it is not only an attainment in itself, but it is the beginning of a habit of right identification and distinction, which when formed may be trusted to carry the mind through any problem. It is a warning, because it shows that a mistake of method made early in education is not simply one mistake, but starts the mind on a wrong and, therefore, confusing and mentally wasteful tack, which will lead to confusion in every subject as long as persisted in.

Number, then, psychologically must be some one case or development of this analytic-synthetic process. Everything in number, all its processes and properties must be implied in the natural, instinctive dealing with things upon which the child has always been engaged. Our first psychological principle, then, regarding the teaching of number is that

Any right method of teaching number involves the greatest amount of continuity with what the child mind has been previously doing it but makes an object of consciousness what the child has uuconsciously done.

But to make any use of this principle, we must see what is the specific connection between number as a synthetic-analytic act and the act of identification and discrimination implied in all attention, and in recognition of an object.

Every act of attention makes a unity, a totality, out of every subject attended to it grasps the varied elements into one. Each of us makes a whole, an integer, every time he does anything. With a shifting of attention the unity changes. If I look at a tree, that tree is the unity for me; all its various elements are integrated; if I want to know more especially about one of its parts, my synthesis changes and a leaf is the unity; put the leaf under the microscope and a cell, perhaps, becomes the whole. Let the physicist get his attention directed to the matter, and the unit changes to an atom. The unity varies with the mental act, because it is made by what is held together in one act of attention. The first unities, for a child, are probably based for the most part on actual physical activity anything is one which can be handled as one. His ball, a hat, a drawer, a cup of milk are each one, because he moves each as a whole. A child, then, long before he begins the study of number has been engaged in the making of unities; of these distinct unities he has a large stock on hand when he begins the study of number as such. Each of these wholes has parts but express attention is not paid to these parts, either in themselves, or in their relation to one another or to the whole. Unities and Units.

So far we have unities but not UNITS. Each of these wholes is a unity, a qualitative interger, that is, an individual through the unity of meaning which pervades it. To conceive of, say, a bean not simply as a whole or unity, but also as a UNIT, means to bring it into relation with a number of similar objects, making up with them a larger whole or unity. Quantity, as distinct from quality, begins psychologically whenever individual things are considered as parts in the composition of a larger individual, or when any individual is regarded as an aggregate of parts or smaller unities; these two statements being the analytic and synthetic way of saying one and the same fact.

Now this distinction between a unity and a unit may seem somewhat fine-spun for practical purposes, yet I venture the remark that the conscious recognition of it, or unconscious acting upon it, makes the difference between a right and a wrong method in beginning the study of number. According to the way the mind naturally works, every unit is always one factor in the composition of a qualitative whole(stated synthetically) or one element into which a whole is divided (stated analytically). Whichever way we state it, the unit is always, psychologically or according to the mind's own method, a member of a whole which has a quality or character of its own. Any true method of instruction will recognize this method of mind. The wrong method will set up an independent, fixed unit, and thus violate the law of mind. An illustration suggests

itself most readily in dealing with fractions. Suppose a child has begun with 1 as an entity by itself, complete and final; then learns 2 as the sum of two of these independent units, and so on: now, all of a sudden he is introduced to fractions. What a tremendous break this is! Where do fractions come from? He is asking this question mentally, even if he does not ask it orally, even if he does not himself know that he is asking it. The whole attitude of his mind is asking it in trying to adjust itself to this intruder. Where did it come from? What does it come for? What is its use? Not a single thing in his previous dealings with number throws a bit of light upon this question. Finally, then, he must fall back on the bare brute fact: There are such things as fractions, as half-apples, as quarter-pies, etc., and it becomes necessary to invent some way of dealing with them. Nine chances out of ten (I hope not ninty-nine out of a hundred), the teacher is in just the same attitude of mind. Fractions seem to him something strange and artificial. I should like to see the results of an honest census on this point. I believe that a large number of teachers would state that while they understand the processes, and can explain the ground, say, for the rule in dividing fractions, still fractions themselves seem unreal,- although necessary, because things themselves are broken into parts, and because, unfortunate ly, not all numbers are even. I can say for myself, that while theoretically I am p rfectly convinced that fractions are rot an external operation performed with numbers, but arexpressions of the essential nature of number, still my early education having been entirely upon the principal that a unit was a thing fixed in itself., 2, a collection of two of these fixed units, and so on. It is with great reluctance that I now deal with anything in a fractional form, and always with a dumb feeling that they are irrational and absurd anyhow. Here is fault number ne; fractions, so far as the attitude of the mind is concerned, mark a complete break with former experience, and therefore seem irrational, out of all harmony with the structure of the mind as already formed. Interest, in the second place, is lacking, since it is a fundamental principle of psychology, that we take interest only when we feel some point of identity, of sympathy, between the subject and ourselves. Attention is highly forced, because of lack of interest and because there are no ideas in the mind with which we can attend. The attitude of mind in dealing with fractions is thus rendered lame and awkward. It may be laid down as an absolute principle that when any subject is uniformly found difficult, it is not because of intrinsic difficulty in the subject, but because the mind is in an unnatural state in regard to it—in a state violating some psychological principle, whether because it has been introduced to the matter too early, or because some defect in previous training has set up a wrong habit so that the mind cannot now work freely. I shall deal with fractions later on, but I will simply call attention to the fact that all this unnaturalness of fractions vanishes, provided the mind has been trained from the start on the psychological principle that a unit is not fixed in itself, but is one of the parts of an individual, varying as the group to which it belong varies. The confusion of the qualitative whole, or unity, with the unit is thus at the basis of all wrong method.

The hand attracts distinct notice before the separate fingers, then each finger attracts attention. The hand is one, an individual or whole, each finger is equally a unity. There are no units yet. If however, we take the fingers as parts of the whole hand, then we have true number, true units. Each finger is now one, in the numerical sense; it is a unit. But it is equally one-fifth. The very moment the individual thing becomes a numerical unit (and not a mere qualitative whole), that moment it becomes either a fractional part of a whole, or such a whole made up of fractional parts. Not a word need be said about fractions, but mentally fractions are already there: the mind is making parts into a whole, a whole into parts, and is considering the relation of these parts to

the whole. When fractions are finally introduced, they are no strangers or intruders. The mind has only to consciously recognize what it has been unconsciously doing ever since it began dealing with number at all.

Thus we get, to return from our illustration to our principle, the second fundamental psychological law: Alwnys deal with numbers on the basis of a whole made up of parts, on the basis, that is, of some group which makes a natural whole, never on the basis of separate fixed units.

This seems a good place to say something about the so-called intuitive or objective method of teaching number. It is better to have the right practice and give a wrong reason for it, than have any number of good reasons and then a wrong practice. But it is better yet to have a good reason for a good method. The method of dealing with things, shoe pegs, tooth picks, beads, no matter what, or with symbolic representations of them as lines, dots, etc. is justified; not, however, because number is an intrinsic quality of the things, which can someway be got out of the things simply by looking at them, but because the mind is first conscious of the products of its own activity, and only afterward of the processes. The objects are necessary to any right method, because and in so far as they present to the mind the results of its own action; they enable it to see what it has been doing, and thus prepare it to get hold of what, after all, is the end of the intuitions, etc.,- namely, the method, or operation. From the process to the product, from consciousness of the product to consciousness of the process is the law. This law shows both the value and the limitations of the reliance upon objects. So far as the objects are merely thrust before the child, the good they do is purely accidental; the child does not learn from them, but from what he himself does. So far, as the objects, or dots, are used in such a way as to represent in objective form what the mind is doing, they are a necessity, and a necessity from which nothing but good can result.

The third psychological principle is, then: Have the child use a natural whole or group of objects in such a way as to develop in him regular habits of breaking up wholes into parts and of putting together parts into wholes, these operations being performed in such a way as to make a child see the relations of the parts to each other and to the whole.

A word or two about how large a group to begin with may be a good illustration.

One is an absurd, three a bad beginning; four is a natural one, both because of the square, and because of the way it divides. But it hardly allows sufficient play to the child's mind in forming combinations, to be an exclusive basis for a long time. Ten, from the number of fingers, is a good beginning as a qualitative whole in itself, but can be divided evenly but once. Twelve is natural or familiar to the child, from its connection with a dozen of many sorts of precious things,- oranges, bananas, cookies, etc. The dozen group should be introduced then as soon as the child is fairly, (not thoroughly) familiar with four and five. The two sixes, the three fours and the four threes may be noted (as groups, of course), and then no attention paid to the numbers from six to twelve for a time. Eight, as two fours, and nine as three threes (groups again) may well be taught long before either seven or eleven, or before all the combinations in five and six are exhausted. Now all this will strike, as a sort of blasphemy against "modern methods," all those who make it first article of creed, that no number above five should be introduced in the first six (some say twelve) months, or above ten in the first year, or even second year. But I want to illustrate the difference between the method of teaching based upon a whole or group of parts, and that based upon a fixed, separate unit, and so I will carry my heresy a step farther. It will be a good thing to introduce the idea of 100 at a comparatively early stage, certainly within the first six months,- not as a number by itself, but as the dollar, stating that a dollar is one hundred cents. The idea of 100,

as a group of ten tens, being given concretely; ten being the unit here, ten tens (presented concretely as with bundles) is as easy to grasp as ten single sticks.

The identity of and of a dollar with 25 and 50 cents respectively follow next, and then the ten dimes or ten cents, using in all cases the terms half, quarter, etc., interchangeably, making no attempt to have the child memorize these facts, but allowing him to pick them up, if, indeed, he has not already picked them up at home, or doing errands, while the teacher was trying carefully to guard his precious mind from acquaintance with such a dreadful thing as a hundred, and was wearing out interest and power of teaching by ringing the changes on the numbers below five. Aside from the fact that a good deal ought to be risked in order to get an early and natural introduction to the fractions, one-fourth, onehalf and the decimal one-tenth, aside from the necessity of variety to continued interest, aside from the natural interest a child has in money, there is another reason, based on the nature of number as a whole of parts, which makes the introduction of one hundred good and not bad psychology. One hundred is just as much one as one apple is: as such, as a dollar, it is just as easily grasped, so with fifty as one half. A silver half-dollar is just as much one as one tooth-pick is. It is pure superstition, based on the original myth of a fixed unit, which supposes that twelve is a much harder idea than say three. In the form of a dozen, it is probably a more familiar, to say nothing of being a more interesting idea, than either seven, eight or nine. In any case it is one, a whole, and that is more than can be said probably of any other numbers, save five and ten if the child has learned to count on his fingers. Twelve inches make one foot, twelve hours one day, twelve oranges one dozen, etc., and to suppose that one foot is not a simpler idea than three inches, or one day than three hours is absurd. If it is absurd, equally absurd is the idea that the existence of twelve must be carefully concealed from the child till he has been sickened of all the lower numbers by monotonous drill. The fact that twelve is connected with such interesting facts as a foot, as the hours on a clock face, with a dozen oranges, should be used as the basis of important knowledge which the child should be getting, but which he does not get, if he is kept confined to the smaller numbers. Six will be taught more easily as half of a dozen than as an entity by itself, and so on. In other words let it once be recognized that units are not fixed things, but differ with the kind of thing which attracts attention, and much of present nonsensical method will disappear of itself, leaving the teacher free to adopt such devices as seem most useful under the particular circumstances. If you do not believe that 12 and 100 can be helpfully introduced, try it.

But to return to our third principle. Analysis and synthesis are here, as in every genuine psychological process, two sides of one and the same operation. At the same time the parts are put together to make the whole, the whole is divided up into its parts. Suppose that the child has a vague idea of a dozen; he knows, that is, that a dozen oranges form a whole, a natural group. Just how much, how large the dozen is he does not know; in other words, his mind is in the same state about that, in which it is about most things. It has a rough, vague idea. There is an equally vague analysis in the indefiniteness as to the number of parts. He does not know just how many there are. Any thing that will answer the question how much will answer that as to how many, and vice versa; the how much is the unity, the whole, stated in terms of its units, or component parts; and how many is the parts stated with reference to the aggregate, or the unity. Here we have the fundamental and cor-relative factors of all quantity - magnitude, how much, and units, how many. Now the common error of beginning with a fixed, or separate unit tries to keep the two sides of number apart — to separate the how many from these how much, and give the former an independent existence. Everyone would recognize that

a dozen had no existence save with reference to its parts; a right method in dealing with number consists in recognizing that 1, 2, 3, etc., have no separate existence, but exist as making some whole, the dozen or, of course, some other group.

Did you ever ask yourself why it is that a child will get the notion of two when he is in his second year, while many intelligent children will not get that of three till the fourth or even fifth year ? Or, to put the same question in another form, did you ever ask yourself why, when you are teaching a small child to count, he is so apt to insist that three, or four, is a certain particular bean? Every teacher can duplicate the following: When teaching my little boy to count with some acorns, he was, I flattered myself, getting on finely, till I called a certain acorn six; when he became indignant, reached out for another one, and insisted that that was six. (In a previous counting, that acorn with some peculiarity had been 8.x.) If you ask yourself the reason, you will see that a unit, 2 or 3, etc., has no fixed value of its own, but owes its value to its relative place or order in the group. 1 is not an absolute 1; it is simply the first 1; 2 is simply the second one taken; 3 the third one, in constructing the whole, or in analyzing it. Three names a sum, so far as it itself is the whole or group; otherwise, it names the place of a mental act in a sequence or series of acts. The difficulty which the child has, in other words, is in getting hold of the idea of order or relation as distinct from the particular thing. A child who has got the idea of three as involving a relation which remains the same, while the things may vary, will learn to count to ten in half an hour; one may count a hundred and if he does not get hold of the relative character of his units, his counting will prove to be mere parrot-like repetition of words.

A child early gets the idea of two, because that does not involve the recognition of order or relation. He is attending to one thing, and he is conscious there are other things. Or, 2 means simply more than 1. It means two unities, not two units. Now the introduction of three means the recognition that each unity is no longer a whole by itself, but is one part in a larger whole, or group, and has, therefore, its value not in itself, but in its relation to this whole, and to the other parts in the whole. It is the recognition of number proper.

What is the bearing of this? Why, precisely that there is no unit in itself, outside of the place which it occupies in a whole, that is, the first in order, second in order, and so on; that means one part or member of an individual, qualitative whole; that "how many" units cannot be separated from "how much" (group of units); that analysis and synthesis always go on together.

We may now develop the latter part of our third principle, - having the child see the relations of the parts to each other and to the "whole" into a fourth principle as follows:

A child must be led to compose and break up groups in such ways (a) that the unit will be seen to be one equal part in a group or whole; (b) that each particular unit will be seen to have its value from the place which it occupies in the series or sequence of acts of composing the whole; and (c) the whole shall be seen to be a unity which has its value not simply as the last term in the series, but as the sum of the acts of taking different units.

In other words, objects in their relations and groupings must be so dealt with that there shall be developed the ideas of a unit, an equal part, of the whole; the idea of place in a series, first, second, third, etc.; and as a sum formed by the repetition of a unit a certain number of times.

As there has been some discussion in the POPULAR EDUCATOR recently (November No., p. 104) about the idea of times in number, it may be well to point out that the idea of times is an absolute psychological necessity to the idea of number as magnitude. If a child when he got to three did not carry along in his mind his previous acts of counting one and two, the object! would not be the third, (as order) and hence, not three

(as sum) but simply another independent unity. The "times" denotes simply the act of the mind in taking different things not as independent, but keeping them together (synthesis) as parts of one whole. I do not know how well it would work practically, but as experiment an occasional sequence of sounds (strokes of bell, or chords on piano) might be alternated with groups in space. As the latter emphasizes the side of the whole or group, the former would emphasize the side of the mental act in repeating a unit to make the group. In any case, the idea of sequence (times) must be present.

If the fingers of the hand form our group, then any one finger, taken as one part of a larger whole, is a unit; what unit it is depends upon what place it has in the series of acts (times) in which the mind takes the parts, first, second, etc., its value is the ratio of this to the whole, namely, one-fifth, two-fifths, etc. In other words, all intelligent counting takes the mind both forwards and backwards. The index finger, say, is 2 not in itself, but because it occupies second place in a series of acts; but this series has a limit, in this case 5 as the fingers on one hand, so that 2 as it looks forward is 2 out of 5, or 3. In the same way, the value of the whole group is the summing of its units. That is, it may be taken either as one whole, in which case we have 1 5 X, or as the summing of a series, (the number of times) in which case we have 55 X 1. There is no arithmetical unit which is not both the multiple of some other unit, and the fractional part of some whole.

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Translating the psychological terms over into arithmetical:The whole or individual taken as made up of parts (the synthesisanalysis) (1) is a definite number, a dozen bananas, five fingers, or whatever. (2) Every such number involves an analytical part into which it may be resolved and out of which it is composed (synthesized). This part, in arithmetical terms, is the unit, that which measures the value of the whole. (3) There is the synthetic act of putting together the parts, or, in arithmetical language, the taking the unit a certaih number of times.

I hope I have now shown (I) what number, as a psychological fact, is:- viz., the ordinary and indispensible act of synthesis-analysis made more definite and accurate; (II) that, therefore, number, as grasped by the mind, is not a thing or a mere quality abstracted from a thing, but is the activity or process of distributing a whole into its parts, and, at the same time, of combining these parts into the whole; and (III) that the necessary factors of the definition of number, namely, (a) an integer, or whole, (b) a unit as standard of reference, or as measure, and (c) value, or how many parts, (repetition of unit a number of times,) are absolutely necessary results from the psychological process.

In a succeeding article I wish to show how the various propertieg of number, addition, multiplication, ratio, and the fundamental operations are all simple developments of this psychological activity, and are, therefore, used from the outset when the method here set forth is followed.

In closing, I wish to guard against one error, though I hope it is not necessary to do so. I do not mean that the child is to be taught the various terms, standard unit, value, etc., involved in the defini. tion of number. What I do mean is that those methods must be used which shall bring out the essential factors of number, and their true relation to each other according to the psychological nature of number, making the child feel them by getting the habit of using them in the right way. Such methods will be used whenever the teacher knows the psychology of number.

Fewer Children in a Room.

It is obvious that the young woman with fifty-six pupils before her is attempting what no mortal can perform. I suppose it is pra ticable for one young woman to hear the lesson out of one book of all of fifty children before her during the hours of the school session, and keep a certain amount of watch over the children who are not reciting their lessons, providing the grading is almost perfect, and we are going to be satisfied with "uniform" results. But the new teaching is of quite a different character. It requires alertness, vitality, and sympathetic enthusiasm. President Eliot.

METHODS

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

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Making Rules Plain.

By ADDA P. WERTZ, Minneapolis, Minn.

O the average pupil, a rule means but very little. It seems merely a task to be learned rather than a principle to be applied.

For example: you may have a class in Mensuration; have you, by models, shown them that the length multiplied by the breadth actually equals the area?

If you have never followed such a plan, try it. You will never regret the effort, and though the children may forget the rule, they will not forget your models, and will apply your teaching. What more can you ask!

Fig. III. is a triangle whose area is to be found by multiplying the base (3 in.) by the altitude (2 in.). Why this rule?

Cut the model of common wrapping paper, divide it at the dotted line by placing the portions of the oblique line, side by side, you have resolved it into an oblong. The length of this oblong is the same as the base of Fig. III. The width of the oblong, the same as the altitude of Fig. III. Its area is the length (base) multiplied by the width († the altitude) or 3x1 = 3 sq. in.

Fig. 4.

Fig. IV. resolves itself into an oblong, thus :

Fig. 6.

Cut the Fig. VII. by the dotted lines and arrange them as in VIII, and you approximate an oblong whose width corresponds to the radius of VII. and whose length is the same as the semi-circumference of VII.

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Fig. 7.

One lesson during which the teacher makes and exhibits models. One, in which the pupils cut the same from paper. One more, in which the circle is re-arranged by teacher and pupils.

The way is then clear. No more trouble with areas. The problems in the text become the mere practice examples for which they were intended.

The omnipresent slow boy may not understand; begin at the starting point and tell it again. Do not undertake to patch an explanation, you do so at the risk of bewildering others in the class.

Preparatory to work of this kind, use the words length, width, breadth, base, altitude, hypothenuse, circumference, semi-circumference, radius, in connection with language, in sentences, in drawing, or spelling, until they are a fixture in the child's vocabulary.

"I had 6 nuts and I gave away 5 to Bessie, and only kept 1 myself." Self righteousness is a sin that children ought to be kept free from.

3.Questions where prices are inaccurate or the statement improbable.

An observer heard this answer to a problem: "Price of the cow 6 cents! To another question the answer was that the boy earned three dollars a day picking berries. Attention paid to truth in every little thing will be sure to influence for truth of character. Prices might sometimes be left to be filled in according to the children's own ideas or from a blackboard list of present market - Primary Education.

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Arithmetic.

By WM. M. GIFFIN, Cook Co. Normal School, Chicago.

ET me begin this article by thanking Mr. Elwood for his ex.

cellent paper "Rules in Arithmetic," which will hold water,'' and not only that, but which is backed up by all the laws of psychology known to man.

The great trouble with most work in arithmetic is, that the fellow who makes the book has all the fun and delight of discovery, while the pupils are simply stuffed with his discoveries.

From the first in all the work in arithmetic, the children should be made to observe and be given independent effort and healthful mental activity. This can be done by dividing the subject of arithmetic into eight parts or subjects, viz (1) Lines; (2) Area; (3) Volume; (4) Bulk; (5) Weight; (6) Time; (7) Values; (8) Single Things.

First, Lines.

The first year pupils can be made happy by such exercises as the following:

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Exercise One.

How long is A? How long is B? How long is C? How long is D?

Mo'ive. To necessitate the use of the ruler by the children.
Exercise Two.

1. How long are A and B together?

2. How long are A and C? B and C? B and D? A, B and C? B, C and D ?

Motive. To create a want in the mind of the child. He knows A, B, C and D. He has discovered some truths about them. Now he is made to want some more information and to find it he must add something to something.

Exercise Three.

1. A equals what part of B? A equals what part of C. Of D? 2. B eqnals what part of C? Of D?

3. A equals what part of B plus C? Of C plus D?

4. What part of B equals A? What part of C equals A? Of D? 5. What part of D equals B?

Motive. To necessitate close observation and comparison. The comparison is the foundation of all future work, and the child can not be given too much of it.

Exercise Four.

1. How many A's can be made of B? How many A's can be made of C? Of D?

2. How many B's can be made of D? Of C?

3. How many A's can be made of B plus C?

Motive. To give the child his first lessons in division through things rather than symbols.

Exercise Five.

1. If from B a line the length of A is taken, how much of B is left?

2. If from C? If from D?

3. If a line the length of B is taken from C, how much of C is left? If from D?

Motive. To teach subtraction.

Exercise Six.

1. How long are two A's? Two B's?

Exercise Seven. (See Fig. 1.)

1. Find the length of A B without measuring.

2. Find the length of A H without measuring.

3. What is the perimeter of Fig. I. ?

4. B C equals what part of A H? Of A B?

5. D C equals what part of HG? Of A B?

Motive. To necessitate the drawing of an inference by the pupils. How few, alas, can do so! To cause the child to be selfreliant, and to search out his own premiss. To lay a foundation for his future work in geometry.

These are but a few of the thousands of excellent exercises that can be had in this interesting subject which has been so badly neglected in the past. Exercises adapted to all the primary and grammar grades can be given in each of the eight subjects.

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Fractions.

[Continued from January.]

ANNA A. DE VINNE.

HEN the children have had plenty of practice in writing fractions, they are ready to be taught the meaning of the terms numerator and denominator.

Cut an apple into four equal parts. Hold up one part. Let the children name it and write it. Divide a piece of chalk into thirds, and a sheet of paper into fifths. Take one piece of each; have them name and write on the blackboard the fraction it represents. "We have now before us&. What does the four tell us about the unit?" You will have no difficulty in getting the answer: That the unit has been divided into four equal parts." Proceed in the same manner with the thirds and fifths.

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Let the children make fractions for themselves by cutting up pieces of paper, and naming the parts until they fully understand that the number below the line designates the number of equal parts into which the unit has been divided; also that it names the fraction. As the word denominator will be new to them, do not give it at once. Take something from their own experience and lead up to the word. "How many children know what we say when we name a man for president?" The answer will readily be given, "We nominate him." Then to nominate means to name, and if we put the prefix de meaning from, before it, we have the word, denominate, which means to name from. "This number below the line is named from the divisions of the nnit, and is called the denominator or namer, so that in the fraction the denominator is two and the denomination or name of the parts, halves." The definition having been developed, let the children give it as you write it, and have them learn it.