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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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