Essays on Mathematical Education

Ginn, 1913 - 139 páginas
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Página 53 - ... waning faith. Every great study is not only an end in itself, but also a means of creating and sustaining a lofty habit of mind; and this purpose should be kept always in view throughout the teaching and learning of mathematics.
Página 59 - The contemplation of xinassailable mental structures such as are found in Mathematics cannot but raise ideals of perfection different in nature from those found in the more emotional creations of literature and art. It must induce an appreciation of intellectual unity and beauty which will play for the mind that part which the appreciation of schemes of shape and colour plays for the artistic faculties, or again that part which the appreciation of a body of religious doctrine plays for the ethical...
Página 68 - ... perceived by direct intuition, symmetry, or superposition. If this condition be violated the pupil's sense of a proof is confused ; it appears to arrive with much painful labour at a result which he and everyone else knew before to be true, and the image of inevitable but unforeseen truth is not induced. Take, for example, the theorems concerning bisected chords of a circle : if a boy cannot perceive the truth of these with, possibly, a little stimulus, his knowledge of space is so meagre that...
Página 58 - ... enlarge upon the value of this consideration, but I would point out that its importance can be realised by the layman and should influence him considerably. Amid the many arbitrary rules and asserted facts which perforce find place in education the presence of schemes of deduction based on statements which find universal acceptance as descriptions of our space impressions must make for good in the child's development. The peculiar nature of the premisses on which Geometry is based having been...
Página 10 - ... about" until they feel a real need for seeking reasons. The old, and unfortunately, too often the present practice of beginning by attempting to prove statements which seem perfectly obvious makes the whole subject of geometry seem trivial to the students and discourages interest. Carson well says, "One of the few really certain facts about the juvenile mind is that it revels in exploration of the unknown, but loathes analysis of the known" ' (Mathematical Education, Glnn & Co., 1913, 242 p.
Página 3 - Accordingly, no geometrical proposition, as, for instance, that any two sides of a triangle are greater than the third side, can ever be derived from the general conceptions of line and triangle, but only from perception.
Página 19 - ... same is true of many other lines of effort. Professor JWA Young ("The Teaching of Mathematics") says, "For direct practical usefulness, mathematics is second only to the mother tongue." Regarding the possible future of mathematics in this line, G. St. L. Carson ("Mathematical Education," p. 51) says, "I believe that the modern theories of pure mathematics are destined to illumine our understanding of the human mind and of cities and nations, just as the pure mathematics of fifty years ago has...
Página 77 - To construct a preliminary course which consists of work concerning angles, lines, triangles, and circles with, perhaps, a passing reference to a few surveying problems, is to place the child suddenly in a new world where things are replaced by abstractions and to give him an occasional glimpse of his own sphere as from behind bars — bars which are not made thinner by assigning numerical measures to the lengths and angles with which he deals. In the alternative which I have suggested the endeavour...
Página 54 - ... dangerous is the plea that every educated man should have some idea of a subject of such wide utility. Apart from the claims of many other branches of knowledge, it has a further demerit in that the object of teaching the subject is implied to be the acquisition of encyclopaedic knowledge, rather than the development of the mental faculties. The old conception of education as the acquisition of information is dead, and it least becomes mathematicians to do anything to revive it. The use of justifications...
Página 67 - ... exercises may not unfitly be compared to an attempt to teach music by explaining the mechanism of a piano, and the relation between notation and keyboard, before the pupil has heard a single tune. It is a crippling of subjective growth at its most sensitive stage by the crudest form of materialism. A mind which has ranged over all its experience and has made these intuitions has gained a sense of power and accepted truth which cannot be induced to an equal extent by any substitute for the process.

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