essential to the real concrete 4 as the other. In counting, subtraction is already just as truly present as addition. We are not only putting together units to see how many of them there are in a given whole; but we are just as certainly pulling to pieces a given whole or group to see how much of it there is; and two processes are absolutely correlative. We tell the how much by finding the how many. The 4 is one short of 5 just as much as it is the third in order after 1. If the teacher begins with the group and teaches number as a process of splitting up a whole into parts and of putting parts together into a whole, the pupil cannot help feeling this; he will learn subtraction as a part of the same mental operation as that by which he learns addition. By the strict Grube method, the child will first learn addition, and then subtraction, as an unrelat d process psychologically. The teacher may, to be sure, teach subtraction along with addition, but they are still logically and psychologically two separate things. I venture the assertion that a child taught by the right method will be able to make up" his subtractions; he won't have to be taught them; there is, in his mind, only one operation. I give the reason once more; when we use the group as the true unity which is being at the same time analyzed into parts and constructed out of parts, every number is considered in relation both to the unit which comp ses it and in relation to the whole, 5 in this case, which it composes. The former is addition, the latter subtraction.

As some teachers beside primary teachers may read this, I will add that almost all the difficulty in teaching addition and subtraction in algebra comes from the fact that the child has got the idea of a fixed unit in his head, and the notion that number is simply a piling up of these fixed units. The minus quantity thus becomes a bugbear. But if number has been so handled from the start that plus and minus are seen to mean direction of movement, the difficulty is non-existent. Direction down from a zero point on a thermometer is just as real as direction up; to have to pay money out is just as simple and real as to have it paid in, etc. Now upon the principle indicted above, the ideas of subtraction, minus and reversed movement (position back from 5 as well as positi n ahead from 1) are associated from the start. The algebraic operations only bring the matter more clearly to consciousness. Indeed, if it were worth while quite young children, educated upon the principle here laid down, could be got to add — and add intelligently — minus quantities.

I shall now proceed to show that multiplication and division are involved in the simple process of enumeration, and when taken as distinct operations only make explicit what is already contained in counting always with a proviso that counting is not regarded as a development of 1" but as th measurement of the value (how much) of a group, by means of stating how many units it contains. All counting implies the idea of times. Suppose the child in counting has reached 3; this 3, as a last term, expresses the value of a certain group. It synthesizes three distinct units; it says we now have so many all put together. It is itself a group, regarded as a sum. But this very idea of a total or sum implies both logically and psychologically, another idea that of repeating the unit a certain number of times" taking" as we say. We cannot get the idea of three as sum until we have got the idea of third in order of enumeration. I referred, in my previous article, to the difficulty children have in getting hold of the idea of 3-how much harder it is than 2, and how apt they are to regard 3 as the name of a particular given object. The difficulty is obviously in getting the idea of order as distinct from the idea of particular objects counted; till a child can see that three means not any particular object, but any object which comes next in order (sequence in

counting) after 2, he may know the names, but he cannot co nt. Now order, or place in the sequence of counting, third, meas clearly the third one. If a child cannot keep this order in his mind, he cannot refer any given object back to its starting point: it is simply another one, with no numerical value. Logically, that is to say, the third in order is a precondition of the idea of three things taken together or a sum; logically, the ordinal value of number precedes its cardinal value. Now the third one means of course, the third time of taking one. What the child‹ounts when he gets the idea of place or order in a series is his own series of acts. Third isn't a property of the object; if it were any given object would always be third. Third means the third time a unit is counted; it refers to the order of the mind's own acts. I dwell at such length on what ought to be the next thing to an axiom because of the attempt to read the idea of "times" out of arithmetic which some professional educators are engaged in. Any one who holds that the idea of times has no business in arithmetic would, if he were logically consistent, be obliged to say that the child is right when he maintains that his thumb is not 5 but 1, because he began with the thumb before and called it 1 then.

The bugaboo which Professor Giffin brought up in the September number of the POPULAR EDUCATOR, will make a good i lustration of the truth. His argument against the use of the "times" idea is that "I may say three times three apples and only have the same three apples: I may take up and put down the same three apples thr e times, and that is three times the same three apples." Suppose we take one apple instead of three; I take it up three times. Now what is the problem? Is it to find out how many apples there are? Or is it to find out how many times I have counted? If the former, the apple evidently is to be taken only once (even here we can't rid of the use of times) and to take it up thr e time is to show that the question is not understood; but if the question is as to th number of times'the hand lifts the apple, 3 is the correct answer. So in Prof. Giffin's illustration; the moment you state the problem, either the operation of picking up the 3 apples three times becomes absurd or else nine is the correct answer. To pick up three apples three times is the same as to pick up one apple nine times- this is the answer to the operation mentioned; provided the operation has any sense in it. And I hope I shall be forgiven if I repeat the matter again; either the child is counting apples, and then it violates the conditions of the problem (is absurd) to pick up all the three apples three times, or else the c i'd is counting times and then nine times is the corr ct answer.

To sum up to find out the sum 3 means first to find out that one of them is the third in order or sequence; and this means that the unit is taken or repeated three times. And this means multiplication. Three is three times one; any integer or sum whatever is, in reality, the unit of measurement applied to different objects a certain number of times; ten is ten times one, and so on. Every number whatever may be expressed as 1. (The idea may be expressed algebraically by saying that any number is not simply a sum of units but is a co-efficient, expressing the number of times the quantity is taken.) Now for division. So far we have taken our unit of measurement for granted. The finger, the bean, the toothpick is evidently there, and requires no explanation. But we must remember that we take the finger simply as one whole thing in itself; while when we count it, we treat it as one of a group; this is the very essence of counting to take the unit not as a whole in itself, but simply as one member or part of a group. Where does this idea of being not a whole in itself but a part of a larger group come from? Evidently, psychologically and logically, the

idea of division is already present. A whole has been broken up into parts; it has been divided, and counting is the act of finding how many such divided parts there are within the whole. The very fact that we have distinct units to enumerate shows that the act of division is implied.

Now it is easy to go astray here. Someone may object: "No, the objects are already divided up; the fingers are already so many entirely distinct things, and we have simply to count the things already there." There is no doubt that physically, the things are already divided; but it does not follow from that that they are numerically divided that they are already distinct units. I should like the objector to explain the difficulty children have in learning to count. If the things are already so many distinct units, why can't we count them as soon as we see them? Surely, there is no great difficulty in learning the mere names, 1, 2, 3, 4, 5, etc. In other words, the very difference between counted or numbered objects and a lot of physically distinct things is that the latter are not and the former are taken as parts of a wh le, as members of a group. Division, the process of getting units of distinct numerical value as portions of a group, is thus logically the fundamental numerical process. This does not mean, of course, that it is the first in consciousness; rather, being the more fundamental, it will be the last to come to consciousness.

But it does mean that any right method of instruction in number will use from the start the process of breaking up a whole into definite parts, and, familiarizing the child practically with division, make it easy for him later to get the conscious idea.

It should be noticed in passing (what we shall come back to in fractions) that up to this point we have been dealing only with crude methods of division, or fixing the value of a unit of measurement. Each finger, as one, equals every other finger, but the value or amount of one would differ from the other if we measured it - that is, one finger is larger than another. Only in exact measurement or the comparison of quantities in space (weighing implies comparison of displacement in space) do our units have exact values. The practical point here is that it is absurd to introduce the child to division and fractions save through measurements which actually break a whole into parts of definite value.* Our arithmetics are mostly absurd enough all the way through, but perhaps their crowning absurdity is putting weights and measures" off as a distinct thing by themselves. Historically, the processes of division and fractions arose from the need of measuring land and weighing objects; logically, no one can understand them save as reached through measurement.

Our next point is, that multiplication and division always go together. One is not simply the reverse of the other, but we cannot perform one without implying the other. The idea of times and of parts are strictly correlative. Three, in counting the fingers, is, as a sum or integer, three parts of the whole; as ordinal, it is the third time of taking one. As times' is always one of the three elements in multiplication, every process in division will have parts' or 'portions' as one of its terms. 6 If the divisor is not parts, then the quotient will be. And this is all there is to the wonderful discovery of "two entirely different kinds" of division, one of which is division, while the other is. something "wholly different ", partition. In division, to sta'e the matter as it is, we must have given the whole or group the thing to be divided. This is the dividend, of course. Our other terms are the number of parts (how many) and the value of each

* If anyone has stil any doubt on the question of 'times' I wish he would test it on measuring space. As an inch isn't an object already physically existing in space, two inches certainly cannot be two such things. What can it be but the result of laying off a ce tain length two times?

part. It ought to be axiomatic that we cannot separate these two things. So many parts of so much value ach make up the whole — nothing short of this is a complete statement. Therefore, (because of this necessary correlation) if we have the whole (dividend) and one of the other terms, we can always find the third. Because of this correlation it is misleading to speak of two kinds of division. There is one kind, but that one kind has two phases, according as the whole and the value of each part is given, to find the number of parts (the multiplier); or as the whole and he number of portions are given, to find the value of each part, to find the multiplicand.*

This seems a convenient opportunity to speak of the question so much agitated of late the question of dividing $12 by $4 and dividing $12 by 4. The first problem is evidently this :- Given a whole $12 and parts of the value of $4 each; how many such parts are there? Or how many times must the value $4 be taken to constitute $12? What is the multiplier? The quotient here is obviously abstract number of parts, or number of times one part, as unit of value, is taken. The second problem is :- Given a whol of the value of $12 and 4 parts in such whole; what is the value of each part? Here the divisor is abstract - not quantity, or amount, or value, but simply the counting off of a number of portions, or sub-groups.

I shall postpone the minute discussion of ratio and fractions to another article; but for the sake of completeness, I wish to point out that ratio and fractions are not further operations performed with number, but that all number involves the processes brought out distinctly in the ideas of ratio and fractions. The analyticsynthetic process which constitutes number, gives the value of some group by telling how many units of measure are found in it. All value is thus essentially ratio; it expresses the ratio between the given number and the unit which measures it. To say that a given stick is 8 inches long, is to state the ratio between the space in question and the unit of length. The one inch has in turn the ratio of 1 to 8; this particular one inch is one-eighth the whole space. So in counting the fingers. Five is the whole or group; in relation to this, 1 is one-fifth; 3 is threefifths; the whole is five-fifths. When we begin with a fixed unit and proceed to "develop one", ratio is a new and irre`evant process. When we recognize that we have from the first a whole or qualitative unity within which counting goes on, the idea of ratio is implied; every number has its value fixed by its place in the whole.

When we recognize that this place has to be related both to the whole and to the unit of measurement we have fractions explicitly before us. In counting the fingers, three has to be placed both in reference to the whole, 5, and in reference to the unit of reference, 1. Its value in the latter relation is 3; in the former it is three out of five; in its whole position it is both, namely, three-fifths. In other words, the denominator always expresses the number of units of measurement found in the whole group taken as itself a unit; the numerator expresses the number of units of measure found in the whole group as unit. The fraction expresses all that it is possible to say (arithmetically) about the quantity. It tells us the value of the whole group which limits the measurement,; it tells us the value of the part taken as unit of measurement, and it tells us the ratio between these two values. It sets before us the entire analysis-synthesis in a single act and expression. That is to say, suppose we have to express the numerical value of a certain amount of space. The

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* If there is still any one not convinced by Dr. McLellan's refuta. tion of the notion that quotient equals dividends, I comm nd the above to his notice.

whole group, the unity is 1, 3, 38, according as we are dealing with foot, yard, or mile. If we divide this unity into a number of equal parts, and take one of them we get the unit of measuremer, 1, 3, 320; if we state, in terms of this unit, the ratio between the value actually given, and the whole group which fixes its value, we get 1, 1⁄2, or whatever the most explicit statement of value possible in pure.y arithmetical terms. The consequence of this, I will show later.

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Our Critic.

NE of the most hopeful of the "New Departures" in educational matters is the movement of several woman's clubs in different parts of the country, toward the uplifting of educational ideas and administration, by introducing prominent educational lecturers to the community; making the occasion either free or open by a very small admission. For many years the various clubs in Boston have regaled themselves in this way, with no care for the public. We believe that there is not even now a teacher's club in our Athens that offers to the general public the opportunity of hearing a distinguished educator on the topic that concerns most intimately the welfare of every family. Even the Massachusetts State Board of Education, of late, in the management of its Teacher's Institutes, has almost entirely neglected the great opportunity on every such occasion, of one grand rally of the people, to listen to a valuable discourse by some educator of commanding ability and reputation. Our western and southern women seem to have "caught on to this opportunity and are now offering to the public what their more exclusive literary sisters of the East have been content to reserve for themselves. We are informed by Rev. A. D. Mayo, that, last year, in his southern journeyings, he was introduced to the people of Lexington, Kentucky, by the Sorosis Club, composed of the f remost women of the city. Recently the woman's Literary Club, of Dayton, Ohio, which owes its origin largely to Mrs. J. B. Marlay, formerly principal of the city training school for teachers, has inaugurated a system of lectures on Education; having already brought to the city Dr. Rice and Mr. Mayo, and proposing further movement in this direction. This revival of interest of the superior women in these Western and Southern cities, is not a day too early to meet an "impending crisis" in the management of the common school system of these great and wealthy communities, a condition which can best be met by the coming to the front of the good women to the defense of the public schools against the special demoralizing influences that now threaten to check their development, if not, in some cases, to side-track this great American Institution.

Every system of public schools in its growth passes through all the diseases of childhood. Within the past century the New England schools have, in turn, been brought to bed" by the successive attacks of its many enemies, personal favoritism, and partisan political intrigues. Having survived all these, they now seem on the eve of being assailed by the even more subtle malady of a high expertism that drives the mental horse in the team in a fashion so fast and furious that other considerations retreat to the background. But in the West and South, the public school system is, just now, involved in mortal combat with a malignant spirit of political partisanship, which insists on planting the Spoils system in the very heart of municipal life, the childrens preserve. We can name a score of important cities in the North West in which the fate of the school superintendent is often

involved in the election returns from one or more hotly contested wards, and the whole dispensation of school appointments depends on the success of one political party. The State Superintendency of Education is made a political office. Within the past ten years, scores of the oldest city superintendents of schools have been bowled down by this wretched game of political ninepins, inaugurated by ward politicians, and tolerated in a sort of hopeless and helpless way by the educational public. In this emergency the coming to the front of the woman reserve appears like a real dispensation of Providence." And when this becomes a permanent and recognized influence in behalf of public Education; as it already is on the side of Religion, the Public Charities, Temperance, and the industrial elevation of women— the drooping hearts of good men, wearied with this protracted conflict against the combined enemies of good schooling, will be chcered, and better things may be hoped for. Meanwhile, the women's clubs of the various cities will confer a great favor on the people by presenting the most eminent Educators, men and women, as advocates of the broadest and best view of the American Common School.

We learn that it is proposed to introduce the "Department system of instruction " by way of experiment, in some of the grammar schoo's of Boston. The experiment will be watched with deep interest, and its result noted, not only in the direction of more thorough instruction, but also in its tendency to inaugurate in the people's commoa school the methods of the university and the policy of extreme expertism in Education. For, concealed as it may be, this movement is the beginning of a thoroughly understood movement from the University and European direction, to take possession of the people's American common school; remove it largely from the control of the people; suppress the growing influence of woman in the school room; and substitute the rage for mental achievment and thoroughness for the roundabout training of character which has been the one central glory of the common school in our country; the saving el, ment of the inst tution, despite its many infirmities in other ways. Of course, it is desirable that all our scholing, from the elementary to the university grade, should be genuine; by the best methods; economical of the time and strength of the pupil; a true beginning for the education for manhood, womanhood and American citizenship. But whether the placing of every leading study in the hands of an expert, chosen especially for aptitude in the mental training in his own beat, is the way out of the presert difficulty, is by no means clear. The moral and character training in our colleges that came from the contact of students with the admirable men that filled the chairs of instruction in past years is now in peril from the substitution of department experts; often young professors or instructors, of narrow, intense, onesided manhood; often inferior in these qualities to many of their pupils. We shall make a prodigious mistake if we remove the children, especially below the high school age, from the personal influence of the majority of women teachers in the schools, both of city and country, and give, in place, the domination of a supervisor, infuriated with zeal to push his own department, with a core of instructors who, if superior, will chafe against this despotism, and, if inferior, will lose that peculiar influence which is now the saving grace in the common school.

OFFICE OF COUNTY SUPT. OF SCHOOLS, CHICAGO.

I have carefully examined the two little volumes entitled ESOP'S FABLES by Mara L. Pratt. I have used them with my own children following the first reader. The books charm the children and through reading them they learn to love to read.

They can be profitably used as supplementary reading in second grade, and for that purpose I know nothing else in book form equal to them. They are Literature.

O. T. BRIGHT, Co. Supt.

ETHODS

The Editors will be pleased to receive contributions for this Department.

First Lesson in Interest.

By WM. M. GIFFIN, Chicago.

Exercise One.

1. If A lends B one dollar for one year on condition that B pays him six cents for the use of it; how much will B owe A at the end of the year?

2. At the same rate how much would C pay A for the use of $2 for one year?

3. How much would D pay for $3? E for $4? F for $5 ? G for $6?

4. Hould pay how much for $1 if he had it two months?

5. I would pay how much for $2 for two months?

1. B paid A 49 cents for the use of some money for seven months. How much did A lend B ?

NOTE.-If B paid 49 cents for seven months, he paid of 49 cents for one month, or 7 cents, and for two months he paid two 7 cents (or if the pupil understands the use of times, 2 times 7 cents or 14 cents.) Then A lent B $14, because the money paid for two months is always as many cents as there were dollars lent.

One more device is worthy of a place in teaching mensuration; observation is our first and best assistant.

To show that the solid contents of a cylinder equals the area of the base the height, have a cylinder of tin 3 in. in diameter and 3 in. in height. Fill it one-third full of water. The quantity of water the area of the base X 1.

Fill it two-thirds full of water; The quantity of water = the area of the base X 2.

Fill it full: The quantity of water = the area of the base X 3.

A tin cone of the same base and height may be filled with water and emptied into the cylinder just three times: hence, the solid contents of a cone the solid contents of a cylinder having the same dimensions.

A ball having the same diameter may be placed in the cylinder which will then hold one cone full of water. Hence: The solid contents of a sphere the contents of a cylinder having the same dimensions.

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How to Make a Mimic Volcano.

All teachers recognize the fitness of illustration in geography and are pleased with every opportunity which offers tangibility to this usually abstruse branch of instruction.

An interesting thing in this line is the mimic volcano which is made as follows: build a miniature mountain, about eighteen inches high, of sand or earth and insert a one-and-one-half inch tube of coarse paper through the center. Fill the crater with granulated sugar and chlorate potassium equally mixed and of the same consistency. A drop of sulphuric acid does the rest. The room should be darkened and proper care taken to avoid igniting any articles near by; although there is but little danger of this and the effect is highly entertaining and profitable. The experiment should be preceded by a talk with the pupils on the subject.

PRIN. JOE E. HERRIFORD.

OLLOWING the study of germination of seeds may very properly be placed that of the growth and development of buds and branches. Let the children observe buds as to their position on the stem, and in their relation to each other.

A few of the larger buds should be dissected before growth begins to discover the growing point. The horse chestnut furnishes one of the best specimens for this work. The pussy willow and the cherry should then be compared with this.

Except in the country, it will perhaps be wise for the teacher herself to gather the twigs for the children to study, as young children show little discrimination in attacking a tree, and the shade trees of our towns should not be exposed to their raids. The following may be used as a guide for the study of buds:

I. Their position.

Buds.

quite large catkins. There are two kinds of pussy willows: the staminate and the pistillate. The staminate has stamens, and the pistillate has no stamens but a pistil. In a few weeks the flowers grow very large, and the stamen can be distinctly seen.

There are hundreds of flowers on every catkin, and every flower has two stamens. When the stamens grow larger the anthers barst, and yellow pollen falls all over the catkin giving it a lovely, yellowish color.

When you look at the pistil through the microscope you can see that behind the pistil is a very dark bract. There is only one pistil on a bract. There is no pollen on the pistils, but the green color of them gives the catkin a greenish hue. HILDA NILSSON Grade VIII. Pleasant St. School. Grade VIII.

Pleasant School.

III. Their arrangement on the stem.

The development of underground stems should be taken up in the early spring, and the reasons for their being classed as stems rather than roots discovered. Select for study such typical specimens as the potato, onion or lily, Jack-in-the-Pulpit or crocus, and the Solomon's Seal.

This work may be made preliminary to the study of the spring flora. As soon as the warm days of May come the flowers will appear in such profusion as to demand all the attention. For this reason it is well to make these preliminary studies this month.

Grade VIII.

The Onion.

The onion is a bulb having a number of thin layers of skin on the outside. The fleshy part is divided into layers and inside of all these layers there are the hearts of the onions. There is no limited number of hearts, some onions have two, others have three or four and sometimes as many as six.

These hearts are round spots lighter than the rest of the onion and are really leaf buds. From these leaf buds there grows a new plant which in time becomes a new onion.

From these hearts, sprouts grow up and get their food from the layers one by one until they get soft and spongy and there is no food left. The outside dries up and fal's off. By and by the roots come and the plant grows and forms new onions.

The cross sections I saw looked like the second drawing. The third one is an onion which I saw where the sprout had come up and the leaves growing up from the sprouts. The fourth one shows the roots that have grown out.

The leaves are long, slender and very smooth. Near the onion they are thick but near the ends they are hollow.

There is a good deal of difference between the potato and the onion, and the onion and Indian Turnip or Jack in-the-Pulpit.

The potato is a tuber and begins at first a little bunch on one of the roots, then it grows larger and larger until it becomes full grown and has a number of little eyes where the sprouts start from.

The Indian Turnip is neither a bulb nor a tuber but a corm, it is much like a bulb only it is solid instead of being arranged in layers or scales as a bulb such as the onion is.

It is also very hard and dry and every year a new bulb is formed and the old one dies down. There are some kinds of cultivated plants which do the same thing. FANNIE E. YOUNG.

Pleasant St. School. Grade VIII. Grade V.

Hilda Nilsson

Pussy Willow.

This twig of pussy willow that I have drawn shows the buds of the pussy willows. They are covered with dark scales that fit right over them, and ca 1 be taken off easily.

Under these scales are white, soft substances that look like bunches of fur. These buds are nearly opposite on the stem. They have no stem of themselves, but they grow closely to the twig.

As they grow larger the scales drop off and the buds become

Pussy Willow.

When you first pick a pussy willow twig there are some dark brown scales covering a little oval shaped body which looks like a