Cognitive obstacles in interiorization of the Riemann’s Sum concept through APOS approach

Abstract

The APOS approach in mathematics learning has been developed by Dubinsky and has been applied in many studies. Instructional to build the concept through the stages of Action - Process - Objects and Schema do not always run well as genetic decomposition that has been built. This study explores students’ cognitive difficulties in learning advanced mathematics on the Riemanns Sum, particularly in the process of interiorization Action into Processes. The subject of research is first year student of UIN Walisongo Semarang and the study was conducted in the first half of 2018. The data were taken using observation, tests and interviews. The results of the qualitative analysis show that students got the Action well but failed to interiorize it into Process due to the inability to connect two-dimensional knowledge with the given partition. It was showed when students could not find rectangular elements on a partition built from the area under the curve with a particular function. Unexpectedly, even though students understand the problem and the student could not use the given function y=f(x) to determine the partition height. Furthermore, students got an obstacle to imagine adding partitions when the number of partitions approach to infinity. The transition from school mathematics to advanced mathematics is perceived when students represent their knowledge partially, instead understanding the problem as a whole interrelated concept. This constraint also indicates the importance of mathematical connections and mastery of prerequisite materials to construct the further concepts.