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Acta Didactica Universitatis Comenianae Mathematics, Issue 8, 2008




Abstract. This article deals with geometrical knowledge. The aim of our work was to des-cribe situation in teaching of geometry at secondary schools in the Slovakia and Hungary. We asked 94 students to solve 7 geometrical problems and analysed their solutions. Rsum. Cet article dcrit gomtrie connaissances. Le but de notre travail tait dcrire la situation de lenseignement de la gomtrie aux coles secondaires en Slovaquie et Hongrie. On a donn 7 problmes aux 94 leves dun lyce et dans la suite on a compar des solu-tions. Zusammenfassung. In unserem Artikel handelt es sich um geometrische Kenntnisse. Das Ziel unserer Arbeit war den Zustand des Unterrichts der Geometrie in den Mittelschulen in der Slowakei und in Ungarn zu beschreiben. Wir haben 94 Studenten gebeten, 7 geometri-sche Probleme zu lsen und die Lsungen haben wir dann analysiert. Riassunto. Questo articolo affronta la tematica delle conoscenze geometriche. Lo scopo del nostro lavoro stato quello di descrivere l'insegnamento della geometria nelle secondary school in Slovacchia e Ungheria. Abbiamo chiesto a 94 studenti di risolvere 7 problemi di geometria e in seguito abbiamo analizzato le loro soluzioni. Abstrakt. V lnku sme sa zaoberali geometrickmi poznatkami. Naim cieom bolo op-sa sasn stav vyuovania geometrie na strednch kolch na Slovensku a v Maarsku. Poprosili sme 94 tudentov o rieenie geometrickch problmovch loh a analyzovali sme ich rieenia.

Key words: spatial ability, spatial geometrical knowledge, imaginary manipulation, reconstruction, teaching and learning spatial geometry, comparative survey



We get most of our knowledge in a visual way so it is very important for us how much we are aware of the language of vision. The definition of spatial abi-lity according to Sra and his colleagues (Sra-Krpti-Gulys, 2002, p. 19) the ability of solving spatial problems by using the perception of two and three dimensional shapes and the understanding of the perceived information and relations relying on the ideas of Haanstra (Haanstra, 1994, p. 88) and others.

In Gardner's (Gardner, 1983) opinion there is no uniform intelligence, everybody has different kinds of insular intelligence. He distinguishes between seven different types of intelligence: linguistic, logical-mathematical, spatial, musical, physical-kinaesthetic, interpersonal and intrapersonal. According to Gardner (Gardner, 1983, p. 9) the spatial intelligence is the ability of forming a mental model of the spatial world and manoeuvring and working with this model.

Gardner's (Gardner, 1983) multiple intelligence theory was improved by Maier (Maier, 1998), distinguishing between five branches of spatial intelligence: spatial perception: the vertical and horizontal fixation of direction regardless

of troublesome information; visualisation: it is the ability of depicting of situations when the components

are moving compared to each other; mental rotation: rotation of three dimensional solids mentally; spatial relations: the ability of recognizing the relations between the parts

of a solid; spatial orientation: the ability of entering into a given spatial situation.


The typical exercises for each types of intelligence are presented below (Figure 1):

Figure 1. The typical exercises (Maier, 1998)

Vsrhelyi's (Vsrhelyi) definition of geometrical spatial ability: the mathe-

matically controlled complex unity of abilities and skills that allows: the exact conception of the shape, the size and the position of the spatial

configurations; the unequivocal illustration of seen or imaginary configurations based on

the rules of geometry; the appropriate reconstruction of unequivocally illustrated configurations; the constructive solution of different spatial (mathematical, technological...etc.)

problems, and the imagery and linguistic composition of this solution. In the classification of the exercises we followed the recommendation of

Sra and his colleagues (Sra-Krpti-Gulys, 2002) who are approaching the spatial problems from the side of the activity.

The types of exercises: projection illustration and projection reading: establishing and drawing two

dimensional projection pictures of three dimensional configurations;


reconstruction: creating the axonometrical image of an object based on pro-jection images;

the transparency of the structure: developing the inner expressive image through visualizing relations and proportions;

two-dimensional visual spatial conception: the imaginary cutting up and piecing together of two-dimensional figures;

the recognition and visualization of a spatial figure: the identification and visualization of the object and its position based on incomplete visual infor-mation;

recognition and combination of the cohesive parts of three-dimensional figures: the recognition and combination of the cohesive parts of simple spatial figures that were cut into two or more pieces with the help of their axonometrical drawings;

imaginary rotation of a three-dimensional figure: the identification of the figure with the help of its images depicted from two different viewpoints by the manipulation of mental representations;

imaginary manipulation of an object: the imaginary following of the phases of the objective activity;

spatial constructional ability: the interpretation of the position of three-dimensional configurations correlated to each other based on the manipula-tion of the spatial representations;

dynamic vision: the imaginary following of the motion of the sections of spatial configuration. The development of the spatial ability has to be started in the early ages of

childhood. The conventions of the spatial representation can be taught effectively at the age of 912. The demand for the visualisation and drawn expression of the three-dimensional space appears at the age of 1214. According to the experience of art teachers, the space representation has to be taught for some children be-cause they would never reach that level by themselves. Therefore our image and definitions of space are not congenital; they are the result of a long developmental and experimental learning process.

This article reports about a survey about the topic of spatial ability and spatial geometry on a Slovakian and a Hungarian university. We surveyed the knowledge of 94 first year mathematics majors at the Komensk University in Bratislava and at the University of Debrecen. We examined the performance of mathematics majors, since we can achieve improvement in the development of spatial ability and the teaching of spatial geometry if the future teachers are competent in this topic.

Shea and his colleagues (Shea-Lubinski-Benbow, 2001) state in their research about the connection between mathematics and spatial ability that the intellectu-ally talented adolescents who has better spatial than verbal abilities are more li-kely to be found in the field of engineering, computer sciences and mathematics.


Although, Danihelov (Danihelov, 2000) and Tompa (Tompa, 2001) reports about the poor spatial ability and spatial geometry of school leavers, we, accor-ding to the researches of Shea and his colleagues (Shea-Lubinski-Benbow, 2001), expect a good result from mathematics majors.

In the first chapter we present the literature about the difficulties of teaching and learning of spatial geometry from both countries. The second chapter contains the numbers and the contents of classes about spatial geometry suggested by the curricula. In the third and fourth chapter we report about the circumstances and the results of the survey then we examine the most frequent mistakes. The last chapter is the summary of the article and our experiences.


GEOMETRY Forty years ago teachers paid more attention to the teaching of spatial geo-

metry, especially to the descriptive geometry in Slovakian secondary schools. Nowadays, these two topics are almost completely disappeared from the curri-cula. We can find the descriptive geometry only among optional courses or some-times not even there. As far as the material of the spatial geometry is concerned, it is limited only to some classes. We often hear explanations for this like the lack of time or teachers don't really like to start a hard topic like that. (Boek, 1990)

The spatial geometry is one of the most problematic parts of teaching mathe-matics. One of the reasons for this is that there are complicated connections among even the basic definitions. The task of the teacher is to motivate the pupils to an adequate extent, to arouse the interest of the pupils in spatial geometry. The teacher has to ponder the opportunities for exercises on classes and pay attention to the former knowledge of the pupils. But the structure of the topic is very important too. (Hejn et al., 1989)

In Hungary, Krteszi (Krteszi, 1972) already wrote about the difficulties of teaching spatial geometry in 1972. The teaching of spatial geometry comes only after the teaching of plane geometry. They deal with the plane geometry first, as if the space around the plane figures did not exist. This is why difficul-ties appear for not developing spatial ability in time. The teaching


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