# The Interpretation of Graphs Representing Situations.

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This work arose out of a concem for those aspects of mathematical competence which are of general value and usefulness to the majority of people We were aware of the widespread use of graphical representations of situations in the sciences, and indeed, in everyday economic affairs; some of these uses in other school subjects appear in Ling [1977] We felt that the treatment of this topic in the mathematics cuniculum was generally underdeveloped, and related too much to specialised mathematical techniques, such as the solution of equations by reading off points of intersection of two graphs. This is pruticularly true of the secondary school curriculum British prirnru:y schools often do exploit some of the practical uses of graphs Indeed, one of the fust guides produced by the Nuffield Mathematics Project in Britain was entitled Pictorial Representation It described work in which data hum various everyday situations ~ such as the takings at a cinema box office over the course of a week were represented by block m line graphs, and the story told by the graph was then discussed and written The graph was used to expose features of the situation not immediately obvious from the numerical data By contrast, the graphical work in currently popular secondary school courses consists mainly of relatively brief treatments of travel graphs, mainly composed of line segments, and, in some comses, more extensive wmk on the rather specialised use of graphs fm optimisation in linear programming situations It might perhaps be considered that the ability to use graphs as a language for these purposes is sufficiently well developed in primary schools, and that further specific teaching is not required in the secondary school; that is, until, say, the introduction of the technical apparatus of calculus fOr determining maxima and minima and fOr sketching curves from equations. ‘The work to be repmted in this paper will show that, in fact, most secondary pupils arÂ·e weak in the ability to interpret global graphical features so as to extract infOrmation about many everyday and scientific situations In undertaking this work, we had the aim of developing some teaching material to fill this gap The material was used initially in exploratory interviews, to clarify the nature of the difficulties experienced by pupils in this field, as well as to see how readily they were able to learn the necessary skills and concepts A full account of this work apperus in the second author’s PhD. thesis [Janvier, 1978a] Discussion of one aspect of the workthe effect of personal experience of the situation underlying the graph -has appeared elsewhere [Janvier, 1981] A description and discussion of the teaching experiment (see below) has appeared in conference proceedings [Janvier, 1978b] The present article attempts to give an outline of the work as a whole and its main outcomes Graphs in the curriculum We shall begin by reviewing in a little more detail the incidence of graphical wmk in current British school courses, and the outcomes of previous general surveys of graphical understanding In the primary school, in spite of the considerable encouragement to emphasise graphical work, which has resulted in increased activity, it appears that block and barÂ· graphs predominate over line graphs in genet al use, and discussion is usually based on point reading_~ , with a little comparison, but rarely treats global features, such as the general shape of the graph, intervals of rise or fall, or of maximum increase. These “global features” became the focus of interest in our work [Bettis and Brown 1976; Read 1970; Ward 1979] An example from the AP:U Primruy Survey of II year olds [Fox man eta/, 1980] shows a typical piece of work. (See table on next page ) 90-95% of pupils here could identify the greatest and least heights of the bars, and 45% could comparÂ·e increases for the srune girls between the graphs. Another question from the same section of the survey shows the well known difficulty of interpolating between the numbered grid-lines on a conversion graph connecting old and new prices. In the extensively used SMP secondary school course (Books A to Z), the use of Cartesian graphs to represent functions is somewhat delayed, on account of the wish to begin the study of functions with relations and arrow diagrams The graphs of relations such as y ~ x + 2, and of the ‘slide-rule’ graph y ~ 2′ come in the second year, well before graphs of practical situations, such as the weight and cost of minced meat, the petrol left in a tank and distance travelled, or the travel graphs of pupils’ journeys to school The earliest work is on plotting points and on identifying regions such as.x > 2, y ~ 3 In the third year, regions such as x + 2y ~ 6 are considered in connection with linear programming problems and their solution sets. When formulae such as A = 3r 2 , y = .x 2 2x are graphed, also in the third year, it is in order to solve equations, not to study the form of the function Later, in the fourth orfifth years, more realistic graphs appear, but this is for the purpose of calculating gradients and arÂ·eas under the graph In the statistics chapters, more real-life data is used, but still the graphs arÂ·e used to display infmmation, not fm the elabmation of the properties of the underlying situation Thus the definition of function as a set of ordered pairs, the technicalities of using graphs to solve equations, and the rather difficult and highly specialised use of graphs for linear programming occupy the bulk of the time, and there is little or no attention to global graph-reading and interpretation