ability to distinguish chair by the eye with great quickness, when mixed up with other forms. By one device or another the eye must be kept busy with it until this end is reached. The next thing in order is to impress the word on the memory by impressing it further on the eye, ear, understanding, etc. The means of doing this are comparison and contrast of form and meaning, speaking and writing, and other forms of building up and tearing down the word. The test is the ability to remember it quickly and readily, and of course to recognize and write it. The last part of the work is the use of the word with other words of the reading lesson. These must be recognized singly before they are to be attempted together. The, dog, is, and black, must be learned before The dog is black. Connected words also need drill.

ARE "TIMES" AND "CONTAINED TIMES"

LOGICALLY CORRECT.

In such expressions as "3 times 4 are (or equal) 12" and "4 is contained 3 times in 12," two questions have been raised; First. Are times and contained times correct? Second. Whether correct or not, should we supplant them by others more easily understood, especially by primary children? Certain critics and progressionists impeach the logical correctness of the expressions. There are two different sects of these reformers of arithmetical nomenclature. Each looks at the language of number through a lens of its own make. The first would say "3 fours are 12," "there are 3 fours in 12," or "12 may be divided into 3 parts or groups of 4 each," etc. This is not new until such expressions as "" 3 four dollars are 12 dollars" is reached. This means that three numbers of the value of four units each equal one number of 12 units, or it means that the number 12 (a complex unit) may be considered as made up of 3 parts, each of which has the numerical value of 4 units. That these fours are dollars has nothing to do with the relations between the 4's and 12. Number is a relation of ideas and objects entirely independent of and distinct from other qualities and relations. For the reason that, in considering number, we do not, at any time, consider the

other attributes of objects, and sometimes ignore them entirely, some numbers are called abstract, and, indeed, it is held by some that all number is abstract. When we say 4 horses," we are utterly indifferent to any attribute, except numerical relation. They may be 4 toy horses worth 5 cents a piece, or they may be four Maud S's worth $20,000 a piece. What then is gained by discarding times and contained times? "4 times 3" means the application of the idea three to some larger conception 4 times in succession, the relation being that of whole and part. If these ideas are pictured (they may or may not be pictured definitely) there will of course be other qualities connected with numerical relation. This gives rise to the claim of other mathematicians that all number is concrete. Numbers are originally formed in the mind by constructing a series of ones, twos, threes, fours, Thus the mind can not put 7 and 5 together and make 12, without first putting I and I and I and I and I to 7 and getting 8, 9, 10, 11 and 12 respectively. So before the child can find 6 twos in 12, he must have made 12 up of twos; the same is true of threes, fours, sixes, etc. This fact is at the bottom of the use of times and contained times. Number, as has been pointed out in this department before, must pass through four stages: (1) Numbered objects (particular present things); (2) concrete ideas of particular things (things themselves absent); (3) abstract ideas of particular things (no definite pictures of definite objects); (4) dealing with symbols (letters, words, figures) without any conscious use of their ideas.

etc.

The objection to times arises in the first two of these stages. If we picture 66 3 times 4 are 12," oooo, we are told that we can not find the times, but that the child readily finds four or a group. Each one of the "groups" is one time the application of the idea four. The child can not go into any of this explanation, but he can see that one of the series above is a time as readily as he can see it a four or a group.

The other sect of nomenclaturists substitute of for times and contained times. They would say "3 of 4 are 12," meaning "3 times 4 are 12," or would say "4 of 12 are 3," meaning "4 is contained 3 times in 12." Of means group of each. There are

serious objections to a new language unless there is real need for it. From anything we have been able to see, neither of the new sets of expressions is able to show any good reason for its existence. Besides, these new expressions emphasize the wrong side of number-teaching. The failure has not been greater to have these expressions 'understood than any of the others of arithmetic-children have not understood any of them.

S. S. P.

PRIMARY DEPARTMENT.

[This Department is conducted by HOWARD SANDISON, Professor of Methods in the State Normai School.]

SPELLING-FIRST GRADE.

OR two months the child is occupied in learning words as wholes, and in this time he should not be called upon to separate words into parts, i. e., to name the letters which compose them. He should, however, be ready at any time to write the name of an object, when the object is presented. (The word having been taught, of course.)

At the beginning of the third month go back to the first word presented and teach the names of the letters. Supposing the first word to have been cat, when c, a and t are learned, bat is easily learned, for a and t are old friends, and b alone is new. The same is true of mat, rat, hat. This fact is a strong argugroups.

ment in favor of teaching words in The child should be able to write every word he knows. If he can read from the board "See my red ball," he should be able to write the sentence from dictation. It follows, therefore, that the spelling lesson proper should be written. The preparation for this lesson may be the copying of words or sentences from the board. Then, having had all work erased from board and slates, dictate short sentences. Examine each slate carefully, correcting all mistakes in capitalization and punctuation. This exercise, if continued daily for six months, will produce valuable results.

SECOND GRADE.

Two new words may be learned each day by second grade pupils, and the teacher should take good care in dictating sen. tences, to review each day the words learned on the day previous. Two new words may seem a small number, but at that rate the pupil will learn four hundred in the course of the year. Each word learned should be learned thoroughly. The pupil should know its written and printed forms, and should be able to use it correctly in a sentence, with proper capitals and punctuation marks.

After the preparation for the lesson, work should be erased from slate and board, and sentences formed by the teacher or pupils should be written on the slates. The slates should be examined daily with care, that no bad habits of penmanship be allowed to strengthen. FANNIE S. BURT.

NUMBER WORK-THIRD YEAR.

THE work in the Third Year should be concrete to a considerable degree. Among the materials which will assist in making the work concrete may be mentioned :

:

One hundred one-inch wooden cubes, which may be obtained from any carpenter for twenty or twenty-five cents; a box of educational toy money, price twenty-five cents; a box of wooden splints, which cost five cents for twenty-five hundred, or a box; a box of elastic bands; a quire of printer's paper cut into two or fourinch squares will last one or two terms and will cost twenty cents. Toy money and splints are valuable in teaching the three kinds of ones necessary to be known in this grade, i. c., a one one or a unit; ten ones, or one ten, ten tens, or one hundred. By viewing one cent as representing one one; a dime, or ten cents, as representing one ten; one dollar, or ten ten-cents as one hundred, the work,will become simple and interesting, hence easily comprehended. Splints are valuable because they furnish means for individual work.

Illustration (On the table are a box of elastic bands and a pile of splints for each child.)

Teacher. Show me twenty-two ones. Children place twentytwo one ones, side by side.

Tr. Show me twenty-two by using two kinds of ones. Children tie two ten ones together, thus having two bundles of tens and two ones.

Tr. What have you now? Children. Two one tens and two one ones. Either of the two one tens is worth ten times as much as either of the two one ones. Either of the two one ones is

worth only one-tenth as much as one of the one tens.

Each child should be able at the end of the year to make any number from 11 to 111 by thinking it in relation to its kinds of

ones.

=

By the skillful use of sticks the principle that when any term of the minuend is less than the corresponding term of the subtra hend, the minuend must be changed, etc., can be easily comprehended. Illustration: 102 13? To be read-The whole number is 102; the known part is 13; what is the unknown part? The unknown part can only be found by separating. As the parts are unequal the kind of separation is subtraction. In subtraction, first separate the ones; then the tens; next the hundreds. Two ones will not contain three as one of its parts and there are no tens to make into units, for the tens are all tied up in the hundred. In order to find the unknown part the whole must be changed by untying the hundred so that it shall become ten tens, and by making one of these tens into ones. The new

whole, nine tens and twelve ones, differ from the first whole in

form but not in value.

Twelve ones less three ones is nine ones.

Nine tens less one ten is eight tens.

The two unequal parts that make the whole are 13 and 89.

The wooden cubes are valuable in teaching the combination and separation of numbers under 100 and in teaching addition, subtraction, multiplication, and division tables. Illustration: Arrange 20 cubes in the form of an oblong prism, 5 in. by 4 in. Lead the pupils to see that by separating the prism you have