28704516938276546397+ 35869427359163827263
and similar long numbers to be multiplied by 2 or by 3 or by 4 or by 5 or by 6, found 851 errors in 28,267 answer-figures, or 3 per hundred answer-figures, or3⁄5of an error per example. The children were 9½ to 15 years old. Laser ['94], using the same sort of addition and multiplication, found somewhat over 3 errors per hundred answer-figures in the case of boys and girls averaging 11½ years, during the period of their most accurate work. Holmes ['95], using addition of the sort just described, found 346 errors in 23,713 answer-figures or about 1½ per hundred. The children were from all grades from the third to the eighth. In Laser's work, 21, 19, 13, and 10 answer-figures were obtained per minute. Friedrich ['97] with similar examples, giving the very long time of 20 minutes for obtaining about 200 answer-figures, found from 1 to 2 per hundred wrong. King ['07] had children in grade 5 do sums, each consisting of 5 two-place numbers. In the most accurate work-period, they made 1 error per 20 columns. In multiplying a four-place by a four-place number they had less than one total answer right out of three. In New York City Courtis found ['11-'12] with his Test 7 that in 12 minutes the average achievement of fourth-grade children is 8.8 units attempted with 4.2 right. In grade 5 the facts are 10.9 attempts with 5.8 right; in grade 6, 12.5 attempts with 7.0 right; in grade 7, 15 attempts with 8.5 right; in grade 8, 15.7 attempts with 10.1 right. These results are near enough to those obtained from the country at large to serve as a text here.
The following were set as official standards, in an excellent school system, Courtis Series B being used:—
Grade.SpeedAttempts.Percent ofCorrect Answers.Addition81280711806107059704870Subtraction81290711906109059804780Multiplication8118071080698057704660Division8119071090688056704460
Kirby ['13, pp. 16 ff. and 55 ff.] found that, in adding columns like those printed below, children in grade 4 got on the average less than 80 percent of correct answers. Their average speed was about 2 columns per minute. In doing division of the sort printed below children of grades 3Band 4Agot less than 95 percent of correct answers, the average speed being 4 divisions per minute. In both cases the slower computers were no more accurate than the faster ones. Practice improved the speed very rapidly, but the accuracy remained substantially unchanged. Brown ['11 and '12] found a similar low status of ability and notable improvement from a moderate amount of special practice.
3562389749796556458234787379378848268298224769856269578523249642729445337999289768964779248469926989——————————
20 = .... 5s56 = .... 9s and ....r.30 = .... 7s and ....r.89 = .... 9s and ....r.20 = .... 8s and ....r.56 = .... 6s and ....r.31 = .... 4s and ....r.86 = .... 9s and ....r.
20 = .... 5s56 = .... 9s and ....r.30 = .... 7s and ....r.89 = .... 9s and ....r.20 = .... 8s and ....r.56 = .... 6s and ....r.31 = .... 4s and ....r.86 = .... 9s and ....r.
It is clear that numerical work as inaccurate as this has little or no commercial or industrial value. If clerks got only six answers out of ten right as in the Courtis tests, one would need to have at least four clerks make each computation and would even then have to check many of their discrepancies by the work of still other clerks, if he wanted his accounts to show less than one error per hundred accounting units of the Courtis size.
It is also clear that the "habits of ... absolute accuracy, and satisfaction in truth as a result" which arithmetic is supposed to further must be largely mythical in pupils who get right answers only from three to nine times out of ten!
The bonds in question clearly must be made far stronger than they now are. They should in fact be strong enough to abolish errors in computation, except for those due totemporary lapses. It is much better for a child to know half of the multiplication tables, and to know that he does not know the rest, than to half-know them all; and this holds good of all the elementary bonds required for computation. Any bond should be made to work perfectly, though slowly, very soon after its formation is begun. Speed can easily be added by proper practice.
The chief reasons why this is not done now seem to be the following: (1) Certain important bonds (like the additions with higher decades) are not given enough attention when they are first used. (2) The special training necessary when a bond is used in a different connection (as when the multiplications to 9 × 9 are used in examples like
7298
where the pupil has also to choose the right number to multiply, keep in mind what is carried, use it properly, and write the right figure in the right place, and carry a figure, or remember that he carries none) is neglected. (3) The pupil is not taught to check his work. (4) He is not made responsible for substantially accurate results. Furthermore, the requirement of (4) without the training of (1), (2), and (3) will involve either a fruitless failure on the part of many pupils, or an utterly unjust requirement of time. The common error of supposing that the task of computation with integers consists merely in learning the additions to 9 + 9, the subtractions to 18 − 9, the multiplications to 8 × 9, and the divisions to 81 ÷ 9, and in applying this knowledge in connection with the principles of decimal notation, has had a large share in permitting the gross inaccuracy of arithmetical work. The bonds involved in 'knowing the tables' do not make up one fourth of the bonds involved in real adding, subtracting, multiplying, and dividing (with integers alone).
It should be noted that if the training mentioned in (1) and (2) is well cared for, the checking of results as recommended in (3) becomes enormously more valuable than it is under present conditions, though even now it is one of our soundest practices. If a child knows the additions to higher decades so that he can add a seen one-place number to a thought-of two-place number in three seconds or less with a correct answer 199 times out of 200, there is only an infinitesimal chance that a ten-figure column twice added (once up, once down) a few minutes apart with identical answers will be wrong. Suppose that, in long multiplication, a pupil can multiply to 9 × 9 while keeping his place and keeping track of what he is 'carrying' and of where to write the figure he writes, and can add what he carries without losing track of what he is to add it to, where he is to write the unit figure, what he is to multiply next and by what, and what he will then have to carry, in each case to a surety of 99 percent of correct responses. Then two identical answers got by multiplying one three-place number by another a few minutes apart, and with reversal of the numbers, will not be wrong more than twice in his entire school career. Checks approach proofs when the constituent bonds are strong.
If, on the contrary, the fundamental bonds are so weak that they do not work accurately, checking becomes much less trustworthy and also very much more laborious. In fact, it is possible to show that below a certain point of strength of the fundamental bonds, the time required for checking is so great that part of it might better be spent in improving the fundamental bonds.
For example, suppose that a pupil has to find the sum of five numbers like $2.49, $5.25, $6.50, $7.89, and $3.75.Counting each act of holding in mind the number to be carried and each writing of a column's result as equivalent in difficulty to one addition, such a sum equals nineteen single additions. On this basis and with certain additional estimates[7]we can compute the practical consequences for a pupil's use of addition in life according to the mastery of it that he has gained in school.
I have so computed the amount of checking a pupil will have to do to reach two agreeing numbers (out of two, or three, or four, or five, or whatever the number before he gets two that are alike), according to his mastery of the elementary processes. The facts appear in Table 1.
It is obvious that a pupil whose mastery of the elements is that denoted by getting them right 96 times out of 100 will require so much time for checking that, even if he were never to use this ability for anything save a few thousand sums in addition, he would do well to improve this ability before he tried to do the sums. An ability of 199 out of 200, or 995 out of 1000, seems likely to save much more time than would be taken to acquire it, and a reasonable defense could be made for requiring 996 or 997 out of 1000.
A precision of from 995 to 997 out of 1000 being required, and ordinary sagacity being used in the teaching, speed will substantially take care of itself. Counting on the fingers or in words will not give that precision. Slow recourse to memory of serial addition tables will not give that precision. Nothing save sure memory of the facts operating under the conditions of actual examples will give it. And such memories will operate with sufficient speed.
TABLE 1
The Effect of Mastery of the Elementary Facts of Addition upon the Labor Required to Secure Two Agreeing Answers When Adding Five Three-figure Numbers
Mastery of the Elementary Additions Times Right in 1000Approximate Number of Wrong Answers in Sums of 5 Three-place Numbers per 1000Approximate Number of Agreeing Answers, after One Checking, per 1000Approximate Number of Agreeing Answers, after a Checking of the First DiscrepanciesApproximate Number of Checkings Required (over and above the First General Checking of the 100 Sums) to Secure Two Agreeing Results960700902164500980380384676120099019065690647099595819975210996768549841659975489599211599838925996809991996299940
There is one intelligent objection to the special practice necessary to establish arithmetical connections so fully as to give the accuracy which both utilitarian and disciplinary aims require. It may be said that the pupils in grades 3, 4, and 5 cannot appreciate the need and that consequently the work will be dull, barren, and alien, without close personal appropriation by the pupil's nature. It is true that no vehement life-purpose is directly involved by the problem of perfecting one's power to add 7 to 28 in grade 2, or by the problem of multiplying 253 by 8 accurately in grade 3or by precise subtraction in long division in grade 4. It is also true, however, that the most humanly interesting of problems—one that the pupil attacks most whole-heartedly—will not be solved correctly unless the pupil has the necessary associative mechanisms in order; and the surer he is of them, the freer he is to think out the problem as such. Further, computation is not dull if the pupil can compute. He does not himself object to its barrenness of vital meaning, so long as the barrenness of failure is prevented. We must not forget that pupils like to learn. In teaching excessively dull individuals, who has not often observed the great interest which they display in anything that they are enabled to master? There is pathos in their joy in learning to recognize parts of speech, perform algebraic simplifications, or translate Latin sentences, and in other accomplishments equally meaningless to all their interests save the universal human interest in success and recognition. Still further, it is not very hard to show to pupils the imperative need of accuracy in scoring games, in the shop, in the store, and in the office. Finally, the argument that accurate work of this sort is alien to the pupil in these grades is still stronger againstinaccuratework of the same sort. If we are to teach computation with two- and three- and four-place numbers at all, it should be taught as a reliable instrument, not as a combination of vague memories and faith. The author is ready to cut computation with numbers above 10 out of the curriculum of grades 1-6 as soon as more valuable educational instruments are offered in its place, but he is convinced that nothing in child-nature makes a large variety of inaccurate computing more interesting or educative or germane to felt needs, than a smaller variety of accurate computing!
The second general fact is that certain bonds are of service for only a limited time and so need to be formed only to a limited and slight degree of strength. The data of problems set to illustrate a principle or improve some habit of computation are, of course, the clearest cases. The pupil needs to remember that John bought 3 loaves of bread and that they were 5-cent loaves and that he gave 25 cents to the baker only long enough to use the data to decide what change John should receive. The connections between the total described situation and the answer obtained, supposing some considerable computation to intervene, is a bond that we let expire almost as soon as it is born.
It is sometimes assumed that the bond between a certain group of features which make a problem a 'Buyathings atbper thing, find total cost' problem or a 'Buyathings atbper thing, what change fromc' problem or a 'What gain on buying foraand selling forb' problem or a 'How many things ataeach can I buy forbcents' problem—it is assumed that the bond between these essential defining features and the operation or operations required for solution is as temporary as the bonds with the name of the buyer or the price of the thing. It is assumed that all problems are and should be solved by some pure act of reasoning without help or hindrance from bonds with the particular verbal structure and vocabulary of the problems. Whether or not theyshouldbe, theyare not. Every time that a pupil solves a 'bought-sold' problem by subtraction he strengthens the tendency to respond to any problem whatsoever that contains the words 'bought for' and 'sold for' by subtraction; and he will by no means surely stop and survey every such problem in all its elements to make sure that noother feature makes inapplicable the tendency to subtract which the 'bought sold' evokes.
To prevent pupils from responding to the form of statement rather than the essential facts, we should then not teach them to forget the form of statement, but rather give them all the common forms of statement to which the response in question is an appropriate response, and only such. If a certain form of statement does in life always signify a certain arithmetical procedure, the bond between it and that procedure may properly be made very strong.
Another case of the formation of bonds to only a slight degree of strength concerns the use of so-called 'crutches' such as writing +, −, and × in copying problems like those below:—
AddSubtractMultiply2361—7924—323—
or altering the figures when 'borrowing' in subtraction, and the like. Since it is undesirable that the pupil should regard the 'crutch' response as essential to the total procedure, or become so used to having it that he will be disturbed by its absence later, it is supposed that the bond between the situation and the crutch should not be fully formed. There is a better way out of the difficulty, in case crutches are used at all. This is to associate the crutch with a special 'set,' and its non-use with the general set which is to be the permanent one. For example, children may be taught from the start never to write the crutch sign or crutch figure unless the work is accompanied by "Write ... to help you to...."
Write - to help you to remember thatyou must subtract in this row.Find the differences:—39236744783656264524Remember that you must subtractin this row.Find the differences:—85632714965138457832
The bond evoking the use of the crutch may then be formed thoroughly enough so that there is no hesitation, insecurity, or error, without interfering to any harmful extent with the more general bond from the situation to work without the crutch.
Another instructive case concerns the bonds between certain words and their meanings, and between certain situations of commerce, industry, or agriculture and useful facts about these situations. Illustrations of the former are the bonds betweencube root,hectare,brokerage,commission,indorsement,vertex,adjacent,nonagon,sector,draft,bill of exchange, and their meanings. Illustrations of the latter are the bonds from "Money being lent 'with interest' at no specified rate, what rate is charged?" to "The legal rate of the state," from "$Xper M as a rate for lumber" to "Means $Xper thousand board feet, a board foot being 1 ft. by 1 ft. by 1 in."
It is argued by many that such bonds are valuable for a short time; namely, while arithmetical procedures in connection with which they serve are learned, but that their value is only to serve as a means for learning these procedures and that thereafter they may be forgotten. "They are formed only as accessory means to certain more purely arithmetical knowledge or discipline; after this is acquiredthey may be forgotten. Everybody does in fact forget them, relearning them later if life requires." So runs the argument.
In some cases learning such words and facts only to use them in solving a certain sort of problems and then forget them may be profitable. The practice is, however, exceedingly risky. It is true that everybody does in fact forget many such meanings and facts, but this commonly means either that they should not have been learned at all at the time that they were learned, or that they should have been learned more permanently, or that details should have been learned with the expectation that they themselves would be forgotten but that a general fact or attitude would remain. For example, duodecagon should not be learned at all in the elementary school; indorsement should either not be learned at all there, or be learned for permanence of a year or more; the details of the metric system should be so taught as to leave for several years at least knowledge of the facts that there is a system so named that is important, whose tables go by tens, hundreds, or thousands, and a tendency (not necessarily strong) to connect meter, kilogram, and liter with measurement by the metric system and with approximate estimates of their several magnitudes.
If an arithmetical procedure seems to require accessory bonds which are to be forgotten, once the procedure is mastered, we should be suspicious of the value of the procedure itself. If pupils forget what compound interest is, we may be sure that they will usually also have forgotten how to compute it. Surely there is waste if they have learned what it is only to learn how to compute it only to forget how to compute it!
The next case of the formation of bonds to slight strength is the problematic one of forming the bonds involved in understanding the reasons for certain processes only to forget them after the process has become a habit. Should a pupil, that is, learn why he inverts and multiplies, only to forget it as soon as he can be trusted to divide by a fraction? Should he learn why he puts the units figure of each partial product in multiplication under the figure that he multiplies by, only to forget the reason as soon as he has command of the process? Should he learn why he gets the number of square inches in a rectangle by multiplying the length by the width, both being expressed in linear inches, and forget why as soon as he is competent to make computations of the areas of rectangles?
On general psychological grounds we should be suspicious of forming bonds only to let them die of starvation later, and tend to expect that elaborate explanations learned only to be forgotten either should not be learned at all, or should be learned at such a time and in such a way that they would not be forgotten. Especially we should expect that the general principles of arithmetic, the whys and wherefores of its fundamental ways of manipulating numbers, ought to be the last bonds of all to be forgotten. Details ofhowyou arranged numbers to multiply might vanish, but the general reasons for the placing would be expected to persist and enable one to invent the detailed manipulations that had been forgotten.
This suspicion is, I think, justified by facts. The doctrine that the customary deductive explanations of why we invert and multiply, or place the partial products as we dobefore adding, may be allowed to be forgotten once the actual habits are in working order, has a suspicious source. It arose to meet the criticism that so much time and effort were required to keep these deductive explanations in memory. The fact was that the pupil learned to compute correctlyirrespective ofthe deductive explanations. They were only an added burden. His inductive learning that the procedure gave the right answer really taught him. So he wisely shuffled off the extra burden of facts about the consequences of the nature of a fraction or the place values of our decimal notation. The bonds weakened because they were not used. They were not used because they were not useful in the shape and at the time that they were formed, or because the pupil was unable to understand the explanations so as to form them at all.
The criticism was valid and should have been met in part by replacing the deductive explanations by inductive verifications, and in part by using the deductive reasoning as a check after the process itself is mastered. The very same discussions of place-value which are futile as proof that you must do a certain thing before you have done it, often become instructive as an explanation of why the thing that you have learned to do and are familiar with and have verified by other tests works as well as it does. The general deductive theory of arithmetic should not be learned only to be forgotten. Much of it should, by most pupils, not be learned at all. What is learned should be learned much later than now, as a synthesis and rationale of habits, not as their creator. What is learned of such deductive theory should rank among the most rather than least permanent of a pupil's stock of arithmetical knowledge and power. There are bonds which are formed only to be lost, and bonds formed only to be lostin their first form, beingused in a new organization as material for bonds of a higher order; but the bonds involved in deductive explanations of why certain processes are right are not such: they are not to be formed just to be forgotten, nor as mere propædeutics to routine manipulations.
The formation of bonds to a limited strength because they are to be lost in their first form, being worked over in different ways in other bonds to which they are propædeutic or contributing is the most important case of low strength, or rather low permanence, in bonds.
The bond between four 5s in a column to be added and the response of thinking '10, 15, 20' is worth forming, but it is displaced later by the multiplication bond or direct connection of 'four 5s to be added' with '20.' Counting by 2s from 2, 3s from 3, 4s from 4, 5s from 5, etc., forms serial bonds which as series might well be left to disappear. Their separate steps are kept as permanent bonds for use in column addition, but their serial nature is changed from 2 (and 2) 4, (and 2) 6, (and 2) 8, etc., to two 2s = 4, three 2s = 6, four 2s = 8, etc.; after playing their part in producing the bonds whereby any multiple of 2 by 2 to 9, can be got, the original serial bonds are, as series, needed no longer. The verbal response of saying 'and' in adding, after helping to establish the bonds whereby the general set of the mind toward adding coöperates with the numbers seen or thought of to produce their sum, should disappear; or remain so slurred in inner speech as to offer no bar to speed.
The rule for such bonds is, of course, to form them strongly enough so that they work quickly and accurately for the time being and facilitate the bonds that are to replace them,but not to overlearn them. There is a difference between learning something to be held for a short time, and the same amount of energy spent in learning for long retention. The former sort of learning is, of course, appropriate with many of these propædeutic bonds.
The bonds mentioned as illustrations are notpurelypropædeutic, nor formedonlyto be transmuted into something else. Even the saying of 'and' in addition has some genuine, intrinsic value in distinguishing the process of addition, and may perhaps be usefully reviewed for a brief space during the first steps in adding common fractions. Some such propædeutic bonds may be worth while apart from their value in preparing for other bonds. Consider, for example, exercises like those shown below which are propædeutic to long division, giving the pupil some basis in experience for his selection of the quotient figures. These multiplications are intrinsically worth doing, especially the 12s and 25s. Whatever the pupil remembers of them will be to his advantage.
1.Count by 11s to 132, beginning 11, 22, 33.2.Count by 12s to 144, beginning 12, 24, 36.3.Count by 25s to 300, beginning 25, 50, 75.4.State the missing numbers:—A.B.C.D.3 11s =5 11s =8 ft. = .... in.2 dozen =4 12s =3 12s =10 ft. = .... in.4 dozen =5 12s =6 12s =7 ft. = .... in.10 dozen =6 11s =12 11s =4 ft. = .... in.5 dozen =9 11s =2 12s =6 ft. = .... in.7 dozen =7 12s =9 12s =9 ft. = .... in.12 dozen =8 12s =7 11s =11 ft. = .... in.9 dozen =11 11s =12 12s =5 ft. = .... in.6 dozen =5.Count by 25s to $2.50, saying, "25 cents, 50 cents, 75 cents, one dollar," and so on.6.Count by 15s to $1.50.7.Find the products. Do not use pencil. Think what they are.A.B.C.D.E.2 × 253 × 152 × 124 × 116 × 253 × 2510 × 152 × 154 × 156 × 155 × 254 × 152 × 254 × 126 × 1210 × 252 × 152 × 114 × 256 × 114 × 257 × 153 × 255 × 117 × 126 × 259 × 153 × 155 × 127 × 158 × 255 × 153 × 115 × 157 × 257 × 258 × 153 × 125 × 257 × 119 × 256 × 158 × 129 × 128 × 25State the missing numbers:—A. 36 = .... 12sB. 44 = .... 11sC. 50 = .... 25s60 = .... 12s88 = .... 11s125 = .... 25s24 = .... 12s77 = .... 11s75 = .... 25s48 = .... 12s55 = .... 11s200 = .... 25s144 = .... 12s99 = .... 11s250 = .... 25s108 = .... 12s110 = .... 11s175 = .... 25s72 = .... 12s33 = .... 11s225 = .... 25s96 = .... 12s66 = .... 11s150 = .... 25s84 = .... 12s22 = .... 11s100 = .... 25sFind the quotients and remainders. If you need to use paper and pencil to find them, you may. But find as many as you can without pencil and paper. Do Row A first. Then do Row B. Then Row C, etc.Row A.11|4512|4525|4515|4521|4522|45Row B.25|5511|5512|5515|5522|5530|55Row C.12|6025|6015|6011|6030|6021|60Row D.12|7511|7515|7525|7530|7535|75Row E.11|10012|10025|10015|10030|10022|100Row F.11|9612|9625|9615|9630|9622|96Row G.25|10511|10515|10512|10522|10535|105Row H.12|6415|6425|6411|6422|6421|64Row I.11|8012|8015|8025|8035|8021|80Row J.25|20030|20075|20063|20065|20066|200Do this section again. Do all the first column first. Then do the second column, then the third, and so on.
1.Count by 11s to 132, beginning 11, 22, 33.
2.Count by 12s to 144, beginning 12, 24, 36.
3.Count by 25s to 300, beginning 25, 50, 75.
4.State the missing numbers:—
A.B.C.D.3 11s =5 11s =8 ft. = .... in.2 dozen =4 12s =3 12s =10 ft. = .... in.4 dozen =5 12s =6 12s =7 ft. = .... in.10 dozen =6 11s =12 11s =4 ft. = .... in.5 dozen =9 11s =2 12s =6 ft. = .... in.7 dozen =7 12s =9 12s =9 ft. = .... in.12 dozen =8 12s =7 11s =11 ft. = .... in.9 dozen =11 11s =12 12s =5 ft. = .... in.6 dozen =
5.Count by 25s to $2.50, saying, "25 cents, 50 cents, 75 cents, one dollar," and so on.
6.Count by 15s to $1.50.
7.Find the products. Do not use pencil. Think what they are.
A.B.C.D.E.2 × 253 × 152 × 124 × 116 × 253 × 2510 × 152 × 154 × 156 × 155 × 254 × 152 × 254 × 126 × 1210 × 252 × 152 × 114 × 256 × 114 × 257 × 153 × 255 × 117 × 126 × 259 × 153 × 155 × 127 × 158 × 255 × 153 × 115 × 157 × 257 × 258 × 153 × 125 × 257 × 119 × 256 × 158 × 129 × 128 × 25
State the missing numbers:—
A. 36 = .... 12sB. 44 = .... 11sC. 50 = .... 25s60 = .... 12s88 = .... 11s125 = .... 25s24 = .... 12s77 = .... 11s75 = .... 25s48 = .... 12s55 = .... 11s200 = .... 25s144 = .... 12s99 = .... 11s250 = .... 25s108 = .... 12s110 = .... 11s175 = .... 25s72 = .... 12s33 = .... 11s225 = .... 25s96 = .... 12s66 = .... 11s150 = .... 25s84 = .... 12s22 = .... 11s100 = .... 25s
Find the quotients and remainders. If you need to use paper and pencil to find them, you may. But find as many as you can without pencil and paper. Do Row A first. Then do Row B. Then Row C, etc.
Row A.11|4512|4525|4515|4521|4522|45Row B.25|5511|5512|5515|5522|5530|55Row C.12|6025|6015|6011|6030|6021|60Row D.12|7511|7515|7525|7530|7535|75Row E.11|10012|10025|10015|10030|10022|100Row F.11|9612|9625|9615|9630|9622|96Row G.25|10511|10515|10512|10522|10535|105Row H.12|6415|6425|6411|6422|6421|64Row I.11|8012|8015|8025|8035|8021|80Row J.25|20030|20075|20063|20065|20066|200
Do this section again. Do all the first column first. Then do the second column, then the third, and so on.
Consider, from the same point of view, exercises like (3 × 4) + 2, (7 × 6) + 5, (9 × 4) + 6, given as a preparation for written multiplication. The work of
483687479
and the like is facilitated if the pupil has easy control of the process of getting a product, and keeping it in mind while he adds a one-place number to it. The practice with (3 × 4) + 2 and the like is also good practice intrinsically. So some teachers provide systematic preparatory drills of this type just before or along with the beginning of short multiplication.
In some cases the bonds are purely propædeutic or are formedonlyfor later reconstruction. They then differ little from 'crutches.' The typical crutch forms a habit which has actually to be broken, whereas the purely propædeutic bond forms a habit which is left to rust out from disuse.
For example, as an introduction to long division, a pupil may be given exercises using one-figure divisors in the long form, as:—
773and 5 remainder7 )541649514926215
The important recommendation concerning these purely propædeutic bonds, and bonds formed only for later reconstruction, is to be very critical of them, and not indulge in them when, by the exercise of enough ingenuity, some bond worthy of a permanent place in the individual's equipment can be devised which will do the work as well. Arithmetical teaching has done very well in this respect, tending to err by leaving out really valuable preparatory drills rather than by inserting uneconomical ones. It is in the teaching of reading that we find the formation of propædeutic bonds of dubious value (with letters, phonograms, diacritical marks, and the like) often carried to demonstrably wasteful extremes.
It will be instructive if the reader will perform the following experiment as an introduction to the discussion of this chapter, before reading any of the discussion.
Suppose that a pupil does all the work, oral and written, computation and problem-solving, presented for grades 1 to 6 inclusive (that is, in the first two books of a three-book series) in the average textbook now used in the elementary school. How many times will he have exercised each of the various bonds involved in the four operations with integers shown below? That is, how many times will he have thought, "1 and 1 are 2," "1 and 2 are 3," etc.? Every case of the action of each bond is to be counted.
THE FUNDAMENTAL BONDS