many addends there are how many groups of four things - and is therefore purely a number; an abstract number if you will, for the conception would not change with change of addend; this pure concept three, would remain absolutely unchanged if the groups of things we rechanged either in number or in kind, indefinitely to groups, (dollars, apples, cows, etc., etc.,) of 1, or 2, or 3, or n. No other meaning for the multiplier can be even conceived by a mathematically sane mind. How then, is the inverse problem connected with this? In the one case we have the group of things and the times repeated to find the absolute quantity. Or, expressed symbolically, $4 × 3 = $12. In the other case, we have two of these things given, viz: $4 and $12, to find the third, viz: 3. And both science and common sense demand that this three shall be found; this three and not a transformed three, as three dollars, or three, four dollars," or three anything else in all the wide realm of the concrete. Yet we have the astounding statement that "the quotient is not an abstract number — it is three, four dollars". Expressed in symbols this would be, when the dividend is recalculated, $4 × 3 ($4) = $12; Or, in words:

66

... $12 = $4 × Q; but Q = $12.

... $12 = $4 × $12 = unadulterated nonsense.

"Get children to see this law and when grown to men and women they will not make such mistakes"; true, perhaps. At all events, they will not be accounted responsible for such mistakes, nor, for any mistakes. But why waste words? The "law" is beyond the reach of argument. It is prima facie proof the absurdity of the premises, or of the—well, the ineptitude of the logician -or of both.

III. The divisor cannot be greater than the dividend. ".88 means how many eights in eight-tenths; how absurd. .8 of a pie to be divided among 8 boys—do you mean to tell me, Doubter, that you are going to find how many 8 boys in .8 pie." By no means, my dear professor. You are too severe on Doubters. You are worse than the impatient Psalmist. There may be some excuse for the hasty inference that "all men are liars' there is none for the deliberate assumption that all men are fools. The direct proposition (multiplication) is, one-tenth repeated eight times, gives eight-tenths. The inverse problem is, given the product, eight-tenths, and one (eight) of the two factors, to find the other factor, one-tenth. Are we to infer that given 8 and .8 in .1 x 8.8, we cannot find .1? By what operation do we find it if not by division ? "We want," he says, "of .8 pie and the answer is given at once, .1; but this differs very widely from division." How, I ask, and how widely, does it differ from division? And suppose the ans. is not "given at once"?

IV. The inferences I., III., and V., are practically involved in IV: The divisor can never be an abstract number; where a uumber is divided into equal parts it is not division, it is partition.' In this the disciple follows his master (see "Taiks on Teaching," p. 105, et seq) and I fear that the "ditch" is the destiny of both. Never an abstract number! Then one inverse of the problem $4 × 3 = $12 must, we suppose, be insoluble given the $4 and the $12, and we get the required result (3) by mere division But the other inverse, given the 3 and the $12 to find the $4, is not division; it is—it is partition some Newtonian thing which is beyond division, but which we hope is within the compass of the Calculus. One-third of $12 is $4 we are told, "and the answer is

given at once." But in the example, divide $209671 by 539, the answer is not "given at once." How then is it to be found? Not by division, it seems, because "divisor and dividend must be of the same name "; not by division because the divisor is abstract, and one cannot take 539 from 209671 dollars." Not by division. "because we are looking, not for the number of groups, but for the number of units in each group?" How then, do we get the answer, since "it is not given at once"? I reply:- By exactly the same process that is employed in dividing by $539. And in every such (possible) problem in "partition" the answer is universally and necessarily obtained by the division process. "Yet, bless your heart, Doubter, that, nevertheless, is not a problem in division, it is something widely (or in the language of the Master radically) different it is "PARTITION"!

(c) Let us now try to look at the problem of division from the stand-point of common sense. It is a fundamental maxim in education that we learn with what we have learned; division is the inverse of multiplication; in learning division we use our knowledge of multiplication. We know that in multiplication, we take a group of units, expressed by the multiplicand, a given number of times expressed by the multiplier that the multiplier must be a pure (i.e. "abstract") number that the law of commutation holds e.g. 3 groups of 4 things is identical with 4 groups of 3 (the same) things, etc. By the adjusting activity of Attention this knowledge - this "apperceiving" mass is brought to bear on the problem. of division.

-

In the one case (a) we are searching for "times" i. e., for the multiplier 3 which with the given multiplicand $4 will give the product $12. In the other case (b) we are searching for the unit-group $4 — the mulitplicand, which with the given multiplier, 3, will give $12. In the former case (a) the divisor is concrete and the quotient necessarily abstract a pure number. In the latter case (b) the divisor is abstract and the quotient necessarily concrete. Speaking somewhat loosely, therefore, we may say that there are two kinds of division: division (a) in the sense that a number ($12) contains a given number (of "same name ") a required number of times; and (b) division in the sense that a number ($12) is to be distributed into a given number (3) of unit-groups of required value ($4)

But these "two kinds" of division are not "widely and radically" different. On the contrary they are essentially CORRELATIVE; the one implies the other; the number of the unit groups cannot be found without their value, i.e. without the number of units in each; nor the value of the unit-groups without their number. In both cases, the "searching" operation, as well as the mental action involved in it is precisely the some; and in both is implied the idea of division into equal parts. The difference, such as it is, is NOT in the number of data, or in the operation, or in the fundamental principles which underly the operation, or in the application of these principles, it is in the INTERPRETATION OF THE RESULT: in one case the result expresses simply "times", i.e. the number of unit-groups, and is a pure (or abstract) number. In the other case the result expresses the number of standard units in the unit-group and is a concrete number, i.e. it expresses the absolut magnitude of the unit-group. For example: $12 $4 = 3 (not "3 four dollars'); here the quotient is a pure number, it is in fact the ratio of the two magnitudes $12 and $4. Again: $12 ÷ $4 = 3; here the quotient not only involves the relation (ratio) of 12 to 4, but also names the standard unit; i.e. it is a concrete number expressing the absolute value of the unit-group. In the first example we have the answer to the question (as it might be put):- what is the ratio of

$12 to four dollars? In the second example we have the answer to the question:-four is the ratio of $12 to what number?

In view of these basic principles it is surprising that these mutually and necessarily related processes should be declared fundamentally different so different in fact that separate names are needed to mark properly the distinction between them.

It is perhaps still more surprising that some generally thoughtful men, e. g. the author of The Philosophy of Arithmetic-positively state that " the divisor can never be an abstract number." Let us venture to illustrate the real processes in the "two" divisions, working (for the sake of example) by partial quotients as in "long" division. (a) Divide $12 by $4: $12 $4 ? i.e. find the multiplier, which with $4 for a multiplicand, will give $12.

The complete quotient is therefore $1 + $1 + $1 = $3 — the required multiplicand which must be used in re-calculating the dividend, $12. In (a) we had to use the preliminary knowledge that once $4 is four dollars, etc.; in (b), the preliminary knowledge that four times $1 is four dollars, etc.

In example (b) we might indeed have reasoned thus: $12 is to be distributed into four equal groups, how many dollars will each group contain? Suppose $1 in each group. Then $1 X 4 = $4. But $12 is found to contain $4 three times. Therefore, since $4 gives four groups of $1 each, $12 will give four groups of $3 each. Whatever may be said of this reasoning it is clearly less direct than that given in example (b).

Since the "fundamental" operations can be performed by simple counting (we speak of commensurable numbers), examples (a) and (b) may be illustrated by actual intuitions.

(a) Divide $12 by $4, i.e. find the number of $4-groups in $12. Representing $12 by three rows of four dots each, the result is arrived at directly by counting:

From the foregoing it appears plain that, as to preliminary knowledge:

1. The idea of "times' must be clearly grasped since it is essential to the conception of number.

2. The multiplier must be abstract; a pure number denoting simply "times".

3. In multiplication the factors may be interchanged; i.e. a × b =bX a. Or in words a groups of bunits each is identical with b groups of a units each.

As to Division.

1. The operation in division is the inverse of that in multiplication. That is, the problem of division is: given the product and one of the two factors which produce it to find the other factor.

2. Therefore there may be, in a sense, two kinds of division corresponding to the search for the multiplier and the search for the multiplicand, i. e. for the number of the unit-groups, and the value of each group.

3. That in searching for results the mental and mechanical processes in the "two kinds" are precisely the same; the only distinction between the "kinds" being in meaning of results - in the one case "times" (repetition of unit-group) being found, in the other, the value of (number of units in) the unit-group. But the results are correlated, it being impossible to find one without the other.

4. That the divisor and dividend may be both concrete in which case the quotient is abstract. Or the divisor may be abstract and the dividend concrete in which case the quotient is concrete.

5. That an utterly false "principle or law" of division cannot simplify (as is alleged) any of the processes of fractional arithmetic.

A School in Utopia.

A dear old doctor whom I know, occasionly raises his voice slightly to say to the circle around the dinner table, "I have something to tell."

I have something to tell; something that is novel in an educational journal.

It was at the end of the second week, and as I went about the rooms I heard the following utterances.

8th Gr. Teacher: "These new pupils have a very sweet spirit, and they are so intelligent, and take hold of the work so well, that I look forward to a very successful year."

7th Gr. Teacher: "Yes, my room was never before so full. Every seat is taken, and four sit at a table, but they have been beautiful. They have good habits in attention and preparation, and annoy me very little by indifference and restlessness. They have had good training before they came here, you see."

6th Gr. Teacher: "I like the room very much, they take right hold of their work as though vacation had not broken into it at all. They are very pleasant and have nice manners in the school room. Another 6th Gr. Teacher: "I thought I could never have another set of pupils as nice in every respect as those of last year, but, truly, these are going to be just as delightful. They are bright and cheerful and helpful in every way. I like them."

"When such a spirit characterizes the teachers of a school, the principal is never made to mourn by complaints of what the new class doesn't know."

Chicago, Oct. 26, 1893.

"The Popular Educator Arithmetic" is of exceeding value to every lower grade teacher because of its large number of very practical problems. These problems are such as children can understand and solve, many of them at sight, because they are based on every-day transactions, expressed in every-day language. L. O. FOOSE,

Supt. of Schools, Harrisburg, Pa.

N

School Examinations.

By HENRY CLARKE.

OW that the "rising generation" is acknowledged to be able to think and to do for itself.... which in old times was not a thing to be thought of.... it is time for some expression of opinion upon the subject by those who have given the matter careful thought and study. In the present article, we can only briefly consider this "persecution of the innocents."

For it must be said, at the outset, that we are crowding too much work upon the growing boy or girl in their school curriculum. This, I believe, is generally granted, although as far as I am aware, no remedy for the evil of a practical nature has as yet been conceived.

But of this in future.

Little has been said of the unwise, not to say clumsy examination of teachers in country districts, by awkward and ignorant school committees. I may therefore be allowed to recall one or two experiences which occurred long ago.

I was fresh from the Normal School. My training, I now feel certain, had been excellent. Almost all the pupils in that school had been either graduates or high-class pupils from high-schools, and my own standing was as fair as the average. I was, then, wellfitted to teach a common district-school, which was the place I applied for.

The well-meaning examination-committee however, submitted me to fully two hours of persistent questioning, which might have been completed, had the questions been at all judicious, in half an hour. He asked me about everything askable; he set me puzzling examples in mathematics, inquired my opinion on measuring wood; went through a large range of subjects which I should never be likely to refer to in the most remote manner in the school-room, and finally, when we were both completely tired, granted me the certificate which he had in mind granted me long before. I have now on file this trophy of a victory which I am proud of having gained. One other document is still in existence, secured in T―, as the outcome of a rather formidable examination of all branches of study taught in a grammar school. There, finally, after a severe test, a full hour in duration, a problem in cube-root was given me to solve, a subject not recently taken up and so half forgotten But the skill gained in the Normal School in the use of the algebraic formula settled the matter in my favor. I was granted the certificate, and as it covered a large territory was by its help, appointed to a good school elsewhere.

Reference to covering territory with the scope of a certificate reminds me of the excellent examination given me at W, by a just examination-committee, whose certificate covered the right to teach throughout the state, when duly countersigned by the Commissioner of Public Schools. The examiner received me, with my committee man, in his study, asked some plain, not to say abstruse questions concerning what I was expected to teach, and in a few minutes knew well enough what I was able to do. Not less gratifying had been the entrance-examination into the Normal School; a strict and faithful examination, testing carefully all the grou d, yet neither long nor puzzling.

In regard to length, a school-boy hands me a set of questions which have been used in P- this year; who, speaking of the common dread of such tests, incidentally told me, that an afternoon was wholly spent in a similar examination by some of the pupils; so that at half past six o'clock, these pupils were still hard at work trying to work out the puzzles set them to do. I have at hand two papers used at this examinatlon. Here is a specimen of such questions as an under-graduate in a high school is expected to answer: "IV. Give an account of the religious ideas of the Greeks. What two kinds of colonies did the Greeks found? Who was Pisistratus?"

"V. Sketch briefly the origin and progress of the Athenian power from the invasion of Xerxes to the Peloponesian war; and show how the history of Athens illustrates the advantages and disadvantages of a democratic form of government.

The pathetic side of this catechism is not more ludicrous than pitiful. Such studies for boys of seventeen and girls not much older, of themselves try their utmost skill. For can it be expected of pupils who attempt theses leaps into literature that they must not only state facts but an opinion requiring mature knowledge and judgment?

A change ought certainly to be made. We should ask such questions as would test their faithfulness and show them how to remedy faults unwittingly made. The examination should be, not a dreaded inquisition, whose result may mean ruin; but a welcome trial of skill and knowledge such as the athlete's race or the rower's contest for first place. Not a fierce challenge, but a fair test; let it cover what the pupil has learned in general; let it not confound but encourage him.

Respecting what may well be undertaken in future examinations, as well as what shall be avoided, I can perhaps do no better than to quote the words of an intelligent young woman who has read over all I have just written. Speaking of a recent examination which she passed in a school of high grade, she told me that her own principal recently gave what seemed to a few of her pupils a rather easy set of questions, and that they were surprised that so little work should have been required to answer them. One of the girls, speaking to the teacher, said she was gratified with the character of the examination. "Did you think, girls," replied this most sensible teacher, "that I ought necessarily to make your examination difficult? It is not what you don't know that I want to find out, but what you do know." And my friend thereupon added, "I don't see the need of asking such questions as show that one or two things have not been heeded as we went over the work in class. When Miss A asks an examinatian question, she does not pick out some little obscure matter which we may have omitted to notice; she does not ask ambiguous questions about Columbus, if she examines in history, but says, "Give as good an account of Columbus as you can." "Yes," I replied, "and the excellence of your account is a test of your faithfulness."

I think it will be a welcome era to the much overworked pupil of the high school when he is assured that his examination will contain nothing finical and difficult to answer. A sensitive, excitable boy or girl approaching the end of the term is under the present circumstances pretty sure to be outdistanced by the cooler, luckier one who may have been not only less faithful in his studies but more dependent upon luck for his final success. What the scholar knows, not what he does not know, should be the real quest of an examination paper. What if he has forgotten the little obscure event which, slipped from his memory, may never be referred to twice in the whole course of his life? If he is able to write some intelligible account of what he has learned, not omitting any important event, that is better than a sharp guess at what may be required of him on examination day.

Penmanship as a separate exercise has no place in primary schools. It is a part of the lesson in reading, yet it loses none of its importance as a subject by this arrangement. It is in the mind of the teacher co-ordinate with the reading, not subordinate to it, and consequently loses none of its importance to the child. There may come a time in the training of high school pupils and there should come a time to all intending to teach when special lessons in movement and form are given with a view of making expert penmen. For such, the lessons by Clarence E. Spayd, of Harrisburg, Pennsylvania, will be found very helpful. Mr. Spayd's lessons are full of life and spirited suggestions. His exercises are well graded and in all respects adapted to make expert penmen. DR. D. J. Waller, Jr.,

Late Supt. Public Instruction, Pa.

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I should like to ask Mr. W. M. Giffin, what is the difference between "Division" and "Partition"? I cannot see any after searching the dictionary. "I have .8 of a pie and desire to partition it equally among 8 boys," is perfectly equal to "I have .8 of a pie and desire to divide it equally among 8 boys." It is a problem in division and whether we say: Divide .8 of a pie by or find the part of .8 of a pie, is perfectly equal and in both is a problem in division. The of anything is obtained by division, whether done by the aid of figures, of rule and compasses or of scissors; even the expression is a case of division the eighth being considered a divisor of the number above the line.

W

Teacher Montreal.

Some More Arithmetic.

By WILLIAM M. GIFFIN.

E have been asked, "Why make a distinction between division and partition ? "Because they are two different operations. From the very start the mind acts differently. When I ask in division, how many $3 in $12? The child's mind knows how many dollars there are to be in each group but does not know how many groups there are to be. When I ask in partition, what is of $12, the mind knows how many groups there are to be, but does not know how many there are to be in each group. Let us take larger numbers. I ask how many seventeen dollars, are ? and I know at once there are there in $3910 ? or $3910 ÷ $17: to be $17 in each group, but I do not know there are to be 230 groups till I find out. When I ask what is of $3910 I know there are to be 17 groups, but I do not know there are to be 230 dollars in each group till I find out. Since, then, there are two different processes, I prefer to give them each a name; calling the first division, i. e., dividing a number into a number of equal numbers, and calling the second partition, i.e., dividing a number into a number of equal parts to find the number in one part.

This

Now ask them the following: off and ninety per cent of them will tell you it is! Which is true but why it is, they do not know. It is a mystery to them, and hence fractions become a puzzle, something to learn rules about and never understand. If 1 = so too does of † = }. pen

4 of

4

pens

Ask a child this question: $2 equal what part of $9 and he is ready at once to say . Again, 3 books equals what part of 7 books and he is ready again to tell you, even had you written them what part of or books what part of would have been just as ready.

thus

7

books, they

But you ask them this: 3 =what part of 3, and they begin to look for paper and pencil. Had you asked them gills = what part of what part of 12-? The pints ? They would have said gills gills

3.

=

answer is ready then at once, or . If taught fractions rightly they will answer the other questions just as soon; as they will say what part of 3? what part of } = 3. Let us then, in our first lessons in fractions, do away with all rules that manipulate the namer or denominator of the fraction, also with the inverting of the divisor in division, and the writing of "X" when we should use "of" and fractions will never give us one-half the trouble in the future that they have in the past.

are not examples in multiplication for two reasons. First, the product (so called) is not the same name as the multiplicand which it should be, and it is not any larger than the multiplicand, which it should be. They are problems in partition. We can not have a problem in multiplication with both multiplier and multiplicand a fraction. All such problems are in partition, and should be expressed with "of" and not "X". If we ask, of $12= ? we both multiply and divide, hence, we cannot call it either one or the other, hence, we call it partition.

DRAWING

In Grammar Schools.

By N. L. BERRY, Sup. of Drawing, Newton, Mass.

AN you tell me what this is? I thought you would know! Yes it is a target. How many of my boys and girls have ever practiced archery?"

"We have a target at home, and a pistol that shoots arrows with rubber tips. My brother hit the bulls-eye five times one day."

"Well, Fred, what have you to say?"

"When we were up country last summer, I made a target on the big barn door."

"Good! Then you may tell us how you did it."

"I drove a tack in the door, just where I wanted the bulls-eye, then I tied one end of a string to a piece of chalk, and made a loop in the other end to slip over the tack, and pulling the string tight, marked a large circle with the tack at the center. Then I shortened the string, and made a smaller circle, and then another, smaller, and then another, and another, until there were five circles."

Thank you, you may take this string and chalk, and show us on the blackboard how you did it. Of course we cannot drive a nail in the blackboard, so you must try to hold the string carefully with your finger at the centre.

Now, you see, we have several circles, with but one centre. When circles are arranged in this way, they are said to be concentric."

The blackboard compass would of course, prove preferable to a string for describing circles.

In reviewing the ellipse and oval, like the circle already learned in primary grades, try to do it in a way you have never tried before. Make the lesson "brand new." What if the method is somewhat unconventional? Your pupils appreciate variety. What do they care for circles, ellipses or ovals unless they find them in some way related to that which they enjoy?

Geometric figures pure and simple, make pretty "dry feed" for live boys and girls.

Try paper and scissors, as suggested in the study of the straight line figures, and teach the following details:

When the above have been taught, try the following as a result sheet

1. Describe a circle. In it draw a diameter, radius, chord.

2. Describe four concentric circles.

6, 2 C 3, and 2 D 4. With

centres 1 and 2, and radius