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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. CERME 9 - Ninth Congress of
the European Society for Research in Mathematics Education, Charles University in Prague, Faculty
of Education; ERME, Feb 2015, Prague, Czech Republic. pp.591-597. �hal-01287028�
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, xhevdet.thaqi@uni-pr.edu, ekremhalimi@hotmail.com 2 Unversity of Barcelona, Barcelona, Spain, quimgimenez@ub.edu
In this paper, we try to show that in the process of un- derstanding of isometric transformations, the mean- ing 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 percep- tion 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 fu- ture 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 imple- ment a practice on learning to teach the mathematics;
to analyze elements of the constructions of person- al meanings of future teachers about mathematics and, to recognize the difficulties of the students to understand, relate and organize mathematical con- tents, terms and properties associated to that content.
During last years with my colleges we tried to contrib- ute in this process focused concretely in geometri- cal 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
difficulties with concept images constructed about geometrical transformation.
Some investigations have highlighted the reasons and advantages that provide the study of geomet- rical transformations (Jackson, 1975; Küchemann, 1980; Jaime, 1993; Harper, 2003; Jagoda & Swoboda, 2011; Thaqi, 2009). A main reason to study the trans- formation is curricular, since “The transformations are applications of the geometrical functions, and this treatment is fundamental for all the mathemat- ics” (Jackson, 1975, p. 554). Other reason is that the transformations provide geometrical dynamic task.
Despite these reasons and advantages that the trans- formations 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 addition- al formation in formal geometry and empathize the conceptual understanding, starting from the analy- sis of the geometrical environment with conceptual explorations. The study of how prospective teachers build a meaning of the concept of isometrics transfor- mation 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 investiga-
tion: inductive, deductive and inventive building of
The meaning of isometries as function of a set of points… (Xhevdet Thaqi, Joaquim Gimenez and Ekrem Aljimi)
592 concept. Later on, the same author tells us that the
concepts are the results of accumulated social expe- riences (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 com- mon characteristics .
The conceptual construction rests on a set of process- es of construction, visualization, exploration of prop- erties, elaboration of explanations and classifications, within others. Among more and better experiences we have, the conceptual image gets closer to the con- cept because, like Vinner and Hershkowitz states:
to acquire a concept means, to acquire a mecha- nism 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 defini- tion” (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 pro- cesses and properties (Tall & Vinner, 1981, p. 152).
Jaime & Gutiérrez (1995) state that in the formation of images of a concept that a person has, the experience
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 men- tal 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 knowl- edge about geometrical transformations. It has also been shown by several authors that pre-service ele- mentary 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 combi- nations 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 transfor-
mations 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.
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 theoret- ical 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 math- ematical education of the teachers, we considered adults who acquire a professional scientific knowl- edge, and our work that has transformed a vision of the processes in the creation of a meaning, we iden- tify 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 con- clusions 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 previ- ous curricular-cultural analysis based on textbooks, official curricular proposals and teachers’ training
materials, of these context is the same as showed in the deep study (Thaqi, 2009), not detailed in this paper.
During the development of the practical sessions dedicated learning to teach geometrical transforma- tions, 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, in- terviews and video records. The collected data was analysed using method described by Strauss & Corbin (1998) along with analytic induction. Such a question- naire is the last step for a more wide developmental study in which group of students have the same train- ing about learning to teach geometrical transforma- tions (Thaqi, 2009).
The activities have been realized in usual classrooms of the Faculty of Education. In the group there were 18 students of 3
rdgrade of the study program of primary education. Some other questions were added to iden- tify 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 geomet- ric 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) 2.2. Activities about didactical sources and geometrical transformations (SRA) Projections and shadow (SP) 3.1. Recognition of work about shadow in primary school (SPP)
3.2. Properties of shadow (SPA) Reasoning, arguing and justi-
fication of geometric transfor- mations (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
The meaning of isometries as function of a set of points… (Xhevdet Thaqi, Joaquim Gimenez and Ekrem Aljimi)
594 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 cen- tral when they begin their professional life as math- ematics 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 pre- pare activities where, the basic knowledge of geomet- rical transformations are based in the intuitive teach- ing and experiences about the search, the discovery and the comprehension from the prospective teacher.
In this way, the prospective teacher learns the knowl- edge and geometrical properties from the everyday world aspects constructing them as the concept im- age. 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 transfor- mations, properties and relations on transforma- tions, 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 transfor- mations 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 proce- dures. 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 transforma- tion 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 stu- dents 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 conceptu- al 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
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 ex- plain 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 transfor- mation as a transformation that conserves the shape and size – conservation of shape and size is the defi- nition 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 con- sidering 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 col- umn 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 deforma- tions. The activity SAA5 shows the dynamic transfor- mation 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 invari- ance 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
The meaning of isometries as function of a set of points… (Xhevdet Thaqi, Joaquim Gimenez and Ekrem Aljimi)
596 and then justify the announced result. We find that
Blerina identifies correctly the elements of the trian- gle that change and the elements of triangle that don´t change. She identifies the change of the shape, perim- eter, 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 sur- face of the triangle does not change using the correct symbolization (the formula for the calculation of the surface of triangle: (S =
bh2) basing on the definition of 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 defor- mation makes it possible that the students approach to recognize and build the concept of geometric trans- formation 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 stu- dents 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 reproduc- tion 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 prop- erties of the symmetry: the axis, the same distances to the axis and the process of the application point-to- point. 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
practical knowledge. In the cases that students recog- nize the transformation as a process of function of points to points, it is easy to identify the functional dependence between positions, the important prop- erties 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 con- sider isometric transformation as a fold, changing position or repetition of an object or figure, they are confronted with difficulties to establish the impor- tant elements of the transformation process. In other words, those who construct the idea of transformation 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.
ACKNOWLEDGMENT
This research and presentation in CERME 9 is support- ed by GIZ (Deutsche Gesellschaft für Internationale Zusamenarbait GmbH), office in Republic of Kosovo.
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