Undergraduate Afghan Students' Understanding Of Concepts In Elementary Group Theory

Most of high school algebra is based on algebraic manipulations and solving equations,while algebra at university level is based on set-theoretic definition and formal proof.The transition to this formal mathematics will not take place easily due to the abstract nature and presentation of group theory concepts in undergraduate algebra.This abstract presentation of group theory concepts chal enges students' understanding as well as their success in the subject.This study explored about undergraduate Afghan students' understanding of key group theory concepts,their misconceptions and difficulties in learning fundamental concepts of group theory,and linked the identified difficulties and misconceptions to its possible sources/causes as well as to some met-before notions,supportive and problematic aspects of students' conceptions of definitions of these formally constructed concepts.The study made use of the ‘three worlds of mathematics' theory of David Tal and problem-solving ability of a student as a way of measuring the level of understanding of a learner solving concept-related problems in elementary group theory.Analyzing the data collected in the form of written questionnaires from 112 students on problems related to elementary group theory concepts has revealed several important issues regarding students' understanding of elementary group theory concepts.This study showed in general that undergraduate Afghan students' understanding of group theory concepts became weaker as the level of concepts became more sophisticated.This study also showed that undergraduate Afghan students still have new misconceptions and many difficulties in understanding group theory concepts.The study specifically showed that the first major difficulty was related to the symbolic representation of the definitions of concepts in group theory.It was more difficult for students to understand the symbolic representation of the definition of a concept in the axiomatic formal world of mathematics compared to the word form definition of the concept in the embodied formal world.The second major difficulty was in dealing with non-routine and abstract problems compared to more concrete problems related to their high school mathematics.Students were more successful in dealing with already-met problems in high school than dealing with problems they had not already sufficiently worked with as they did in high school.It was also found that providing the definition of a concept for some students may not help them successfully deal with the problem that only needed to apply the given definition on the problem to be solved.The third major difficulty of these students was in applying the steps and axioms of group theory concepts.Majority of these students could not recognize the important aspects of the definitions and were unable to apply the steps and axioms of the formal definitions of group theory concepts.The study also identified some specific supportive and problematic met-befores and conceptions of group theory concepts that influenced students' judgment in the process of solving concept related problems.In a nutshell,this study showed that a deep gap exists between students' personal meanings and formal definitions of group theory concepts.To overcome these difficulties and misconceptions,we propose that these undergraduate Afghan students' understanding of definitions and their personal meanings of group theory concepts may be improved and strengthened by dealing frequently with various types of problems related to group theory concepts,and one has to take into consideration the link between the identified misconcept io ns and difficulties with its possible sources/causes in learning group theory concepts.