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Showing papers in "Mathematics in School in 1978"



Journal Article

15 citations





Journal Article

5 citations


Journal Article

4 citations




Journal Article

1 citations


Journal Article
TL;DR: In this paper, the authors identify three distinct ends which a study of mathematics serves: to illuminate situations taken from everyday life, to help in the development of some other discipline, and to find useful applications may be found of one branch of mathematics to another.
Abstract: The phrase "applications of mathematics" implies the existence of connections between mathematics and other disciplines. The existence of such connections also implies that something of "practical" usefulness may be expected to emerge from these applications. Until very recently applications of mathematics, in the context of the school curriculum, have been confined to the field of theoretical mechanics. Furthermore, the way in which mechanics has been taught has been criticised. The criticism has been made chiefly on the grounds that the examples chosen to illustrate the concepts of mechanics were artificial. For instance, problems in school texts frequently refer to "inelastic strings" and "smooth planes". These references make explicit the kind of assumptions which are being made. A consequence of teaching by means of such problems was that no consideration was given to how, when faced with a real situation, a precise mathematical formulation of the problem could be constructed. Pollak' identifies three distinct ends which a study of applications of mathematics serves. First, to illuminate situations taken from everyday life. Secondly, to help in the development of some other discipline. Thirdly, useful applications may be found of one branch of mathematics to another. At primary school level the first of Pollak's objectives is readily attained. The activities suggested by Dienes and Piaget, involving apparatus designed to give concrete representation of elementary mathematical concepts, can also be turned to advantage for relating elementary mathematics to real life. For example, children weigh, measure, pour liquids to compare volume, estimate areas by counting squares, and so on. Numeric data is obtained and manipulated by arithmetic techniques for the purpose of drawing conclusions. Motivating and illustrative examples become more difficult to find as we proceed through secondary school. This, to some extent, is because the curriculum becomes more specialised and different subjects have their own logical development. Yet mathematics is nowadays applied to areas such as the social and life sciences where hitherto it was unknown. It is therefore appropriate for teachers to seek changes in the content of the school mathematics curriculum which will reflect this growing diversity of application. A variety of situations in everyday life can be the source of interesting mathematical investigations. Applications of mathematics need not involve situations which are complex or remote from everyday experience. Honest applications begin with a situation in a field outside mathematics which is not well understood and which we desire to understand better. By resorting to mathematics we hope to formulate a mathematical model which we can manipulate and from which we can improve our predictions concerning what will happen when corresponding changes occur in the real situation. Current philosophies concerning the teaching of applications stress the importance of introducing students to different kinds of mathematical model. There is evidence to suggest than an appreciation of the use of mathematical models and the development of the abilities needed to construct them are not adequately developed by secondary school courses. Students need to develop the ability to make order from apparent chaos and to ask meaningful questions. Hence the current concern and interest in "open-ended" problems posed in a general form. Discussion of mathematical modelling tends to avoid the basic problem of what comes first, the study of mathematics or the study of situations. Most teachers would probably favour making a study of a piece of mathematics first and then applying it. Yet the difficulty encountered by pupils taught in this way is that if the mathematics is not understood its relevance to the application is not appreciated either. It can be argued that our own natural powers of reasoning could lead us to develop mathematical ideas in response to something else. An advocate of this approach to mathematics teaching is Engel2 who has attempted to make a systematic use of applications in order to develop pieces of mathematics. His article contains several interesting examples of applications which are, however, presented at too abstract a level for the majority of secondary school children. At secondary school level we require applications which can