On formal and informal notation of calculation during the early learning of arithmetic by young students

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.