Acta Didactica Universitatis Comenianae

Mathematics, Issue 14, 2014, pp. 71-86

DEVELOPING GEOMETRIC SKILLS THROUGH

ACTIVITIES IN LESSONS OF MATHEMATICS

LUCIA RUMANOVÁ, EDITA SMIEŠKOVÁ

Abstract. The paper deals with problem of geometric constructions on secondary school

level. There were 34 pupils who participated in our activity. It was focused on the practical

use of constructions in school mathematics aiming to enhance pupils’ motivation and

improve their geometrical skills in teaching geometry through construction of stone marks

known from the past.

Key words: fine arts, education, lesson, pupils, theory of didactic situation, solution

1 INTRODUCTION

Geometry in the past was based on practical needs of life. Development of

geometric knowledge most contributed to architectural activity, such as land

surveying, construction of homes, forts etc. Similarly, geometric knowledge was

required for orientations in the field - i.e. transport by sea or desert, manufacture

of tools, weapons, and also in shipbuilding. Gradually, geometry became a science.

Geometry has been well studied and developed because of these practical reasons,

but also philosophical and theoretical reasons. (Surynková, 2010)

The situation with arts was analogical. The arts have always accompanied

mankind and it is a means of communication. For example, a full perception of

a work of arts is unthinkable without vision. But it is not by any vision, but the

conscious vision. Vision itself is intricate, as well as imaging what is seen. This

is true even if it is only a view of the subject without any artistic ambitions. This

fact is known from school practice. It must be taken into account how much

work pupils have in order to sketch geometric objects into a picture correctly.

(Šarounová, 1993)

The study of geometry is demanding, therefore it is often circumvented at

various levels of education, e. g. when there is not enough time left, geometry is

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72

omitted. Geometry should be taught illustratively. Geometric constructions

should never be taught as procedures under which pupils do not see anything in

the picture or cannot imagine anything.

Geometrical construction problems seem to be difficult for pupils. Those

who have already encountered these constructions try to recall the algorithm,

which in many cases they do imprecisely. Generally, pupils do not think about

mathematical properties when doing geometrical constructions, they take only

two approaches: remembering the algorithm, or drawing (not constructing)

the required “picture”. They use many mathematical notions incorrectly.

(Marchis, Molnár, 2009)

One of the main aims of mathematical education as such is preparing the

pupils for dealing effectively with the real-life situations. (Švecová et al., 2014)

Geometry should be taught in an interesting and logical way, and also the use

of geometry in practice should be emphasized. Geometry stems from practical

activities. Use of geometry in practice is most evident in the architecture and arts.

Creating a perfect piece of art, which is in harmony, it is possible to achieve use

of some mathematical and geometric relationships. Architecture is primarily

intended for practical use, but can also be "pleasing for eyes".

Incorporating art in geometry inspires pupils to observe the world around

them with the eyes of a mathematician. Pupils have an opportunity to demonstrate

their knowledge of geometric shapes, as well as produce interesting works of art

that can be displayed to show their understanding. Why arts? Activities for

teaching geometry lesson for young pupils can often be bland. Adding an art

component to lesson plans provides pupils with a hands-on medium for

learning. Classroom teachers may want to coordinate their efforts with the art

teacher; however, the following instructions can easily be accomplished without

collaboration of the art teacher. (Neas, 2012)

In modern educational theories necessity of pupils’ activity during lessons

is highlighted. There is also demand for development of not only pupils’

knowledge but all key competencies. Therefore we need teaching methods

which could meet these needs. (Vankúš, 2008)

2 THEORETICAL FRAMEWORK

The activities presented in the article and the research brings together concepts

and principles of the theory of didactical situations (TDS) and the analysis methods

of problem solving processes.

The core of TDS issued from this didactic school is analysis of problem in

particular levels of didactic situations. A didactic situation according to the TDS

consists of three main parts: devolution, a-didactical situation and institutionalization.

DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS

73

A framework to this approach is based on the works by Brousseau (1998),

Chevallard (1992), Sierpinska, (2001).

Brousseau (1998) defines didactic situation as a situation for which it is

possible to describe the social intention of student´s knowledge acquirement.

This situation is realized in a system called the didactic system (didactic triangle)

that is composed of three subsystems: learner (student), learning (teacher),

information and relations between them. The relations – the didactic contract

represents results of intervention explicitly or implicitly defined relations between

students or group of students. It is the environment and the educational system

that prepare students to accept completed or nascent knowledge. They are

exactly the rules of game to activate the student.

The basic notion of TDS is the Didactic milieu, following Piaget´s theory

the milieu is source of contradictions and non-steady states of learner (subject) by

process of adaptation (by Brousseau (1986) it is assimilation and accommodation).

The environment is specific for every piece of knowledge. Different levels of

milieu are embedded one inside another, a situation at one level becoming a milieu

for a situation at a higher level – action at an upper level presumes reflection on the

previous level. (Regecová, Slavíčková, 2010)

We can see this structure of milieu in Table 1.

Table 1

Structure of the milieu in didactical and a-didactical situation (Földesiová, 2003)

M3

Constructional

milieu

P3

Teacher -

didactic

S3

Noosferic situation

M2

Project

milieu

P2

Teacher-

constructor

S2

Constructional

situation

M1

Didactic

milieu

E1

Reflective student

P1

Teacher- designer

S1

Project

situation

M0

Milieu of learning

E0

student

P0

Teacher- designer

S0

Didactic

situation

M-1

Modelling

milieu

E-1

Cognizant

intellect student

S-1

Learning situation

M-2

Objective

milieu

E-2

Active student

S-2

Modelling situation

M-3

Material

milieu

E-3

Objective student

S-3

Objective situation

L. RUMANOVÁ, E. SMIEŠKOVÁ

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According to Brousseau (1998), the a-priori analysis is one of the tools that

teacher can use in lesson planning. Analysis a-priori is necessary to do before

finding a solution of a particular problem. It is useful for a teacher to prepare

background for various possibilities that can be observed in lessons.

A good preparation of a-priori analysis is a condition for successful devolution

and a-didactical situation. Therefore it helps one prepare better a-didactical

situation, a situation where children get the knowledge on their own. (Novotná et

al., 2010)

We apply the principles of this theory confirming the importance of the

analysis a-priori and a-posteriori results.

2.1 DETAILS OF PRESENTED ACTIVITY AND RESEARCH

The activity was realized in March 2014 with 16 – 17 year old pupils. They

were 34 pupils of grammar school in Nitra.

In school practice the ability of pupils to solve an open problem

independently develops poorly. Pupils mechanically apply earlier acquired

geometric skills and algorithms when solving these problems. The main aim of

the submitted paper was to determine whether pupils can apply and use

effectively their knowledge and expertise from different parts of mathematics in

solving a particular geometric problem. Pupils looked for the beauty of arts,

became acquainted with the rules of formation, so the arts was not only

admired, but also understood. In the activity, including the research, the main

focus was on the practical use of constructions in school mathematics, aiming to

enhance pupils’ motivation and improve their geometrical skills in learning

geometry through construction of stone marks known from the past.

In some parts of mathematics it is difficult to invent such a “game” between

the teacher and pupils which would meet qualified conditions of TDS so that it

would be possible to assemble a set of problems and activities. Here definitely

belongs geometry and certainly the constructional problems. Therefore, we have

created problem tasks that are based on formerly acquired knowledge

of geometry. The following activities, without naming the topics which we want

to deal with, it is also possible to route of TDS.

Before meeting the pupils we prepared a detailed a-priori analysis of

activities later on realized with them, and we thought about what we expected

from the pupils (Chapter 2.2). In the context of devolution, we thought about

how pupils would explain important rules and then we would not interfere in the

individual activity. When the pupils understood what to do, our help (work of

the teacher) was terminated. Devolution was completed and followed by a-

didactical situation. We also thought that during the activity pupils would go

through all three phases of a-didactical situation (action, formulation, validation).

Pupils would obtain new knowledge during these three phases. We expected that

DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS

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pupils would acquire knowledge of stone marks, principles of their constructing,

history and arts. We would finish the didactic process with the last part of TDS -

institutionalization. In our case the institutionalization would be a discussion with

pupils, no matter how pupils would solve the problem. We would try to explain

the solution of the problem and to affirm their acquired knowledge.

We proceeded with the pupils as follows:

- pupils filled in the initial test – we wanted to determine their basic

geometric knowledge needed to solve the problems;

- we familiarized pupils with the activity which was then carried out

with them;

- pupils individually solved a given problem on the topic.

After realization of activity we compared our projection from a-priori

analysis of pupils´ solutions and made results in a-posteriori analysis.

2.2 A-PRIORI ANALYSIS OF PUPILS´ ACTIVITIES

We formulated analysis a-priori before pupils’ solutions (the research), and

the analysis includes descending analysis (analysis of the teacher’s work) and

ascending analysis (analysis of pupils’ work).

Since we had not known the pupils in the experimental group before, we

decided to check their geometric knowledge. The initial test was focused on the

basic concepts related to the settlement of problems. We were interested if these

pupils had necessary knowledge to comprehend the situation and to solve the

given task.

In accordance with the tenets theory of didactic situations frame: within the

frame of the didactic situation S3 (Noosferic didactic situation) we made an

analysis of math textbooks for secondary schools and an analysis of various

mathematical materials. The end of Noosferic situation would be the milieu for

the following situation. Then we chose geometric notions that the pupils needed

to follow solutions during the activities. The initial test is listed in Appendix 1.

Relating to the pupils' knowledge, we formulated possible responses to the

individual questions of the test. The teacher should attempt to “read the pupils

minds” and get at the same level of their thinking. Similarly, we took into

account the pupils when compiling tasks in other planned activities. We were

prepared for the situation that the initial test would indicate insufficient pupils´

knowledge and then it would be needed to improve their knowledge through

other tasks.

In the context of problem tasks we solved the task with pupils in order to

use the geometric constructions in creating the so-called stone marks. Activities

with pupils were situated into the story of builders in the middle Ages, who

signed those marks and sculpted them into the building. Relating to the

knowledge of pupils, we compiled problems and tried to engage pupils into the

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76

process of their solving. Problems and sequence of activities with pupils are

listed in Appendix 2.

Pupils would get acquainted with the problem and with the material milieu.

Social component of material milieu would be minimal because other help

would not be allowed in our experiment (self – activity of pupils). Pupils should

be able to solve the given tasks using drawing aids – compasses or triangular

rulers. Within problem solving pupils would become familiar with several types

of stone marks and procedures for their constructions. They would have to

know the construction principles of three basic keys to stone marks. Finally, we

would want them to give the key construct its own mark. Pupils could then

realize that they as stonemasons in the past would try the uniqueness and

originality of their own signature.

We expected that pupils would:

- be able to work correctly with the drawing aids, such as compasses,

triangular rulers;

- know the geometric construction of basic geometric figures, such as

a square and an equilateral triangle;

- be able to find constructively the middle point of the diagonal (and

sides) of the object, construct the perpendicular led by point to the side,

construct an angle axis and the axis of side;

- be able to construct inscribed and circumscribed circle of a triangle and

a square;

- know the construction of triangle height, medians and diagonal of

a square.

Finally, in the end of the activities pupils would solve the problem alone

and they could choose from two options (Appendix 3). While addressing these

tasks pupils could use the knowledge which we would also use. Pupils would

receive a text of problem in which we put them in the position of medieval

stonemasons. They would have to demonstrate their drawing skills and geometric

skills. We assumed that the biggest problem would be with the accuracy of

pupils´ construction. Therefore, if the mark of pupils would not be accurate

enough, they would not get the correct final key.

We would observe the proposed aims and also pupils’ solutions. It would

be a situation where the analysis of teacher’s work and analysis of pupils’ work

meet, and this situation would be the result of the teaching process. In didactic

situation the work of pupils would be affected by teachers and their advice in

form of institutionalization, which could help the pupils to solve the given tasks,

but teacher must first of all take into consideration pupils’ solution. The teacher

could help with individual elements of construction marks.

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77

2.3 A-POSTERIORI ANALYSIS

At the beginning of the lesson the pupils attended the initial test. We

wanted to found out their knowledge before the problem task. As we expected

within the ascending analysis (a-priori analysis) pupils have necessary

knowledge to comprehend the situation and then to solve the given task.

The initial test consisted of 8 questions. The most successfully answered

question was question 6: “A median of a triangle is ...”; while the least

successfully answered question was question 4: “According to length ratio we

divided triangles …” The results are summarized in Graph 1.

Graph 1. Pupils´ results of initial test

The most frequent incorrect answers to question 4 were:

- isosceles triangle, equilateral triangle;

- isosceles triangle, equilateral triangle, irregular triangle.

Answers to question 3 “Write how many axes of symmetry figures in the

picture below have.” were surprising; some of the pupils’answers were:

- a circle has 360 axes of symmetry;

- a circle has 1 axis of symmetry;

- a pentagon has 1 axis of symmetry;

- a pentagon has 3 axes of symmetry.

We did not consider the misunderstanding of terms in the test. We worked with

pupils of grammar school, so according to the National Program of Education

(2011) and also having in mind the type of school the pupils should have known

these terms. We think that the incorrect and unusual answers of pupils could be

the result of inattention or less knowledge of mathematics.

After the initial test pupils were given the problem task which was inspired

by the work of Vienna architect Franz Rizha who discovered the secret of the

stone marks. He searched for the geometrical construction of keys of stone

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marks. In accord to the TDS Objective situation was the same in every strategy.

Pupils acquaint with the problem task (see Appendix 3) and material milieu (basic

writing tools, drawing aids – compasses or triangular rulers, context of the task

...). Social part of material milieu was minimalized because pupils solved problem

task separately. In the Modelling situation is the pupil active and he try to solve

the given problem task with material milieu. In the next Learning situation is

pupil in the position of solver of the problem task, he start to formulate to the

own initial findings and conclusions – we (as teacher) did not helped pupils. So

realization of research do not assume the interruption by teacher to the solution

of pupils, pupils do not reach Didactic situation.

Now we give a few specific observations and the results of the pupils'

solutions.

Pupils were provided with the text by which they were put into the role of

journeymen in the middle Ages. The middle Ages journeymen had to show their

competencies in geometry. Pupils who could not solve the problem task often

made mistakes at the beginning; therefore we think it was caused by inattentive

reading of the text. Other mistake was inaccurate and confusing drawing which

resulted in impossibility to find out the secret of the key of their stone mark.

We were interested in the connection between pupils' knowledge and their

ability to solve the problem task.

Pupils could obtain 1 point for each correct answer in the initial test.

The following graph depicts the frequency of pupils awarded 0 – 8 points in the

initial test, and correct or incorrect solution of the problem task (one of the

variants A or B).

Graph 2. Frequency of pupils according to solution the problem task A and B

DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS

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The contingent table for group of 34 pupils was made. Pupils' results of the

initial test were divided into two groups. Pupils who had 0 – 4 correct answers

were included in the first group; pupils who had 5 – 8 correct answers were

included in the second group. Then the solutions of the problem task were

marked by 1 if they were correct, by 0 if they were incorrect. Table 2 shows that

pupils who did better in the initial test were also more successful when solving

the problem task. On the other hand pupils who are in the first group of the

initial test were not so successful with the problem task.

Table 2

The contingent table of pupils' results

Questionnaire

Total

1. group 2. group

Tasks A or B 0 15 5 20

1 4 10 14

Total 19 15 34

A solution sample of pupils' who solved the problem task correctly:

A solution sample of pupils' who solved the problem task incorrectly:

L. RUMANOVÁ, E. SMIEŠKOVÁ

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3 CONCLUSION

The education process attempts to teach pupils reading comprehension, so

that pupils would be able to follow the instructions, and/or should accept

mathematics as a part of human culture as an important society tool through

cross-curricular relations. (NPE – ISCED 3A, 2011)

Basing on the reactions of pupils in the questionnaire on the activities we

write about in the article, we conclude that pupils:

- are interested in such activities;

- learned something new or deepened their geometric knowledge;

- by drawing more complicated units (how pupils named it) they

learned patience and precision, the lesson was enjoyable for them and

they felt free to apply their creativity (often in mathematics they do not

feel like that);

- learned something from the history and discovered connections

between well-known facts which they had not been aware of before;

- described the activities as very interesting and the procedure of

drawing was clear and understandable for them;

- had time for individual work and moreover a good deal of freedom in

solving the problem tasks.

Therefore, we think that including such kind of tasks in teaching process is

useful, although we are aware of the fact that their preparation takes a lot of time

and space. We believe that giving pupils this type of tasks remains an open

problem. That is why teachers diagnose the formal knowledge of their pupils by

these tasks, encourage and develop their geometric skills. This is the reason why

the teacher should not forget that geometry should be taught from the very basic

level.

In the future, this problem will be a subject for next research.

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Chevallard, Y. (1992). Concepts foundamentaux de la didactique: perspectives apportées par une

approache antropologique. Recherches en Didactique des Mathématiques, Vol. 12/1.

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1592-4424, p. 33-42

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Marchis, J. – Molnár, A. É. (2009). Research on how secondary school pupils do geometrical

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via natural differentiation in mathematics. Rzeszów: Wydawnictwo Uniwersytetu

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Regecová, M. – Slavíčková, M. (2010). Financial literacy of Graduated students. In: Acta Didactica

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Švecová, V. - Pavlovičová, G.- Rumanová, L. (2014). Support of Pupil's Creative Thinking in

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Vankúš, P. (2008). Games based learning in teaching of mathematics at lower secondary school.

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1338-5186, p. 103-120

LUCIA RUMANOVÁ, Department of Mathematics, Faculty of Natural Sciences, Constantine the

Philosopher University in Nitra, 949 74 Nitra, Slovakia

E-mail: lrumanova@ukf.sk

EDITA SMIEŠKOVÁ, Department of Mathematics, Faculty of Natural Sciences, Constantine the

Philosopher University in Nitra, 949 74 Nitra, Slovakia

E-mail: edita.smieskova@ukf.sk

L. RUMANOVÁ, E. SMIEŠKOVÁ

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Appendix 1

INITIAL TEST FOR PUPILS

1. Which dimensional figure could be hide under ground?

_____________________________________________________________

_____________________________________________________________

2. Mark, which the pair of figures in the picture are axially symmetrical?

a) b) c) d)

3. Write, how many axis of symmetry have figures in the picture below?

cccc_____ ______ _______ ______ ______

4. According to length ratio we divided triangles …

_____________________________________________________________

_____________________________________________________________

5. A height of triangle is ...

_____________________________________________________________

_____________________________________________________________

6. A median of triangle is ...

_____________________________________________________________

_____________________________________________________________

7. The center of circumcircle of a triangle is … (mark only one answer)

a. the intersection of the angle bisectors,

b. the intersection of the axis of sides,

c. the intersection of the medians of triangle and we called it the centroid,

d. the intersection of heights of triangle and we called it the orthocenter.

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8. The center of inscribed circle of a triangle is (mark only one answer)

a. the intersection of the angle bisectors,

b. the intersection of the axis of sides,

c. the intersection of the medians of triangle and we called it the centroid,

d. the intersection of heights of triangle and we called it the orthocenter.

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Appendix 2

Introduction of the activity – motivation speech:

„ Nowadays, the architect or designer is signed for his project. How was it in the past?”

In the time of Greece and Rome the builder was signed by a stone mark that engraves directly to

masonry construction and architects made it in the same way in the middle Ages. As the builders

were many, as well as stone marks were numerous and each mark was a different. We showed to

pupils how stone marks look like and gave them examples where they might see them.

The construction of the key to stone marks:

"Every stone mark made up to a certain key which the building company strictly guarded. All

marks should be able to put into such key. Since now it is known 14 keys, which we call the basic

root of the mark. „After then we showed 12 roots of the marks to pupils.

We then showed 12 pupils roots marks:

The use of the stone mark by journeyman:

„If the journeyman received his own stone mark, he should signed by it. The condition of use was

that he had to know its details so that he was able to put it into the key. Unauthorized use was

punishable.”

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We offered pupils some examples of stone marks from different period of history:

The organization of building companies:

„Builders worked together in the communities which we called building companies. Builders had

their privileges and their master who took care about apprentices and journeymen. Each of them

got his own mark with the key of construction. The key was a pattern created by repeating of

simple root of the building company or developing the root to network of parallel lines. After

promotion of the apprentice to journeyman, the mark was more developed. The mark of

journeymen contained of lines, which at least one intersect other. The intersection was at right

angel. If the journeyman was older and experienced, he got a new mark. This mark consisted of

intersection of diagonal lines. The mark of master consisted of whole circle.”

The legend by which our activities were inspired:

„The journeyman had to travel a lot and so that to expanded their experiences and education.

Supposedly it was so, if they came to other building company, they had to three times knock on

the door and answered on three questions. And then the door of the company was opened, but

they were not still received. They had to make an exam of the geometry of which part was the

construction of their stone mark and the explanation of the construction. If they were successful in

this exam the building company received them.

The construction of three roots of stone marks:

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Appendix 3

The problem for pupils

Imagine that you are a journeyman of some Slovak building company from Middle Age. This

building company granted you your own building mark. Your building company sent you to

disseminate knowledge and education to the other parts of world. After few days of walking you

stopped before a gate of Prague building company. How says a legend, you three times knocked

on the door and gave a right answer on three questions. Gate of the building company is opened,

but you are not still received. There is an exam from geometry. Demonstrate yourself by your

building mark and explain the principles how was your building mark constructed.

Choose one variant A or B and write a reason why you have decided so.

Variant A

You are experienced and skilled journeyman and you know the principles of many geometrical

constructions, however the way to the Prague building company was long. Only one thing you

remember about the construction your building mark is that the construction of the mark’s key

begin with the construction of two equilateral triangles. Their shapes created a six-pointed star

and lie on the circle to which is your building mark inscribed. The longest mark’s line is height of

one that equilateral triangle. More than you know, that you have to construct some different

equilateral triangles and their heights. Parts of your building mark lie on these triangles and

heights. How to do so? Discover the secret of mark’s key and prove to Prague building company

that you are worthy to become their new journeyman.

Variant B

You are experienced and skilled journeyman and you know the principles of many geometrical

constructions, however the way to the Prague building company was long. Only one thing you

remember about the construction your building mark is that the construction of the mark’s key

begin with the construction of two equilateral triangles. Their shapes created a six-pointed star

and lie on the circle to which is your building mark inscribed. More than you know, that you must

inscribe six the same smaller circles which are touched from inside of the biggest circle. These

circles intersect each other and some parts created the line of your building mark. How to do so?

Discover the secret of mark’s key and prove to Prague building company that you are worthy to

become their new journeyman.