The Various Uses of Graphs [and Discussion]
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... SYLLABUS. 265 Cambridge Joint Board Examinations. I think the Committee will naturally press that upon the Joint Board, and I should think they would consider it very favourably. It is a little unfortunate, I think, that, although some examining bodies have recognised the principle, they have not chosen the particular propositions that we should like them to have chosen-they have not chosen the propositions, or at least not all the propositions, that the Committee recommended, and they have put on the list some that we did not ask for; but still it is their acknowledgment of the principle which may be of great use in the future, and we hope that they will come to recognise which are the ones which we want to be on that list. If there are no other proposals about this Report, then I shall put it to the meeting that the Report be approved. (Carried.) THE VARIOUS USES OF GRAPHS. IT cannot be said too often that each portion of a mathematical curriculum must be chosen with some definite purposes in mind, and that these must be the controlling influence in the scheme of development. Such purposes may be roughly divided into two classes: the first includes all desires to illustrate or elucidate general ideas and principles, and the second includes all desires to relate one part of the subject to another, or to some other domain of thought. It is not here implied that possible fulfilment of some such purposes enforces the inclusion of any particular branch of mathematics; its difficulty may be too great for the students in question, and in any case only a limited amount can be included in a given time-table. But it is very definitely intended that no branch shall be included merely because it happens to be sufficiently easy or to tickle the fancy. There is ample real and human interest to be found without sacrifice of purpose; and while undue difficulty is to be deprecated, it must always be remembered that definite achievement resulting from continued effort is of the very essence of the learning of mathematics. The object of this paper is to consider the purposes which may be served by the inclusion of graphs in the curriculum, and in particular to show that they fall into each of our two classes; that is, some of them are concerned with the presentation of general ideas and principles, while others are concerned with the interrelation of various subjects. The uses of graphs extend far beyond the approximate solution of equations, finding of maxima and minima, and the like. Properly developed, they provide the best introduction to the ideas of functionality and elementary analysis; and further, they show the relation of algebra to these ideas in a manner peculiarly clear and convincing. When one looks at a graph of any particular phenomenon for the first time, what is the first consideration which enters the mind ? Despite the laborious efforts of examiners and writers of books, a man who understands what a graph is does not first look at the details of axes and scales. He places it right way up-I admit that this involves a momentary consideration of the directions of the axes-and examines its general shape, thus gaining at one glance an outline of the whole sequence of events which is being considered. The graph is not the only way of gaining this information, any more than the formula 47rr3 is the only way of learning how to find the volume of a sphere; but it is the most vivid and enlightening method, just as the formula is more vivid and enlightening than an involved verbal statement of arithmetical processes. We may say of a certain plant that it grew slowly at first, then more rapidly, then more slowly, and finally reached a maximum height, but the appropriate curve gives this information more rapidly and effectively. 3 DISCUSSION ON THE SYLLABUS. 265 THE MATHEMATICAL GAZETTE.