Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA

Abstract

Our goal in this paper is to study students' understanding of integration by parts based on two theories, APOS and OSA. We make an epistemic configuration (EC) of primary objects that a student activate for solving tasks in relation to the integration by parts, and then we design a genetic decomposition (GD) of mental constructions that he/she might need to learn the integration by parts. We then describe the EC and GD in terms of the levels of development of Schema (i.e., intra, inter and trans). Three tasks in a semi-structured interview were used to explore twenty three first-year students' understanding of integration by parts and classify their schemas. Results showed that students had difficulties in integration by parts, especially in using this technique to obtain a simpler integral than the one they started with. Using APOS and OSA gave us a clear insight about students' difficulties and helped us to better describe students' understanding of integration by parts.

What might be frequently overlooked is actually still beneficial: Learning from post national-standardized school examination

“…The complexity of the process was coupled with the need for students to perform algebraic manipulations in order to obtain algebraic forms corresponding to the available options. A number of previous studies (e.g., Borji & Font, 2019;Borji et al, 2021) have revealed that when solving integration by parts, students found it difficult to determine u(x) and dv such that the determination could make it easier for them to determine v(x) and ∫ vdu. Other than that, Li et al (2017) have found that techniques determination was one of the difficulties experienced by students in solving integral problems, and students were often overwhelmed with many techniques of integration.…”

The Analysis of Primary Students’ Learning Obstacles on Plane Figures’ Perimeter and Area using Onto-Semiotic Approach

“…An onto-semiotic approach considers the nature or meaning of mathematical objects and includes three aspects of mathematics: problem-solving, symbolic language, and organized logical and conceptual systems (Godino, Batanero, & Roa, 2005). The onto-semiotic approach has been used to analyze the obstacles to learning mathematics based on previous research: (1) analyze students' understanding of algebra using the onto-semiotic approach, which focuses on language components, descriptions, procedures, propositions, and arguments (Amin et al, 2018); (2) examine learning obstacles related to sequences and series, with a focus on situation components, descriptions, and procedures (Rachma & Rosjanuardi, 2021); (3) examine students' difficulties with the Pythagorean theorem, focusing on language elements, situations, descriptions, procedures, propositions, and arguments (Rudi, Suryadi, & Rosjanuardi, 2020); (4) analyze the understanding of integration focusing on language components, problems, concepts, procedures, propositions, and arguments (Borji & Font, 2019).…”