# The Teaching of Algebra (including Trigonometry)

Longmans, Green and Company, 1914 - 616 lappuses

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5. lappuse - The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.
350. lappuse - The essenljials of the ideas connoted by the terms " indeterminate value " and " singular points " find their place here. In Ex. LXXVI an inquiry into the properties of a few functions of two variables is made the occasion for extending the method of rectangular coordinates to the analysis and description of curved surfaces. The investigation is undertaken in the spirit of ch. iv., § 9 ; that is, the surfaces are treated as tri-dimensional graphs to be studied not so much for their own sake as for...
378. lappuse - To wit, when we seek to subject them to numeration ... we find that they flee away perpetually, so that not one of them can be apprehended precisely in itself. . . . Now that cannot be called a true number which is of such a nature that it lacks precision. . . . Therefore, just as an infinite number is not a number, so an irrational number is not a true number, but lies hidden in a kind of cloud of infinity.
79. lappuse - It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.
46. lappuse - ... is less compact and less easily reproduced. Its message is frequently inarticulate and obscure. For these and similar reasons it should be regarded as a subsidiary algebraic instrument which fulfils its best office when it either leads up to a formula by which it may itself be superseded, or serves to unfold more fully the implications of a formula whose properties have been only partially explored.
17. lappuse - One never has existed and probably never will exist apart from the other. The view that they represent wholly distinct forms of- intellectual activity is partial, unhistorical, and unphilosophical. A more serious charge against it is that it has produced an infinite amount of harm in the teaching of mathematics. Our purpose in teaching mathematics in school should be to enable the pupil to realize, at least in an elementary way, this two-fold significance of mathematical progress. A person, to be...
19. lappuse - ... or a in the algebra course and exclude sin x or tan x. All alike are pieces of symbolism invented for the description and interpretation of facts of the external world. Each represents a typical kind of function.
16. lappuse - Mathematical truths always have two sides or aspects. With the one they face and have contact with the world of outer realities lying in time and space. With the other they face and have relations with one another.
277. lappuse - What were before values of x and y are thus converted into values of y and x. With the direct and inverse graphs before us the following points at once become clear : (a) Each of the direct functions is a singlevalued function of x ; that is, to every value of x there corresponds one and only one value of the function, (b) The inverse linear and the inverse hyperbolic functions are also...
19. lappuse - ... perfecting the technique of his subject, finds it natural as well as most effective to take a special group of allied methods or allied problems and to develop them as far as he can without concerning himself too greatly about the practical value of his work. Now this systematic exploration of special arts of mathematics is, no doubt, of vital importance for the continued growth of the science.