Symbolic manipulations related to certain aspects such as interpretations of graphs
This thesis describes an investigation into university students' manipulation of symbols in solving calculus problems, and relates this to other aspects such as drawing and interpretation of graphs. It is concerned with identifying differences between students who are successful with symbol manipUlation and those who are less successful. It was initially expected that the more successful would have flexible and efficient symbolic methods whilst the less successful would tend to have single procedures which would be more likely to break down. Krutetskii (1976) noted that more successful problem-solvers curtail their solutions whilst the less able are less likely to acquire that ability even after a long practice. This suggested a possible correlation between success and curtailment. An initial pilot study with mathematics education students at a British University showed that in carrying out the algorithms of the calculus, successful students would often work steadily in great detail, however, they were more likely to have a variety of approaches available and were more likely to use conceptual ideas to simplify their task. However, the efficiency in handling symbolic manipulation may not be an indication that the students are able to relate their computational outcome to graphical ideas. A modified pilot test was trialed at the Universiti Teknologi Malaysia before a main study at the same university in which 36 second year students were investigated in three groups of twelve, having grades A, B, C respectively in their first year examination. The findings of this research indicate that there is no significant correlation between ability and curtailment, but ability correlates with conceptual preparation of procedures where there is an appropriate simplification to make the application of the algorithm simpler. The more able students may have several flexible strategies and meaningful symbolic mathematical representations but these may not always relate to visual and graphical ideas. On the other hand the less able students are less likely to break away from the security of a single procedure and liable to breakdown in getting the solutions for the calculus problems.