The construction processes of factor and multiple concepts of 6th grade students in realistic mathematics education based learning environment

This research aims to examine constructing the concepts of prime number, prime factor, common factor/divisor, and common multiple of 6th-grade students in a teaching environment designed according to the Realistic Mathematics Education (RME) approach. The research was carried out in a 6th class of 15 students studying in a state secondary school in the Black Sea region and was designed with a case study as a qualitative research method. A readiness test was prepared to reveal the pre-knowledge of the concepts that formed the basis of factor/divisor and multiple and was applied to the students. The students were divided into heterogeneous groups according to the results of this test, the researcher's observations, and other course teachers' opinions. Students were asked to solve three posed contextual problems in the RME environment, and individual ideas were supported by in-group and inter-group discussions throughout the teaching process. Three participants, whose readiness level was advanced, upper-intermediate, and intermediate, were determined by the purposeful sampling method from the groups. Four individual interviews were conducted with each of them. The data obtained from the transcripts of the camera and audio recordings of the teaching process, group discussions and interviews, individual and group worksheets, and observations were evaluated with content analysis within the framework of APOS theory. The results demonstrate that all three participants constructed the concepts of prime number, prime factor, common multiple and common divisor/factor at the object stage and even constructed the concepts of the greatest common divisor and least common multiple as objects. In addition, the participant with upper-intermediate readiness level constructed as an object by associating the prime number with the greatest common divisor and the least common multiple, which is in the following class achievements. Also, the participant with advanced readiness level completed the process stage and started to form an object. It has been revealed that coordination with factor/divisor, multiple, odd-even number, factor tree, divisibility rules and algorithm processes are significant in constructing of prime numbers, prime factor, common factor/divisor, and common multiple concepts. According to the results, suggestions for future research in teaching the concepts above are presented.