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The Aim and Methods of School Algebra1
Published online by Cambridge University Press: 15 September 2017
Extract
§ 1. The rhythm, of mathematical progress. To the question how the aim described in the preceding lecture is to be realised, the whole of the rest of the course is proffered as an answer. Here, it, will be sufficient to summarise the main features of the. methods which are later to be given in detail.
The professed aim is to make school mathematics a reproduction, as faithful as the difference of the situations permits, of the mathematical activities of the great world. The method must, then, be based upon an analysis of the general course of development which these activities display. Remembering, as before, that this kind of analysis always introduces an artificial simplicity into a naturally complicated matter, we may yet lay it down that mathematics advances by the constant repetition of a normal cycle of progress—a cycle in which three typical phases may be distinguished. In the first, the heuristic phase, the mathematician, face to face with a new type of problem, devises a new notation or mathematical method to deal with it. In the second, the formal phase, the new notation or method is studied apart from the immediate needs of practical application. The conditions and the range of its validity are investigated, its wider possibilities are explored, and its relation to other methods examined. In the third, the application phase, the extended notation or the perfected method becomes once more an instrument for the solution of problems, and is found to be applicable successfully over a field often many times wider than the area in which it originated. Any one of these problems may, in turn, become’ the starting point of a new cycle.
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- Copyright © Mathematical Association 1912
Footnotes
1
The substance of the two introductory lectures of a course on the teaching of algebra.
References
page 214 note 2 The present writer has given an account of this book in the Mathematical Gazette for December, 1910, and January, 1911.
page 214 note 3 It must be remembered that at this time only integral indices were used.
page 215 note 1 A too familiar instance has just been given me by a student as a piece of his personal experience when a schoolboy. His class in woodwork were taught a new joint which was subsequently to be used in making an Oxford frame. All did the work so imperfectly that they were not allowed to proceed to the frame. A second attempt produced equally unsatisfactory results. In despair the teacher gave them the frame to make—and almost every boy turned out a creditable article !
page 216 note 1 Reference may be made to the chapter on this subject (Ch. III.) in the Report of the L. C. O. Conference on the Teaching of Arithmetic (1906-8).
page 217 note 1 Accumulating experience is demonstrating ita soundness in the parallel cases of the teaching of reading, of number, of needlework, etc.
page 217 note 2 E.g. Euclidi. 4.
page 217 note 3 A simple instance from geometry : Let APQ be an angle in a semicircle, AQ being the diameter. Let P approach Q. Then as the angle A grows smaller the angle Q grows nearer to a right angle. When A is zero Q will be a right angle ; i.e. the tangent PQ is at right angles to the diameter. From the purely logical standpoint this argument is quite insufficient. For the italicised sentence contains the assumption that a truth (the angle Q=90° - A) which holds when A, P, Q form a triangle continues to hold good when they are no longer a triangle. From the utilitarian standpoint it could, however, be accepted. No one could harbour any doubt that the fact is so, nor hesitate to use it for the most important practical purposes.
page 218 note 1 Or else, like Hilbert (Grwndlagm der Geometrie), place the proposition among his axioms.
page 218 note 2 Principia Mathematica, Preface.
page 219 note 1 See also Section VI. of the article on Wallis in the Mathematical Gazette for February, 1911.