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The article concerns the research on students' perception of visual structures in van Hiele's sense (1986). The empirical % 1986 studies described in the paper make use of a combined methodology -- eye-tracking and a written study questionnaire. There was an analysis of the results of 14 pupils in the paper the 1st grade of a middle school and 19 pupils from a secondary school. The subjects were shown slide on a computer screen with task and were requested to solve them. Further eye-tracking parameters suggest that the ability of switching over from one structure to another, as described by van Hiele (1986), between the geometric and arithmetic structures of the presented objects guarantees success in solving the task. Constructing various structures is not only fundamental for solving problems, but also for the sake of the development of many mathematical concepts and their properties in relation to mathematics, as well as in the context of developing other key competences.
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This article refers to one of the important objectives in teaching mathematics – shaping its formal language. It raises the question of teaching methods in crossing the decimal threshold in addition and subtraction, used in Poland for over 50 years, that are too formal for students of the first educational stage, and that boil down to successive transformations on a symbolic level. On one hand, there will be presented a theoretical analysis of literature concerning the formation of the language of mathematics and inefficient methods associated with a too early transition to the level of symbolic writing, that can lead to the danger of formalism in teaching mathematics. On the other hand, based on fragments of the research, there will be presented extreme difficulties of a second grade student in a primary school in the process of crossing of the threshold of ten in symbolic writing and examples of the use of less formal writing for children, adapted to their natural development. Considerations in the article suggests leaving a rigid school methodology, while encouraging the use of less formal writing in computational mathematics in early childhood education, as a transitional stage in the development of a proper formal language for students.
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This article summarizes the third bi-annual conference in the series Chil-dren’s Mathematical Education (CME)1, held from the 2nd of July tillthe 5th of July 2012, in Rzeszów, Poland. It was organized with scientific andadministrative support by the University of Rzeszów, under the chairmanshipof Ewa Swoboda. The theme of the conference was Generalization in mathematicsat all educational levels.
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