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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 describe 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.

Résumé. Cet article décrit géométrie connaissances. Le but de notre travail était décrire la situation de l’enseignement de la géométrie aux écoles secondaires en Slovaquie et Hongrie.

On a donné 7 problèmes aux 94 éleves d´un lycée et dans la suite on a comparé des solutions.

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 geometrische Probleme zu lösen und die Lösungen haben wir dann analysiert.

Riassunto. Questo articolo affronta la tematica delle conoscenze geometriche. Lo scopo del nostro lavoro è 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 článku sme sa zaoberali geometrickými poznatkami. Našim cieľom bolo opísať súčasný stav vyučovania geometrie na stredných školách na Slovensku a v Maďarsku.

Poprosili sme 94 študentov o riešenie geometrických problémových úloh a analyzovali sme ich riešenia.

Key words: spatial ability, spatial geometrical knowledge, imaginary manipulation, reconstruction, teaching and learning spatial geometry, comparative survey Á. BOSNYÁK, R. NAGY-KONDOR


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 ability according to Séra and his colleagues (Séra-Kárpáti-Gulyás, 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):

–  –  –

Vásárhelyi's (Vásárhelyi) definition of geometrical spatial ability: the mathematically 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 Séra and his colleagues (Séra-Kárpáti-Gulyás, 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 projection 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 information;

• 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 threedimensional configurations correlated to each other based on the manipulation 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 9–12. The demand for the visualisation and drawn expression of the three-dimensional space appears at the age of 12–14. According to the experience of art teachers, the space representation has to be taught for some children because 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 Komenský 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 intellectually talented adolescents who has better spatial than verbal abilities are more likely to be found in the field of engineering, computer sciences and mathematics.


Although, Danihelová (Danihelová, 2000) and Tompa (Tompa, 2001) reports about the poor spatial ability and spatial geometry of school leavers, we, according 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.


Forty years ago teachers paid more attention to the teaching of spatial geometry, especially to the descriptive geometry in Slovakian secondary schools.

Nowadays, these two topics are almost completely disappeared from the curricula. We can find the descriptive geometry only among optional courses or sometimes 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. (Božek, 1990) The spatial geometry is one of the most problematic parts of teaching mathematics. 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. (Hejný et al., 1989) In Hungary, Kárteszi (Kárteszi, 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 difficulties appear for not developing spatial ability in time. The teaching and learning of this topic causes several different problems for the teachers and the pupils as well. The spatial ability of the pupils – because of the small number of classes dealing with spatial geometry – is poor. There are very few teachers who know how to illustrate things quickly and properly on the board. The division of plane and spatial geometry in time and topic in the curriculum, results in missing the development of spatial ability in the optimal age. Although, the spatial ability can be developed to a great extent. The school's job is to develop these abilities, and the teacher's duty is to acquire the classical and modern methods that are Á. BOSNYÁK, R. NAGY-KONDOR suitable for that. Because the undeveloped spatial ability cannot be replaced by a high level of theoretical knowledge. Vásárhelyi (Vásárhelyi) writes that the analysis and the generalisation are obstructed by mistakes that appear in the

pedagogical practise of demonstration:

• the split of the observation and the


way of thinking;

• the over crowdedness of the demonstrational material;

• special situation...etc.

In Hungary the primary aim of teaching geometry in grammar schools is the development of the geometrical approach, the imagery thinking and the ability of plane and spatial orientation. For the successful acquiring of geometrical notions, constructs and transformations, it is essential to have a talent in drawing – according to the age. A survey made on primary school pupils (Frei, 2004) shows that the geometrical knowledge of the pupils is continuously augmenting and deepening in the higher classes. Although, the role of the spatial ability is growing in higher classes, it does not reach the 40 percent of the expected developmental level even in the 8th year classes, the 5th year pupils have reached a very low level in the test of spatial ability. Therefore the development of the spatial ability would deserve more attention in terms of teaching geometry.

Tompa (Tompa, 2001) reports about the results of school-leavers on the mathematics school-leaving exam in 1995-98. The topics of the school-leaving

exams were chosen from the following ones:

• percentage, first- and second-degree equations,

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