The Parents' Review
A Monthly Magazine of Home-Training and Culture
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Teaching Arithmetic Part 2
by C.H. Wilkinson
[Note—this article had mathematical symbols and formatting that did not transfer properly. LNL]
Teaching Arithmetic Part 2: Some weaknesses I have found and their remedy. (continued from page 578.)
I would now pass from general ideas to show how I would teach arithmetic from its earliest stage. I begin with a child at six years of age. The infant work below that age is such that there is nothing vital as regards method in it. The latest developments of kindergarten, which are well understood and well realized by infant teachers, need no suggestion for their improvement. There are continuous advances being made in this work, and they simply need the teacher to be on the alert to gain the advantage of utilizing them. The chief principle adopted is to teach the child to bring all his counts to ten. In adding he would make ten of his first figures before proceeding to the next figures beyond ten; and this he would do for every ten in succession. Subtraction, as it is miscalled, is worked on the same method. At this stage it is equally easy to break any object into ten parts, and give the child to understand that each of those parts is a tenth; and that there are ten tenths in a whole number as well as ten units in one ten. It is likewise possible to teach him that a tenth may be represented in two different ways, viz., 1 or 1/10. You teach the child that 1 is a unit, and that "t" is one unit and one-tenth, and I think you might teach that "i i" is one-tenth + one-tenth of one-tenth of a unit. This must be shown to be of value before doing it. The addition and subtraction and multiplication of decimals (or the two former at any rate) can be taught at the same time as you teach ordinary addition and subtraction; and so the child is at home at once. You can tell him also that 1/10 means one whole number to be divided by 10 or into tenths. You can put it that the top figure is to be divided by the bottom. You could only do this at the age where you begin division.
But you can tell even at the earliest stages that one over two, as "1/2" is the half, one over four, as "1/4," is the quarter, and so on, of every simple fraction; and let them add halves together and see how many they have, and how many whole ones they make, and so on, with quarters and fifths and tenths, etc. At the stage where division comes in you would say that 1/10 means one to be divided by 10. This is not mathematically accurate, but it is accurate enough to get the underlying idea. Of course in the case of 3/10 it would not always be true that it was a tenth of three articles, or that 3 were to be divided by 10 and one-tenth of each taken. In the cases of like values it would be true. Take money, 3/10 of 13, or 1/10 of 3s is the same thing; but 3/10 does not mean the 1/10 of 3s. Still the general mathematical principle is here, that where you have a number on top it has to be divided by the bottom in order to get its value and relationship to the standard unit. That is the principle you want to convey, and the distinction referred to is readily made clear to the child at a more advanced age. In improper fractions it is more nearly true. Thus 13/10 of an orange is an impossibility, for an orange only has 10/10. But yet in calculations we get figures of this sort to deal with as a short and ready way of getting to a desired result. What is wanted at an early age is to get into the mind of the child of the idea of how a fraction is represented and how it is easiest manipulated, and later on he will regard it actually as what it is, viz., a known portion of a unit. I sometimes go into a class of six-years old children and say, what is the half of a half? The teacher regards me for a moment with horror, and the children look blank. I know beforehand they will do so. Then I put it in the concrete and say, if I have an apple and divide it in half and give it to two of you, how much each will the two have? They see that. Now suppose I take one half and divide it into two and give to two of you, how much each will the two have? One quarter they say right away. Then what is the half of a half? Answer: 1/4. Then I ask, what did I do when I cut the half into two equal parts? I get from them, "I halved it," or, "I divided it." Then if I divide a half by two, what do I get? Answer: 1/4.
In Standard I., where division is going on, he drops to the fact that division can be stated in fractional form. The teaching of both fractions and decimals can be carried much further in this class. Having utilized the fractional form in Standard II to the extent of making him illustrate his mental arithmetic principles by it, and shown him how to cancel out, I should have the foundation for algebra right at the start of Standard III, or at least at 8 years of age for the average child.
When in Standard II, let him take this sum: six dozen cost 6s.; what will one article cost? If he tells me rightly, I want to know how he has done it, and when he has done it in one way, I want the other way also. First, ix dozen into 6s = one dozen for 1s. Multiply one dozen by 1s, and 1s. by 12. Get the reason for this. 12 singles into 12 pence = 1d. for one single. The second way is to reduce six dozen to single ones = 6 x 12 =72. Reduce 6s. to pence = 6s. x 12 = 72d. Divide 72 pence by 72 singles; 72d. divided by 72 singles = 1d. Then I ask why one penny? Why not 1s.? Why divide pence by single ones? Why not divide single ones by pence? It is astonishing how little of this searching is done. Hence the mind of the child is hazy, and his statements are indefinite. Now, let him state the sum in fractional form. Next put 6 [British pound] for 6s. and let them give the five ways in which the answer may by obtained and their reasons for each step. In a little sum like this you have thus given them division, multiplication, and several other principles incidentally, and while they are doing something which to them is new, they are practising what is old, and recapitulating without knowing it. By stating it in fractional form, and showing how cancelling out can be done, you get them on another stage. You can show them that, if you have (4x12)/6 it is the same thing, whether you regard it as (xx)/12 = 8 or as (4xx)/8 = x/.4 = 8. [exact figures were mostly illegible]
After this at Standard III., at the start, I would show a child that any hieroglyphic may represent value or number. I lead up to it by saying that some people do sums in letters instead of figures. Taking him from the known to the unknown, I say, "x dozen cost 6s., what is the cost of one single article?"
State it fractionally, (6x12)/(3x12) Let some child give a figure for x and work it out as above. Next take x dozen and y shillings = (?x12)/(xx12) Let two different children give figures for y and x, and work it out as before. You need to know from the child why y and x have to be multiplied by 12 each. If you multiply one y by 12, how many y's do you get? You want to put all sorts of questions as, Why not put x on the top, or why not put y at the bottom? In this way the child learns that letters (any letters) may have a value when they represent numbers, or money, or measure, etc. You are familiarizing them with letters as a means of calculation, while making yourself more positive that they know the principles of their figures properly. This work can be done as blackboard work and mental arithmetic. The children do the work in their heads and tell you what to put down. A little done each day before the regular arithmetic lesson will improve the written work of the boys themselves, and carry them on very quickly, and make them confident and accurate in their work. The converse is good in the higher classes. You can ask a boy to square figures and work out problems mentally which would be hopeless unless this style of training were made a feature. Take the formula (a + b), (a—b), and from it he can square quite larger figures. Take 372. He would do it—
37 + 3 = 40
Of course you do not tell him what formula to use. A boy of this sort at this stage will select the right formula for the particular sum. This is just one kind as an example. In this way you make one subject help another, and get greater brilliancy in both. I went into a very poor school looked at from the point of view of social standing and funds. It was in the slums of a large midland town. In Standard III., when three moths' work had been done in that standard this was the work they were doing when I went in. How many slates at Bd. Each can I buy for £C: They work it on the lines I have indicated, viz., C x 240/B. Boy gave a figure for C., viz., £25 Another boy gave 3d. for B. (25 x 240)/3 = (25 x 80)/1 = (100 x 80)/4 = (100 x 20)/1 = 2000/1 = 2000 slates. The third fractional equation was given to show that as you could cancel out, so you could also multiply top and bottom by the same figure without destroying the value of the fraction, or altering it in any way. Also because at that school decimals were coming on, two or three standards higher up. Therefore they prepared the boys somewhat for them, by bringing 5's and 10's, and 20's and 25's to 100's. It also showed them the value of decimals, inasmuch as they leaned that to multiply by 10's or 100's was much easier. They could tell you why they did everything, and it was quite easy to them and they liked it.
When teaching numeration I think few teachers show that in the same way as you have 80 single articles you may have 80 items. If they are dealing with 809, they will show that you have 8 hundreds and 9 single units. They do not show that you have 80 tens and 9 units. When you come to longer lines of figures, as 76,956, they show that they are 76 thousands and 9 hundreds and 5 tens, and 6 units. They never show that there are 769 hundreds, or 7,695 tens, which is the very think the children ought to know before starting long division. When teaching multiplication, many children are never taught to know that it is merely addition being done in the shortest way, and that 4 times 4 only mean 4 + 4 + 4 + 4 added together. Nor in division are they led to realize that division is only subtraction; the divisor being deducted from the dividend a given number of times, and the result being that there are so many of the divisor as are indicated by the quotient contained in the dividend. 16 divided by 2 goes 8 times is only another way of saying that 2 is contained 8 times or there are 8 twos in sixteen. This is often not shown.
Even in numeration I have never known a teacher explain why 769 should not mean 967. That is to say, that they do not say why tens are put to the left of units and not to the right; nor why hundreds are put to the left of the tens and units and not to the right of them. The child should be shown that it is merely an arbitrary arrangement dogmatically adhered to in order to avoid confusion, but that it could quite as well have been arranged the other way. Of course the history of the question could also be introduced. It should be pointed out to the child that on the railway tickets of some lines the opposite order of arrangement actually exists without causing any confusion to the company. In all teaching of arithmetic the shortest method should be taught to the child. He should know that the way he is taught is the shortest way, and that the reason for doing it in that way rather than according to some more lengthy method is to save time and unnecessary work. As is the shortness of the day so must be the shortness of our work in order to get as much done as possible. This is one important royal road to success and highest usefulness in life. For this and many other reasons, what is taught as subtraction should never be taught as subtraction; nor should addition always taught as a column of figures. Take subtraction. A boy does not in reality subtract anything at all He finds the difference between two numbers. He should learn at the time he begins his fractions, if not before that time, such signs as +, -, ./., etc., His so-called subtraction should be done simultaneously by the use of minus signs and by placing the figures under one another. If the figures are placed one under the other as in finding the difference between 17 and 42, there is no reason why they should not be put with 17 above the 42 as well as with the 17 under the 42, thus 17 or 42. I should put them sometimes one way and sometimes the other. The child at present is taught almost that the sum is necessarily a subtraction sum because the 17 is underneath. If he is taught that he has o find the difference wherever it is he becomes much more alert and much less mechanical. In any case the method of working should always unalterably be by adding on to the smaller number enough to make the larger, and the child should be clear that the amount thus added makes the difference between the two numbers. This is what is known as the Italian method. The teachers lose a lot of time and give themselves much extra work by not adopting the Italian methods of subtraction and division. Here is a sum I saw a clever master give his class.
The five lines of figures in bracket marked "A" had to be added together and deducted from the top line B, column by column as the boys came to them. The sum was on the blackboard, and the boys stated what had to be done, and gave the figure to be put down. Thus 6 + 5 + 2 + 4 + 9 + x in lines A = 2 in line B, viz, 6 as shown in line C. Thus column after column was done till you got the difference in line C of 15,923,386. Its correctness is proved by adding the lines A to the line C, and these will make the total in line B.
Just here is where saving in time of one description comes in, besides altering the sum and making it more difficult. He drew the chalk through the second and fourth columns o figures to the left of the units, i.e., through the "tens" columns and the "1,000's" column, and put Ls.d. over the top and added some fractions of 1d.
Again A lines were added and taken from line B, giving the difference in line C between the other amounts. See the time saved just to run chalk through and go on with the same sum and see the new notion the boys get of the different values of figures, used in different relationships. The advantages of subtraction done in this way are manifold. First, it results in better method-training for the Italian way of doing long division. I have found also that subtraction by addition (or the Italian method) is quicker and more accurate as a rule.
Again, technically, it is more logical from an educational point of view. A child is first taught addition; then under the old system he would be taught something that appeared to be the antithesis of the rule for addition. But under the system I advocate he goes on to subtraction and does it under the guise of addition with which he has just previously been made familiar. He simply adds on to a number another number which shall be big enough to make the smaller number the same as the larger. His first lesson had been to add several numbers on the one to the other; or in other words to find the sum total of a given set of numbers. Now he simply adds on to certain numbers enough to make a given sum total. The process is intelligible to the child and follows in natural sequence. Then again this process appeals to the child because it is part of his daily life. If he is sent to purchase goods the tradesman in giving change never subtracts. He adds on to the cost of the goods the amount of change necessary to make up the amount of coin the child tenders.
The chief reason of value other than these educational ones is that at the present time the business books in smart houses are printed with three columns on one page, Dr., Cr., Balance. The credits and debits are worked on to the balance column right away. Suppose the balance at your bankers to be a credit, all the credits for the day will be added up (pence columns first of course) and the first column will be added on to the pence in the balance column, and then in like manner the shillings. The debits will be treated in a similar way, only they will be deducted as you go along, the pence from the pence and so on, the only figures appearing on the ledger being the entries and the balances. No totals of credits or debits and no pencil entries. This can only be done in one way, viz., by addition. The utilization of this Italian method in doing long division I think invaluable. Take 679382 ./. 348. Done in the ordinary way by long division it would be:—
348) 670 3,8,2, (1952?
But the Italian way would be:—
The advantages are many. It is shorter. It is quicker. It is finer for mental gymnastics. It tends to ensure more care, for if the child makes a mistake he has to go through the whole sum again to find it; and so it leads to greater accuracy. It is in harmony with previous methods of teaching the ordinary elementary rules as subtraction. In long division usually the greatest number of mistakes occur in the subtraction. By this method multiplication and subtraction are done simultaneously.
With the long division the advance of knowledge in decimals is readily extended and increased.
Suppose 214,705 is to be divided by 203.
In this instance I have put the quotient over the dividend instead of at the side. I think it makes it clearer to the child. I should do this in ordinary long division. The sum was done first as an ordinary long division sum. Then a decimal point was put in the middle and the boys asked where the point should be in the quotient. It was shifted from point to point in the dividend and the boys shifted it readily in the quotient. The reason was given for the 1 going over the 4 and the 0 over the 7 and so on.
There is much I could refer to on this subject, but space and time suggest my confining to one other matter, and that is the nature of the sums set. I have said I like the principle of the Government B scheme, because of its problems and mode of progression, which suits well the training of the child. Some books have excellent little problems in them. I think, however, there is a field for more definite purpose in setting sums. I would like all arithmetic to represent some facts in relation to other subjects which should unconsciously impress those facts on the mind of the child.
Take history, for example, I would set a sum in this form. The battle of Waterloo was fought in 1815, Wellington had for his army so many each of English, Dutch, and Belgian troops. How many had he altogether? Napoleon had such a number. How many men were in the field under the two leaders? Blucher brought so many more to Wellington's aid. How many did this make? Then deal with the losses by the flight of the Belgians, the deaths on each side, the prisoners taken from time to time, and you get your sums. Then say the total on each side were so many, how many more would Wellington have needed if the French had had 10 per cent more than they had, and so on for proportion sum. Divide the force into divisions under their generals. Give one general x number, another one y number, and another one z number of men. Tell them that x stands for so many, and y for another quantity, and z for another number, thus—x/47,000 y/47,000 z/47,000. Ask them how many altogether. So at this early stage you get them familiar with the use of letters. Use actual numbers as far as history reveals them.
Take another example in geography. A steamer takes 125 hours to go from Liverpool to Philadelphia, and she steams 25 miles an hour. How many miles are the ports distant? The sum should be so set as to the time taken and the rate per hour, as to give an actual distance for the answer. You might even take the time of the record steamer to date. My example is illustrative and the figures are not actual in any sense.
You could take all the various ports and take the distances they are apart in various parts of the would. Then take the various rates of sailing and steaming; and work out from these data the length of time it would take the different vessels. Problems set on these lines would be most valuable. Mountains and their height, rivers and their length, towns and their distances apart, can all be utilized for like purposes. Physics give scope for lovely, simple, and interesting arithmetic problems, and in the more advanced arithmetic, astronomy could to some extent be impressed into our service. I have not touched on many important features of arithmetic dealing with factors, etc. I think we all realize the importance of attention to these. The sole idea of this paper is to suggest some newer and more satisfactory method of dealing with all rules and at all stages, so that the teaching may be more regularly and gradually progressive—more thorough in the ground-work and more usefully co-ordinated with other subjects. What I have suggested in regard to the elementary principles of the most elementary educational work should, I think, possibly apply, as far as the principles are concerned, to more advance work. In any case, I am certain that the advanced work of our rising generation would be greatly improved, and the children would be more successful if the elementary work were done with greater attention to detail and with a more minute application to the development of underlying principles.
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