Математическая деятельность как наиважнейшая цель математического образования

Mathematics is one of the oldest sciences which dates back to the beginning of the human society. There were many attempts over the years to define the term “mathematics” but etymologically it is derived from the Latin word mathematicus .

Bertrand Russell was right to give the following definition of the term mathematics describing it as a process “when we get out of the most well-known parts of it we can be led to one of the two opposing directions. The one which is more common is the structural direction which leads to things still more and more complex: from integers to fractions of the real numbers, complex numbers; from addition and multiplication to differential calculus, integrals and further to more advanced mathematic.” [Russell 2003: 17].

Mathematics, just like the mother tongue, is taught from the time of birth. The famous mathematician Leopold Kronecker (1823–1891) said: “God created the natural numbers, all else is the work of a man.” Long before we had written language and schools, people learned to count and make other simple uses of math.

U. Oszwa created the tabulate showing the child’s fundamental achievements at the early phases of the development of his arithmetical skills, which provides the consecutive stages of the child’s education in a chronological way and shows the way how he gains his skills.

Table 1

The child’s age (years)	Arithmetical achievements
0;0	Distinguishing the sets which are not numerous
0;4	Addition and subtraction of one object
0;11	Differentiating the rising and diminishing sequence of objects
2;0	The beginning of getting to know the sequence of numbers, the ability to assign them one by one while trying to share something out (e.g. building blocks)
3;6	Addition and subtraction of one element while playing with objects and numbers; applying the rules of cardinal numbers in order to determine the number of elements in a set.
4;0	Counting on fingers spontaneously when there is the need to add something fast
5;0	Adding the sets which are not numerous
5;6	Understanding that addition is alternate and starting counting from the bigger number of objects
6;0	Understanding the permanence of the particular number
6;6	Understanding the rule of how addition and subtraction is alternate; counting correctly even up to 80
7;0	Getting out of memory the correct facts about numbers (e.g. 2+2=4 without the need to count it every single time)

The source: U. Oszwa (2006), The Development and Assessment of Mathematical Skills Amongst 6-Year-Old Children, Methodical Centre of Psychological and Pedagogical Help, Warsaw, p.3.

Teacher should take into consideration the child’s good, “he should know how to organize the children’s classes. They ought to be filled with play, interesting tasks and games. It is necessary to talk to the child because it is in favour with the development of his thinking” [Gruszczyk-Kolczynska, Zielinska 1997: 10]. Anna Klim-Klimaszewska while describing the basic functions of the teacher states that he “functions as the organizer and guide and also the children’s partne” [Klim-Klimaszewska 2010: 160].

J. Karbowniczek states that “From among many works and classes the most enjoyable and satisfying are cheerful, educational, attractive and interesting games and plays. While choosing them there should be taken into consideration the versatile development of the child’s personality, i.e. physical, intellectual, social and emotional spheres and also interests, abilities, needs and his engagement” [Karbowniczek 2006: № 6].

Mathematical activity is one of the most important aims of mathematical education. It is the activity of the mind typical of the mathematician. According to Anna Z. Krygowska “learning mathematics is the organized activity which covers:

• 1)    taking over and assimilating information obtained from various sources,

• 2)    using the information directly in order to:

• a)    do the standard exercises which have the practice character,

• b)    gain new information independently,

  • 3)    creating subjectively new for the learner elements of knowledge, subjectively new concepts, statements and methods while solving problems formulated by others or the learner himsel” [Krygowska 1979].

Erich Wittman presents mathematical activity as the aims of learning in the process of teaching: mathematizing, classifying, putting in order, reasoning through analogy, generalizing and specifying, justifying and proving, formalizing, affective and creative behaviour [Siwek 2005:139].

The initiator that releases the child’s creative activity is the teacher who chooses the methods which make it easier to create the climate of mathematical environment. His actions should lead to the releasing reactions in his pupils, such as: interest, being focused on receiving some information, stimulating their willingness to get good results, satisfaction from participating in classes. While talking about methods which are used at a nursery school when it comes to mathematical education I often wonder: Why in the process of introducing the children into the world of the abstract mathematical language not to take advantage of the methods which are used to teach a foreign language? The reason is their creators usually based on the principle of creating such a climate by the teacher which wouldn’t be stressful in the teaching/ learning process, that would let the pupils relax during their classes, overcome the fear and get rid of inhibition regarding speaking and also undoubtedly makes the learning process more attractive. My attention was focused on the educational project method.

Projects are usually worked out by a small group of children from a class, sometimes it is the whole class and sometimes an individual child. The fundamental feature of a project is its research character and in addition research actions should be intentionally focused on finding the answer to the question given regarding the topic. The question may be asked by children themselves or the by the teacher and it may appear during the mutual work of the teacher and his students” [Katz 1994].

On the basis of literature and some scientific experience it can be stated that it is not every task given to students (pupils) that can be named the project. The task can be defined as an educational project only if the following conditions are fulfilled: it has a research character, it includes various areas of education, research activeness is focused on asking questions by children themselves and trying to find the answer on their own, it initiates unaided making decisions concerning types of executed tasks and types of organized episodes, it engages children practically, intellectually and emotionally, it creates the possibility for developing the interests of ‘the researchers’, actions taken provide intellectual vigour, children have their influence on choosing the topic and they accept the project aims, children work together or in groups, children are familiar with deadlines of the task execution, it involves with its rage outdoor activities even in initial stages, it integrates educational institution with a child’s home and local environment, the participants are familiar with the rules of assessment, all of the research group members know the ways of presenting their work.

Thus, the educational project is the set of planned actions. Students take them and solve the problems on their own, while the teacher is supposed to monitor and give essential hints, he also assesses their work on specific stages. He does not criticize or impose his will. It is advised for the teacher to praise his pupils the more the better because it motivates them to take even more ambitious actions. The prize for each child, the small and the big one, is not only the mark, it is enough to exclaim: ‘what a brilliant idea’, ‘you are so smart’, ‘that’s great’, etc. Educational process which is carried out like that enables parents to join in it in a natural way, without the compulsion from above.

Briefly, the project in a mutual adventure of children (students), the teacher and parents while getting to know the world.

The whole process of the project execution depends on the age of its participants, especially the second stage (working on the project’s outline). When it comes to small children there can be omitted creating the instruction (contract), because often some of the decisions written there, the teacher may place in so called teaching practice journal . In this document there should be systematically written down crucial information about children and decisions made by them while taking cognitive actions.

The project structure can be divided into five stages:

• 1.    Initiation of the project (choosing the topic).

• 2.    Working out the plan of the project.

• 3.    Carrying out the project (taking up the research activeness).

• 4.    Finishing the project, the presentation.

The topic of the project may be given both by children (pupils) and by the teacher or parents depending on needs or the teaching programme.

The teacher draws up the preparatory network. Here, he can refer to the content and goals of the executed programme. He can place at his drawing only words, terms and expected children’s questions.

In the phase of graphic form creation there should be various matters taken into consideration, e.g.: the time devoted to execute the project, the shape of the final product, chances to receive some financial resources for possible needs, joining parents in, the way of documenting events and archiving collected materials, the possibility of having the experts’ help, preparing the room for the project (classroom), collecting the necessary materials and equipment, sharing responsibility concerning actions taken regarding the project, solving conflicts in the group of little explorers etc.

Next, the teacher along with children create the initial network (thematic) or the list of questions. Children ask some questions and the teacher writes down their ideas and remarks on the piece of paper. For younger children the poster may be complemented with pictures, drawings.

In order to eventually finish that stage there ought to be collected materials and equipment.

The execution of this stage usually lasts the longest and it depends on the fact that children conduct their own research works, collect materials, sort them and then work them out. In this phase the teacher discreetly supervises actions taken by ‘the explorers’, he advises them and offers literature, makes it easier to let them get to the source of information. At this stage the teacher along with children may come back to the project’s network and complement it.

The works can be finished in three different ways:

• •    the project was carried out – the intended effect was achieved and its executors absorbed new knowledge and abilities. The results of their work can be presented to parents or other people interested in that undertaking;

• •    although the project was formally finished, the group continues its current activity (e.g. there can be organized a trip to the forest in order to deepen the knowledge regarding plants and its inhabitants). It is so called ‘open’ variant of finishing the project;

• •    the project was not executed – the achieved effects are compared with intended ones and with both thematic and preparatory networks. There are analyzed individual stages of work in search of the reason of failure. Not completing the project does not have to mean that there was also the teacher’s failure. With reference to social learning it may turn out that the project method was more effective than in the other two variants (e.g. there was nobody who did the work of the other person). It should always be taken into account, that there may be various reasons, due to which the project can not come into being.

• 5.    The assessment, which is not with regard to the final product, but concerning all of the activities while executing the project.

The teacher and his little explorers analyze the project’s course, taking into consideration any mistakes and failures and then they can draw the conclusion for the future. Children are encouraged to share the gained knowledge. The teacher’s task is not only assessing the whole process with its final product, but also assisting the children while presenting the results to parents or other people interested in that undertaking.