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Universal Journal of Educational Research 4(9): 1944-1949, 2016 http://www.hrpub.org
DOI: 10.13189/ujer.2016.040903
Geometrical Transformations in the Mathematics
Textbooks in Kosovo and Catalonia
Xhevdet Thaqi1,*, Joaquim Gimenez2
1Faculty of Education, University of Gjilan "Kadri Zeka", Kosovo
2Faculty of Teacher Training, University of Barcelona, Spain
Copyright?2016 by authors, all rights reserved. Authors agree that this article remains permanently open access under the
terms of the Creative Commons Attribution License 4.0 International License
Abstract The rationale behind this study concerns the student's post learning (or intended learning). It is a more
issues school administrators and teachers/prospective accurate diagnostic measure to access the student's probable
teachers face over the progress of these students in mathematical skill from the "old" curriculum, and it also
mathematics classrooms in two different countries provides more insight into the "new" curriculum which may
(Spain/Catalonia and Kosovo). This study aimed to present a assist in a smoother transition from a curriculum to another
method of examining primary school mathematics textbooks curriculum. The quotation by Robitaille (1995) below further
with the purpose of evaluating students' expected past stressed the importance of textbook analysis from the
learning and comparing students' expected geometrical mathematics education perspective: "Mathematics textbooks
transformations learning across the different curricula. The exert a considerable influence on the teaching and learning
analytical procedure of the examination of text content as of mathematics, so understanding of how textbooks vary in
presented in this study is itself a diagnostic technique for their content and approach across countries is an important
assessment of the students' past learning, which addressed area of investigation. An analysis of textbooks makes an
the main objective of the study. As a main result, we found important contribution to understanding curricula in
that the organization of content is based upon a disconnected particular country..." (p.5&61).
way of presentation about transformations, and a few In this presentation we intend to examine Primary
attentions paid to variability issues. mathematics curricula in the two countries (Catalonia and
Keywords Geometry, Transformation, Mathematics
Kosovo) and to formulate a comparison of the curricula
Education, Mathematics Textbooks
across nations based on the textbooks geometrical
transformation content. The significance of this study is
two-fold: At one level, the study provides information about
cognitive requirements of geometrical transformations
content in elementary textbooks and this information can be
1. Introduction used to improve mathematics curriculum and teaching in
other countries. At another level, it explores new ways of
We know that it's loose treatment of geometric gaining information from these comparative assessments of
transformations in primary education and teacher training in mathematics achievement and, possibly, limitations of
many European countries (Svoboda 1999). Faced with the detailed inference from such testing. Textbook is being used
challenges of globalization of education systems and teacher in mathematics classrooms very frequently all over the world
training in mathematics in Europe, we need to know even (Johansson, 2005). However, the importance of textbook
more recent origins of the students' previous training. was not realized as much as today. To emphasize that
Recently there have been developed some comparative textbooks both reflect national curriculum and influence the
studies to show the differences between countries from East teaching practices critically, it has become natural to put a
and West Europe with different geometric traditions. From a level as "potentially implemented curriculum" between
semiotic and anthropological approach, we consider that the intended curriculum and implemented curriculum
textbooks analysis is a tool that realizes the institutional (Johansson, 2005). When we consider curricula of countries
situation between countries. An analysis of textbooks can all around the world we also see that geometrical
make important contributions to an understanding of transformations are very important in school mathematics
curriculum in a particular country (Kunimune, and et altr.
2009). It would serve to provide a window into the
educational system, which might lead to an indication of 1Robitaille, 1995,
Universal Journal of Educational Research 4(9): 1944-1949, 2016 1945
curricula (NCTM 2000). the geometrical transformations concept image (Thaqi,
Our hypothesis is that the notion of geometric 2009).
transformation is untreated in previous training of the According to the Catalan official curriculum, the number
prospective primary teachers, because it appears barely on of math activities is not determined and it is the teacher who
their primary education as conceptual/relational construction. is responsible for planning and developing activities and
In particular we plan to observe in Primary school textbooks, classes as other conditions and circumstances for each
what kind of transformations are, and how are they subject. So there is no a priori determined number of classes
structured? And what is the image of geometrical devoted to math to do in school. Within mathematics classes
transformations concept that emerges?. We accept the fact are planed developing of geometry classes with few
that the visual is of significant importance in the first level of activities devoted to geometric transformations.
geometrical cognition (Jagoda & Swoboda, 2011), and
according to this theory all perceived regularities are
classified as visual structures. The things that inspire 3. Geometric Transformations in
children, propel them to action and which undergo control Curricula and Textbooks
and are reflected upon are: rhythm, order and regularity.
First we present some benchmarks of the curriculum,
compared with those who had at that time in the NCTM
2. Methodology standards that began to influence Europe, but not yet
actively.
For most teachers of mathematics, the textbook is their Teaching some geometric transformations has been
primary guide for curriculum implementation. Current present in all curricula of primary education, though in
prospective primary teachers were formed in the plans of the different ways. It is based primarily on the study of isometric
90s in the case of Catalonia (Spain) and with new programs transformations (symmetries, rotation and translation).
of 2002 in Kosovo. Therefore we decided to analyse Intuitively arises only something on similarity and
textbooks associated with that period. The criteria for the congruence and projective transformation, but only as
selections of textbooks were based on the selected textbooks construction of plane representation of an object from
being written in a language comprehensible by us, and that three-dimensional representation of the object (NCTM) and
the textbooks were used in countries with a relatively high understand that a" figure is different depending of the point
number of schools institutions. It is anticipated that such an of view that we look "(Catalunya). Little is said about the
investigation will be of value to those responsible for the variation and types of transformations, as well about
correct level of placement of these students. Two sets of transformation as process or change and even less about
textbooks from these countries, Catalonia and Kosovo, were communication and reasoning with transformations and
selected for this study. The textbook geometrical cultural and historical elements in transformations (Svoboda,
transformations content of each country was analysed in 2007; Thaqi 2009). Referred only to the "use of geometric
terms of strand weighting and content details. The findings ideas to describe situations and to communicate information
facilitated a comparison of the learning paths offered by the "(Curriculum Catalunya) but generally as changes of
various textbooks, fleshed out the differences and position and not speaking specifically of geometric
similarities of the various curricula and made available transformations as functional relationships. Structurally
detailed comparisons of the textbooks' geometrical speaking, the processes are applicable verbs like, identifying,
transformations content in terms of topics covered. This will recognizing, and building relationships. There are very few
decide to choose a text that for some time has had only a connecting verbs and reasoning. Few actions are indicated in
Catalan edition, the "Text - la Galeria". It is different in the taxonomy of establishing hierarchies and when talking
Kosovo. The Ministry of Education of Kosovo (MASHT) about construction, these actions are related to geometric
decide editor of the textbooks to be used by public schools. property of congruence. A transformation is indicated only
In recent years, the editor of the textbook "Dukagjini" were as change in their intuitive manipulative aspect. Mistakenly
chosen for the subject of mathematics in primary schools, used the term generator to find the source element of given
and for this reason we consider these books as reference for transformation and end point, and there is talk of finding the
the Kosovar context. result of a transformation in construction way by different
The results were analysed and compared among the artefacts.
textbooks of two countries in terms of cognitive domains that About communication and reasoning, it insists on
these content required. The results were interpreted in the recognizing that different points of views are associated with
context of the textbooks and relevant curriculum. This different transformations, and it comes in the search for
interpretation is descriptive in nature and serves as a means explanations of regularities. No one knows exactly what
to identifying the nature of similarities and differences across value are phrases like "Recognize turns and complex
the textbooks from Spain/Catalonia and Kosovo in terms of symmetries."
1946 Geometrical Transformations in the Mathematics Textbooks in Kosovo and Catalonia
Table 1. Comparative analysis of the curricula about geometric transformations
Geometric Catalonia Kosovo
transformations
About the object Symmetry, turns, shadows Symmetry, rotation, translation
transformation, Identify figures with symmetry axis, Differentiate symmetrical figures from unsymmetrical
terminology and types of Understand that figure looks different according to the Identifying relation between two isometric figures
transformations (in the viewpoints that look. Recognize, name and define different changes of
plane) Identify the transformation that relates two given figures. positions of figures.
About the relations and Composition and Decomposition of figures Identifying relation between two symmetrical and
isometric figures.
hierarchies in the notion of Recognize rotations and complex symmetries. Construct congruent with the figure given in relation of
transformation Identify the transformation that relates two given figures. symmetry, rotation and translation.
About transformation as a Transformations of geometric figures in a manipulative Recognize, describe, classify, name and define the
changes positions of different figure
process or change Recognize what the object that generated a particular Construction of symmetric line and bisector
shade Construct the congruent figure of given figure.
About communication and Understand that the figure looks different according to Regularities of figures from symmetries
reasoning by the viewpoints that look. Describe, name and define changes of shape positions.
transformations Recognize complex symmetries and turn.
About cultural and Differentiating and symmetrical objects from
historical elements in Identifying geometric figures using models. unsymmetrical.
transformations
Catalunya Kosovo
Textbook: Matematiques 2, Textbooks: Matematika, 2,
Editorial: "Text - la Galera", 2007 Editorial "Dukagjini", 2004
Page 68, /Activity 2/ Page 36: / Activity 2'/
"Now make symmetries in a graph paper; just go counting squares and set "do (build) symmetrical figures to the given figures"
up very well if the lines were to the right, left, diagonally, up or down "
Figure 1. Comparative examples of 2nd grade textbooks of Primary School
4. Results about Textbooks in Catalonia Kosovar textbooks properties are analysed according to the
and Kosovo Soviet tradition, and change processes are recognized in
Catalonia. No conceptual hierarchies are established in the
By looking at the analysed textbooks, we can see that in sense of showing that there are "other changes" to different
the first cycle (1st and 2nd grade) symmetry is identified as symmetry, nor made explicit the "no symmetry". As shown in
an important element, though never referred to it as a Figure 1, both referred to as graph paper simplest form of
transformation. Kosovo textbooks only refer to "build the execution. In both countries are privileged contexts of action
symmetrical". As instructional standards, handling is over observation of the real-social.
privileged as the folded paper in Catalan curriculum, In the textbooks for the Second Cycle of Primary school of
emphasizing it as a process. In Kosovar textbooks, seems to two countries, finding axes of symmetry is the most obvious
distinguish between buildings and transformation. In common element. Only in Kosovo textbook the symmetry is


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