The influence of multiple representations on the learning of calculus by ESL students

Abstract

The goals of this study were to research the learning difficulties among a group of four pre-university introductory calculus students who were mainly international students studying English as a Second Language (M). The intention was to create a constructivist-style classroom environment in order to determine if it could improve students` knowledge about the use and management of multiple representations (that is, graphical, rum~ symbolic, pictorial, linguistic or diagrammatic approaches for problem representation), increase their classroom communication as a means to improving ability in the modelling of calculus word problems, and to develop, implement and evaluate a teaching package that encouraged the use of multiple representations as a means of improving conceptual understanding. The achievement of these goals was sought by means of the development, implementation and evaluation of a number of calculus extended tasks that encouraged the use of multiple representations. These activities facilitated the compilation of a menu of approaches to the solution of mathematical problems, while the longitudinal nature of the study allowed for the monitoring of student changes in their preferred approach. A traditional calculus curriculum was used for the study, but the instructional emphasis was based more on students' understanding of concepts in a classroom environment utilising a constructivist approach rather than on their memorising computational techniques. Reading, writing, and discussion were emphasised m small group settings to develop language skills and to foster an appreciation of the alternative solution strategies of individual students.The study was conducted at an International College north of Perth in Western Australia, and the majority of students in the sample were from Non-English-Speaking-Backgrounds (NESB). A range of methods was used to collect qualitative and quantitative data in order to increase the credibility of the research. These methods included audio recordings of structured task-based interviews with each of the four students in the sample; teacher analysis of student worksheets; my classroom observations; the analysis of alternative student conceptions on assessment tasks obtained through post-test interviews, and my personal reflections. Quality controls were employed to ensure the credibility of the data collected. As classroom teacher and principal researcher, it was possible for me to treat each of the four students involved as an individual case study. Descriptive questionnaires were used in order to gain information regarding the course and the use of graphics calculators. The results are applicable to ESL introductory calculus students only, and the nature of the sample implies a number of study limitations detailed in Chapter Five. There was extensive evidence of the benefits of the use of a multi-representational mode and evidence also of the benefits of encouraging the use of a diversity of modes of classroom instruction. Outcomes of the study were qualified by the difficulties ESL students face in coordinating conflicting information and interpreting the language demands of problem presentation. It is expected that this study will assist m extending the knowledge and understanding of the learning difficulties faced by ESL students in the am of pre-university calculus.Results of this study suggest that instructional material has an important influence on ESL students’ use and management of multiple representations. However, there are often limitations to the influence of the material due to student preferences, mathematical ability and firmly held beliefs as well as on the amount of detail presented in a problem Secondly, small group learning environments based on a constructivist approach were found to influence student ability to model calculus word problems in a positive manner, provided there is teacher support to overcome cognitive obstacles. Finally, it was established that an effective teaching package could be developed to assist ESL students in calculus learning. The teaching package's evaluation highlighted the need for matching language use in problem presentation with the current mathematical language register of each student.