## The Teaching of Elementary Mathematics |

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algebra America angle applied arith arithmetic axiom beginning 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 Geschichte give given grades greatest common divisor Greek Greek Mathematics Hence highest common factor Hindu numerals interest knowledge Laisant Leipzig logic mathe mathematician mathematics Mathematique matics matter means measure merely method Methodik metic metric system modern multiplication negative number number space objects Paris Pestalozzi plane geometry postulate practical present problems proof propositions proved pupil quadratic quadratic equations question reason recent Rechenunterrichts Rechnen recognized rectangle result rules Schmid schools scientific simple solid geometry solution solve spherical polygon 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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Page 289 - ... 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...

Page 27 - Children should be led to make their own investigations, and to draw their own inferences. They should be told as little as possible, and induced to discover as much as possible.

Page 241 - 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...

Page 154 - 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...

Page 21 - ... billions (the English thousand millions), (2) addition and multiplication of integers, of decimal fractions with not more than three decimal places, and of simple common fractions, (3) subtraction of integers and decimal fractions, and (4) a little of division. Of applied arithmetic we need to know (1) a few tables of denominate numbers, (2) the simpler problems in reduction of such numbers, as from pounds to ounces, (3) a slight amount concerning addition and multiplication of such numbers,...

Page 44 - 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.

Page 230 - 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.

Page 92 - 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.

Page 27 - 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.

Page 31 - 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...