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Proceeding of the International Scientific Colloquium: The level of technical language use among teacher training college students A matematika nyelv hasznalatanak szintje a tanitokepzos hallgatoknal The development of science and education is a part of a long-term Educati- on Sector Development Plan Following the example of Europe and the rest of the world, special attention in the field of education is given to mathematical literacy of children PISA programme as well as to mathematics teacher training quality insurance in higher education.

Mathematics teaching in Croatia faces modified strategic, organizational, social and technical conditions. Introducing one-shift classes in primary scho- ols, including children with special needs talented ones and those with diffi- culties in regular classes, filozofijj day program for all students, two teachers per class, greater mobility of children and teachers in schools and new teaching technologies demand changes in the methodology of mathematical education of both children and future teachers of mathematics.

It is important to develop life-long learning programme filozociju teachers of mathematics that includes docto- ral studies. Research in the field of mathematics teaching implies multi- and interdis- ciplinarity. Therefore a cooperation with scientists outside the field of mathe- matics psychologists, special-ed teachers, educators is an imperative, although we strongly believe that improvements in mathematics teaching should be en- couraged within the field of mathematics. A precondition for developing new approaches and methodologies in mat- hematics teaching in Croatia is a first-hand experience with the results of inter- national research and standards in mathematics teaching and defining doctoral studies within the same field.

We would also like flozofiju this event to initiate the start of doctoral studies in the field of mathematics teaching in Croatia following the examples from Europe and worldwide. We are very grateful to numerous Croatian and international scientists who have recognized the importance of this event and managed to find the time to attend this gathering.

We would also like to thank the heads and entrepreneurs of the local community who financed this event for the most part. On behalf of the Organizational Committee, I express my deepest gratitude.

Razvoj znanosti i obrazovanja dio je dugorocnoga prioriteta razvoja Jvod ske – Po ugledu na Europu i svijet u okviru obrazovanja posebna se pozornost pridaje, kako matematickom opismenjavanju djece PISA program tako i izo- brazbi ucitelja matematike osiguranje kvalitete u visokom obrazovanju. Nastava matematike u Hrvatskoj je pred izmijenjenim strateskim, organi- zacijskim, socijalnim i tehnickim uvjetima.

Uvodenje jednosmjenske nastave u osnovne skole, inkluzija djece s posebnim potrebama talentirane i one s tesko- cama u redovite odjele, produzeni boravak za sve ucenike, dva ucitelja u odjelu, veca pokretljivost djece i nastavnika u skolama, nove tehnologije u nastavi izi- skuju promjene u metodologiji matematickoga obrazovanja, kako djece tako i buducih ucitelja matematike. Vazno je osmisliti cjelozivotnu izobrazbu ucitelja matematike koja ukljucuje filozfoiju doktorske studije.

Istrazivanja u nastavi matematike pretpostavljaju multidisciplinarnost i in- terdisciplinarnost. Stogaje pri istrazivanjima u nastavi matematike neophodna suradnja sa znanstvenicima izvan podrucja matematike psiholozima, defekto- lozima, pedagozima, istrazivacima iz podrucja informacijskih znanostiiako drzimo da razvoj metodike nastave matematike filozotiju njegovati u okvirima ma- tematicke struke.

Pretpostavka je iznalazenju novih pristupa i metodologija u nastavi mate- matike u Hrvatskoj upoznavanje izprve ruke rezultata inozemnih istrazivanja i strane prakse, kako u nastavi matematike tako i definiranju doktorskih studija iz metodike matematike. Ta- koder zelimo da ovaj skup pridonese brzem zazivljavanju doktorskih studija iz metodike nastave matematike u Hrvatskoj po ugledu na vec postojece u Europi i svijetu.

Vaznost ovoga skupa prepoznali su brojni domaci i inozemni znanstvenici, a neki od njih uspjeli su odvojiti dio fiilozofiju vremena za zajednicko druzenje. Zahvaljujemo im, ali i vrijednim celnicima i poduzetnicima lokalne zajednice koji su u najvecem dijelu sponzorirali odrzavanje ovoga skupa.

U ime organizacijskoga odbora svima od srca zahvaljujem. Upon completion of a three-year study programme in mathematics, students can continue with their studies by enrolling in one of the 4 branches offered k the curriculum. Upon graduating from one of these branches, a candidate is granted an MSc degree in Mathematics Education.


T he paper gives an overview of the curriculum referring to the branch Teaching Mathematics. After successful completion of the first cycle study programme in mathema- tics at the University of Sarajevo, a graduate is awarded a BSc degree, and by completing the additional two years of study the second cycle study program- me a graduate is filozodiju an MSc degree with the branch indicated. There exist rel branches: After successful completion of the Master level programme in the first three branches, a gra- duate is awarded an MSc degree in Mathematics with the branch indicatedwhereas by completing the study programme in teaching mathematics a gradu- ate is awarded an MSc degree in Mathematics Education.

Lectures referring to the branch Teaching Mathematics in the 4 th and 5 th year of study will be conducted as follows: Mirjana Vukovic, Full Professor Out of eight courses that are offered, students take six.

Students should pass at least five of the courses they attended. One of the courses might be replaced by some other Master level course offered by some other related faculty or uni- versity with prior approval issued by the Doctoral Study Committee. Sefket Arslanagic was appointed head of the branch Teaching Mathe- matics.

Entry requirements include a completed undergraduate study program- me in mathematics or any related science with a GPA of min. The aforementioned provision might also be applied to any interested appli- cant who obtained a BSc degree in mathematics with a GPA less than eight 8. After passing the required examinations, an applicant defends the Masters thesis done in co-operation with the thesis advisor.

A representation is something that stands for something else.

Each representation should consist of the following aspects: The idea of representation is continuous with mathematics itself. We distinguish between external representation environment and internal representation mind.

External representation refers to all external media, whi- ch has as its objective to represent a certain mathematical idea. We mainly use representation with concrete material, graphical representation and mathemati- cal symbols when teaching mathematics to young children.

This paper discusses the role of using different external representations in the process of learning and teaching mathematics. The importance of establishing relations between different representations is stressed with a model of representational mappings.

Within this theory we have defined two concepts: A child can give meaning to a particular representation if he or she is able to perform a required transformation within a representation. We differentiate between internal mental images and external envi- ronment representations.

h Cognitive development is based on a dynamic pro- cess of intertwining mental images and environment. This means that a vilozofiju ssful process filozzofiju learning is an active formation of knowledge in the process of interactions between external and internal representations. Internal representations are defined as mental images or mental presentations not representations: In mathematics classes pupils are introduced to three different types of symbolic elements or external representations: In the following sections, we will be dealing with the role filozofjiu different representations of mathematical concepts for learning with understanding.

We are going to present each of external representation in mathematics very briefly. For someone a concrete representation stands for a particular structured representation which is used only in the process of teaching and learning ma- thematics, and does not have any special meaning out of that process. We will call such fklozofiju as structured material, e. However we also 20 Proceeding of the International Scientific Colloquium understand any other concrete material, let us call it unstructured material that a child uses in order to learn a particular mathematical concept, as a concrete representation.

Children in Slovene use filozoiju Multilink that can be struc- tured into sticks of 10 and individual cubes to illustrate the place value nature of numbers. There is a common view of teachers and parents that children learn filozofiiju hematics more easily if they have the possibility to manipulate with concrete material. Research in this matter is not unitary.

For example, during the s and s Dienes blocks were widely used in the Netherlands, but criticism of their use as being helpful for the representation of abstract number structure, but weak in the representation of number operations when they become more complicated Beishuizen, has lead to the use of bead frame and bead strings Anghileri, Let us list some other of the authors who researched the role of structured apparatus and unstructured material in the process of tea- ching and learning mathematics.


Fennema and Fridman showed positive role that relate more closely to images of a counting strategy of using concrete material at primary level but not in secondary school, while Suydam and Higgins found manipulating with concrete material useful in whole elementary school.

Labinowicz observed young children using Dienes blocks and came to the conclusion that children had problems establishing re- lations between these blocks and the place value ffilozofiju of integer numbers. Again, on the other hand, Fuson and Briars found very positive role of these blocks uvof learning adding and subtracting integer numbers. These contradictory conclusions make us aware that fi,ozofiju material itself does not ensure successful learning.

In other words, the process of teaching and learning mathematics is very complex, and one part of it is also manipulating with concrete material.

We believe that manipulating with concrete material without thoughtful reflection on the process of manipu- lating and without relating concrete representations to other representations in mathematics is not sufficient for successful learning of mathematical concepts. The nature of a mathematical concept, the way of using concrete material, and the material itself define how the learning is going to take place. Mathematical textbooks, workbooks, and other material for children are full of graphical representations which differ filozofijh originality and correctness.

A concrete representation of the number is all countable objects around us. But we do not count everything together.

We can only count objects which have a certain characteristic in common and differ in some way at the same time what makes this group of objects countable. Graphical repre- sentations of numbers are mostly illustrations of objects, animals and persons which pupils write down with mathematical symbols or numerals.

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Graphical representations are not used only for illustration of mathematical concepts but for illustration of certain mathematical symbols as well. These representations are there to help pupils remember a certain mathematical symbol easier.

Pupils adopt the concept represented by a symbol even before it is introduced as a symbol. Adopting the concept and learning how to write it down with a sym- bol is going on at the same time.

It is not possible to exclude the fact that by adopting a mathematical symbol for a certain concept, pupils learn about that mathematical concept as well.

Let us illustrate this idea we adopted from Heddens with Figure 1. Graphical representations as a bridge between concrete representations and mathematical symbols 22 Proceeding of the International Scientific Colloquium Graphical representation, drawn cars Figure 1is semiconcrete represen- tation, representation of subtraction with rectangles is an example of uvov nbstract drl more distant from experience world in our case.

Representation with rectangles shown in the picture above Figure 1 could be a semiconcrete representation in any other situation. As already mentioned, in the teaching and learning process we use a variety of different graphical representations.

A graphical representation depends on the nature of a mathematical concept and on a representation with concrete representation. We have to mention a drl line as a special case of semia- bstract representation in mathematics. A number line causes many problems to children because its interpretation includes both ordinal and cardinal aspects of integer numbers. On one hand, a number is presented with a position on the line, and on the other hand, a number stands for the number of movements on the line.

According to Anghileri It is hard to believe that these ideas could be accepted in our curriculum in learning arithmetics because teachers and also parents strongly believe that filozlfiju without concrete representa- tions and learning the place value system is just not possible.

In terms of resear- ch these ideas are worth challenging in practice. Let us briefly mention also manipulation with mathematical symbols. The number of symbols is small but there are many combinations of these symbols with special rules whi- ch hold for particular combinations of symbols.

This is what makes handling with mathematical symbols difficult for many children. In many cases children manipulate with mathematical symbols mechanically without understanding.