ARITHMETIC lowing, which would be quite out of his reach To find what each payment must be in order 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. Principles and maxims to be kept in view I. There are two while teaching arithmetic. distinct and strongly marked general aims in arithmetical study: (1) To master the rationale of the processes, and (2) To acquire facility and accuracy in the performance of these operations. The means which secure one of these ends are not necessarily adapted to secure the other. Thus, to secure the first, for example, in reference to addition, the steps are, learning to count, learning how numbers are grouped in the decimal system, learning how to make the addition table, and, finally, by means of a knowledge of the sum of the digits taken two and two, learning to find the sum of any given numbers. In regard to the latter process, the pupil needs to know why we write units of a like order in the same column, why we begin at the units' column to add, why we "carry one for every ten," as the phrase is, etc. But all this may be known, and yet the pupil make sorry work in practical addition. In order to secure a knowledge of the rationale, each step needs to be clearly explained and fully illustrated, and then the pupil must be required to repeat the whole, "over and over again," in his own language. For this purpose, much class drill on the blackboard, in having each pupil separately explain in detail 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 These three things repeated in about the slates. 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 them in a general way, and (3) a careful and con- *The signs of division, multiplication, etc., are not used with strict propriety in this specimen exercise; they are applied to the result of all the preceding been included in a parenthesis. Thus in this case it is 5+3, or 8 which is meant to be divided by 2, giving 4, operations in each case as though all before them had to this 3 added, giving 7, this squared, giving 49, etc. 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 90 86 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 occur in actual life, and that in expressing them, the usual phraseology be employed. For example, 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? (2) I have a 7 per cent note for $500, dated Feb. 6th, 1873, and due July 10th, 1876. Mr. Smith proposes to 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 per annum on his investment. What must he pay me? other words, what is the present worth of this note Sept. 18th, 1874? The first supposes a transaction which could rarely, if ever, occur, and even disguises that. Most pupils who have gone through discount in the ordinary way, if asked, "What interest does 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 interestin 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 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 ing 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 fif-ty 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, t Educational History.-An act was passed by the territorial legislature in October, 1863, authorizing the establishment of common schools; and was enacted. Nothing, however, of any importhe next year, another and more complete law tance 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 and in consequence thereof, a law was passed in common schools was awakened among the people; 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 addinance of free schools in their respective districts. tional taxes for the establishment and mainteBy this law, the governor was made ex officio superintendent of public instruction, and the was not until 1872 that, in pursuance of these judges of probate, county superintendents. It 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 taught exclusively in English, and had made In 1873, the total school speak the English language; but they had been satisfactory progress." population between the ages of 6 and 21, was 824 females. Of these there were only 482 atreported as 1,660, of whom 836 were males, and 343. The whole amount paid for school purtending public and private schools, the former, In February, 1873, the poses was $11,060. school law was amended, constituting the system as it now exists. School System. The governor of the territory is ex officio superintendent of public instruction, and apportions the school fund among the several counties, according to their respective school population, consisting of children between the ages of six and twenty-one years. It is made his duty to visit and inspect the schools as often as once in each year. The probate judges of the several counties are ex officio superintendents of public schools for the same. They are appointed by the governor, and hold their respective offices for two years. A tax of 35 cents on each $100 is levied in the several counties for the maintenance of schools, and a tax of 15 cents on $100 for the whole territory. The money is divided in proportion to the school attendance. Each district may levy additional taxes by a vote of two thirds of the district. Education is made compulsory; that is, parents or guardians can be compelled to send their children sixteen weeks during the year to some school, when within two miles of their residence, or have them instructed at home. Educational Condition.-The schools of Arizona are all of a primary grade; and teachers receive from $100 to $125 a month, males and females receiving an equal salary. According to the report of Gen. Safford, of Dec. 21st, 1875, there were in the territory 2,508 children between the ages of six and twenty-one, of whom 598 attended public schools. The receipts for the preceding year were $28,759.92, and the disbursements were $24,151.96. This report stated that, under the existing school law, the free school system had been made a success, and that ample means were afforded by which every child in the territory might obtain the rudiments of an education. ARKANSAS. This state was originally a portion of the territory of Louisiana, purchased from the French government in 1803. It remained a part of that territory until 1812, when Louisiana being admitted as a state, it became a part of the Missouri territory, which was organized in that year; and so continued till 1819, when it was organized as a separate territory. It was admitted into the Union as a state in 1836. Educational History.-The constitution of 1836 contained a declaration in favor of education to the effect that " as knowledge and learning, generally diffused through the community, are essential to the preservation of free government," it should be the duty of the general assembly to provide for the sale of lands donated to the state by the general government for educational purposes, and to apply the money received therefrom, to the establishment and support of schools. In accordance with this provision of the constitution, the legislature passed certain acts prescribing the manner of disposing of the school lands, which acts are, substantially, still in force. Two provisions of this law are worthy of special notice, on account of their disastrous consequences. The first was, that, upon the petition of a majority of a township, the county commissioner should sell the sixteenth section, in forty-acre tracts, to the highest bidder, one-fourth of the purchase money being payable in cash, and the balance, within eight years, in installments. The second was, that the county commissioner should loan the school moneys in his hands to parties who would give satisfactory notes to secure their payment with interest. The practical operation of the law was as follows: A, B, and C purchased a sixteenth section, say January 1st; A and B being security for C's notes for deferred payments, B and C for A's notes, and A and C for B's notes. Each party paid the school commissioner, say five hundred dollars, as his first payment, and took his receipt. The same day, they each borrowed five hundred dollars from the school fund of the county, thereby virtually borrowing from the school commissioner the money to make the first payment on the lands. The notes given were made payable in "lawful money of the United States"; but, after the secession of the state, payments were made in confederate money, and purchasers of school lands were not slow to complete their payments in that currency at par. During this period, the state auditor was the chief executive school officer, and made his report to the governor. The last school report, under the ancien régime, was made by William R. Miller, state auditor, to Governor Rector, who held office at the time of the secession of the state. In its printed form, it consisted of one leaf of a book about as large as Webster's Spelling Book, and states that there were then but two public schools in the state. Evidence from other sources shows that, by the peculiar system of financiering described above, by loss in confederate money and Arkansas war bonds, and from the usual casualties incident to a state of civil war, a very large proportion of the sixteenth-section and other school lands of the state was squandered, without creating any considerable permanent school fund. Of that which was created, the sum of $8,000, the last remnant, was invested in the purchase of medicines for the confederate troops; and the medicines were lost on a steamer which was wrecked on Brazos river, in Texas. Two provisions of the Constitution of 1868 related to public schools. Section I. of Article VI. provided that "The executive department of this state shall consist of a governor, etc., and a superintendent of public instruction, all of whom shall hold their several offices for a term of four years." Article XI. related to education, and its several sections provided, (1) that the general assembly should establish and maintain a system of free schools for the gratuitous instruction of all persons between the ages of five and twenty-one years; (2) that the supervision of such schools should be intrusted to a superintendent of public instruction; (3) that a state university should be established; (4) that a school fund should be created from the sales of school lands, escheats, estrays, grants, gifts, one dollar capitation tax, etc.; (5) that no part of the ARKANSAS school fund should be invested in the bonds of any state, city, county, or town; (6) that the distribution of the school fund should be limited to such districts as had kept a school for at least three months in the year for which the distribution was made; and that each child should be required to attend school at least three years; (7) that, in every district in which the school fund should be insufficient to support a school for at least three months in the year, the general assembly should provide by law for levying a tax; (8) that all lands, moneys, etc., held in the various counties for school purposes, should be reduced into the general school fund; and (9) that the general assembly should be empowered to raise money by taxation for building schoolhouses. In addition to these provisions, a section of the article on finance, etc., made the purchase money for school lands payable into the state treasury, and obligated the state to pay interest at the rate of six per cent per annum, upon the same. This constitution was adopted in February, 1868; and, upon the 13th day of March succeeding, an election for state officers was held, General Powell Clayton being elected governor, and Hon. Thomas Smith, superintendent of public On the 2d day of April ensuing, instruction. the first legislature under the new constitution met, and, in due time (July 23d), enacted the school law, which with certain modifications, few in number but very important in character, has ever since been in force in the state. In all, the claims of the Total.. Oct. 1, 1870. U. S. Currency. .$ 2,691.98 $58,954.95 .$22,201.37 12,991.12 $35.192.49 During this period, the school revenues were subject to depletion from three causes: (1) The taxes on sixteenth-section lands were merged into the general revenue of the state; (2) The "fines, penalties, and forfeitures," levied by the various courts, were loosely handled by the collecting officers; (3) In many cases, the electors of the various school districts refused to authorize the levying of the local tax for school-houses; and (4) by an act approved March 2d, 1869, school-taxes were made payable in interest-bearing certificates issued by the state treasurer. Notwithstanding all these obstacles, the school system was able to present, in 1870, considerable progress since the preceding year, as will be seen from the following statistics: 44 attending school " schools 66 66 66 "teachers "teachers' institutes.. 1870 1869 180,274 176,910 107,908 67,412 2,537 1,489 2,302 1,335 41 12 This law provided for the appointment of circuit superintendents, one in each of the ten judi- Number of children of school age. cial districts of the state, whose duties in their several circuits were analogous to those of the state superintendent, in supervising, making reA school trustee was appointed in ports, etc. each school district, with the same duties as those already specified. The report of the school trust ees were made annually to the circuit superintendents, who transmitted the information to the state superintendent, to be used in his biennial report. Under many difficulties and embarrassments, Superintendent Smith organized his department in August, 1868; and in December following, the trustees of the various districts were elected. In September, 1869, a special session of the state board of education-composed of the state and circuit superintendents held. At this time the only free schools existing in the state were a few for persons of color, established by the United States, through the agency of the Freedmen's Bureau. The resources of the school department consisted of (1) saline Lands, about 20,000 acres; (2) seminary lands, about 1,000 acres; (3) sixteenth-section lands, The original quantities of about 841,000 acres. these lands, which were donated by the United States government for common school purposes, were two sections, each of the first two classes, and 928,000 acres of the third class. Of the saline and seminary land funds, about $12,000 in specie, war-bonds, confederate money, etc., had been transferred, after March 6th, 1861, to the general revenue fund of the state; and about The whole number of school-houses built prior Amount of money paid teachers.. $405,748 $188,397 to 1868, was 632; in 1869 and 1870, it was 657. -1869 was $377,919.94, and the district tax, The apportionment of the state fund for 1868 $215,348.79. In addition to these evidences of progress should be mentioned the organization of the State Teachers' Association, July 2d, 1869; and the commencement of the Arkansas Journal of Education, Jan. 1st, 1870. The institutions for the blind and for deaf-mutes were also rehandsome buildings erected for their accommoorganized during the period referred to, and dation. |