Abstract
Countries in all stages of development have long expressed concern over what they perceive to be a low level of achievement attained by students of mathematics. The concern over mathematical deficiencies has general educational import viz. application of the mathematics that has been learnt. "They can do a thing in mathematics but cannot do the same thing in physics". Through consideration of this situation, I formulated my research problem: the search for a feasible improvement, however partial, in this achivement through modifying the current methodology of mathematics teaching-learning situations. I have confined this study to secondary school mathematics in Kuwait. This notwithstanding, if this study appears to be convincing, then implications could be drawn for curriculum design, methods of teaching and the setting of tests and examinations as well. This is at both secondary school and university levels of education. In satisfying the research problem, it was indicated that, if the construct 'achievement' is assigned a numerical value A, then A is a function of the independent variable R consisting of the variant component of the 'contents' of a teaching-learning task. Nevertheless, it was viewed that a true change in R is due to a change in the 'structure' S underlying R. However, a modification may result from the interaction of S with another structure H or some part, U, of H in underlying the methodology of mathematics teaching-learning situations. S was identified as a set of instances which do not explicitly make use of a constant of implicit form, 'a' say, other than the variables if any: H on the other hand contains only such instances that make use of one or more of such constants. Empirical studies showed that behaviour is not necessarily consistent in two instances - governed by a unique behavioural objective - one in S and the other in H. In this context a concept of space of knowledge (W,0) was introduced consisting of a set W of structured knowledge - identified by a finite set of rules - which is expected to diffuse a particular piece of knowledge through a certain normal population 0. It was also noted that a space of knowledge (S,0) - the learning space of current methodologies - is influential but insufficient for diffusing the required knowledge either in achievement or application. However, it was argued that a part U (of H) - identified by instances in H which include a unique constant - could feasibly contribute, with S, to a modification in the methodology. Furthermore, it was viewed that learnability in U should be subject to certain conditions that make U 'docile' i.e. applicable. Finally, a modification in the methodology of teaching-learning situations was proposed which relied on S + U as a learning space in the form of a model based on an algorithm. Evidence based on experimentation supports the notion that the achievement of students in mathematics under a tehcnique based on S + U is superior to the conventional based on S.