Promoting versatile understanding in integration using a computer

Abstract

Traditionally in teaching, mathematics educators have attempted to provide a range of experiences that develop mathematical conceptions in a cognitive manner so that the learner both knows and understands. However, in practice, many teachers may put an undue emphasis on the learning of procedural methods in their mathematics lessons rather than stressing both procedures and concepts. This may be due to many factors and constraints, including available time, pressure for good results in external assessment, etc. Hence, students' preferred methods in approaching calculus problems often appear to involve a process-oriented style of thinking, where they apply known, standard procedures or algorithms, but they may have difficulty with their understanding of important concepts. The aim of this research was to investigate the possibility of improving student learning by promoting versatile thinking, which enables one to understand and utilise both the concepts or objects of mathematics and the processes from which they have been encapsulated. In the pilot study, the results of an integral calculus questionnaire which was supplied to senior secondary school students and designed to gauge their understanding of concepts associated with the Riemann integral are described. The initial aims were to verify the validity and suitability of the questions for the main study, and consider student thinking on conceptual questions. A number of misconceptions in the students' understanding, along with the instrumental, process-oriented thinking underlying them emerged from the questionnaire and these misconceptions are described. The first phase of the main experiment involved a single group design where 6th and 7th form school students used spreadsheets and symbolic manipulator modules supplementing a traditional approach to integration to investigate the processes and concepts of integration. The questionnaire developed was employed as the instrument to ascertain any changes in skills or understanding. The second phase of the study involved an attempt to use a controlled experimental design with stage 1 university students in order to confirm the results and triangulate the data from phase one. The results provide clear evidence of how the students obtained a significant improvement in versatile understanding from the additional computer investigations, particularly with regard to the misconceptions present in the pre-test. In contrast, the control group students, with their traditional learning of calculus, often experienced no change in their understanding, displaying the same misconceptions in both pre-test and post-test. A computer enables students to experience many possibilities with respect to the relationship between numerical, graphical and symbolic representations. Thus the computer tutorial used in the research may encourage an improved cognitive base for a versatile understanding of the concepts associated with integration, making it possible for the student to develop a perception in terms of both process and concept. The results of this experiment provide evidence that such appropriate use of the computer can aid students both at school and university to construct a more relational, concept oriented understanding of integration.