How fifth-grade pupils reason about fractions: a reliance on part-whole subconstructs

Fractions are an important part of the primary school mathematics curriculum, being introduced to children in the Slovenian educational system during the second grade (at age seven). For clarification of terms it is worth mentioning that the term fraction is only introduced in the sixth grade. Before that, the term equal parts of a given whole (In Slovene language, there are two different terms which are used in teaching about fractions: one is equal parts of a given whole as deli celote and another is fraction as ulomek. Fraction in our elementary school has the meaning of all other subconstructs proposed by Kieren (1976) except part-whole subconstruct. This subconstruct is covered by another term, namely deli celote. For easier understanding, we use in this paper only the term fraction.) is used. When we speak about fractions in primary school, we actually speak about parts of a given whole where the whole could be a set of objects or a single object that can be divided into equal parts. We used the fractional scheme theory for the theory framework of our research, focusing mainly on parts within the whole, the part-whole fraction scheme, and, to some extent, the partitive unit fraction scheme. Ninety fifth-grade pupils took a test we developed with the aim of achieving a better understanding of the field of pupils' understanding of the part-whole aspect of fractions. More precisely, we were interested in how pupils understood the idea of dividing a whole into equal parts: is the idea of dividing mainly connected to dividing into congruent parts, are pupils able to reason about equal parts as parts of equal area, and do they adapt their reasoning according to the demands of the problems. The test consisted of four problems, each of them divided into parts (a total of eight problems to solve altogether). We used descriptive statistics and qualitative research based on the grounded theory method to process the data gathered from how participants solved the problems. We used the inductive process to determine how to categorize the responses and find relationship between the different categories. The findings showed that at the end of the fifth grade, pupils did not yet master the part-whole subconstruct of fractions and had problems related to linking congruency and parts of the whole. The mistakes pupils made clearly showed that a number of types of misunderstandings were prevalent: for example, not being able to connect the idea of fractions to equal parts of the whole or connecting the idea of fractions only to congruent parts of the whole as well as mistakes related to determining what part of the whole is presented.