GEOMETRICAL TRANSFORMATIONS AS VIEWED BY
PROSPECTIVE TEACHERS
Xhevdet Thaqi*, Joaquin Giménez**; Nuria Rosich**
*University of Pristhina; **University of Barcelona.
An empirical study is developed to find attributed meanings for geometrical
transformations with prospective primary teachers in Kosova and Spain. The study
reveals the influence of students’ previous background, more than cultural
differences.
Key words: Geometrical transformations, prospective teachers, comparative study
INTRODUCTION
One new challenge for teacher training on European Higher Education is to reduce
international differences, face to immigration processes, exchanges, and globalization.
There is a continuous interest of CERME community to understand how mathematical
practices are developed by using cultural writings or lesson marks in different
countries (Stigler et al, 2000) as culturally situated mathematical practices (Llinares &
Krainer 2006); and how mathematics are construed by participants as a hidden
variable in researching mathematic knowledge for teaching (Andrews, 2009). In
particular, some authors explain differences in the use of geometry: From natural
perspective in Latin countries instead of more soviet or German axiomatic perspectives
(Girnat, 2008). There are many researches about knowledge and use of geometrical
transformations in Secondary School (Hoyos, 2006) but less study have been
conducted for Primary Schools. It’s also shown that difficulties at Primary students’
conceptualizations (Williford, 1972) depend on a weak knowledge of teachers (Law,
1991 quoted by Yanik & Flores, 2009), in particular on geometrical transformations
(Pawlik, 2004).
Such studies reveal that teachers’ lack of students Mathematical Knowledge and
confidence in mathematics are contributory factors to the low standard of mathematics
attainment of their pupils in many countries. It’s also known from several authors that
pre-service elementary teachers have difficulties in determining: (1) the correct
transformation and motion attributes 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 required specification of
inputs but instead as particular actions, each with given ‘default’ or prototypic
parameters. It was also observed that the use of technological devices has strong
advantages facing the use of isometrics, because of possibilities of variability analysis
(Harper 2003). A recent study about the prospective teachers’ knowledge of
translations and other rigid transformations (Yanik & Flores 2009), revealed that
scholars (1) started by referring about transformations as undefined motions of a single
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object, (2) followed by using transformations as defined motions of a single object,
and (3) understanding about transformations as defined motions of all points on the
plane.
In Spain, some studies using Van Hiele’s levels, found prospective Primary teachers’
difficulties in using symmetrical notions by answering isometrics tasks (Jaime &
Gutierrez 1995), but a few proposals were drawn to analyze qualitatively what are
teacher’s ideas before developing professional tasks. In fact, while European
mathematical tradition uses Klein’s perspective about transformation geometry, just
French and German curriculum is explicit in doing it for Secondary School. Even the
term “transformation” is mentioned only at the end of secondary school and don’t
solve the problem of transition from the use of natural environmental geometry in
Primary Schools into Secondary school axiomatic perspectives (Kuzniak & Vivier
2008). Instead of it, in many Latin countries it’s observed just in Analytic Geometry
tasks from 15 years-old pupils (Bulf, 2008). Therefore, in our research study, we focus
on analyzing influences of previous prospective teachers’ cultural background before
developing training activities about learning to teach geometrical transformations. We
studied and compared the results in Kosova and Spain in a bridging collaborative
international framework (Jaworski 2006), where we expect to find different
conceptualizations in their responses.
METHODOLOGY
It was planned an ethnographical research as a case study, with two separated groups
of future teachers: 13 students of a 2nd year course at Faculty of Teacher Training at
Barcelona University (UB) in Spain, with only one mathematical/didactical subject and
15 students from Faculty of Education at University of Prishtina (UP) in Kosova, with
two previous geometry courses based on classic Euclidean geometry, but no previous
didactical training. They are 18-22 years old students. A previous curricular-cultural
comparative analysis based on textbooks, official curricular proposals and teachers’
training materials, showed deep differences among both previous preparatory and
culturally different frameworks (Thaqi, 2009), not detailed in this presentation.
The results of a starting questionnaire, was the basic data considered in this paper, in
order to analyze beliefs, meanings and prototypes, from students’ texts transcriptions.
Such a questionnaire is the first step for a more wide developmental study in which
both groups of students have the same training about transformations in geometry
(Thaqi, 2009).
A semi structured questionnaire is designed by using 14 open (mainly contextualized)
written questions, plus consequent interviews for we considered necessary to capture
students ideas about the topic (see the main ideas in Table 1). Some other questions
were added to identify reasoning and specific ethnic-cultural elements about
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geometrical transformations, ideas about teaching and learning, and about their
thinking about future classrooms on geometrical transformations.
Aspect of meaning of geometrical
transformation
Identified Activities
Terminology. Types of transformation.
1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 14
Properties. Relations and hierarchies
1, 3, 4, 5, 6, 7, 11, 12, 14
Transformation as a process or simple change
1, 2, 3, 4, 5, 6, 7, 9, 12
Others aspects (reasoning, teaching, etc.)
8, 10,13, etc.
Table 1: Sets of questions related to mathematical ideas about transformations.
Further analysis about learning to teach transformations professional activities was
given from videotaped transcriptions not included in this article. Furthermore, first
sessions serve as confirmation of attributed meanings found in the questionnaire,
because it gives information about training practices (Stigler, Gallimore and Hiebert,
2000: 87) showing us cultural objects with specific languages or symbolic systems.
Data collected through developmental process were analyzed using ongoing analyses.
During the ongoing analysis phase, the researcher tried to understand the participant's
way of thinking not presented in this article. After each teaching episode, the
researcher-team coded and analyzed the video records of students’ interactions to the
given tasks. The main purpose was to find patterns and create descriptions of the
students’ mathematical knowledge development over time as an hypothetical learning
trajectory of participants. During ongoing analyses the researcher tested his initial
hypotheses and generated new conjectures to be tested in the following teaching
episodes.
ABOUT ATTRIBUTED MEANINGS
Based upon students answers in both countries, we divide the results into three parts:
(a) about the meanings and use of geometric transformations as a mathematical object
and associated examples; (b) definitions, conceptual structure, and (c) representations
and non isometric transformations.
Transformation as mathematical object.
To identify degrees of knowledge (table 2) we assume they build more or less pseudoconceptual perspective (according Vinner, 1997) by analyzing their justifications,
argumentation, properties and use of examples and counterexamples. It was not
surprised that none of the students has shown a consolidated knowledge about the idea
of transformation, or the idea of transformation as a function, even in the case of
projection (usually defined as a function). The majority (64%) belong to an
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intermediate level in the case of University of Prishtina (UP). The main class of
mathematical objects they identified is symmetry, as we expected.
Degree of knowledge about transformation as a math object
Barcelona Prishtina
N=13
A) They are able to build complete images, using terminology and
justifying interpretations carefully with good statements.
B) They show some conceptual images by using prototypical examples
46%
N=15
64%
including some relevant properties Identify the transformation of the
figure without any explicit explanation about properties.
C) No answer or no meaningful explanations. Poor images, based upon
54%
36%
examples and visual prototypical examples.
D) Blank or without any sense
--
--
Table 2: Results compare between Barcelona (UB) and Prishtina (UP)
We analyze student’s texts by observing their answers to find their ideas about the set
characteristics to find semiotic conflicts.
We deduce that Kosovar students (UP) assume a “transformation perspective” by
using deep mathematical expressions. For instance, the student Vj when talking about
tiles associates rotation as the only movement to when associate different type of
isometrics. For instance, when we ask for tiles, Vj indicates: “…they design the part
of the figure through the paper to be turned in order to obtain the whole figure. Thus,
it will show the rotation” (Vj, p5:3, UP). In some other cases pupils identify the
expression “through displacements...” as a way for describing the transformation that
generates figures from a module.
In the case of Spanish students, transformation is mainly associated as a simple
relation between objects and their transformed by changing some characteristics
(called undefined motion in Yanik & Flores, 2009). The change of the position is not
always taken into consideration: “... the movement does not mean a change of form,
but only the position, while the transformation involves change of the form” (Al, p.9:
8, UB).
A few students told us about the invariant terms of a transformation, and give
interesting properties about transformations as the knowledge of repetition by period
T=2π (student Pe in UP) when they explain the rotation of a door. When we ask “tell
us some statement to give meaning for rotation”, just some students from UP spoke
about invariance of shape and size when talking about isometrics. They used the
expression “change of a same thing”, and others used the expression “without
changing…” even when they talked about projectivity.
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A lot of mathematical inconsistencies appear with Spanish students, associating the
example of rotation with the class of isometrics, and telling that a rigid movement is
not a transformation. In some cases, the question seems to promote intuitive or
pseudo-conceptual knowledge, more than structured, as it’s the case of tiling, in which
students explain rotation as the only isometric transformation. In fact, “…I understand
a movement to take some object or image and displacing it, without any change...·”
(Mc, p5: 2 – 3, UB).
About definitions, properties and structure
Only in two cases we find that the (isometric) transformation is seen as defined
motions of all points on the plane. This is the case of Ad (UP) after naming the
vertices of the triangle, express the functional dependence between the positions of the
same:”... first has been the displacement of point A, and during the movement of
point A, points B and C takes position presented on figure (see Fig. 1a), so the points
have changed columns ... the point B has moved to the place of point A …“ (Ad, p12:
7-8, UP). Few UP students adequately affirm that the movement is only one type of
geometric transformation and established a correct relation between the property of
conservation of size and form on isometric transformations.
b)
Fig.1 Isometric representation as a product of two symmetries by Ad
(UP) and Jo (UB)
a)
In the case of UB, most of the students identified isometrics transformation as
repetitions, but generally symmetry is not identified in that set. Another group of UB
participants are limited in identification of visual characteristics of symmetry, rotations
and translations. The main image for a transformation is based upon visual
understanding (transforming=deformation), and isometrics movement is interpreted as
displacement. Few students visually identified the translation vector, and confirmed
that the translation of a figure is equal to the product of two axial symmetries with
parallel axes (Figure 2b): “The translation is the product of two symmetries because
this is the symmetry of the symmetry….figure a2 is the translation of initial figure a1.
“(Jo, p8: 4, UB).
Many students of UB identify similarities as a common sense word, without
mathematical explanations. Sometimes they accept “I don’t know the properties of
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similarity”. In general in both countries, not all necessary elements for definitions have
been stated, as change of direction in relation to some movements. When talking about
rotation, only angles are observed, but no reference point, and equal distance. Some
Kosovar students (UP) establish a right relation between conservation properties and
corresponding transformation, but in the case of UB there is only one, because they
talk about repetitions or similar figures as common sense. As an example, student Al
says: “It can be when you open an orange, because it can appear equal parts, but
they are not” (Al, p5:2, UB).
When they try to establish a functional approach, there was carried out just a figural
perspective, by arguing with artefacts as paper folding, or sun rays. Just two students
from Kosova (UP), identified a functional idea for similarity by using conic sections
(Figure 2) presenting their previous mathematical knowledge.
Figure 2. Drawings showing the transformational idea for similarity
Conservation of shape is observed by almost all participants of UP students, but not in
UB, as it’s viewed as a figural phenomenon. A nice description was given by student
Ar who said: “the shadow will be different in a human body example,… the focus of
light is considered the centre of the projection, and light arrows are the right lines
for the projection.” (Ar, p10: 7, UP).
Figural explanations are typical in UB, and they only explain light/shadows
phenomena in terms of dependence, but without any explanation about the transformed
elements, indicating the change, not the application. Some UP students told us about
dependence, indicating the main variables: “the shadow depends upon the place where
is observed, if the stair seems to be broken, on the contrary it’s a plane “(Ad, p10:35, UP).
In UB, some students explain the transformation as a relation among two different
stages of an object. Other students see the transformation as a radical change of a
physical object. We are really surprised that many of the students feel that projections
are not transformations recalling us the natural properties, not the mathematical trends.
Let’s say Na comment: “There are not transformations. The shadows are the
projections of an image, because of light focus” (Na, p10: 4, UB).
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About representations and non-isometric transformations.
This part is not analyzed in many studies, In fact, visualization of on isometric
transformations is usually trained by using problem solving activities. Nevertheless,
they are not enough for being recognized for UP students in Kosova. We have more
consolidated answers in UB because it’s more usual for them to be confronted with
isoperimetric manipulative tasks. It’s also expected to find that deformations are a not
considered globally as transformations. Some students identify projection as
deformation, without explanation about the conservation of the shape. Let’s observe
the example: “…when we work with shadows, we say it’s a work about projections
because the figures have been deformed. An image is obtained from another, as we
see in the overhead projector. Projected image is deformed, it stretched or enlarges”
(Ol, 10:9, UB)
One possible explanation is that Klein’s perspectives are not introduced in the
curriculum and many teachers do not know about it. Figural judgements are based
upon a few amount of prototypical examples, by comparative wrong arguments. Let’s
see an example in which Da says: “the transformation converting a rectangle 3cm X
7cm with a 20 cm string into another using the same string, is a conservation of
perimeter and area” (Da, p14:5, UP).
CONCLUSION
The common general low results in both countries about transformations reveal a lack
of previous background, not only because of a lack of mathematical knowledge, but a
lack of confrontation tasks in which transformations plays usually a restricted
mathematical role. In Spain, it was found the influence of natural images to build a
functional idea of projection, but a lack of deep understanding about the role of
properties in definition processes. This aspect was better in case of Kosovar students,
as it was expected because of such a German-Russian tradition. In general, the fact
that similar results appear, reveal that mathematical content knowledge background is
not enough to build these concepts. None of the students in both countries have a
complete concept of transformation and structure as a function and they show figural
images different in each task.
Variability construct is needed to understand the Klein’s meaning of invariance for
associating geometries to transformations. It also means that simple visualization is not
enough to understand such concepts. More emphasis on a wider sense of
contextualization and confronted dialogue is also needed to discuss about types of
transformations in a functional feature. Nevertheless the Euclidean orientation of
Kosovar curriculum for low secondary school, gives possibilities to find some students
relating their previous theoretical framework with didactical purposes as future
teachers. In fact their comments are based not only by intuitions about transformations,
but relating mathematical knowledge. We also found that in both countries, students
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had no enough time to build powerful images about types of transformations, and we
suggest the need for having experiences of other transformations than isometrics.
During the developmental activities, we could reinforce such a didactical research
conjecture (Thaqi, 2009) by doing professional tasks in which we insist on invariance
as a phenomenon. It’s possible because representing a transformation with a function
notation requires more abstract thinking and is crucial for understanding
transformations as one-to-one mapping of the points of the plane (Hollebrands, 2003).
Our results are coherent with the emergent dialectics global/punctual (Bkouche 1991,
Jahn, 1998) as a semiotic conflict.
We also found the importance of using interactive environments to analyze invariance
(Harper 2003) and the need of visualizing when doing global transformations (Olkun et
al 2009).
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APPENDIX
Some items of the questionnaire.
• Observe
the
typical
Kosovar
embroidery as it’s shown in the figure.
There is a repeated part. Find it and
draw it in the figure. Explain how to
use a transparent paper to show the
rotation as a transformation conserving
the size. How do you call the
transformations which conserve size
and shape of an object but change the
position of the object?
• Observe the tiling in the following
figure. There is one repeated part
Find it and draw it. Explain how to use
a transparent paper to show the
translation as a transformation
conserving the size. How do you call
the transformations which conserve
size and shape of an object but change
the position of the object?.
• Present three tasks you can use to explain symmetry and three examples to
explain similarity (homothetic).
• Explain the meaning of the statement “Translation is a product of 2 symmetries
“
• Transformation and movement is the same? Explain it.
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