The Construction of the Natural Number’s Concept in Kindergarten: The “Go!” Game

Natural number is present in daily activities and children’s lives. From the early age, they memorably and repetitively count small quantities and express various situations (age, date of birth, family members, measurements, orderings, etc.). Numbers represent one of the fundamental pillars of mathematics, as well as one of its main topics, as highlighted in the Italian National Guidelines for kindergarten and the first cycle of school (MIUR, 2012). The concept of natural numbers is highlighted as being a very sophisticated and complex concept (Mann, 2014; NCTM, 2000). In the Italian National Guidelines, it is suggested that it should be addressed and built up gradually in a vertical perspective from the early years of school and preschool to grade 13 (MIUR, 2012, p. 49). It often happens that the need to count and use natural numbers by memorizing the numerical sequence is not accompanied by equally in-depth work on the understanding of numbers. This may be a more frequent situation for children who experience difficulties, as they may find comfort and security in reproductive and schematic learning, rather than more complex learning. The consequence of these learning choices means that they risk exclusion from more meaningful learning experiences. Our didactic assumption is that the process of formalizing leading so that it characterizes natural number, through understanding three aspects, cardinal, ordinal and recursive, can be achieved through the mediation of suitable artefacts. This can be done in accordance with a specific teaching and learning model, based on the framework of the Theory of Semiotic Mediation (TSM) (Bartolini Bussi & Mariotti, 2008). The TSM provides a reference for the design and the implementation of a teaching sequence, as well as the analysis of the data coming from the experiment. The main aim of the experimental educational path described in this chapter is to support children’s development of their sense of natural numbers, using games and artefacts, as outlined in the TSM. In particular, games are selected as a means to approach the discovery and construction of new mathematical concepts, while artefacts are chosen based on the problem situations presented to the children, in accordance with the principles of semiotic mediation theory. The analysis of the transcriptions of children’s talking during the game reveals the evolution of the concept of one-to-one correspondence, the equipotency between sets (thus an initial idea of cardinal numbers), the representation of numbers, and their ordering. Additionally, in the final phase of the game, comparing the cardinality of plastic bags and ordering them based on their cardinality evokes the idea of a number’s successor through the +1 operator. These ideas lay the foundation for building the ordinal, cardinal, and recursive aspects of natural numbers. The cardinal aspect of Natural number refers to the idea of biunivocal correspondence between elements belonging to two equipotent sets (Piaget, 1997). In mathematical literature (Kunen, 2011), the term equipotent refers to sets that have the same cardinality, meaning there exists a bijective function between them. Two sets A and B are equipotent if there is a one-to-one correspondence (bijection) between their elements. The ordinal aspect of Natural number concerns the comparison of cardinal numbers establishing which set contains the greatest number of elements. This aspect focuses on sorting, gradually leading to the construction of the number line (Piaget, 1997). The recursive aspect of the Natural number, stated in Peano’s axioms, fixes a starting point by assuming zero as the first Natural number, and constructs the next number of n, through adding 1 to any number. In summary, the cardinal aspect allows the elements of a set to be counted; the ordinal aspect allows them to be reordered, according to an ascending or descending order; the recursive aspect avoids “jumps” in the Natural number’s sequence (Butterworth, 1999). This idea is linked to the construct “number sense” that is seen as essential for learning arithmetic. Many researchers also believe that the development of number sense is a fundamental skill for acquiring awareness of the structural aspects of number and for promoting pre-algebraic thinking (Fuson, 2012; Griffin et al., 1994). The educational path we have designed, identified different phases related to each aspect of Natural numbers (Usmanov, 2020). The design, implementation and analysis of results in the educational path are based on the theoretical framework of TSM. This framework provides an important reference for studying the relationship between artefacts, the actions they enable, and the construction of mathematical concepts by the learners who use them. Finally, the analysis focuses on the semiotic potential of the artefact used.