Role of concrete visual representations for understanding of the long division algorithm in public and Montessori primary schools

Concrete visual representations are indispensable in mathematics lessons, as they help students to develop their numerical ideas and understanding of abstract mathematical concepts and procedures. Many experts agree on the importance of concrete visual representations in knowledge acquisition, but their opinions on their use are not unanimous. Most agree that concrete visual representations are necessary in the first three years of primary school, but few emphasize their use in the higher grades. The long division is one of the subjects taught in mathematics at grade level, which requires a high level of abstract thinking. The way this content is taught in a Montessori primary school is quite different from the way it is taught in a public primary school. The main difference is that in a Montessori environment, this content is dealt with in a very concrete way, with the handling of concrete materials, whereas in a public primary school, it is mostly dealt with on a symbolic level. In this master's thesis, we compared the understanding of the long division algorithm between students in the two schools, focusing on the impact of the use of concrete visual representations. We conducted a qualitative study in which we individually observed students in public and Montessori primary school while they were solving long division tasks. We subsequently analysed their products and observations on the use of concrete visual representations. Analysis of the results shows that Montessori primary school students use concrete materials more often and perform better on long division tasks than students in public primary school. In the sample, only students with learning difficulties used concrete visual representations. Gifted students did not use them, as they had probably already reached an abstract level of thinking. Students with learning difficulties in mathematics who used concrete visual representations in their calculations scored on average better than other students with learning difficulties who did not use representations, indicating the positive impact of representations for this group of students. An analysis of the errors that occurred during the problem-solving tasks showed that two types of errors that do not occur in the gifted group are prevalent in the case of students with learning difficulties, namely errors related to the recall of arithmetic facts and zero in the result. The results of the survey are intended to encourage teachers to use concrete representations more often when teaching procedurally demanding mathematical content, such as long division