Investigating the understanding of Tehran’s second elementary school students of the fraction’s part- whole subconstruct, based on APOS and SOLO theories with using an unusual task

The aim of this research was investigating understanding of Tehran’s second ‎elementary school students of the fraction concept (part-whole subconstruct) based on APOS and ‎SOLO theories. This study’s method was descriptive-survey ‎method, its statistical population was Tehran’s second elementary school students in the academic ‎year 1397-1398 and its sample was 598 people of the statistical population that selected by multistage sampling method. A test with an unusual task used for collecting data. The content validity of the research tools were confirmed by experts and scholars of mathematics education, and the reliability of the research tools was obtained based on Cronbach's alpha of 0.7. Results of this ‎study showed that students have a limited understanding of fraction’s concept and they have common misconceptions. The most common misconceptions ‎found include: (1) Disregarding the requirement of ‎equal parts in part- whole ‎subconstruct; 2- Understanding fraction in part- whole subconstruct as ‎the part-to-part ratio & 3- ‎Using approximate partitioning ‎to determine the exact fraction of the specified part.‎ Analyzing the responses based on the APOS theory also revealed students did not have ability to use this concept in dealing with unusual situations. In analyzing ‎the responses based on the Solo model, more than 60% of the responses were ‎in the multistructural level. The suggestions of this research ‎are not to over-emphasize the part -whole subconstruct, creating ‎equal opportunities for developing other subconstruct‎s of fraction and emphasizing on ‎conceptual learning of procedures and algorithms relative to fraction ‎concept in content and ‎opportunities presented to students.‎