ARITHMETIC 41 bers be selected at first, as will cause difficulty plicable the principle involved in this may be. in education there is really no need of it. If the demands of actual life are so meager, that we must make a large part of our discipline in arithmetic consist in unraveling such manufactured puzzles, is it not well to ask the question whether there are not other branches of science which will afford the needed discipline by dealing with the actual and useful, instead of wasting time and strength on the purely fictitious? The arithmetics of to-day, however, are a great advance, in this respect, on those in use fifty years"Mr. A put out $759, on 7 per cent interest; ago; but no editor of a text-book on arithmetic has yet felt at liberty to cut out entirely these superfluous problems. Undoubtedly, the demands of science and of business life furnish abundant resources in this direction; but these more abstruse problems do not fall within the purview of an elementary course, nor come within the demands which actual life makes upon the great majority of persons. There are a great number and variety of intricate questions which do actually arise in discounting negotiable paper, as well as in the abstruse questions which insurance and annuities present; but it is not the aim of our elementary courses to train pupils for such specialties; and when in any properly co-ordinated course of study such topics are reached, their solution will then come in the regular line of the application of general principles, and the stu lent will have acquired sufficient maturity to comprehend the business, economical, or political relations which give rise to them. What should constitute the course in arith metic. In the first place, there should be a thorough unification of the processes of mental and written arithmetic. There is but one science of arithmetic; and every thing that tends to pro duce the impression in the pupil's mind that there are two species, the one intellectual and the other mechanical, is an obstacle to his true progress. What is valuable in the methods now peculiar to mental arithmetic, needs to be thoroughly incorporated with what is practically convenient or necessary in written arithmetic; so that the whole may be made perfectly homogeneous. The basis upon which this is to be effected is, that principles should be discussed first by the use of small numbers which can be easily held in the mind, and which do not render the difficulty or labor of combination so great as to absorb the attention, or divert it from the line of thought; and that we should pass gradually, in applying the reasoning, to larger numbers and more difficult an 1 complex combinations, in which pencil and paper are necessary. The rationale should be always the same in the mental (properly, oral) arithmetic and in the written, pencil and paper being used only when the numbers become too large, or the elements too numerous, to render it practicable to hold the whole in the mind. For example, suppose the pupil to be entering upon the subject of percentage. The first step is to teach what is meant by per cent. In order to this, small numbers will be used, and the process will not require pencil and paper, nor will such num 64 what was the interest for a year?" After the principle to be taught is clearly seen, larger numbers should be introduced, and such as require that the work be written. But the same style of explanation should be preserved; and great care should be taken to have it seen that the method of reasoning is the same in all cases. To illustrate still farther; as, in practice, the computer ordinarily uses the rate as the multiplier, the form of explanation, when the whole is given orally, should be adapted to this fact. At first, such an example as the first above will naturally be solved thus: "If Mr. A lost 5 sheep out of 100, out of 3 hundred he lost 3 times 5, or 15 sheep." But before leaving such simple illustrations, the reasoning should take this form: "Since losing 1 out of 100 is losing .01 of the number, losing 5 out of 100 is losing .05 of the number. Hence, Mr. A lost .05 of 300 sheep, which is 15 sheep." Thus, in all cases, the form of thought which will ordinarily be required in solving the problem, should be that taught in the introductory analysis. A farther illustration of this is furnished by reduction. At first, the question, "How many ounces in 5 lb.?" will naturally be answered, Since there are 16 oz. in 1 lb., in 5 lb. there are 5 times 16 oz., or 80 oz." But in practice the 16 is ordinarily used as the multiplier, and it is better that the introductory (mental) analysis should conform to this fact. Hence, the pupil should be led to see, at the outset, that, as every pound is composed of 16 ounces, in any given weight there are 16 times as many ounces as pounds; and he should be required to analyze accordingly. Apart from this use of what are called mental processes, there is no proper well-defined sphere for their employment. In practical applications, it is quite unphilosophical to classify the examples, by calling some mental and others written. We do not find them so labeled in actual business life. The pupil needs to discriminate for himself as to whether any particular example should be solved without the pencil or with it. It should also be borne in mind that business men rely very little upon these mental operations. They use the pen and paper for almost every computation. In the second place, in constructing our course in arithmetic, we need to give the most careful attention to the condition and wants of the youth found in our public schools. Perhaps it is no exaggeration to say, that from eighty to ninety per cent of the pupils disappear from these schools by the close of the seventh school year; and not more than one in one hundred takes a high school course. Since all pupils of the common schools have need of the rudiments of number, as counting, reading and writing small numbers, the simple combinations embraced in the addition, subtraction, multiplication, and division tables, the simpler forms of fractions, and the more common denominations of compound numbers, an elementary text-book is deemed to be needful for many schools. The objections often urged to having these primary lessons entirely oral are, that it makes an unnecessary draft upon the time and energy of the teacher, renders the pupils' progress very slow, does not so readily supply the means of giving them work to do in their seats, and more than all, begets in their minds a dislike for study and self-exertion, and a disposition to expect that the teacher must do all the work, and thus carry them along. But whatever disposition may be made of primary arithmetic, as usually understood, there is an imperative demand that the course in arithmetic for the masses should be so arranged that the more important practical subjects can be reached and mastered by a majority of our youth during the comparatively short time which they can spend in our schools. In order to effect this, three things will be found necessary: (1) a rigorous exclusion of all topics relatively unimportant, (2) a judicious limitation of the topics presented, and (3) care that, in the laudable desire to secure facility in fundamental processes, -adding, multiplying, etc., the teacher does not consume so much time that the great mass of the pupils will never advance beyond the merest | rudiments of the subject. The range of topics to be included in the common school course, will be the fundamental rules; common and decimal fractions; denominate numbers (care being taken to reject all obsolete or unusual denominations, and to give abundant exercises calculated to insure a definite conception of the meaning of the denominations); percentage, including simple, annual, and compound interest, with partial payments, common and bank discount, and some of the more common uses of percentage. If, after this, the course may be extended, the next subjects in importance are ratio, proportion, and the square and cube roots. Much more than this cannot be embraced in a course which the masses of our youth are able to master; and in treating these, constant care will be necessary to introduce problems which occur in actual life, and as far as possible to exclude all others. Something of common mensuration should be introduced in connection with the tables of measures of extension; and the more common problems in commission, insurance, taxes, stocks, etc., will be readily introduced in percentage without occupying either much space or time. For the few who can take a more extended course, a thoroughly scientific treatment of the subject of arithmetic is desirable: and this quite as much for its disciplinary effect, in giving breadth and scope to the conceptions, and inlucing a disposition to systematize and gener alize, and thus to view truth in its relations, as for the amount of mere arithmetical knowledge which may be added to the pupil's stock. Here we may introduce an analytical outline of the subject, presenting the topics in their philosophical relations, rather than in their mere practical and economic order and connection. Thus, in treating notation, the various forms of notation can be introduced, as of simple and compound numbers, other scales than the decimal, various forms of fractional notation, the elements of the literal notation, etc. Then, as reduction is but changing the form of notation, this topic will come next, and will embrace all the forms of reduction found in common arithmetic, as from one scale to another, of denominate numbers, of fractions common and decimal, etc., showing how all arithmetical reductions are based on the one simple principle: If the unit in reference to which the number is to be expressed is made smaller, the number must be multiplied, and if the unit of expression is made larger, the number must be divided. Passing to the combinations of number, under addition all processes thus designated in arithmetic will be treated, and the general principles out of which they all grow will be developed. In this method of treatment, the pupil will not find himself merely going over the clementary subjects through which he plodded in the days of his childhood,but new ranges of thought will be presented, at the same time that all the principles and processes of the elementary arithmetic are reviewed; the very first sections, even those on notation, reduction, and the fundamental rules, bringing into requisition most of his knowledge of arithmetic, and giving vigorous exercise to his mind in grasping new truth. in addition to all this, which pertains to the method of presentation, there will be much of practical arithmetical knowledge to be gained. in the business rules, discount needs a much fuller treatment than it has usually received in any of our text books. Many problems, of frequent occurrence in modern business circles, are not provided for in these books; and, in fact, some of the most common have had no solution at all which has been made public. The wonderful development of the insurance business demands that its principles and methods receive a much fuller treatment than they can have in an elementary course this is especially true of life insurance. Foreign exchange, customs, equation of payments, etc., are other topics suitable for this advanced course, which are quite impracticable in an elementary course within the reach of the masses. Two other ends will be subserved by this method: (1) It will be a leading purpose to teach the pupil how to investigate, and to this end he should be put in possession of the great instrument for mathematical investigation, namely, the equation. Of course, only the simpler forms of the equation can be introduced; nevertheless, enough can be given to enlarge very greatly the student's power to examine new questions for himself. By means of the equation, he may be taught the solution of such problems as the fol But ARITHMETIC lowing, which would be quite out of his reach without this instrument: To find what each payment must be in order to discharge a given principal and interest in a given number of equal payments at equal intervals of time. To find the present worth of a note which has been running a certain time, and is due at a future time, with annual payments on the principal, and annual interest; so that the purchaser shall receive a different rate of annual interest from that named in the note. These and many other important business problems are quite within the reach of the simple equation, and are scarcely legitimate questions to propose to a student who has not some knowledge of this instrument. (2) The second general purpose which we shall mention as being subserved by this course is, that by grouping all the arithmetical processes under the fewest possible heads and showing their philosophic dependence, the whole is put in the best possible form to be retained in the memory. Thus, if it is seen that a single principle covers all the cases in reduction, that another simple principle covers all the so called "problems in interest," that all the common intricate questions in discount are readily solved by the simple equation, etc., these processes will not be the evanescent things which they have often been. 43 principles by the pupil so that he can state them in a general way, and (3) a careful and continued repetition of them in the class, in application to particular examples, will secure the first of these general ends of arithmetical study. To secure the second, namely, facility and accuracy in applying these principles, so as to be able to ald with ease, rapidity, and accuracy, long continued drill, with the mind quite unencumbered by any thought of the reasons for the processes, will be indispensable. It will not be sufficient that pupils solve accurately numerous examples, in the slow plodding way to which they are accustomed in their private study, but large numbers of fresh problems should be furnished in the class, which the pupils should be required to solve with the utmost promptitude, and with perfect accuracy. In respect to all mere numerical combinations, as addition, subtraction, multiplication, division, involution, evolution, etc., oral drills like the following will be of the greatest use and should be continued until the combinations can be made as rapidly as we would naturally read the numbers: Teacher repeats while the pupils follow in silence, making the combinations, "5+3÷2*+3, squared, -7÷7x 3+7, square root, etc." These oral drills may be commenced at the very outset in regard to addition, and extended as the other rules are reached, and should not be dropped until the utmost facilPrinciples and maxims to be kept in view ity is secured. A similar drill exercise can be while teaching arithmetic. I. There are two secured by pointing to the digits as they stand on distinct and strongly marked general aims in the board, or on charts, and simply speaking the arithmetical study: (1) To master the rationale of words which indicate what combinations are the processes, and (2) To acquire facility and ac- required. Any figures which may chance to curacy in the performance of these operations. stand on the board may be used in this way to The means which secure one of these ends are not secure an indefinite amount of most valuable necessarily adapted to secure the other. Thus, to drill. This latter exercise, making the combinasecure the first, for example, in reference to ad- tions at sight - is of still greater practical value dition, the steps are, learning to count, learning than the former, in which the ear alone is dehow numbers are grouped in the decimal system, pended upon; for it is a singular fact that learning how to make the addition table, and, facility in one method does not insure it in the finally, by means of a knowledge of the sum of other, and the latter is the form in which the the digits taken two and two, learning to find the process is usually to be applied. Again, in the sum of any given numbers. In regard to the business rules the principles underlying the prolatter process, the pupil needs to know why we cesses must be clearly perceived. and the pupil, write units of a like order in the same column, by continued practice in explaining solutions why we begin at the units' column to add, why we written upon the board, must become able to give carry one for every ten," as the phrase is, etc. in good language the reason for each step. But But all this may be known, and yet the pupil when all this is secured, there will be found need make sorry work in practical addition. In order of much drill on examples to the answers of to secure a knowledge of the rationale, each step which he cannot have access, and which he must needs to be clearly explained and fully illustrated. take up and solve at the moment. In this departand then the pupil must be required to repeat the ment, much valuable exercise may be given by whole, "over and over again," in his own language. handing the pupils written notes or papers in due For this purpose, much class drill on the black- form, and requiring them to compute the inboard, in having each pupil separately explain interest. or discount, or make the required comdetail the reasons for each step of the work which he has before performed, will be necessary. Pupils may be required to bring into the class practical exercises solved on their slates, and then sufficient time be given to explanation from the slates. These three things repeated in about the same way, (1) a clear preliminary explanation of principles either given in the text-book or by the teacher, (2) a thorough mastery of these putation at sight. But the illustrations now given will suffice to show that there are, as above stated, two general purposes-the theoretical and used with strict propriety in this specimen exercise; they are applied to the result of all the preceding operations in each case as though all before them had been included in a parenthesis. Thus in this case it is 53, or 8 which is meant to be divided by 2, giving 4, to this 3 added, giving 7, this squared, giving 49, etc. *The signs of division, multiplication, etc., are not the practical-which must run parallel through all good teaching in arithmetic, and that they are generally to be attained by different means. II. In order to realize the above, a careful discrimination needs to be made between simply telling how a thing is done, and telling why it is done. Very much of what we read in our text-books, and hear in class-rooms, under the name of analysis, in explanation of solutions, is nothing more than a statement of the process a telling how the particular example is wrought. This vice is still so prevalent as to need the clearest exposition and the most radical treatment. Indeed, it has become so general as to be mistaken by the masses for the thing it purports to be; and pupil and teacher frequently seem to think that this parrot-like way of telling what has been done is really a logical exposition of the principles involved. The following example, clipped from a book not now a candidate for popular favor, will serve to illustrate our meaning: ".0017)36.3000(21352 34 23 17 60 51 69 90 Commencing the division, we find that 17 is contained in 36, 2 times. We place 2 in the quotient, and subtract 2 x 17 from 36. The remainder is 23. 17 is contained in 23, 1 time. Place 1 in the quotient, and subtract 1 × 17 from 23. To the remainder 6 we annex one of the Os, and find that 17 is contained in 60, 3 times with 9 re16 mainder. We continue this process, annexing to each remainder a new figure of the dividend, until we find a final remainder 16, which does not contain 17, but the division by 17 may be expressed by writing the divisor underneath." 85 50 34 Compare this with the following: Reasons for the Rule in Short Division.The divisor is written at the left of the dividend, simply that we may be able to see both at once conveniently. We begin at the highest order to divide, because by so doing we can put what remains after each division into the next lower order and divide it; and thus we get all that there is of any order in the quotient as we go along. We write the quotient figures under the orders from whose division they arise, because they are of the same orders. The process ascertains how many times the divisor is contained in the dividend, by finding how many times it is contained in the parts of the dividend and adding the results. This may be readily illustrated by an example. For this purpose let us divide 1547 by 4. The following is an analysis of the operation: 1547 equals 12 hundreds, 32 tens, 24 units, and 3 units; III. There should, also, be a careful discrimination between pure and applied arithmetic, in order that they may be so taught as to secure the proper end of each. Pure arithmetic is concerned solely with abstract numbers, and the breadth of discipline to be secured by its study is not great; but the applications of arithmetic are almost infinitely varied, and give a far wider scope for mental training. In the latter, the questions are not how to multiply, add, subtract, etc., but why we multiply, add, or subtract. Thus, in solving a problem in interest, it would be quite out of place to cumber the explanation with an exposition of the process of multiplying by a decimal, but it is exactly to the purpose to give the reason for so doing. The most important object in applied arithmetic is to acquaint one's self so thoroughly with the conditions of the problem-if in business arithmetic, with the character of the business-as to discern what combinations are to be made with the numbers involved. Many of these applications are quite beyond the reach of the mind of a mere child. Thus, to attempt to explain to very young pupils the commercial relations which give rise to the problems of foreign exchange, or the circumstances out of which many of the problems in regard to the value of stocks grow, would be perfectly preposterous. IV. In teaching applied arithmetic, it is of the first importance that the problems be such as the usual phraseology be employed. For example, occur in actual life, and that in expressing them, compare the following: (1) What is the present worth of $500 due 3 yr. 7 mo. 20 da. hence, at 6 per cent per annum? 1873, and due July 10th, 1876. Mr. Smith proposes to (2) I have a 7 per cent note for $500, dated Feb. 6th, buy it of me Sept. 18th, 1874, and to pay me such a sum for it as shall enable him to realize 10 per cent In other words, what is the present worth of this note per annum on his investment. What must he pay me? Sept 18th, 1874? rarely, if ever, occur, and even disguises that. The first supposes a transaction which could the ordinary way, if asked, "What interest does Most pupils who have gone through discount in the $500 bear, in the first example?" would answer, "6 per cent." Of course, it is understood that the money is not on interest. Moreover, we find no such paper-no notes not bearing interest-— in the market. Again, the assumption seems to be that the note-if even a note is suggested at all-is discounted at the time it is made. Thus, it is obvious that the first form is calculated to give the pupil quite erroneous impressions; whereas the second brings a real transaction into full view. ARITHMETIC ARIZONA propositions, and statements of processes may be given first, and illustrated, demonstrated, or applied afterward. (See ANALYTIC METHOD, and DEVELOPING METHOD.) ARIZONA was organized as a territory Feb. 24th, 1863, being formed from the territory of New Mexico. Its area is 113,916 square miles; and its population, excluding tribal Indians and military, in 1870,was 9,581. 45 V. From the beginning to the end of the course, in which the faculties chiefly exercised are obserit should be the aim to teach a few germinal prin- vation, or perception, and memory, and in which ciples and lead the pupil to apply them to as great the pupil is not competent to formulate thought, a number of cases as his time and ability may or to derive benefit from abstract, formal statepermit. Thus, at the very outset, a good teacher ments of principles, definitions, or processes; will never tell the child how to count; but hav- (2) An intermediate stage, in which the reasoning taught him the names of the numbers up to ning faculties (abstraction, judgment, etc.) are fourteen, will show him the meaning of the word coming into prominence, and in which the pupil fourteen (four and ten); then he can be led to go needs to be shown the truth, so that he may have on to nineteen by himself. No child ought to be a clear perception of it, before he is presented told how to count from fifteen to nineteen; and with a formal, abstract statement, the work, howafter twenty, he needs only to be shown how the ever, not being concluded until he can state the names of the decades, as twen-ty, thir-ty, for-ty, truth (definition, principle, proposition, or rule) and fifty are formed, to be able to give the rest intelligently, in good language, and in general himself; nor does he need to be told how to count (abstract) terms; (3) An ultimate stage, or that through more than one decade. In reference to in which the mental powers are so matured and the fundamental tables, it may be suggested that trained, that the pupil is competent to receive no pupil should be furnished with an addition, truth from the general, abstract, or formal statesubtraction, multiplication, or division table ready-ment of it. At this stage, definitions, principles, made. Having been taught the principle on which the table is constructed, he should be required to make it for himself. As preliminary to practical addition and subtraction, the combinations of digits two and two which constitute any number up to 18 (9+9) should be made perfectly familiar. Thus the child should recognize 1+4, and 2+3, as 5; 1+5, 2+4, and 3+3, as 6; etc.; and this should be made the foundation of addition and subtraction. He should be taught, that if he knows that 3+4=7, he knows by implication that 23+4=27, 33+4=37, etc. Passing from the primary arithmetic, he should be taught common fractions by means of the fewest principles and rules consistent with his ability. Thus in multiplication and division, To multiply or to divide a fraction by a whole number, and To multiply or to divide a whole number by a fraction, are all the cases needed; and these should be taught in strict conformity with practical principles. Thus, to multiply a whole number by a fraction is to take a fractional part of the number; and to divide a number by a fraction is to find how many times the latter is contained in the former. To cover all the forms of reduction of denominate numbers, nothing is needed but the principle or rule, that to pass from higher to lower denominations, we multiply by the number which it takes of the lower to make one of the higher; and to pass from lower to higher we divide by the same number. These simple principles should be seen to cover all cases, those involving fractions as well as others. In like manner, by a proper form of statement of examples, and an occasional suggestion or question, most of the separate rules usually given under percentage may be dispensed with. In dealing with the cases usually denominated problems in interest, all that is needed is the following brief rule: Find the effect produced by using a unit of the number required, under the given circumstances, and compare this with the given effect. This should be made to cover the cases usually detailed under six or eight rules. VI. There are three stages of mental development which should be carefully kept in view in all elementary teaching: (1) The earliest stage, Educational History.-An act was passed by the territorial legislature in October, 1863, authorizing the establishment of common schools; and the next year, another and more complete law was enacted. Nothing, however, of any importance was accomplished toward the establishment of a system of common schools in the territory until the appointment of A. P. K. Safford as governor in 1869. Through the most laborious efforts on his part, a public opinion in favor of common schools was awakened among the people; and in consequence thereof, a law was passed in 1871, which levied a tax for the support of schools, of ten cents on each one hundred dollars of the taxable property of the territory, and authorized the supervisors of counties and the trustees of the school-districts to levy additional taxes for the establishment and maintenance of free schools in their respective districts. By this law, the governor was made ex officio superintendent of public instruction, and the judges of probate, county superintendents. It was not until 1872 that, in pursuance of these provisions, schools were established. In July of that year, the governor stated that "a free school had been put in operation in every school-district where there was a sufficient number of children." The larger portion of the children, he further stated, "were of Mexican birth, and few could speak the English language; but they had been taught exclusively in English, and had made satisfactory progress." In 1873, the total school population between the ages of 6 and 21, was reported as 1,660, of whom 836 were males, and 824 females. Of these there were only 482 attending public and private schools, the former, 343. The whole amount paid for school purposes was $11,060. In February, 1873, the |