Obstacles to students’ learning of the limit concept: A comparative study

Abstract in English

This thesis explores obstacles to students’ learning of the limit of a function as students learn the concept in post-compulsory schools in two culturally different contexts, namely Sweden and Ethiopia. The study investigates obstacles to students’ learning of the intuitive limit concept (the limit concept that is based on the intuitive definition) taught in upper secondary schools of these two contexts and characterises the nature of these obstacles. Additionally, the study compares the obstacles identified in the two contexts.

The study uses a collective instrumental case study approach. Data were collected through teacher interviews before and after lessons, lesson observations, student group interviews, and analysis of textbooks and official documents such as curricula and syllabi.

The study employed an eclectic conceptual framework that considers the mathematical content, the didactical aspects (teaching) and the students. The study identifies three main categories of obstacles: epistemological obstacles (EOs) arising from the nature of the mathematical concept of limit and other concepts important in limits, didactical obstacles (DOs) arising from teaching practices, and cognitive obstacles (COs) arising from incompatibilities between what is encountered in the lessons and students’ previous knowledge. Within each category, the study lists specific obstacles within each case and then compares and contrasts these. The analysis reveals the dependence of some obstacles on the culture of teaching and the context, particularly DOs and COs. The study also highlights the interplay between different specific obstacles and how the presence of one obstacle can contribute to the occurrence of another. Some of the interplays between some of the specific obstacles seem more common in one of the contexts than the other. For example, in the Swedish case, it was more common that the didactical choices acted on the epistemological obstacles in such a way that didactical obstacles were generated, which then played a role in the occurrence of some cognitive obstacles. This was summarised symbolically as EO→D→DO→CO. On the contrary, in the Ethiopian case, it was more common that the didactical choices impacted on how students understood an idea, which then resulted in the occurrence of cognitive obstacles. This was summarised symbolically as EO→D→C→CO. Thus, the study finds that EOs can be reinforced by the didactical choices reflected in the teaching. Furthermore, the specific EO can have different sources and triggers in the two contexts.

Considering two culturally different contexts, this study sheds light on the obstacles to students’ understanding of the limit of a function in their learning of the concept. It highlights the influence of the culture of teaching and the context dependence of obstacles, providing insights into how the nature of the concept, teaching practices, and students’ prior knowledge impact on their learning. The study’s findings contribute to filling the gap in previous research of a lack of evidence concerning the interplay between the three forms of obstacles, the role of teaching culture and context on the occurrence of obstacles, and classroom-based comparative studies on the teaching and learning of the limit concept and calculus. The result of the study may also contribute to informing teaching practice and to the development of teaching strategies and curriculum design for improving students’ understanding of the limit concept and other mathematical concepts in general.