Without claiming a scientific definition and using a more simple language, we could say that by a geometric constructions problem we understand the problem of drawing of a/some geometric figure(s), starting from specific given elements, using specific geometric tools, respecting some clear rules and following a logical route in a finite number of steps.
Beginning with the general methods of mathematics problems solving, our ancestors’ experience highlighted a certain staging (a solving scheme for geometric constructions problems), which involves the following stages:
1. Analysis » 2. Construction » 3. Demonstration » 4.Discussion
This presentation doesn’t have to be considered ‘dogmatically’, the four steps may intertwine, can be reordered, or they can even be skipped.
So, the steps in solving the compass-and-straightedge geometric constructions problems start with the Analysis, which supposes the already existence of a geometric figure with the notices …., checking the conditions… and having the following particularities… As opposed to the regular stages of solving a mathematic problem in general, that begin with ”the whole enunciation of the problem is carefully read“ and „the known and the unknown data are extracted“ etc, this approach seems to start suddenly, skipping some initial steps, and the phrase „we consider the problem solved“ turns into „haven’t you drawn it already?!“
Apart from studying the text of the problem, any geometric problem and especially the geometric constructions problems begin with the imperative „DRAW“. For this, it is supposed that the person who is solving the problem has the necessary geometric knowledge, masters the geometric tools and the constructions technique well enough for the whole attention and inspiration to be focused on building a geometric configuration clearly, suggestive and precisely enough, that can be the basis of the problem solving.
Like any scout, who before exploring an unknown territory, apart from a set of knowledge, rules and techniques, must have a minimal, indispensable kit in his adventure, we could say that any insight in the Euclidian geometry’s field needs to be done by holding the geometric tools. But let’s go back to the „scouts’ school” to see what it can offer us.
Studying the necessary curriculum for teaching mathematics in the Romanian secondary school, we notice that the geometry elements shyly appear, at an intuitive level (as it is normal, according to the stage of the student’s psychological development) as a pre-geometric stage during which the introduction of some geometric representations is highly connected to the shape and the dimensions of some real bodies in the surrounding environment. At this stage (classes I-IV), the pupil manages already to classify, to hierarchize, synthesize and compare, to preserve distances, areas, volumes, weights and to realize intuitively when the shape of some suffers changes.
Still shyly and under the direct guidance of the primary school teacher and the Math teacher the students have in the fifth form, the geometric tools (different patterns, graded ruler and the triangle ruler for the beginning) make their appearance as a necessity to represent different elementary geometric figures. The notion of „building“ (competence 2.3 „Using the geometric tools for measuring or for building geometric configurations“) appears only in the new curriculum for the 5th grade (to which I will come back later).
The purpose of these elementary geometric acquisitions of the students in the 1st -5th grade ensures the transition from the stage of „concrete operations“, prepared through the perception, observation, analyzing and generalization of the space properties of some real objects, to „the stage of the formal thinking“(beginning with the 6th form and during the entire adolescence) and finally to the geometry based on logically-deductive judgments.
It is important for the introduction of demonstration not to be rushed until the end of the 6th grade. The psychologists consider that, up to this age, the child’s mental evolution doesn’t allow the axiomatically introduction of geometry yet, and there is a risk of losing almost completely the idea of the existence of a link between geometry and objective reality. This opinion is better highlighted by the Mathematician H. Freudenthal: „One day the child will ask «Why?» and it is useless to start the systematic geometry before that moment had come. Moreover, it could really be harmful. If we agreed overteaching geometry as a means of making the children feel the force of the human spirit, of their own spirits, we mustn’t deprive them of their right to make discoveries themselves. The key of geometry is the expression «why». Only the happiness killers will pass the key on earlier“.
Any middle-school Math teacher can say that the introduction in the 6th form of the Geometry as a distinct branch of Mathematics has a great impact among the students accustomed with a specific Math judgment pattern recently assimilated from the Arithmetic lessons. Since the very first geometry classes there can be noticed a difference between the students who understand and enjoy this school object and the students who show difficulties and/or seem unconcerned. In this case, the teacher’s role is overwhelming and beyond competences, strategies and educational ideals, syllabus and a number of dedicated classes, the success is ensured by the experience, training, calmness, style, education, commitment and, why not, by those ”mathematical tricks“, that the teacher makes available for the students.
And, if some notions like point, straight line, semi-straight line, plane, angle and some relations as congruence, parallelism, perpendicularity, have already been established, suddenly the idea of compass and ruler geometric constructions makes its appearance (without trying to remember when the notion of compass has been used until then), under the form of building a congruent segment with a given segment, the building of the mediator of a segment and the building of a bisector of an angle. Then, a well-deserved break is taken, for appearing again in the 7th grade, when it comes about the similarity of triangles, the circle, or regular polygons, after which the eternal sleep lays over the compass-and-ruler geometric problems. From time to time, here and there, a passionate teacher dares to disturb the tranquillity proposing an optional school subject for some of the 6th- 8th forms, or for technology high schools, when studying technical drawing.
The initiative of those in Waldorf schools needs to be praised, because, through the ideas exposed in the Mathematics curriculum, they make the transition from the geometry of the surrounding environment to the geometry based on logical-deductive judgments through solving different geometric constructions problems, allowing the students to discover notions, relations, axiomatic sentences and all this because in their vision „the geometric constructions represent a combination of imagination and actual handy activity“ and „…this thing corresponds to the children’s abilities at this age“, with the idea of „…making geometry more accessible, so that it would stop being considered an arid and too abstract field“.
After we took notice about H. Freudenthal’s opinion about the early introduction of the geometry based on judgments and after we read the curriculum for the applied mathematics in Waldorf school, I return, as I promised, to the new compulsory Mathematics curriculum, proposed in Romania beginning with the school year 2017-2018 and referring strictly to the geometry field for the 5th grade, we notice a „decongestion“ of the matter, giving up on some amount of scientific content (a strong requirement of Maths teachers, parents and students), through its congestion introducing two new notions, in contrast with the old curriculum: The straight line (including the line axiom) and The angle – representing a large part of the geometry for the 6th grade. Not to mention that the dedicated number of classes didn’t change and the amount of geometry to be taught in the 8th grade didn’t suffer any modifications, all that was obtained was the decongestion of the matter for the 7th grade and the cascade transferring of more abstract notions towards the 5th and 6th grades (students of 10-12 years old).
This „flexibility of the curriculum“ took into consideration (it is written in the document) among other things: ”the adjustment of the curriculum to the expectations of the society and to the realities of the learning system, having as a purpose the students’ training for life and profession“, the compliance with ”…the differences between students of the same age (rhythm of learning, level of previous acquisitions, inner motivation, cultural and community specificity)“ and it was thought that ”…it can be browsed in the 75% from the allocated time to Maths classes, the rest of them being at the teacher’s disposal for remedial activities, consolidation and progress classes“. If Freudenthal’s words had been taken into consideration, it would have been wonderful.
We comfort ourselves with the fact that for the first time, the following affirmation can be found in the text of a curriculum for the 5th grade: „The approach of the geometry elements particularly targets the development of the skills necessary to use the geometric tools and the formation of the abilities needed for identification, investigation and building the geometric bodies and shapes“.
Considering all the above, we are stubborn and dwell upon the role and the importance of the geometric constructions in the secondary school as a supportive element in passing towards the geometry based on reasoning and understanding this field as an important way of solving the geometric problems as a relevant activity in the daily life; and if the idea of those from Waldorf school doesn’t match the educational ideal of the today’s Romanian society and we can’t filter more the notions of geometric constructions in the school curriculum, we can at least help the students with an offer of an optional subject matter at the level of Optional school subject.
1. * * * Programa şcolară pentru disciplina Matematică, www.ise.ro
2. * * * Programa şcolară pentru matematică, clasele V-VIII, Alternativa educaţională Waldorf, Bucureşti 2001, pagina 2, www.waldorfcluj.ro/files/2/Matematica%20V-VIII.pdf;
3. Constantin, O. – Îndrumări metodice privind predarea geometriei în gimnaziu, www.math.uaic.ro/~oanacon/depozit/Pred_geo_sp.pdf;
4. Păduraru, Vasile – Construcţii geometrice cu rigla şi compasul – abordări metodice, Editură Ştef, Iaşi, 2018.
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Written by Vasile Păduraru, Mathematics teacher,
Mărgineni School, Neamţ County, Romania
Translated by Nicoleta Orza, English teacher,
Dumbrava Roşie School, Neamţ County, Romania