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Acta Didactica Universitatis Comenianae Mathematics, Issue 8, 2008

THE SPATIAL ABILITY AND SPATIAL GEOMETRICAL KNOWLEDGE OF UNIVERSITY

STUDENTS MAJORED IN MATHEMATICS

ÁGNES BOSNYÁK, RITA NAGY-KONDOR

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. 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 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 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 opí-sať 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 2

1 INTRODUCTION

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 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 SPATIAL ABILITY AND SPATIAL GEOMETRICAL KNOWLEDGE 3

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

Figure 1. The typical exercises (Maier, 1998)

Vásárhelyi's (Vásárhelyi) 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

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 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 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 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 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 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, Danihelová (Danihelová, 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.

2 THE DIFFICULTIES OF TEACHING AND LEARNING SPATIAL

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. (Božek, 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. (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 difficul-ties 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


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 abstract 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 abi-lity 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 deve-lopmental 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, • logarithmic, trigonometric equations, functions, • number theoretical problems, • plane geometry, co-ordinate geometry, geometrical calculations, • spatial geometry, location, • vectors, trigonometry, • titles to be proven.

According to the report, pupils performed so poorly that if they had to pre-sent one certain competence from each and every topic to pass their exams many of them would fail.

A similar survey was made in Slovakia. Danihelová (Danihelová, 2000) in her collection volume reports among others about the results of the school-leaving exam from mathematics in 320 Slovakian secondary schools in 1999–2000. The topics of the school-leaving exam were chosen from eight categories in Slovakia: • set theory and logic, • number theory, • combinatorics and calculus of probability, • serials and functions, • equations, inequalities and equation systems, • analytical geometry,


• plane geometry, • spatial geometry.

From the survey it has turned out that the level of knowledge of the pupils is very low. The efficiency in each topic was between 38,56 % and 61,11 %. Most of the deficiencies can be observed in the solving of spatial geometrical tasks.

It is worth to think about why is always the spatial geometry the hardest task on all school-leaving exams?

3 THE TEACHING OF SPATIAL GEOMETRY IN SLOVAKIA AND HUNGARY

Both in Hungary and Slovakia in lower classes of primary school, due to

the age of the pupils, there is a key role of experimental learning and manipula-tion. In secondary schools the theoretical plane and the deductive character of the object is coming more and more to the front. Of course the processing built on experience and activity still cannot be neglected, especially in the case of pupils who are learning mathematics quite slowly. The differences in the abstraction level of pupils are huge, that is why it is important for the objective and imagery visualizations to come on. The professional recommendation on the school ma-terials in mathematics for the teaching of 9th–12th year pupil’s orders minimal demonstrational tools for the topic of geometry, like wall pictures, slides for the over head projector, compasses, rulers and conic sections. As minimal mathemati-cal instruments pupils are required to use compasses, rulers, plane- and spatial geometrical models, and calculator; and they are advised to use computer pro-grammes, the Internet and didactic CDs for there better understanding.

In the Slovakian secondary schools, both in the Slovakian-speaking ones and in the Hungarian-speaking ones, generally the same course-books are used, of course the ones that suit the schools' character the most. According to the curriculum of mathematics the number of the compulsory mathematics classes a week 4–4–3–3. (Bálint-Sádecká, 1999) (Gavora, 2001) (Hanula, 2001)


The number of classes that can be used for teaching certain topics in Slovakia (Učebné osnovy gymnázia – štvorročné štúdium, 1997, 1999) (Table 1):

Table 1. The number of classes that can be used for teaching certain topics in Slovakia

SECONDARY SCHOOL 1st year 2nd year 3rd year 4th year

Introduction to mathematics 20 0 0 0

Number theory 14 0 0 0

Functions, equations, inequalities 40 54 0 46

Geometry 20 60 83 0

Combinatorics, calculus of probability and statistics 24 0 0 20

Complementary curriculum 6 10 8 16

Tests 8 8 8 8

The number of classes recommended for the teaching of spatial geometry

in Slovakian secondary schools is 45 classes. Out of this 20 classes are held in the second year and the rest 25 is held in the first half of the third year.

The material suggested for the teaching of spatial geometry in Slovakia (Table 2):

Table 2. The material suggested for the teaching of spatial geometry in Slovakia

1st year 2nd year 3rd year 4th year

The mutual position of spatial elements.

The mutual position, distance, and angle of spatial elements with calculations and compiling.

The surface area and volume of the prism and

the roll; the Cavalieri-principle.

The surface area and volume of the pyramid, the cone, the truncated pyramid and the truncated cone.

The surface area and volume of the sphere;

regular solids; interlocked solids.

There are five series of course-books in Hungary for the 9th–12th year classes.

When the course-books are being chosen the teacher or the school has to decide what kind of methodology they wish to follow. They can choose books that are more like activity books, or they can stay with classical course-books that contain the necessary knowledge and ready-made exercises. They can even choose books


(Czeglédy et al., 2005) that allow the preparation for both the basic and the advan-ced level school-leaving exams.

In Hungary, according to the curriculum of the secondary schools the mini-mum compulsory number of maths classes a week in the 9th–12th year classes is 3–3–3–4. In the curriculum they pay extra attention to emphasize the spiral structure of the material. Besides they find the use of the inductive method very important in making definitions. The teaching order of each topic is not determi-ned by the curriculum, it is regulated in the syllabus based on one of the curri-culum. The table below shows the number of classes that can be used for teaching certain topics based on one of the curriculum (Kosztolányi, 2000). During the making of the syllabus these topics can be rearranged, where it is needed, with the materials that belong to it.

The number of classes for each topic in Hungary (Table 3):

Table 3. The number of classes for each topic in Hungary SECONDARY SCHOOL 9th year 10th year 11th year 12th year

Methods of thinking 6 6 10 15

Arithmetic, algebra 38 40 31 23

Functions, series 15 12 14 25

Geometry 39 39 40 45

Calculus of probability and statistics 5 8 10 10

Review at the end of the year 6 6 6 10

Altogether 111 111 111 128 In Hungary the recommended number of classes the teaching of geometry

(Czapáry-Czapáry, 2002), (Kosztolányi, 2003), (Számadó-Békéssy, 2002), (Urbán, 2004), (Vancsó, 2005), (Vincze, 2001) (Table 4):

Table 4. In Hungary the recommended number of classes the teaching of geometry 9th year 10th year 11th year 12th year

Kosztolányi 39 39 40 45 Czapáry 21+3 31 40 43 Hajdú 36 36 40 ?1

Hajnal 24 26<2 33<2 30<2 Vancsó 29 33 35 ?1

1 These two course-books do not published handbook for teachers for the 12th year. 2 This syllabus determines only minimum number of mathematics classes.


Within this, the recommended topics and number of classes for teaching spatial geometry (Table 5):

Table 5. The recommended topics and number of classes for teaching spatial geometry 9th year 10th year Kosztolányi 2 3–8 Czapáry 1 3 Hajdú 1 4–5 Hajnal 1 4–5 Vancsó 1

The mutual position of spatial elements; Notable set of points in the space.

3–4

The relation of the surface area and volume of similar solids; Spatial calculations with the help of trigonometric functions; The use of vectors in space.

11th year 12th year

Kosztolányi 0 21

Czapáry 0 21

Hajdú 1 ?3

Hajnal 0 16<4

Vancsó 0

–

?3

The mutual position, distance, and angle of spatial elements; The surface area and volume of the prism and the roll; the Cavalieri-principle; The surface area and volume of the pyramid, the cone, the truncated pyramid and the truncated cone; The surface area and volume of the sphere; regular solids; interlocked solids.

In the comparison to make things a bit clearer we used 1st, 2nd, 3rd and 4th

year instead of 9th, 10th, 11th and 12th. The minimal compulsory number of maths classes a week is more in the 1st and 2nd year and less in the 4th year in Slovakia than it is in Hungary, but in the four years it is altogether 163. The following diagrams show the rate of geometry and spatial geometry correlation to mathe-matics and geometry. The percentage rates were calculated in every case accor-ding to the curriculum. For the calculation of the rate of spatial geometry we considered the number of classes in the Kosztolányi curriculum.

3 These two course-books do not published handbook for teachers for the 12th year. 4 This syllabus determines only minimum number of mathematics classes.


The first diagram within the maths class shows the rate of geometry in both countries (rate within mathematics) (Figure 2):

Geometry

15,15

45,45

83,84

0

35,14 35 36 35,16

0

20

40

60

80

100


Rat

e w

ithin

mat

hs (%

)

Slovakia

Hungary

Figure 2. Rate of geometry within the maths class5

The rate of geometry compared to the other maths topics in Hungary is

higher in the 1st and 4th year and lower in the 2nd and 3rd year than it is in Slovakia. There is a big difference between the 3rd years (47,84 %), since pupils are taught geometry during almost the whole year. In Hungary the numbers of geometry classes are almost the same in all four years, while in Slovakia the numbers of geometry classes are gradually growing from the first year to the third and then they do not have any geometry classes in the fourth year.

The recommended number of classes for teaching spatial geometry in the two countries (Table 6):

Table 6. The recommended number of classes for teaching spatial geometry5

1st year 2nd year 3rd year 4th year Altogether

Slovakia – 20 25 – 45

Hungary 1–2 3–8 – 16–21 21–31

5 In the 4th year the students review spatial geometrical knowledge at the end of the year in both

countries.


The next two diagrams show the rate of spatial geometry first in maths then in geometry based on the curriculum in the two countries (Figure 3) (Figure 4):

Spatial geometry

0 15,15

25,25

01,8 4,95 0

16,4

0

20

40

60

80

100


Rat

e w

ithin

mat

hs (%

)

Slovakia

Hungary

Figure 3. Rate of spatial geometry within the mathematics (Kosztolányi, 2000)67

Spatial geometry

0

33,33 30,12

05,13 14,1

0

46,66

0

20

40

60

80

100


Rat

e w

ithin

geo

met

ry (%

)

Slovakia Hungary

Figure 4. Rate of spatial geometry within the geometry (Kosztolányi, 2000)6

6 In the 4th year the students review spatial geometrical knowledge at the end of the year in both

countries.


In Slovakia there are 14 more days devoted for teaching spatial geometry in secondary schools even if we count the maximum number of classes recommen-ded for this topic in Hungary. In case we count the minimum number of classes devoted for spatial geometry in Hungary, then the difference between the two countries will be 24! The suggested material for both countries is the same, the difference is that in Slovakia it is taught in the 2nd and 3rd years and in Hungary it is taught in the 1st, 2nd and 4th years.

All in all, we can admit that there is little time devoted for the teaching of spatial geometry in both countries.

4 THE COMPARATIVE SURVEY We made our comparative survey in September 2005 among mathematics

majors at the Komenský University and the University of Debrecen. We exami-ned and compared the students' spatial ability and spatial geometrical knowled-ge. At the university in Bratislava 43 students at the university in Debrecen 51 students took the test.

We prepared the test in a way that it contained the important components of the spatial ability. With the last two tasks we also wanted to examine the stu-dents' problem solving abilities. The first five tasks can be found in the book of Séra-Kárpáti-Gulyás: The spatial ability (Séra-Kárpáti-Gulyás, 2002), we made a slight change on the fifth task.

The survey:

1. We cut the solid on the task sheet with a plane. We marked the lines of intersections with a thick continual line on the surface of the solid. Draw the lines of intersections on the solid's laid out surface (web)! We draw one point of the line of the intersection on the web of the solid.


2. There is an axonometric picture of a wire framework built inside of a cube. The vertices of the figure are the same as the vertices of the cube or the midpoints of the sides of the cube. How can this figure be shown from the front, top and the right side?

3. On the figure we can see the projection of a solid truncated by planes.

Reconstruct the solid by drawing the visible picture of it! Draw only the visible edges!

4. The edges of the solid seen on the task sheet are of the same length. A ladybird wants to get from point A to point B on the shortest way it is possible. Which one of the edges marked by x does it have to go through? Explain your solution!


5. The points drawn in the cube are the vertices, midpoints of the edges or the sides or the diagonal line. What kind of geometric figure will we get if we connect the given points? Draw the geometric figures! Sample task: SCD – right-angled triangle

6. There is a cube ABCDEFGH (length of the edge: a). Determine the distance of point E from the plane BDG.

7. The measures of an ABCDEFV regular six-sided pyramid: cmaAB 4== , cmbAV 6== . Determine the mesure of the angle between the sides and

the base of the pyramid.

BAF

SEF

BSD

FDC

EAC

= ?


Following the theory of Séra and his colleagues (Séra-Kárpáti-Gulyás, 2002) we made the task sheet from the more important types of tasks. The first task focuses on the imaginary manipulation of the solid. The task is to follow such phases of the objective activity that consist of the complex spatial transformation of the solid. The second task belongs to the types of tasks that deals with repre-sentation and reading of the projection. By mobilizing the experience of the motion, changing the inner viewpoint, imaginary rotation, manipulation of mental representations, and the task is to produce and draw the two-dimensional projec-tion picture of a three-dimensional solid. This type of task is characterized by ana-lytical operations from concrete to abstract. The third task is a reconstructural task. We have to create the axonometrical picture of the solid based on the projection pictures. During the reconstruction the student synthesizes the visual information by studying the projection pictures. The map will be constructed by the series of changing the inner viewpoint by harmonizing three channels. The fourth and fifth task focuses on the checking of the comprehension of the struc-ture. It tests how the inner picture of view reflects the real spatial relations of the solid. The level of the task refers to the punctuation of the correction of distortion caused by the viewpoint and the illustration. The result of the fourth task can point to the punctuation of the “objective cognitive map”, whereas the result of the fifth task can point to the balance of cognition and understanding and to the cooperation of cognition and definitional operations.

The sixth and seventh task shows the problem solving ability of the students in the field of solving spatial geometrical tasks. Some preliminary knowledge of spatial geometry is needed for solving these tasks. They have to be aware of the definitions used in this topic and the characteristics of points, lines and planes. They have to know the relations between them. These two tasks give an answer to our question to what extent and how can the students comprehend the spatial relation and how much they can imagine the inner side and the structure of the cube and the regular six-sided pyramid.

5 RESULTS AND THE MOST COMMON MISTAKES

We gave one point to each task, but only in case of correct part tasks. We did not deduct a point in case of slight inaccuracy and slightly incorrect rates. In the fourth task there are two ways of solution in the case of the second cube, but we did not give extra points for getting both solutions. In the last two tasks it is the students' task to name the vertices of the solids by the given letters. In the case of these two tasks we gave points only to the correct calculations deduced logically. The next diagram shows the performance of the students on the test (Figure 5):


Results

72,09

44,18 51,16

4,65

32,55

18,6 18,6

60,78 58,82

21,1715,69

21,57

5,88

27,45

0

20

40

60

80

100

1. 2. 3. 4. 5. 6. 7.

Tasks

Perf

orm

ance

(%)

Students of Komenský UniversityStudents of University of Debrecen

Figure 5. Students’ performance

We are listing the typical mistakes in the following sections. We call a mis-

take typical if it was committed by at least the 10 % of the students.

1. The students of the Komenský University were 11,31 % better in the tasks of manipulating the imaginary solid. Both groups managed to solve these tasks the best of all. The 27,91 % of the students of the Komenský University and the 39,22 % of the students of the University of Debrecen draw the line of intersections to a wrong place.

2. In the exercises of the representation of projections the students of the University of Debrecen scored 14,64 % better than the other one. The 34,88 % of the students of the Komenský University and the 21,57 % of the students of the University of Debrecen could not imagine the perpendicular projections of some parts of the wire framework figure. The 20,93 % of the students of the Komenský University and the 19,61 % of the students of the University of Debrecen did not draw all the views or they left some parts of the figure out.


3. In the reconstructional exercise was the biggest difference between the two groups. At the Komenský University there were 29,99 % more students who knew the correct solution. The 46,51 % of the students of the Komenský University and the 78,83 % of the students of the University of Debrecen reconstructed the solid incorrectly or incompletely based on its projection picture. The 45,1 % of the students of the University of Debrecen gave the solution that can be seen on the middle figure.

4. The students of the Komenský University scored their worst in the task related to the punctuation of the “objective cognitive map”, they managed to solve the task 11,04 % worse than the students of the University of Debrecen. The students of the University of Debrecen also reached a low level at the test of the comprehension of the structure. Therefore, it is true for both of them that the inner view of the picture does not reflect the object's real spatial relations punctually enough. The 51,16 % of the students of the Komenský University and the 45,1 % of the students of the University of Debrecen in the case of one or both cubes chose that point, through which the way would be the quickest if it was a plane figure.

The 11,63 % of the students of the Komenský University and the 13,73 % of the students of the University of Debrecen gave only one solution in the case of the second cube.


The 18,6 % of the students of the Komenský University and the 41,18 % of the students of the University of Debrecen suspected that the shortest way is not through point C, but they chose the nearest point to that as a solution – without explanation.

Some of the “explanations”:

“Those distances that are the shortest in each square make the shortest way to get from A to B”

“a is shorter than b” “in the right-angled triangle the side is shorter than the hypotenuse”


5. From the students of the Komenský University there were 10,98 % more people who solved correctly the tasks on the balance of perception and thinking. The 27,91 % of the students of the Komenský University and the 19,61 % of the students of the University of Debrecen without drawing anything they named each of the plain figures – incorrectly. Those who tried to draw each of the plain figures gave a correct solution for them in 14–11 cases. The 13,95 % of the students of the Komenský University and the 11,76 % of the students of the University of Debrecen tried to put all the figures in a square or tried to connect each point in the cube – as if all the points in the cube were in the same plane- and they named the plane figures they got.

The 20,93 % of the students of the Komenský University and the 37,25 % of the students of the University of Debrecen wrote line as a solution for the BSD instead of figure.

6. The students of the University of Debrecen scored their worst in this exer-cise 12,72 % less than the students of Komenský University. The 27,91 % of the students of the Komenský University and the 29,41 % of the students of the University of Debrecen tried to solve the task by drawing, but incorrectly, 13,95 % and 19,61 % could not even draw the BDG plane properly or did not even start the task at all.

7. The smallest difference between the two groups was in the problem-solving task: the students of the University of Debrecen solved the task 8,85 % better than the other ones. The low rate of fulfilling the sixth and seventh task shows that the problem solving ability of the students is poor in the field of spatial geometry.


Except for the third task the difference between the students of the two universities did not go beyond 15 %. The students of both universities, except for 2–2 tasks, scored below 50 % on the test. The imaginary manipulation of the object, the description and reading of projection, reconstruction went well – although at the reconstructional task the students of the University of Debrecen were much worse. The students of the University of Debrecen were better at the description of projection, but the students of the Komenský University were better at the reconstruc-tional tasks. Based on the comparison of the curricula and the results of the fourth and the last tasks we can conclude that the students of the University of Debrecen remember better to definitions and titles, since they have learnt them in the not too far past, they have learnt spatial geometry also in the 4th year. In general, though, the students of the Komenský University scored better, maybe because on the 3rd year they learnt only geometry and they spent more time with spatial geometry so they can see the connections better and the have more practice in task solving. On the whole, the fourth, fifth, sixth and seventh tasks were managed to be solved very poorly, less than 35 %. On the basis of this we can conclude that the comprehension of the structure is not perfect, the inner demonstrational picture is inaccurate, it does not reflect the objects real spatial relations, the correction of the distortions of depiction is not accurate enough. The poor scores at the last two tasks show that the problem solving ability of most students in spatial geometry tasks is quite poor. Some of the students are even not aware of the idioms used in the topic of spatial geometry (e.g. the angle between the lateral face and the base). Even though the plane and spatial geometrical notions and titles are important for the spatial ability as well as the development of analogical way of thinking. We can talk about mathematical knowledge only in case of those students who can use the defi-nitions and titles in practice as well. (Kosztolányi, 2000)

At the solving of the seventh task the 41,86 % of the students of the Ko-menský University and the 27,45 % of the students of the University of Debrecen committed the mistake of not knowing exactly which angle's mesure they have to calculate. 18,6 % and 29,41 % found the right angle, but could not calculate the mesure of it.


6 SUMMARY

In contrast to Shea and his colleagues (Shea-Lubinski-Benbow, 2001) we found that the students of our survey applied for maths courses for nothing after graduating from the secondary school and they might want to find employment in this field later, they still do not have really good spatial abilities, spatial ability. Unfortunately, the results of Danihelová (Danihelová, 2000) and Tompa (Tompa, 2001) were confirmed with the maths majors as well, because the re-sults in most of the tasks in our survey were less than 50 %.

During the teaching of spatial geometry in Slovakia and Hungary we often meet with the shortcomings of the students' spatial ability and spatial knowledge even in case of those students who achieved good results in mathematics. This generally results in the students loosing their sense of security, which makes their antipathy towards this topic even worse. But what can be the solution? How could we develop the spatial ability?

The research of Kirby and Boulter (Kirby-Boulter, 1997) presents a way of teaching of geometry on projections, projections and rotations that made a po-sitive effect on the visual abilities of 13–14 year old Canadian children. Based on the results, they concluded that those who had disadvantages according to the tests of spatial ability, their abilities can be improved if the teaching of geo-metry is deep enough.

The role of an exact logical-theoretical background of the stereometrical theory: a long time teaching experience in the stereometry and descriptive geo-metry shows that no adequate spatial ability can be reached without a complete command of theoretical foundations of stereometry. It is very important a routi-ne command of a geometrical mapping method of the space into a plane, for example of the free parallel projection.

Lord applied a 30 minutes practise on a 14 weeks course with first/second year students where they had tasks in which they had to cut three-dimensional solids in their mind and then they had to draw the area of the two-dimensional planes they got. (Lord, 1985) In the post-test the spatial awareness and efficien-cy became better.

According to Horváth and his colleagues (Horváth-Kiss-Horváth, 1991) in the teaching and learning of geometry the three-dimensional models can be a great help. It is much easier to imagine and represent the different views of a solid when we can see the formal characteristics. The proper use and frequent study of spatial visual aids can result in such an inner spatial vision that makes the individual imagination of the spatial relations possible. (Vásárhelyi)

The results of the survey prove that it causes a problem for many students to imagine a spatial figure and this way it affects the solving of spatial geo-metrical tasks as well. Therefore it would be very useful to start the teaching of spatial geometry in secondary schools in both countries with spending more


time with the models of spatial solids and on the university courses some review-systematisation classes or even a whole semester should be devoted for the summa-ry of spatial ability, spatial geometry and solving spatial geometrical tasks. The effectiveness of teaching spatial geometry can be influenced to a great extent by using several different models, manipulation activities with spatial models, especially dynamic, in the demonstration of relations between spatial models and operations with them (even at the university level). When they had acquired the basic characteristics and relations by the help of traditional models then they could be introduced to the new definitions. Since there is a huge difference between the students' level of abstraction, it is important that the objective visuali-sation should be appear when it is needed. Vásárhelyi (Vásárhelyi) calls the attention to the use of computers besides the traditional models, because they pro-vide help for the motivation of the students. The interactive worksheets that can be made by dynamic geometrical systems remind us to the concrete model and they lead us to the iconic representation. After acquiring the direct experiences through concrete objective manipulation, a properly prepared interactive worksheet can help to separate the characteristics of the model from the general characteristics, different objects can be created and be observed in varied size and position.

According to Vásárhelyi (Vásárhelyi) it is practical to make four periods in the usage of models: • with models, • with models and their pictures (e.g. by computer animation), • with pictures and their prepared models or animated pictures, • problem solving only with the help of pictures.

Bakó (Bakó, 2006) was studying the conditions of the beneficial usage of computer in the teaching of spatial geometry. On the grounds of her researches she concluded that in acquiring the basic definitions of spatial geometry and the cognition of solids, the usage of models are still needed. After the development of basic abilities, the computer can get a role in developing the spatial intelligence from the 7th year of the primary school. She found that the help of the computer could develop all of the skills given by Maier (Maier, 1998). At the same time, she experienced that we should not overdo the use of computers, because the explanation of the teacher, the usage of models and individual work are needed as well.

We want to call the attention to one of the computer programs for pupils made at the University of Veszprém (Sík Lányi-Lányi-Tilinger, 2003) that helps the development of spatial ability. It is very useful that the program is inter-active so we can rotate the solids by the mouse or we can use animations. It gives a hand in, among others, the comprehension of the structure, the imagina-ry transformations of the solids, the recognition and representation of the spatial figure and it the field of making projections.


We all agree in that the development of the spatial ability is a very important task because we have to understand and develop the geometry knowledge of the students in the unity of the theoretical knowledge and the spatial abilities. Every skill, like the spatial ability as well can be developed at the right age with the suitable teaching strategy.

The results of the students in this survey were assessed on the basis of written documents. In the future we plan to make task based student interviews and further surveys by questionnaires to reveal the problems and their results.

ACKNOWLEDGEMENTS

We wish to thank Professor András Ambrus his valuable advice and help.

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ÁGNES BOSNYÁK, Selye J. University, Roľníckej školy 1519, 94501 Komárno, Slovakia E-mail: [email protected] RITA NAGY-KONDOR, University Of Debrecen, P.O. Box 11, H-4011 Debrecen, Hungary E-mail: [email protected]

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