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Long-sight seems to be less prevalent, but pupils suffering from this defect are also liable to be misunderstood, as they sometimes hold the book close to the eyes to make the print look larger.

A child with a normal eye should be able to read a page of an ordinary school reader at the distance of forty inches and all intervening distances down to four inches. If the child cannot do this the parent should be informed and advised to consult a competent oculist. The recruit for the British army is required to recognise clearly at a distance of fifty feet a spot one inch in diameter. To adapt this test to the school-room, print letters (not in words, but thus, SPDQRT) three-eighths of an inch high and require the pupil to read them at a distance of twenty feet. Where defects of vision are overcome by the use of proper glasses, their use should be insisted on. Young children especially avoid their use, and frequently the fear of ridicule on the part of their schoolmates makes them lay their glasses aside. I have had cases in which I had great difficulty in enforcing their constant use.

But after all this has been accomplished the teacher should observe the following rules:

I. Have the light fall from behind, or over the left shoulder, if possible, but never come from the front.

2. Study must be interrupted, for youngest children, every fifteen minutes, for older ones every thirty minutes, by change of position. 3. Writing on the blackboard should be an inch high for the main body of the script letters and two inches high for numbers.

4. Greasy slates and pale ink are to be avoided.

5. Drawing maps on a small scale should be forbidden.

6. Children with known defects of vision should be free to approach as near as they like all blackboards, wall-maps, &c.

Fractions! What Are They?

The examination of teachers has brought home to me, with a force that is much more powerful than pleasant, the very unwelcome conclusion that either my comprehension and idea of fractions are radically wrong, or very many of the teachers are mistaken as to their nature and properties. This unexpected and troublesome difference of opinion as to what really constitutes a fraction has caused me to think a great deal about the matter, to examine the authorities as to the correctness of my own opinions, and to search for the probable cause of error in others. A desire to get this thing perfectly straight in my own mind has led me to attempt the preparation of this paper, in which I propose to discuss what fractions are, and also what they are not. One thing is certain: the explanations of fractions given in some of our text-books are ambiguous and contradictory, and teachers and pupils have mistaken their real teaching, or I have gone thus far through life in blissful ignorance of what a fraction really is.

One way to ascertain what a thing really is, is to make sure what it is not. Let us try this method with fractions. Webster defines a fraction to be "a portion, a fragment, a division or aliquot part of a whole number." Now, let us see whether these definitions are arithmetically correct. The same dictionary defines a "portion" to be "a separated part of anything." The number 20 is divisible into two parts, as 10+10; or into four parts, as 5+5+5+5. Now, in this case 10 or 5 is a separated part or portion of 20, but neither 10 nor 5 is a fraction. Eight is a whole number, and either 4 or 2 is a division or aliquot part of 8, but neither 4 nor 2 is a fraction. And these examples make it very evident that the common or dictionary meaning of fraction, which makes it synonymous with portion or part, is not the arithmetical meaning, which is only synonymous with fragment.

Chambers's Encyclopædia defines a fraction thus: "In Arithmetic a fraction is any part or parts of a unit or whole, and it consists of two members, a denominator and a numerator, whereof the former shows into how many parts the unit is divided, and the latter shows how many of them are taken in a given case." Unit and whole are here used as explanatory of one another. A unit is a single whole thing. This definition of a fraction agrees so well with the definitions given in the best text books on Arithmetic, that we may safely accept it as correct. Now, then, let us consider this definition carefully. A frac

tion must consist of two members or terms, the numerator and the denominator, and any number that may be properly expressed by one term is necessarily not a fraction. A fraction also consists of a part or parts of a unit, and as the sum of all the parts is only equal to the whole, it follows that a true fraction can never be greater than the unit, and as the unit in Arithmetic is always expressed by 1, a true fraction can never be greater than 1. To determine what any arithmetical expression really is, it is necessary to reduce it to its lowest and simplest terms; and as the simplest form of expressing the sum of all the parts of a unit is notor, but 1, it follows that a true fraction must always be less than the unit 1. Take the two arithmetical expressions, and 4. Now, being incapable of expression in simpler form than , which is an arithmetical expression consisting of two terms, a numerator and a denominator, is a true fraction; but, being capable of expression in the simple and single term 4, is not a fraction at all, though each of the sixteen, of which is composed, is a true fraction.

There is also another method by which we may determine with ab solute certainty whether any arithmetical expression is a fraction or not. Decimal fractions differ from common fractions only in two points-first, in that the denominator of a decimal fraction is always 10 or some power of 10; while the denominator of a common fraction may be any number. And second, in that the denominator of a decimal fraction, being easily understood, is never written, but is expressed by the decimal point, while the denominator of a common fraction must always be written, because it cannot be known in any other way. But there is no decimal fraction which may not at once be converted into a common fraction by simply writing its denominator under its numerator, as .25 %, a common fraction; and there is no common fraction which may not be converted into its decimal equivalent by simply annexing as many ciphers as may be necessary to the numerator and dividing by the denominator, prefixing to the quotient thus obtained the decimal point, as .25, which result is obtained thus 190.35. The circulating and repeating decimals, which are apparent exceptions to this rule, do not affect its correct

ness.

25

Still another test is the grammatical test, which requires us to use a singular noun with a single number and a plural noun with a plural number. If, then, we must use a plural noun in connection with any arithmetical expression, we may know that it is not a fraction, not

being a part or parts of one, but more than one. We then have these four tests by which we may prove whether any arithmetical expression is a fraction or not: (1st) In its simplest form it must consist of two terms, neither more nor less; (2d) It must express something less than 1; (3d) It must be capable of conversion into a decimal equivalent; and (4th) it must stand the grammatical test.

Now, to show what a fraction is not, let us take up the explanations of fractions given in Venable's Arithmetic, the text-book in use in this county, and also used extensively throughout the State. There are some so-called fractions which by their very names indicate plainly that they are not true or simple fractions, as "compound fraction," which is a fraction of a fraction, as 1⁄2 of 3; a "complex fraction," which is an arithmetical expression having two terms, one of which is itself a fraction, as; and an "improper fraction,” which is always something more than a fraction. But there is not any necessity for confusion or misunderstanding about these, as the very names given to them indicate the exact nature of the so-called fractions, and wherein they differ from a true or simple fraction. Section 93 of Venable's Practical Arithmetic defines a fraction to be "an expression which denotes one or more of the equal parts of the unit or whole; and section 97 says: "A fraction is commonly expressed in figures by two numbers, one above the other, with a line between them. The number below the line is called the denominator, and expresses into how many equal parts the whole or unit is divided. The upper number is called the numerator, and shows how many of these equal parts are taken to form the fraction." These sections very nearly coincide with the standard definition of a fraction taken from the Encyclopædia, and may therefore be considered absolutely correct. But in section 99 of the same arithmetic we have, “A fraction expresses the division of the numerator by the denominator. Three-fourths expresses 3÷4; for we get the same, whether we divide a unit, as a yard, into four equal parts, and take three of these parts, or divide a line three yards long into four equal parts and take one of them. Similarly, 3/4 of a dollar is 4 of three dollars. Hence, in every fraction the numerator is the dividend and the denominator is the divisor." Now, the thing divided is always the dividend, and we are told in section 97 that it is the unit or whole which is divided to form a fraction, so that these statements in sections 97 and 99 directly contradict one another. Three fourths is equal to 3÷4, but it is very far from expressing 34. Mr. Venable says they express the same because

the result is the same; but the results are often the same, though the means be very different. A man dies all the same, whether he be hanged, or shot, or drowned, and as it is no proof that a man was hanged because he is dead, so it is no proof that a fraction expresses the division of the numerator by the denominator because we get the same. A man "gets the same " whether he steals a horse or buys him; but it by no means follows that the two processes are the same. Neither do we always "get the same," for 34 of one apple might very easily be a different thing from 4 of three apples, if the apples were not of equal size and quality; and to sell a man 3/4 of a house is quite a different thing from selling him 4 of three houses. Then in section. 100 of the same arithmetic we are told, "Hence a fraction expresses a part of any number, or the sum of any equal parts into which a number is divided. Thus,,,, are all fractions. And the whole number 15 is the fraction, 30, or 5." Could any explanation be more mistaken or further from the truth? If is a fraction, surely 9 is the denominator, showing into how many parts the unit is divided, and 19 is the numerator, showing how many of these parts are taken; and although there are but 9 parts there, we may take 19-an evident absurdity. Nor does Mr. Venable mean that is an improper fraction, for he gives it with, which is not an improper fraction. And if "a fraction expresses a part of any number, or the sum of any equal parts into which a number is divided," then, as there is no number which is not a part of a greater number, and no number which is not exactly the sum of the equal parts into which it may be divided, it follows that every number is a fraction; and if " the whole number 15 is the fraction, or any other sort of a fraction, there is no number that is not a fraction of some sort, and arithmetic is not the science of numbers, but the science of fractions.

It seems to me that the foundation for these errors is laid in the method of numeration taught; for if it is proper to read units, tens, hundreds, units of thousands, tens of thousands, hundreds of thousands, units of millions, &c., then a fraction, being one or more of the equal parts of a unit, and 500 being five of the ten equal parts into which a unit of thousands may be divided, 500 is a fraction, which it is not at all, for it will not stand any one of the four certain tests previously formulated. It is not composed of two terms, it does not express less than 1, it cannot be converted into a decimal equivalent, and it cannot be used with a singular noun. The truth is, that thousands, millions, &c., are not different orders of units at all, there being

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