Teaching and Learning limits at the Secondary Level in Lebanon. (c2015)

Abstract

The concept of “limit of a function” plays a fundamental key role in comprehending basic calculus notions and its application is also essential in other disciplines. This makes it necessary for students to develop a firm conceptual understanding of limits. The main purpose of this study is to investigate the types of difficulties that Grade 11 scientific track students face while learning limits of functions and the possible sources of those difficulties as well as students’ perceptions of limits. The study took course over two academic years in a Lebanese school. The participants were grade 11 class (24 students and teacher) in a chosen Lebanese school. The conducted research is a qualitative study based on curriculum and textbook analysis, class observation log analysis, analysis of two tests prepared by the teacher, as well as a questionnaire and a post-test analysis administered to 36 grade-12 students to analyze their conceptual understanding and retention of the concept, a year from instruction. Findings show that the difficulties that students face while learning the concept of “limit of a function” are due to several reasons. The analysis of curriculum and textbook shows that the difficulties that students face relate to the representations used in the textbook as well as the way the concept is developed in the Lebanese National Curriculum and textbooks. The analysis of the observation log shows that the difficulties are mainly related to the metaphysical aspect of the notion of limit, and the elusive concept of infinity, as well as to the fact that the limit process cannot be performed by merely simple arithmetic or algebra. The analysis of the questionnaire shows that the most adopted conception of limit adopted by students is that the “limit describes how a function moves as x moves toward a certain point”, followed by the two ideas, the first one being that “a limit is a number that the y-values of a function can be made arbitrarily close to by assigning specific numbers to the x-values” and the second one that “a limit is determined by plugging in numbers closer and closer to a given number until the limit is reached”. The analysis of the three tests shows that the difficulties are mainly related to the fact that the limit process cannot be performed by simple arithmetic or algebra, to the concept of infinity, the concept of function, as well as to previous knowledge. This study gives an incentive to all stakeholders to work collaboratively to rethink the curriculum and textbooks they are using and to create a suitable learning context to develop and apply the concept of limit.