The Teaching of Mathematics in the Elementary and the Secondary School |
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abstract algebra analytic analytic geometry angles applications arithmetic attained axioms binomial theorem calculus child circle class exercise class-room Committee concrete course decimal deductive reasoning definition demonstration difficulty discussion drill Education elementary algebra equal equations Euclid example experience exponents fact formulas fractions fundamental give given grade grasp heuristic high school idea important inductive instruction interest laboratory Leipzig logical mathe mathematicians matics means measure ment mental method metic metric system mind mode nature pedagogic physics plane plane geometry possible practical problems processes proof propositions pupil quadratic equation question rational numbers reasoning regular polygons relations Report right triangle root secondary school segment side simple solid geometry solution solved square statement straight line study of mathematics subject matter sufficiently symbols taught teacher Teaching of Mathematics text-books theorem theory things thought tion topics triangle trigonometry
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Page 416 - It must be conceived throughout as a means to an end, not as an end in itself.
Page 297 - The four fundamental operations for rational algebraic expressions. Factoring, determination of highest common factor and lowest common multiple by factoring; fractions, including complex fractions, and ratio and proportion; linear equations, both numerical and literal, containing one or more unknown quantities; problems depending on linear equations; radicals, including the extraction of the square root of polynomials and of numbers; exponents, including the fractional and the negative.
Page 79 - And therefore, while in all things that we see or do, we are to desire perfection, and strive for it, we are nevertheless not to set the meaner thing, in its narrow accomplishment, above the nobler thing, in its mighty progress; not to esteem smooth minuteness above shattered majesty; not to prefer mean victory to honourable defeat; not to lower the level of our aim, that we may the more surely enjoy the complacency of success.
Page 129 - With the algebraists, however, who are Pagans themselves, the 'Pagan fables' are believed, and the inferences are made, not so much through lapse of memory, as through an unaccountable addling of the brains. In short, I never yet encountered the mere mathematician who could be trusted out of equal roots, or one who did not clandestinely hold it as a point of his faith that x.- + px was absolutely and unconditionally equal to q.
Page 65 - Do you see, Meno, what advances he has made in his power of recollection? He did not know at first, and he does not know now, what is the side...
Page 108 - General formulas which men have devised to express groups of details, and which have severally simplified their conceptions by uniting many facts into one fact, they have supposed must simplify the conceptions of a child also.
Page 297 - ... and geometric progressions, with applications. It is assumed that pupils will be required throughout the course to solve numerous problems which involve putting questions into equations. Some of these problems should be chosen from mensuration, from physics...
Page 412 - ... 5. Finally, among the practical aims to be served by the study of mathematics should be listed familiarity with the geometric forms common in nature, industry, and life; the elementary properties and relations of these forms, including their mensuration ; the development of space-perception; and the exercise of spatial imagination.
Page 416 - ... the acquisition of mere facility or skill in manipulation. The excessive emphasis now commonly placed on manipulation is one of the main obstacles to intelligent progress. On the side of algebra, the ability to understand its language and to use it intelligently, the ability to analyze a problem, to formulate it mathematically, and to interpret the result must be dominant aims.
Page 441 - Any exterior angle of a triangle is equal to the sum of the two opposite interior angles.