Abstract:

Among the most difficult concepts that children must learn in Primary school we can find the concept of rational number, and finding ways to help them to build on this concept is a relevant contribute to their number sense development. Literature has suggested a first approach to fractions through equal sharing, as it helps students understanding the meaning of the numerator (the number of whole items to be distributed among the containers), denominator (the number of containers), and their relationship, facilitating the construction of the concept of fraction. Equal sharing, despite of its relevance and as provided in the Portuguese official curriculum for Mathematics at these levels, does not yet seem to be comfortably mastered by Primary school students. So, we conducted a case study using a qualitative methods approach to analyse 3rd- 5th graders (N=60) understanding of fractions, when solving tasks on equal sharing presented by their teachers, and thus trying to identify students’ ideas of fractions, their difficulties and misconceptions when solving the tasks. Data were collected during classes were students worked in pairs/triads to solve the tasks. In general, the results suggest that age influences the resolution of these tasks, indicating that younger students seem to have greater success in carrying out these tasks, as they easily rely on their informal knowledge about fractions, while older students revealed that they have more difficulties in articulating with understanding the iconic and symbolic representation of fractions. A more detailed analysis of the data allowed us to verify that students rely on the iconic representation of the problem situation, whether representing all the items to be shared or just one of them to support their reasoning, and to justify their written answers. However, there were differences found in results among students from different school years. In this process, 3rd year students sometimes presented a correct iconic representation associated with an incorrect symbolic representation, while 4th year students were successful in almost all tasks. Regarding the results of 5th year students, they revealed difficulties in solving equal sharing problems, as only 2 of the 6 pairs of students solved the problems correctly, despite all students having used the iconic representation to support their reasoning, they revealed problems with symbolic representation, failing to make conceptual change when moving from whole numbers to fractions.

Keywords:

Rational number, learning of fractions, equal sharing, math classroom, preservice teachers.