Chapter VI
Reading edition
THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF PRACTICE AND THE ORGANIZATION OF ABILITIES
THE AMOUNT OF PRACTICE
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
1 + 12 − 11 × 12 ÷ 1
1 + 22 − 22 × 12 ÷ 2
1 + 33 × 1
1 + 44 × 1
1 + 53 − 15 × 13 ÷ 1
1 + 63 − 26 × 13 ÷ 2
1 + 73 − 37 × 13 ÷ 3
1 + 88 × 1
1 + 99 × 1
4 − 14 ÷ 1
4 − 24 ÷ 2
11 (or 21 or 31, etc.) + 14 − 31 × 24 ÷ 3
11 " + 24 − 42 × 24 ÷ 4
11 " + 33 × 2
11 " + 44 × 2
11 " + 55 − 15 × 25 ÷ 1
11 " + 65 − 26 × 25 ÷ 2
11 " + 75 − 37 × 25 ÷ 3
11 " + 85 − 48 × 25 ÷ 4
11 " + 95 − 59 × 25 ÷ 5
6 − 11 × 36 ÷ 1
2 + 16 − 22 × 36 ÷ 2
2 + 26 − 33 × 36 ÷ 3
2 + 36 − 44 × 36 ÷ 4
2 + 46 − 55 × 36 ÷ 5
2 + 56 − 66 × 36 ÷ 6
2 + 67 × 3
2 + 78 × 3
2 + 87 − 19 × 37 ÷ 1
2 + 97 − 27 ÷ 2
7 − 37 ÷ 3
7 − 41 × 47 ÷ 4
12 (or 22 or 32, etc.) + 17 − 52 × 47 ÷ 5
12 " + 27 − 6and so on7 ÷ 6
7 − 7to 9 × 97 ÷ 7
and so on toand so onand so on to
9 + 9to 18 − 982 ÷ 9 83 ÷ 9, etc.
19 (or 29 or 39, etc.) + 9
If estimating for the entire series is too long a task, it will be sufficient to use eight or ten from each, say:—
3 + 213, 23, etc. + 27 + 217, 27, etc. + 2
" 3 " 3 " 3 " 3
" 4 " 4 " 4 " 4
" 5 " 5 " 5 " 5
" 6 " 6 " 6 " 6
" 7 " 7 " 7 " 7
" 8 " 8 " 8 " 8
" 9 " 9 " 9 " 9
3 − 37 − 79 × 763 ÷ 9
4 " 8 " 7 × 964 "
5 " 9 " 8 × 665 "
6 " 10 " 6 × 866 "
7 " 11 " 67 "
8 " 12 " 68 "
9 " 13 " 69 "
10 " 14 " 70 "
11 " 15 " 71 "
12 " 16 "
TABLE 2
Estimates of the Amount of Practice Provided in Books I and II of the Average Three-Book Text in Arithmetic; by 50 Experienced Teachers
Arithmetical FactLowest Estimate Median EstimateHighest Estimate Range Required to Include Half of the Estimates
3 or 13 or 23, etc. + 2 25 15001,000,000 800-5000
" " 3 24 1450 80,000 475-5000
" " 4 23 1150 50,000 750-5000
" " 5 22 1400 44,000 700-5000
" " 6 21 1350 41,000 700-4500
" " 7 21 1500 37,000 600-4000
" " 8 20 1400 33,000 550-4100
" " 9 20 1150 28,000 650-4500
7 or 17 or 27, etc. + 2 20 12502,000,000 600-5000
" " 3 19 11001,000,000 650-4900
" " 4 18 1000 80,000 650-4900
" " 5 17 1300 80,000 650-4400
" " 6 16 1100 29,000 650-4500
" " 7 15 1100 25,000 500-4500
" " 8 13 1100 21,000 650-3800
" " 9 10 1275 17,000 500-4000
3 − 3 25 1000 100,000 500-4000
4 − 3 20 1050 500,000 525-3000
5 − 3 20 11002,500,000 650-4200
6 − 3 10 1050 21,000 650-3250
7 − 3 22 1100 15,000 550-3050
8 − 3 21 1075 15,000 650-3000
9 − 3 21 1000 15,000 700-2600
10 − 3 20 1000 20,000 600-2500
11 − 3 20 1000 15,000 465-2550
12 − 3 18 1000 15,000 650-2100
7 − 7 10 1000 18,000 425-3000
8 − 7 15 1000 18,000 413-3100
9 − 7 15 950 18,000 550-3000
10 − 7 15 950 18,000 600-3950
11 − 7 10 900 18,000 550-3000
12 − 7 10 925 18,000 525-3100
13 − 7 10 900 18,000 500-2600
14 − 7 10 900 18,000 500-3100
15 − 7 10 925 18,000 500-3000
16 − 7 10 875 18,000 500-2500
9 × 7 10 700 20,000 500-2000
7 × 9 10 700 20,000 500-1750
8 × 6 10 750 20,000 500-2500
6 × 8 9 700 20,000 500-2500
63 ÷ 9 9 500 4,500 300-2500
64 ÷ 9 9 200 4,000 100- 700
65 ÷ 9 8 200 4,000 100- 600
66 ÷ 9 7 200 4,000 100- 550
67 ÷ 9 7 200 4,000 75- 450
68 ÷ 9 6 200 4,000 87- 575
69 ÷ 9 6 200 4,000 87- 450
70 ÷ 9 5 200 4,000 75- 575
71 ÷ 9 5 200 4,000 75- 700
XX 40 5501,000,000 300-2000
XO 20 500 11,500 150-2000
XXX 15 450 12,000 100-1000
XXO 25 400 15,000 150-1000
XOO 15 400 5,000 100-1000
XOX 10 400 10,000 100- 975
Having made his estimates the reader should compare them first with similar estimates made by experienced teachers (shown on page 124 f.), and then with the results of actual counts for representative textbooks in arithmetic (shown on pages 126 to 132).
It will be observed in Table 2 that even experienced teachers vary enormously in their estimates of the amount of practice given by an average textbook in arithmetic, and that most of them are in serious error by overestimating the amount of practice. In general it is the fact that we use textbooks in arithmetic with very vague and erroneous ideas of what is in them, and think they give much more practice than they do.
The authors of the textbooks as a rule also probably had only very vague and erroneous ideas of what was in them. If they had known, they would almost certainly have revised their books. Surely no author would intentionally provide nearly four times as much practice on 2 + 2 as on 8 + 8, or eight times as much practice on 2 × 2 as on 9 × 8, or eleven times as much practice on 2 − 2 as on 17 − 8, or over forty times as much practice on 2 ÷ 2 as on 75 ÷ 8 and 75 ÷ 9, both together. Surely no author would have provided intentionally only twenty to thirty occurrences each of 16 − 7, 16 − 8, 16 − 9, 17 − 8, 17 − 9, and 18 − 9 for the entire course through grade 6; or have left the practice on 60 ÷ 7, 60 ÷ 8, 60 ÷ 9, 61 ÷ 7, 61 ÷ 8, 61 ÷ 9, and the like to occur only about once a year!
TABLE 3
Amount of Practice: Addition Bonds in a Recent Textbook (A) of Excellent Repute. Books I and II, All Save Four Sections of Supplementary Material, to be Used at the Teacher's Discretion
The Table reads: 2 + 2 was used 226 times, 12 + 2 was used 74 times, 22 + 2, 32 + 2, 42 + 2, and so on were used 50 times.
2 3 4 5 6 7 8 9Total
2 226 154 162 150 97 87 66 45
12 74 53 76 46 51 37 36 33
22, etc. 50 60 68 63 42 50 38 26
3 216 141 127 89 82 54 58 40
13 43 43 60 70 52 30 22 18
23, etc. 15 30 51 50 42 32 29 30
7 85 90 103 103 84 81 61 47
17 35 25 42 32 35 21 29 16
27, etc. 30 23 32 29 24 23 25 28
8 185 112 146 99 75 71 73 61
18 28 35 52 46 28 29 24 14
28, etc. 53 36 34 38 23 36 27 27
9 104 81 112 96 63 74 58 57
19 13 11 31 38 25 14 22 11
29, etc. 19 17 27 20 32 32 19 18
2, 12, 22, etc. 350 277 306 260 190 174 140 1041801
3, 13, 23, etc. 274 214 230 209 176 116 109 881406
7, 17, 27, etc. 148 138 187 164 141 125 115 911109
8, 18, 28, etc. 266 183 232 185 126 136 124 1021354
9, 19, 29, etc. 136 109 170 154 120 120 99 86 994
Totals1164 9211125 972 753 671 687 471
TABLE 4
Amount of Practice: Subtraction Bonds in a Recent Textbook (A) of Excellent Repute. Books I and II, All Save Four Sections of Supplementary Material, to be Used at the Teacher's Discretion
MinuendsSubtrahends
1 2 3 4 5 6 7 8 9
1 372
2 214 311
3 136 149 189
4 146 142 103 205
5 171 91 92 164 136
6 80 59 69 71 81 192
7 106 57 55 67 59 156 80
8 73 50 50 75 50 62 48 152
9 71 75 54 74 48 55 55 124 133
10 261 84 63 100 193 83 57 124 91
11 48 31 50 36 41 32 46 35
12 48 77 57 51 35 80 30
13 35 22 40 29 35 28
14 25 37 36 49 32
15 33 19 48 20
16 16 36 26
17 27 20
18 19
Total excluding 1 − 1, 2 − 2, etc.1258 755 565 713 571 558 327 569 301
TABLE 5
Frequencies of Subtractions not Included in Table 4
These are cases where the pupil would, by reason of his stage of advancement, probably operate 35 − 30, 46 − 46, etc., as one bond.
MinuendsSubtrahends
1 11 21 etc. 2 12 22 etc. 3 13 23 etc. 4 14 24 etc. 5 15 25 etc. 6 16 26 etc. 7 17 27 etc. 8 18 28 etc. 9 19 29 etc. 10 20 etc.
10, 20, 30, 40, etc. 11 29 16 52 32 51 7 30 22 60
11, 21, 31, 41, etc. 42 14 22 32 12 26 19 52 17 10
12, 22, 32, 42, etc. 47 97 5 13 9 21 11 24 19 17
13, 23, 33, 43, etc. 7 40 7 14 15 13 19 19 22 3
14, 24, 34, 44, etc. 8 28 14 58 13 16 14 26 19 7
15, 25, 35, 45, etc. 21 28 29 54 51 15 21 12 24 8
16, 26, 36, 46, etc. 5 18 12 27 35 69 13 17 19 2
17, 27, 37, 47, etc. 5 9 12 40 32 54 24 12 12 1
18, 28, 38, 48, etc. 2 16 10 23 22 36 18 47 16 0
19, 29, 39, etc. 5 7 7 10 13 28 14 23 16 0
Totals153286134323234329160262186108
TABLE 6
Amount of Practice: Multiplication Bonds in Another Recent Textbook (B) of Excellent Repute. Books I and II
MultipliersMultiplicands
0 1 2 3 4 5 6 7 8 9Totals
1 299 534 472 271 310 293 261 178 195 99 2912
2 350 644 668 480 458 377 332 238 239 155 3941
3 280 487 509 388 318 302 247 199 227 152 3109
4 186 375 398 242 203 265 197 163 159 93 2281
5 268 359 393 234 263 243 217 192 197 114 2480
6 180 284 265 199 196 191 168 169 165 106 1923
7 135 283 277 176 187 158 155 121 145 118 1755
8 137 272 292 175 192 164 158 157 126 126 1799
9 71 173 140 122 97 102 101 100 82 110 1098
Totals1906341134142287222420951836151715351073
TABLE 7
Amount Of Practice: Divisions Without Remainder In Textbook B, Parts I And II
DividendsDivisors
2 3 4 5 6 7 8 9Totals
Integral multiples of 2 to 9 in sequence; i.e., 4 ÷ 2 occurred 397 times, 6 ÷ 2 occurred 256 times, 6 ÷ 3, 224 times, 9 ÷ 3, 124 times. 397 224 250 130 93 44 98 23 1259
256 124 152 79 28 43 61 25 768
318 123 130 65 50 19 39 19 763
258 98 86 105 25 24 34 20 650
198 49 76 27 22 30 33 16 451
77 180 69 54 91 46 36 50 37 31 38 24 28 17 12 27 13 17 16 22 16 9 16 15 278 427 236
Totals1753 809 817 499 275 217 319 142
TABLE 8
Division Bonds, With And Without Remainders. Book B
All work through grade 6, except estimates of quotient figures in long division.
Dividend2345
Divisor12123123412345
Number of Occurrences41386271892402639766185231364353135
Dividend67
Divisor1234561234567
Number of Occurrences2125622468438323725538463254
Dividend89
Divisor12345678123456789
Number of Occurrences1731830250222839911950124492515183038
Dividend1011
Divisor2345678923456789
Number of Occurrences258384612019924243221163711143
Dividend1213
Divisor2345678923456789
Number of Occurrences19812315229939167451615117453
Dividend1415
Divisor2345678923456789
Number of Occurrences772013584486699816798846
Dividend1617
Divisor2345678923456789
Number of Occurrences180191301469983619151466123
Dividend1819
Divisor2345678923456789
Number of Occurrences694913628772321610534104
Dividend2021
Divisor34567893456789
Number of Occurrences24866511323554128543105
Dividend2223
Divisor34567893456789
Number of Occurrences17161581361578118632
Dividend2425
Divisor34567893456789
Number of Occurrences91761850561111131055653
Dividend2627
Divisor34567893456789
Number of Occurrences56334634681042625
Dividend2829
Divisor34567893456789
Number of Occurrences43683193768051123
Dividend3031
Divisor456789456789
Number of Occurrences2127256713431142
Dividend3233
Divisor456789456789
Number of Occurrences501136395877261
Dividend3435
Divisor456789456789
Number of Occurrences835211103152453
Dividend3637
Divisor456789456789
Number of Occurrences37162226191287539
Dividend3839
Divisor456789456789
Number of Occurrences787115437431
Dividend404142
Divisor567895678956789
Number of Occurrences38923426637572830103
Dividend434445
Divisor567895678956789
Number of Occurrences7510327645024671020
Dividend464748
Divisor567895678956789
Number of Occurrences33222622037174332
Dividend49505152
Divisor56789678967896789
Number of Occurrences472792463823125553
Dividend53545556
Divisor6789678967896789
Number of Occurrences43221251165342013168
Dividend5758596061
Divisor678967896789789789
Number of Occurrences031322312303391125
Dividend626364656667
Divisor789789789789789789
Number of Occurrences461175952201101214011
Dividend6869707172737475
Divisor789789898989898989
Number of Occurrences13206162101610753353
Dividend7677787980818283848586878889
Divisor898989899999999999
Number of Occurrences3230410241524120327
Tables 3 to 8 show that even gifted authors make instruments for instruction in arithmetic which contain much less practice on certain elementary facts than teachers suppose; and which contain relatively much more practice on the more easily learned facts than on those which are harder to learn.
How much practice should be given in arithmetic? How should it be divided among the different bonds to be formed? Below a certain amount there is waste because, as has been shown in Chapter VI, the pupil will need more time to detect and correct his errors than would have been required to give him mastery. Above a certain amount there is waste because of unproductive overlearning. If 668 is just enough for 2 × 2, 82 is not enough for 9 × 8. If 82 is just enough for 9 × 8, 668 is too much for 2 × 2.
It is possible to find the answers to these questions for the pupil of median ability (or any stated ability) by suitable experiments. The amount of practice will, of course, vary according to the ability of the pupil. It will also vary according to the interest aroused in him and the satisfaction he feels in progress and mastery. It will also vary according to the amount of practice of other related bonds; 7 + 7 = 14 and 60 ÷ 7 = 8 and 4 remainder will help the formation of 7 + 8 = 15 and 61 ÷ 7 = 8 and 5 remainder. It will also, of course, vary with the general difficulty of the bond, 17 − 8 = 9 being under ordinary conditions of teaching harder to form than 7 − 2 = 5.
Until suitable experiments are at hand we may estimate for the fundamental bonds as follows, assuming that by the end of grade 6 a strength of 199 correct out of 200 is to be had, and that the teaching is by an intelligent person working in accord with psychological principles as to both ability and interest.
For one of the easier bonds, most facilitated by other bonds (such as 2 × 5 = 10, or 10 − 2 = 8, or the double bond 7 = two 3s and 1 remainder) in the case of the median or average pupil, twelve practices in the week of first learning, supported by twenty-five practices during the two months following, and maintained by thirty practices well spread over the later periods should be enough. For the more gifted pupils lesser amounts down to six, twelve, and fifteen may suffice. For the less gifted pupils more may be required up to thirty, fifty, and a hundred. It is to be doubted, however, whether pupils requiring nearly two hundred repetitions of each of these easy bonds should be taught arithmetic beyond a few matters of practical necessity.
For bonds of ordinary difficulty, with average facilitation from other bonds (such as 11 − 3, 4 × 7, or 48 ÷ 8 = 6) in the case of the median or average pupil, we may estimate twenty practices in the week of first learning, supported by thirty, and maintained by fifty practices well spread over the later periods. Gifted pupils may gain and keep mastery with twelve, fifteen, and twenty practices respectively. Pupils dull at arithmetic may need up to twenty, sixty, and two hundred. Here, again, it is to be doubted whether a pupil for whom arithmetical facts, well taught and made interesting, are so hard to acquire as this, should learn many of them.
For bonds of greater difficulty, less facilitated by other bonds (such as 17 − 9, 8 × 7, or 12½% of = 1⁄8 of), the practice may be from ten to a hundred percent more than the above.
UNDERLEARNING AND OVERLEARNING
If we accept the above provisional estimates as reasonable, we may consider the harm done by giving less and by giving more than these reasonable amounts. Giving less is indefensible. The pupil's time is wasted in excessive checking to find his errors. He is in danger of being practiced in error. His attention is diverted from the learning of new facts and processes by the necessity of thinking out these supposedly mastered facts. All new bonds are harder to learn than they should be because the bonds which should facilitate them are not strong enough to do so. Giving more does harm to some extent by using up time that could be spent better for other purposes, and (though not necessarily) by detracting from the pupil's interest in arithmetic. In certain cases, however, such excess practice and overlearning are actually desirable. Three cases are of special importance.
The first is the case of a bond operating under a changed mental set or adjustment. A pupil may know 7 × 8 adequately as a thing by itself, but need more practice to operate it in
285 7 ——
where he has to remember that 3 is to be added to the 56 when he obtains it, and that only the 9 is to be written down, the 5 to be held in mind for later use. The practice required to operate the bond efficiently in this new set is desirable, even though it is excess from a narrower point of view, and causes the straightforward 'seven eights are fifty-six' to be overlearned. So also a pupil's work with 24, 34, 44, etc., + 9 may react to give what would be excess practice from the point of view of 4 + 9 alone; his work in estimating approximate quotient figures in long division may give excess practice on the division tables. There are many such cases. Even adding the 5 and 7 in 5⁄12 + 7⁄12 is not quite the same task as adding 5 and 7 undisturbed by the fact that they are twelfths. We know far too little about the amount of practice needed to adapt arithmetical bonds to efficient operation in these more complicated conditions to estimate even approximately the allowances to be made. But some allowance, and often a rather large allowance, must be made.
The second is the case where the computation in general should be made very easy and sure for the pupil except for some one new element that is being learned. For example, in teaching the meaning and uses of 'Averages' and of uneven division, we may deliberately use 2, 3, and 4 as divisors rather than 7 and 9, so as to let all the pupil's energy be spent in learning the new facts, and so that the fraction in the quotient may be something easily understood, real, and significant. In teaching the addition of mixed numbers, we may use, in the early steps,
11½ 13½ 24 —— rather than 79½ 98½ 67 ——
so as to save attention for the new process itself. In cancellation, we may give excess practice to divisions by 2, 3, 4, and 5 in order to make the transfer to the new habits of considering two numbers together from the point of view of their divisibility by some number. In introducing trade discount, we may give excess practice on '5% of' and '10% of' deliberately, so that the meaning of discount may not be obscured by difficulties in the computation itself. Excess practice on, and overlearning of, certain bonds is thus very often justifiable.
The third case concerns bonds whose importance for practical uses in life or as notable facilitators of other bonds is so great that they may profitably be brought to a greater strength than 199 correct out of 200 at a speed of 2 sec. or less, or be brought to that degree of strength very early. Examples of bonds of such special practical use are the subtractions from 10, ½ + ½, ½ + ¼, ½ of 60, ¼ of 60, and the fractional parts of 12 and of $1.00. Examples of notable facilitating bonds are ten 10s = 100, ten 100s = 1000, additions like 2 + 2, 3 + 3, and 4 + 4, and all the multiplication tables to 9 × 9.
In consideration of these three modifying cases or principles, a volume could well be written concerning just how much practice to give to each bond, in each of the types of complex situations where it has to operate. There is evidently need for much experimentation to expose the facts, and for much sagacity and inventiveness in making sure of effective learning without wasteful overlearning.
The facts of primary importance are:—
(1) The textbook or other instrument of instruction which is a teacher's general guide may give far too little practice on certain bonds.
(2) It may divide the practice given in ways that are apparently unjustifiable.
(3) The teacher needs therefore to know how much practice it does give, where to supplement it, and what to omit.
(4) The omissions, on grounds of apparent excess practice, should be made only after careful consideration of the third principle described above.
(5) The amount of practice should always be considered in the light of its interest and appeal to the pupil's tendency to work with full power and zeal. Mere repetition of bonds when the learner does not care whether he is improving is rarely justifiable on any grounds.
(6) Practice that is actually in excess is not a very grave defect if it is enjoyed and improves the pupil's attitude toward arithmetic. Not much time is lost; a hundred practices for each of a thousand bonds after mastery to 199 in 200 at 2 seconds will use up less than 60 hours, or 15 hours per year in grades 3 to 6.
(7) By the proper division of practice among bonds, the arrangement of learning so that each bond helps the others, the adroit shifting of practice of a bond to each new type of situation requiring it to operate under changed conditions, and the elimination of excess practice where nothing substantial is gained, notable improvements over the past hit-and-miss customs may be expected.
(8) Unless the material for practice is adequate, well balanced and sufficiently motivated, the teacher must keep close account of the learning of pupils. Otherwise disastrous underlearning of many bonds is almost sure to occur and retard the pupil's development.
THE ORGANIZATION OF ABILITIES
There is danger that the need of brevity and simplicity which has made us speak so often of a bond or an ability, and of the amount of practice it requires, may mislead the reader into thinking that these bonds and abilities are to be formed each by itself alone and kept so. They should rarely be formed so and never kept so. This we have indicated from time to time by references to the importance of forming a bond in the way in which it is to be used, to the action of bonds in changed situations, to facilitation of one bond by others, to the coöperation of abilities, and to their integration into a total arithmetical ability.
As a matter of fact, only a small part of drill work in arithmetic should be the formation of isolated bonds. Even the very young pupil learning 5 and 3 are 8 should learn it with '5 and 5 = 10,' '5 and 2 = 7,' at the back of his mind, so to speak. Even so early, 5 + 3 = 8 should be part of an organized, coöperating system of bonds. Later 50 + 30 = 80 should become allied to it. Each bond should be considered, not simply as a separate tool to be put in a compartment until needed, but also as an improvement of one total tool or machine, arithmetical ability.
There are differences of course. Knowledge of square root can be regarded somewhat as a separate tool to be sharpened, polished, and used by itself, whereas knowledge of the multiplication tables cannot. Yet even square root is probably best made more closely a part of the total ability, being taught as a special case of dividing where divisor is to be the same as quotient, the process being one of estimating and correcting.
In general we do not wish the pupil to be a repository of separated abilities, each of which may operate only if you ask him the sort of questions which the teacher used to ask him, or otherwise indicate to him which particular arithmetical tool he is to use. Rather he is to be an effective organization of abilities, coöperating in useful ways to meet the quantitative problems life offers. He should not as a rule have to think in such fashion as: "Is this interest or discount? Is it simple interest or compound interest? What did I do in compound interest? How do I multiply by 2 percent?" The situation that calls up interest should also call up the kind of interest that is appropriate, and the technique of operating with percents should be so welded together with interest in his mind that the right coöperation will occur almost without supervision by him.
As each new ability is acquired, then, we seek to have it take its place as an improvement of a thinking being, as a coöperative member of a total organization, as a soldier fighting together with others, as an element in an educated personality. Such an organization of bonds will not form itself any more than any one bond will create itself. If the elements of arithmetical ability are to act together as a total organized unified force they must be made to act together in the course of learning. What we wish to have work together we must put together and give practice in teamwork.
We can do much to secure such coöperative action when and where and as it is needed by a very simple expedient; namely, to give practice with computation and problems such as life provides, instead of making up drills and problems merely to apply each fact or principle by itself. Though a pupil has solved scores of problems reading, "A triangle has a base of a feet and an altitude of b feet, what is its area?" he may still be practically helpless in finding the area of a triangular plot of ground; still more helpless in using the formula for a triangle which is one of two into which a trapezoid is divided. Though a pupil has learned to solve problems in trade discount, simple interest, compound interest, and bank discount one at a time, stated in a few set forms, he may be practically helpless before the actual series of problems confronting him in starting in business, and may take money out of the savings bank when he ought to borrow on a time loan, or delay payment on his bills when by paying cash he could save money as well as improve his standing with the wholesaler.
Instead of making up problems to fit the abilities given by school instruction, we should preferably modify school instruction so that arithmetical abilities will be organized into an effective total ability to meet the problems that life will offer. Still more generally, every bond formed should be formed with due consideration of every other bond that has been or will be formed; every ability should be practiced in the most effective possible relations with other abilities.