Approximate Answers.

An important drill is furnished in the "approximations." (See Arts. 521, 669, 719, etc.) Pupils should be required in much of their written work to estimate the result before beginning to solve a problem with the pencil. Besides preventing an absurd answer, this practice will also have the effect of causing a pupil to see what processes are necessary. In too many instances, work is commenced upon a problem before the conditions are grasped by the youthful scholar; which will be less likely to occur in the case of one who has carefully "estimated" the answer. The pupil will frequently find, also, that he can obtain the correct result without using his pencil at all.

Indicating Operations. It is a good practice to require pupils to indicate by signs all of the processes necessary to the solution of a problem, before performing any of the operations. This frequently enables a scholar to shorten his work by cancellation, etc. In the case of problems whose solution requires tedious processes, some teachers do not require their pupils to do more than to indicate the operations. It is to be feared that much of the lack of facility in adding, multiplying, etc., found in the pupils of the higher classes is due to this desire to make work pleasant. Instead of becoming more expert in the fundamental operations, scholars in their eighth year frequently add, subtract, multiply, and divide more slowly and less accurately than in their fourth year of school.

Paper vs. Slates. To the use of slates be traced may much very of the poor work now done in arithmetic. A child that finds the sum of two or more numbers by drawing on his slate the number of strokes represented by each, and then counting the total, will have to adopt some other method if his work is done on material that does not permit the easy obliteration of the tell-tale marks. When the teacher has an opportunity to see the number of attempts made by some of her pupils to obtain the correct quo

tient figures in a long division example, she may realize the importance of such drills as will enable them to arrive more readily at the correct result.

The unnecessary work now done by many pupils will be very much lessened if they find themselves compelled to dispense with the "rubbing out" they have an opportunity to indulge in when slates are employed. The additional expense caused by the introduction of paper will almost inevitably lead to better results in arithmetic. The arrangement of the work will be looked after; pupils will not be required, nor will they be permitted, to waste material in writing out the operations that can be performed mentally; the least common denominator will be determined by inspection; problems will be shortened by the greater use of cancellation, etc., etc. Better writing of figures and neater arrangement of problems will be likely to accompany the use of material that will be kept by the teacher for the inspection of the school authorities. The endless writing of tables and the long, tedious examples now given to keep troublesome pupils from bothering a teacher that wishes to write up her records, will, to some extent, be discontinued when slates are no longer used.

Counting.

III

EARLY ARITHMETIC TEACHING

While the majority of children are able, upon entering school, to repeat the names of the first ten or more numbers, they are not always able to count things. The first duty of the teacher is to secure correct notions of the first nine numbers, and this can best be done by the employment of objects, such as beans, splints, shoe-pegs, blocks, etc. A numeral frame is very useful for this purpose.

In counting, it is very important to have the child understand that the second splint is not two splints. This may be made clear to a child by having him put on his desk one bean, then near it two beans, three beans in another place, etc. After the pupil can count understandingly to nine, he should be taught the figures. The notation and numeration of numbers of two or more figures will be discussed in later chapters.

Primary Arithmetic. After children have learned to count readily, experts disagree as to the best method of procedure. Many excellent teachers believe that work should be commenced at once upon numeration and notation, followed by the fundamental operations in the usual order. Some of the advocates of this method favor the completion of each topic before proceeding to the next; that is, numeration and notation are taught at least to billions; then addition is taken up, beginning with small numbers and gradually increasing to examples containing numbers of eight or nine figures. Subtraction, multiplication, and division are each studied to this extent before the next is commenced.

The more intelligent advocates of teaching operations at the

outset, recognize the fact that it is neither necessary nor advisable to defer the addition of small numbers until children are able to write those of three or more periods, nor to postpone finding the sum of and until after the properties of numbers have been studied in the fifth school year. Their plan is to follow such simple examples in the addition of small numbers as involve no carrying, by corresponding ones in subtraction. More difficult examples in both of these operations come next, followed by simple ones in multiplication and division. Easy work in fractions is introduced at an early stage, and problems involving the more common denominate units are brought in from time to time.

The Grube Method. A growing number of educators believe that early arithmetical instruction should be based upon the study of numbers, rather than upon that of processes, that the former should be the prominent feature of the early instruction, and the latter incidental, at least for the first two years.

This method, called after its inventor, Grube, requires the teaching of all of the processes in the case of each number before proceeding to the next. Thus, when the number 4 is studied, the pupil measures it by all numbers smaller than itself. Using 4 beans, he measures by 1, by arranging them as follows: 0 0 0 0. In this way he sees that 1+1+1+1=4; that there are 4 ones in 4, or 1 x 4 = 4; that 4—1-1-1=1; that 4÷1=4.

Measuring by 2, 00 00, he sees that 2+2=4, 2×2=4, 4-2-2, 4÷ 2 = 2.

Measuring by 3, 000 0, he sees that 3+1=4, 1+3=4; 4-31, 4-1=3; that (1 x 3) + 1 = 4, and that 4÷3=1 and 1 over.

The pupil then answers questions given by the teacher, first using the counters and afterwards without them:

Four is how many more than 3? Than 1? Than 2? Three is how many less than 4? Two is how many less? One is how many less?

How many ones in 4? How many twos? Threes? One-half of 4 is what? Two is of what number?

Problems containing the foregoing combinations are then given. in great variety by the teacher until all of the facts about the number 4 in its relation with the smaller numbers are fully mastered.

In teaching any number, no larger number must appear in any way whatever. During the study of 4, it is not permissible to ask 4 twos, or that 4 is 1 less than what, etc., etc.

The work proceeds slowly and thoroughly, at least a year being devoted to the numbers below 10. The second year is given to the numbers from 10 to 20, and the third year to those from 20 to 100.

in this country.

This is probably as far as the method is carried

In the greater number of the schools using this method, systematic instruction in the fundamental processes is commenced by the beginning of the third year; while in some, the Grube method is used for oral work, and the teaching of slate addition is carried on at the same time, even during the first year.

Slate Problems. When, instead of receiving oral instruction for some time, children are taught processes from the outset, it frequently happens that many of them show little ability in solving problems. While some attention should be given in the early years to this side of arithmetic, it should not be permitted to retard too much the advancement of pupils. Many of them have to leave school soon, and they should be taught as rapidly as is consistent with real progress to perform accurately the ordinary operations in whole numbers, simple fractions, and decimals. Being familiar with these tools, greater maturity will, of itself, show which is to be used in such questions as are likely to come up in ordinary avocations.

The teacher should exercise much care to give only such problems as can readily be understood by the pupil, and which do not contain too many conditions or numbers that bewilder