Abraham Kumsa Beyene. Photo: Private.

What is your thesis about? 

– Many students, whether at upper secondary school or university, find it difficult to understand and use calculus– a branch of mathematics that deals with how things change together or vary together (co-variation). Even basic concepts in the field and the understanding and use of these concepts in mathematical analyses is difficult for students.

–    In my study, I investigate obstacles to students’ learning of the limit concept taught in upper secondary school in Sweden and Ethiopia, and compare the results between the two culturally different contexts. 

– The limit concept is a core concept upon which calculus is based.  A contributing reason why this is difficult for students is also due to the fact that there are many other mathematical concepts that are important for the limit concept – and which are also difficult for students to understand. Some examples of such concepts are the function concept and the infinity concept. In my study, I try to get a better understanding of the obstacles and what teaching can consider to facilitate students’ learning.

How come you chose to focus on this particular subject?

– It´s not unusual to hear students and adults say that "mathematics is difficult". I often ask myself the question – why? And especially when it comes to calculus. What is it that makes this particular area so difficult to grasp and understand?

– As someone who is an educated mathematician, I know that everything is rooted in the understanding of the so-called limit concept, which is the very basis of calculus. Other concepts in calculus are also based on this concept. Therefore, I thought it was important to deepen my understanding of what obstacles students encounter when learning the limit concept.

– In addition, there are limited studies on obstacles to students’ learning of the limit concept at upper secondary school level, and how the obstacles might look in different cultural contexts. Most of the studies that exist are conducted at university level and also focus mainly on the cognitive aspect, i.e. the student's own ability to think and understand the concept. Therefore, I wanted to examine this from a broader perspective – and also look at the role the “culture of teaching" might have in the occurrence of obstacles.

Were any results surprising?

– In my study, as said, I examine not only the obstacles that are connected to the student's own ability (cognitive obstacles), but also Obstacles that could arise from the method of teaching (didactical obstacles) and a series of other knowledge that can hinder the student's learning of the limit concept and other mathematical concepts that are connected to the limit concept (epistemological obstacles). One of the things that surprised me was how strongly the different obstacles depended on each other. The presence of one obstacle plays a big role in the occurrence of other obstacles. The interaction between the various obstacles also proved to be of great importance. 

– As an example, I can mention that the didactic (the content and design of the teaching) sometimes can reinforce Epistemological obstacles that affect students’ understanding of the issue. In this regard, among students in Sweden, there are obstacles that emerge from the use of technology to support teaching, while in the Ethiopian case, students did not have the opportunity to benefit the use of technology and what it could offer. 

– Another thing that surprised me was hearing the students say that they did not understand why they needed to learn about the limit concept and did not know what use it would have for real life. It seemed like they had an ambition to use limits in the same way they do with addition, subtraction, multiplication and division.

What do you hope your thesis will contribute to?

– I hope that current teachers and prospective teachers will consider the list of obstacles I came up with and find better ways to teach and disseminate the limit concept and other related mathematical concepts. There are many obstacles that a teacher needs to be aware of in order to create good opportunities for students' learning.

– Even curriculum designers and textbook developers can benefit greatly from my study. I have developed a model that I believe can highlight important aspects that play a role in students' knowledge development and the obstacles that arise from these aspects. The model can also be used to examine knowledge development in other subject areas by modifying the model according to the specific need.

What are your plans ahead? And has your thesis influenced you in any direction?

– It has been very educational and exciting to do research in the area of my interest and profession. It has given me many lessons and perspectives on what is important when teaching mathematics, calculus and the limit concept in calculus.

– My plan, from now on, is to engage in teaching and to continue doing research in mathematics education, with a special focus on teacher education and students’ concept development at the upper secondary and university levels. 

– In societal development, it is important, in many ways that students understand and learn mathematics, not least because mathematical knowledge of upper secondary level is often the basis for being able to continue studying at a higher level. Calculus has also a lot of applications in science and technology.

Download and read Abraham´s thesis

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