Show tbs new the fol following lowin article was ivri writ t ten by goo geo E dibble of layton utah and appeared on the front page of the university chronicle may alay 3rd ard 1898 tho the majority of those who h have been brought up in our common schools will remember the diall callies experienced in mastering za the simple processes of arithmetic the meaningless definitions definition S to be memorized the various rules to bo be mechanically fol followed made of the study anything but a promoter of true mathematical thou thought glit the fundamental idea of number thus imperfectly tau taught C ht was that of tile how individuals number as an abstract relation was not thought of f or at least was believed to be beyond the comprehension compre hention of the child when the ago arrived that true mathe mathematical mathematics matica 1 thinking was necessary the youth found himself with no preparation for the work before him but really hampered by the mechanism to which his whole time had been devoted this difficulty in making the child understand the principles of arithmetic lias has been noted by various educators who have sought to remedy the matter not by alian changing ging the essential idea of 0 num number ber already taught but by reforming the methods of presen presenting individual number by making them more concrete thus most of the late reforms in the methods of teaching primary arithmetic have consisted in tile introduction of objects to illustrate all of the fundamental processes in this way the child has been able to get a clearer conception of the numeration idea of number but lie he receives no greater culture in real mathematical thinking chinkin than he li edidin did in the beginning everything that has savored of abstractness abstract noss lias has been carefully excluded from tile tho childs mind with tile idea that all learning must proceed from the concrete to the abstract abs tact and thus number as a relation has been completely overworked the ithe years that the manure mind devotes to eathem mathem mathematics a ties are given wholly wl bolly to abstract think i ing ado and economy doin demands ands that the mind bo be trained in that kind I 1 of thinking ta while in the most I 1 plastic state individual number is used only in the mechanical part of mathematics and is not of sufficient importance to merit tile the whole attention during the best years of mathematical math e training proficiency in the use of individual number can be acquired with comparatively little effort it is used as a means to a higher hi her end the finding ca of abstract relation the fundamental principle and the whole end and aim of mathematics these should be sufficient seasons why number as relation should be taught from the first with special care and emphasis some may claim that the idea of number as relation cannot be rasped grasped by the child mind but it seems to me that it can amen among 1 the first acts of judgment in the mind of the child are those made by comparing comparin g objects with regard to magnitude in fact the child knows magnitude only by comparison it has a dim consciousness ness of abstract relation from tho the first when it measures one object by another and this consciousness may be brought into greater reater vividness by the proper training mr ir speer is Is the author of a system of number training 0 that brinis brings out the idea of abstract number in the mind of the be beginner gin he bases his system upon number as a ratio and claims that this idea of number should be emphasized from the first to develop this abstract number idea ho he introduces abarge a lare lar e variety of objects in sets those those ineich iii each set bearint bearing za the sanio same relative ma magnitude with these relations before it the child is led to build up his concept of abstract number b e r froni from his individual notions of the relations existing c between the objects in the respective sets tt it I 1 sees 2 not as two individual objects but as tho the relation of one ma magnitude to another one half as large it sees 1 as the relation of two equal magnitudes mag nit udes A as tile ratio of one magnitude to another twice as large air spoor speer interprets the various processes of arithmetic to agree with tho the ratio idea of number with this interpretation they partake of a simplicity scarcely thought of by the student of a few years ago ag 0 one almost wonders vendors why sue lucli a work as the speer arithmetic has not been advanced before and why individual number has so long iong held the sole place in primary number work this is probably due to the fact that individual or concrete number yields the more readily to illustration by objects since that which can be seen and handled grouped and separated can be more easily comprehended but concrete number cannot take the place of abstract number in any process however simple abstract number is fundamental fund amenta and should bo be taught from the first many a y edue educators abhors have been lee led to false conclusions by taking the numeration idea as the true and anc only basis of number among them is the eminent col parker in liis his chapter what can be done don 0 with numbers though n it may seem pretentious on oil my part 1 I shall undertake to criticism criticise critic ise some of the statements of mr aft parker in the light of th the e new ari arithmetic thine in the chapter above referred to the author bases eve everything grything upon individual number and wit this conception attempts to explain processes in which ab relation is involved division is defined as a process of dividing 0 a number into a number of equal numbers instead it is a comparison of two magnitudes to find the relation ex existing astin between them rao for r instance according to mr air parker 12 divided by 4 equals 3 as or the number 12 is separated into a number I 1 of as Vs in arriving it at this I 1 conclusion lie ho has rem regarded 12 as A 4 not us as ma magnitudes but as I 1 can concrete crete numbers ind and from this standpoint shows the absurdity of calling tile tho quotient 3 an abstract number both 12 and 1 I lare lire magnitudes 1 I k nit udes I 1 the quotient J 3 3 IS is not 3 as but is the relation found to exist when the ma magnitude n 1 12 0 is is compared with the magnitude magnitude agni tude 4 multiplication is termed by mr parker tile the process of unit ing c equal numbers in reality jl it is tile the process of findan finding a magnitude that bears a 0 given relation to a given magnitude C it is not the un uniting itic 0 of 3 as that cons constitutes t I 1 a process of multiplication but the finding of a magnitude that bears the same relation to 4 that 3 bears to 1 which in magnitude af agni would be 3 times thries 4 or 12 mr parker has used the term number where with the conception of abstract relation he would have used magnitude true number cannot be combined and separated it may become great or small only as relations between magnitudes aro are increased or diminished addition is a process of combining magnitudes which have been compared with a unit magnitude and is not a uniting of numbers subtraction is a comparison of two ma magnitudes ni for tile the purpose of finding ZD the excess of one over the other and is not as mr parker teams terms it a process of dividing g a number into two numbers the result of the comparison in subtraction is magnitude while in a process of division number is produced there is no such a process as the expression I 1 of 12 indicates that some magnitude has been compared with the magnitude 12 12 and the relation found to exist is i that magnitude which bears the relation J to 12 is i of 12 oy oi 3 thus there is a great unity in all eathem mathem mathematical work from the beginner 0 to the college graduate there is no diversion from tile fundamental fundamental ideas of comparison and ratio fractions ft present no 0 greater reater difficulties to the learner than simple numbers in fact they become one and the same thin thing the new arithmetic will doubtless meet with much conservative criticism but a careful investigation vesti gation of its merits will show it to bean improvement on the old system whatever may inay be said for or against it its true worth will in time be recognized ca by every progressive teacher |