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

Symbols and the bifurcation between procedural and conceptual thinking

Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to d o mathematical problems and to think about mathematical relationships. In this presentation we consider the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra and calculus, then on to the formalism of axiomatic mathematics. This is taken from a number of research studies recently performed for doctoral dissertations at the University of Warwick by students from the USA, Malaysia, Cyprus and Brazil, with data collected in the USA, Malaysia and the United Kingdom. All the studies form part of a broad investigation into why some students succeed yet others fail

A Technique of Fuzzy C-Mean in Multiple Linear Regression Model toward Paddy Yield

In this paper, we propose a hybrid model which is a combination of multiple linear regression model and fuzzy c-means method. This research involved a relationship between 20 variates of the top soil that are analyzed prior to planting of paddy yields at standard fertilizer rates. Data used were from the multi-location trials for rice carried out by MARDI at major paddy granary in Peninsular Malaysia during the period from 2009 to 2012. Missing observations were estimated using mean estimation techniques. The data were analyzed using multiple linear regression model and a combination of multiple linear regression model and fuzzy c-means method. Analysis of normality and multicollinearity indicate that the data is normally scattered without multicollinearity among independent variables. Analysis of fuzzy cmeans cluster the yield of paddy into two clusters before the multiple linear regression model can be used. The comparison between two method indicate that the hybrid of multiple linear regression model and fuzzy c-means method outperform the multiple linear regression model with lower value of mean square error