The Teaching of Elementary MathematicsMacmillan, 1906 - 312 ページ |
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algebra America angle applied arith arithmetic axiom Beman and Smith's Cantor century chapter child circle common considered counting course definition Diophantus elementary algebra elementary geometry equal equation Euclid example exercises factoring figures Fitzga German Geschichte give given grades greatest common divisor Greek Greek Mathematics Hence highest common factor Hindu numerals History idea interest Jänicke Laisant large number Leipzig logic mathe mathematician Mathématique matics matter measure merely method metic metric system modern multiplication negative number number space objects Paris Pestalozzi plane plane geometry postulate practical present problems propositions proved pupil quadratic quadratic equations question ratio reason recent Rechenunterrichts Rechnen recognized rectangle result rules Schmid schools simple solid geometry solution solve square root statement straight line student symbols taught teacher teaching text-books theorem theory things tion to-day triangle true Unterrichts usually write
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277 ページ - ... of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth ; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth ; or, if the multiple of Book V.
229 ページ - Here, then, is the dominating value of geometry, its value as an exercise in logic, as a means of mental training, as a discipline in the habits of neatness, order, diligence, and, above all, of honesty. The fact that a piece of mathematical work must be definitely right or wrong, and that if it is wrong the mistake can be discovered, may be made a very effective means of conveying a moral lesson. Without this aim well in mind, the teacher is like...
142 ページ - Roots and squares are equal to numbers: for instance, 'one square, and ten roots of the same, amount to thirty-nine dirhems'; that is to say, what must be the square which, when increased by ten of its own roots, amounts to thirty-nine? The solution is this: you halve the number of the roots, which in the present instance yields five. This you multiply by itself; the product is twenty-five. Add this to thirty-nine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from...
32 ページ - Now, what I am venturing to maintain is that the individual should grow his own mathematics, just as the race has had to do. But I do not propose that he should grow it as if the race had not grown it too.
218 ページ - I judge that the sum of the interior angles of a triangle is two right angles, the "is" of my judgment expresses the necessary intellectual connection between the two terms connected.
80 ページ - Grube considers the numbers from 1 to 10 sufficient to engage the attention of a child (of six or seven years) during the first year of school. " In regard to extent, the scholar has not, apparently, gained very much — he knows only the numbers from 1 to 10. But he knows them."* The Germans " make haste slowly," but in elementary education they beat us in the race.
17 ページ - In the first place, it guarantees a vividness and permanency of impression which the usual methods can never produce. Any piece of knowledge which the pupil has himself acquired, any problem which he has himself solved, becomes by virtue of the conquest much more thoroughly his than it could else be.
21 ページ - And here give me leave to take notice of one thing I think a fault in the ordinary method of education; and that is, the charging of children's memories...
vii ページ - Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father but by me.
81 ページ - ... number before advancing to the next one is as unnecessary as it is illogical, as impossible as it is uninteresting [Two of] the chief defects of the system are these: 1. It carries objective illustration to an extreme, studying numbers by the aid of objects for three years, until 100 is reached. 2. It attempts to master each number before taking up the next, as if it were a matter of importance to know the factors of 51 before the child knows anything of 75, or as if it were possible to keep...