Main Page -> Working Seminars -> Children 3-6 years old

Guided by:
Edyta Gruszczyk-Kolczynska - University of Warsaw, Poland
Regarding mathematically gifted children all we know is that they acquire knowledge and mathematical skills without any difficulties and they use them in school and everyday life with pleasure. A. W. Krutiecki1 claims that the beginnings of the inborn mathematical abilities can be observed in the children’s behaviors. If those predispositions are nurtured and developed they take such a form, which Krutiecki presented in his famous model of mathematical abilities of older students and adults.
However it is not clear, which stage of children’s lives Krutiecki had in mind – whether preschool children or younger students. Furthermore it is hard to implement Krutiecki’s wise idea because of the following reasons:
  • Teachers and parents believe that during preschool and early school years it is too early to develop children’s mathematical abilities.
  • There are no diagnostic tools adapted to the very young learners’ mental abilities. That is why it is impossible to determine whether they are endowed with mathematical talent or not.
  • Children’s mathematical abilities are only mentioned in the psychological and pedagogical literature, so we do not know how they can manifest themselves and how they can be nurtured and developed.
That is why adults – parents and teachers – are convinced that the ease of learning mathematics is an indicator of outstanding but somehow general intellectual performance. Adults also see no need to deal with mathematically gifted children, because they are coping well in the school and everyday life situations.
It is widely accepted that mathematical talent is rare and can be only presented by some of the older students, for example one or two in the class. At the same time it is considered that the mathematics in the school dimension is so difficult that mathematical abilities are needed to learn it. This way of thinking is convenient for everyone, because:
  • Teachers do not have to blame themselves for not being effective in teaching because what can be done if the class is not able to learn mathematics.
  • Students are excused from working hard, because abilities are heritable and if they were not endowed with them, there is nothing they can do.
  • Parents have no sense of guilt, even when they did not watch the children to do their homework, because the lack of the mathematical abilities cannot be fixed by anything.
The consequence is a silent approval of the lower level of students and adults’ mathematical skills. This also applies to teachers: we tolerate the fact that their students know embarrassingly little of mathematics and we praise them for educational performance, when several students from their class represent a slightly higher level of mathematical skills.
Meanwhile, the research2 on the effectiveness of assisting children development with the mathematical education shows that more than a half of Polish children (58%) in preschool age present a mathematical focus and amazing ease of learning mathematical skills. The fact that more than a half of the children, participating in the experiment, showed the mathematical attitude of mind suggest that the mathematical abilities are not as rare as has been assumed.
Research findings mentioned above are contrary to the widespread conviction that talent for learning mathematics is rare and appear only in some older students. I explain this contradiction as follows:
  • The beginnings of the mathematical abilities are manifested in the preschool age. According to me it happens in the fifth and sixth year of child’s life and sometimes even earlier.
  • Adults and teachers rarely see them, because in this period of life children do not use the number and measure. Adults and teachers do not associate emerging mathematical talents with such characteristics of mind as ease of learning, curiosity, capacity for intellectual effort and precision in thinking.
  • Mathematically gifted children make a lot of problems at school, because they ask difficult questions, solve tasks too quickly and in the non-routine way and then because they are bored, they disturb. That is why gifted children are underestimated and sometimes even punished.
  • Teachers are not prepared to support the mental development of children, who are gifted, particularly in mathematics.
The effect of such suppression of cognitive and creative mathematical activities is that the gifted children are pushed to the average level. Only those children whose parents believe in their more than ordinary mental capacity and perceive in them mathematical skills retain the motivation to learn mathematics and demonstrate high mathematical skills in the senior grades in primary school. Unfortunately only a few students in the class are in such a privileged position. That is why it is considered that the abilities to learn mathematics are so rare. It is ignored that in many cases mathematical talent was simply destroyed.
The first period of wasted opportunities for gifted children will be the last year of kindergarten education and first year in primary school. This cannot be undone in the next few years of school education. It is really worth to take care of problems of children that reveal skills in mathematics in kindergarten and at the beginning of primary school education. During the seminar Recognizing mathematically gifted children and assisting their development at home, in kindergarten and school we shall focus on three issues:
  • the manifestations of special ease of acquisitions information and mathematical skills by older preschoolers and young pupils;
  • methods of recognizing children’s mathematical skills;
  • assumptions and methods to assist children in developing their mathematical abilities at home, in kindergarten and school.
We invite all interested in such issues to present their achievements. In the introduction to the scientific discussion there will be a presentation of the results of research and pedagogical experiences relating to each problem mentioned above. The methods of diagnosing and assisting older preschoolers and young pupils’ development with mathematical education will also be presented and made available.

1 A.W. Krutiecki has developed a useful diagnostic tool to determine the mathematical abilities of older students, detailed descriptions of the mathematical model can be found in his book Psichołogia matiematiczeskich sposobnostiej szkolnikow (Izdatielstwo "Prosfieszczenieje"), Moscow 1968.

2 In the years 1996-2004 E. Gruszczyk-Kolczyńska and E.Zielinska conducted experimental research based on leading workshops for children according to the program Children's mathematics program in kindergartens. Then the future school career of these children were observed and based on the educational outcomes the conclusions were drawn about their mathematical education, there were also interviews conducted with their teachers and parents. It turned out that all of them had no difficulties in mathematics (only some of them – around 8% – had trouble with the acquisition of literacy.) More than a half of the children showed interest in a specific mathematical operations, the willingness of thinking in the mathematical way, in drawing the attention to the mathematical problem, emphasizing the spatial and numerical relations, etc. This characteristic of the mind Krutiecki (op. cit) called mathematical attitude of mind – view of the world through the mathematical eyes – and he believes that it has a leading role in the structure of mathematical abilities.

Guided by:
Paola Vighi - Mathematics Department, University of Parma, Italy
Mathematical competence

According to recent cognitive theories, children can perform mathematical procedures starting from the age of six months (Dehaene, 1997). So it is an opportunity to propose logical and mathematical activities in kindergartens and it could be an important stimulant suitable for their cerebral activity.
In this context, some questions appear: is it possible to promote the development of mathematical competences? In addition, what does the locution "mathematical competence" mean?
In Italian curricula for kindergartens we can read: "Development of competence means to learn thinking about experience through the exploration, the observation and the practice of comparison; to describe own experience and to transpose it through personal and shared traces, recalling, recounting and representing significant events; to grow the attitude to pose questions, to think, to negotiate meanings".
It may appear excessive to speak about "mathematical competences" in kindergarten. In some contexts the word "competence" means "mastery" or "control", but the Penguin Dictionary of Psychology write: "Competence is the ability to realize a work or to achieve something with success". In this way, we involve the concept of ability, that can be understood in practical sense (dexterity) or in cognitive meaning (intelligence). Usually the word "competence" is related to operating aspects, to be able to make something, while the "comprehension" is also related to knowing the reason to make something" (Godino, 2003). In other words, competence takes care of practical aspects, while the comprehension is strictly related to knowledge. The competence is a secure base for comprehension, versus the comprehension facilitates the development of competence.
With pupils 3-5 years old, firstly we can encourage this development, but it is also important to observe and to promote mathematical thinking and reasoning in children.
Plays and their role
In this context, a play has a fundamental role: every play requires different competences, in particular logic and mathematical competences. Sometimes it is difficult to find a good equilibrium between a free and a spontaneous play and a guided play. When the teacher proposes a play finalized to promote particular abilities, he risks to force, in some way, the child and to impose directions of work connected with the play finality. It could be more interesting and opportune to observe the gestures and to listen the talks of the child. This is an aspect related to teacher competences that must seize mathematical aspects and attitudes in child behaviour.
Working Seminar activities
We propose the participants an activity, named "The play of colored houses", executed with pupils 5 years old in an Italian kindergarten. The first idea of this play was suggested by prof. Ewa Swoboda and we decided to experiment it in Italy with particular attention to mathematical competences and attitudes.
After a brief description of the play and some play sessions, we suggest to analyze mathematical concepts involved in it (also not basic and unproblematic!).
Mainly, the participants of working seminar will be asked to predict possible behaviours of children, winning strategies or difficulties, and so on. Afterwards, we will propose some short films and their analysis.

AA.VV. , 2009, Penguin Dictionary of Psychology, Penguin Books, LTD.
Dehaene, S., 2000, La Bosse des Maths, Editions Odile Jacob, Paris, France.
Godino, J., 2003, Competenza e comprensione matematica: che cosa sono e come si ottengono, in La Matematica e la sua Didattica, n. 1, Pitagora, Bologna, Italy.

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