Learning obstacles and the didactical design for teaching the limit of function in a Calculus course

Students often face difficulties in understanding the concept of limits in functions, a challenge that arises due to the abstract nature and complexity of the topic. Despite being familiar with the procedural steps, students may fail to grasp the underlying meaning of limits. This gap in comprehension leads to significant learning obstacles. As such, there is a critical need for effective didactical designs that can enhance the teaching and learning of this concept. This study aims to address this issue through a Didactical Design Research (DDR) approach, which is structured into three phases. The first phase involves a preliminary didactical design analysis, followed by administering a diagnostic test on the limit of functions to 26 third-semester students (Group 1) who have already completed a differential calculus course. This diagnostic test helps identify the initial learning obstacles. In the second phase, a didactical design is developed to address these obstacles, and it is then implemented with 33 first-semester students (Group 2) enrolled in a Differential Calculus course to evaluate the impact of the design. Data analysis is conducted based on the scores from the written diagnostic test, categorizing them into three levels of ability. The findings reveal that the primary learning obstacle for students is the formal definition of limits, and the identified obstacles are epistemological, psychological ontogenic, instrumental ontogenic, and conceptual ontogenic. The results of implementing the didactical design demonstrate a significant improvement in students' understanding of limits, as evidenced by a reduction in the learning obstacles encountered. This research contributes to the development of more effective didactical approaches for teaching complex mathematical concepts, offering a potential model for addressing similar learning challenges in other abstract topics.