification of the prospective teachers’ concept
images and the way they make use of their images and
the mathematical definition of certain concepts for
geometric transformations that they will find central when they begin their professional life as mathematics teachers. The selection of questions in the
questionnaire is closely related to the realized, the
same four sessions of didactic practice on learning
to teach geometric transformations, in the Faculty
of Education in Gjilan.
In the sessions showed in Table 1, there are presented
activities that one of the investigation goal was to prepare activities where, the basic knowledge of geometrical transformations are based in the intuitive teaching and experiences about the search, the discovery
and the comprehension from the prospective teacher.
In this way, the prospective teacher learns the knowledge and geometrical properties from the everyday
world aspects constructing them as the concept image. In the first sessions (SI and SR) there is realised
the treatment of the isometrics, the development of
activities of using various resources. In the session
SP there is developing activities to learn how to teach
the non-isometric transformations (deformations,
projections), while the session SA holds activities of
the development of the capacity of arguing, justifying
and reasoning in the process of personal construction
of the meaning of geometrical transformation.
The students were asked to give an explanation to the
following aspects: terminology and type of transformations, properties and relations on transformations, processes of changes, and other aspects about
geometric transformations as reasoning, teaching,
attitude, etc. We analyzed how the prospective teacher
approaches to recognize isometric transformation
and construct concept images during the process of
a practice of learning isometric transformation; how
is the level of recognition of the relationships and the
hierarchy between different properties, and levels of
difficulties in reasoning about the isometric transformations and communication of the results.
RESULTS
In the analysis of the context (Thaqi, 2009) we have
seen that the Program of Geometry in Faculty of
Education, talks about the formative teaching, and
plans the contents as a set of knowledge and procedures. The study of geometry in this program has as
a goal the mathematical knowledge, that basing on the
qualified posture as formalist it can be understood as
the rules that from some affirmations logically are
followed some others. We present now the analysis of
the moments of progress or difficulties of the student
for future teacher of Primary school, in the process of
building the idea of geometrical transformation that
figure A is transformed in figure B, the usage of the
adequate terminology in every case and identification
of different types of transformations. Few students
talk explicitly about the isometrics as a transformation that conserve the size and shape. Instead, they do
identify the symmetries, rotations and translations as
transformation as such property. Anyhow, in some
cases, the activity makes the intuitive go ahead the
structured knowledge. In that way, when we are in
front of the observation of the embroidery, some students show rotation as a unique isometric, since they
identify it as the only transformation that acts on the
module that is marked (Figure 1). So, the conceptual image of the geometrical transformation is build
basing on visual properties (transform=deform), and
movement (Isometric=displacement).
Firstly, let´s say that in some cases the transformation
is seen as a function of a set of points to the other one,
but that function isn´t identified between the posi-
Figure 1: Reproduction embroidery using mirrors
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The meaning of isometries as function of a set of points… (Xhevdet Thaqi, Joaquim Gimenez and Ekrem Aljimi)
tions of the objects in two different places. This can
be explained with the symmetric figures, because it
gets established easily the correspondence between
the two parts of the figure or object. Instead, for other
types of transformations, they should imagine the
initial and final position of the transformed figure
to be able to establish the idea of transformation as
function or correspondence between parts of figure.
We got convinced that it´s that way when we analyzed
the answer to the problem where there is asked to explain the transformation of the figure A in the figure B,
when Blerina (participant of investigation) explains
it using transformation of set of points in the figure
A on the correspondent points of the figure B. Only
one of all participants’ talks about isometric transformation as a transformation that conserves the shape
and size – conservation of shape and size is the definition of function; while the others identify it as a
repetition - which is associated with difficulties on
identifying properties of transformations. We find
that considering transformation with identification
of invariants (form and size) is better level than considering transformation as simple repetitions.
Indeed, in the cases of considering transformation as
simple repetition (that is equivalent with considering
transformation as a function of whole figure to whole
figure) they do not reach to precise the angle of the
rotation, around what point or axis, etc. Actually, as
the Figure 2 shows, to consider transformation as a
function of set of point of figure A to set of points of
figure B, they feel the need of naming the vertex of the
triangle, and later on, they expresses the functional
dependence:
Blerina: .... firstly there has been a displacement
of the point A, and during the movement
of the point A, the point B and C get the
position presented as in the figure.....
In that way, it is constructed the idea of rotation as a
function of three points in other three points under
the condition:
Blerina: … so the point C has gone out of the column of the point A one row lower, the
point B has gone on point´s A place.
The second step of this process will be to indicate the
axial symmetry with the axes of the symmetry instead
of the mirror: “…we would see it better if we would
imagine a mirror placed in the column of the point A,
where the point A is not reflected (moved) and instead,
the points B and C are reflected” (explain Blerina). We
observe now the findings in the case of the deformations. The activity SAA5 shows the dynamic transformation of a triangle – two stable vertexes and the third
one move horizontally inside a segment (Figure 3a).
The students have to explain the properties that are
observed in this transformation.
About the transformation with the property of invariance of the surface and the variables; it´s interesting
for us that the students first identify these properties
Figure 2: Transformation of triangle as function
Figure 3: Dynamic transformation of triangle. Invariance and variable
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The meaning of isometries as function of a set of points… (Xhevdet Thaqi, Joaquim Gimenez and Ekrem Aljimi)
and then justify the announced result. We find that
Blerina identifies correctly the elements of the triangle that change and the elements of triangle that don´t
change. She identifies the change of the shape, perimeter, position and what does not change like surface,
base of the triangle and the height of the triangle. As
an illustration we show the part of the dialogue with
Blerina, who does a correct justification that the surface of the triangle does not change using the correct
symbolization (the formula for the calculation of the
surface of triangle: (S = bh
) basing on the definition of
2
the surface of triangle as a product of the base and the
height (Figure 3b):
Blerina: The triangle gets converted in a
rectangle triangle...
Tutor: What does change in this process? What
does not?
Blerina: The height of the triangle doesn´t change.
Tutor: What other changes do we have?.
Blerina: The angles and the sides change and, the
surface and height do not.
Tutor: How can you argue that the surface does
not change?
Blerina: The surface of the triangle is in function
of the base and the height. As said, the
base and height does not change and
either does the height … so the surface is
constant independently of the position
of the upper vertex.
With the development of this and of the activity SAA4
we identify that the dynamic presentation of a deformation makes it possible that the students approach
to recognize and build the concept of geometric transformation as a function of variables and constants
S=b·h/2 ( this is to identify the elements of deformation
that are conserved and variables). In the activities
SRA3 there is asked to draw a symmetric image of the
figure, having as a symmetric axis the straight line
drawn (activity SRA3, Figure 4). The students have
mirrors as a didactic resource for their activity. We
haven´t noticed in the observation that any of the students didn´t reach to reproduce the symmetric figure
from the given one. In the video recordings we have
found important to describe the process of reproduction for some students. Before Blerina started to do
the construction of the symmetric figure gives the
comment: “I draw any figure in the plot of points. Later
I draw a straight line in that way that it touches one
vertex of the given figure. After that, I count that the
straight line has to be the axis of the symmetry. Is it
so? …. This means that I do an function of each vertex
of the figure counting little squares in the other side of
the axis...” Blerina does the symmetry using the properties of the symmetry: the axis, the same distances
to the axis and the process of the application point-topoint. It is not needed to determine each point of the
figure, she finds the vertexes of the figure and later,
she uses the property of the aligned points: aligned
points get transformed in aligned points (Figure 4).
High grade students recognize the transformation
if they recognize the relevant elements of geometric
transformation using the process of the function
point-to-point, and the property of the aligned points:
aligned points get transformed in aligned points.
CONCLUSION
Analyzing this findings we can say that in the program
of mathematics for prospective teachers the main goal
of the teaching of geometrical transformation is the
informative knowledge, while is necessary program
for the teacher education, intended to cultivate and
practice the logical reasoning. We consider that the
best is an equilibrating education between thinking
and acting, or between the cultivated knowledge and
Figure 4: The process of construing symmetric figure
596
The meaning of isometries as function of a set of points… (Xhevdet Thaqi, Joaquim Gimenez and Ekrem Aljimi)
practical knowledge. In the cases that students recognize the transformation as a process of function of
points to points, it is easy to identify the functional
dependence between positions, the important properties of the transformation such as symmetry axis,
vector of translation, center and angle of rotation, etc.
and it established the complete concept image about
isometric transformation. In cases when students consider isometric transformation as a fold, changing
position or repetition of an object or figure, they are
confronted with difficulties to establish the important elements of the transformation process. In other
words, those who construct the idea of t​​ ransformation
as correspondence between sets of points do not find
it difficult to have the complete concept image about
isometric transformation, correctly identifying the
properties and elements of that transformation as
the orientation of the image, the axis of symmetry, the
angle of rotation, the translation vector, invariance
and variables etc.
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ACKNOWLEDGMENT
formation as viewed by prospective teahers. In Pytlak, M.,
Rowland, T., & Swoboda, E. (Eds.), Proceedings of CERME8
This research and presentation in CERME 9 is supported by GIZ (Deutsche Gesellschaft für Internationale
Zusamenarbait GmbH), office in Republic of Kosovo.
(pp. 578-587). Rszeszów, Poland: ERME.
Thaqi, X., & Gimenez, J. (2012). Prospective teacher’s understanding of geometric transformation. In 12th International
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