1 + 12 − 11 × 12 ÷ 11 + 22 − 22 × 12 ÷ 21 + 33 × 11 + 44 × 11 + 53 − 15 × 13 ÷ 11 + 63 − 26 × 13 ÷ 21 + 73 − 37 × 13 ÷ 31 + 88 × 11 + 99 × 14 − 14 ÷ 14 − 24 ÷ 211 (or 21 or 31, etc.) + 14 − 31 × 24 ÷ 311 " + 24 − 42 × 24 ÷ 411 " + 33 × 211 " + 44 × 211 " + 55 − 15 × 25 ÷ 111 " + 65 − 26 × 25 ÷ 211 " + 75 − 37 × 25 ÷ 311 " + 85 − 48 × 25 ÷ 411 " + 95 − 59 × 25 ÷ 56 − 11 × 36 ÷ 12 + 16 − 22 × 36 ÷ 22 + 26 − 33 × 36 ÷ 32 + 36 − 44 × 36 ÷ 42 + 46 − 55 × 36 ÷ 52 + 56 − 66 × 36 ÷ 62 + 67 × 32 + 78 × 32 + 87 − 19 × 37 ÷ 12 + 97 − 27 ÷ 27 − 37 ÷ 37 − 41 × 47 ÷ 412 (or 22 or 32, etc.) + 17 − 52 × 47 ÷ 512 " + 27 − 6and so on7 ÷ 67 − 7to 9 × 97 ÷ 7and so on toand so onand so on to9 + 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 ÷ 94 "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
ArithmeticalFactLowestEstimateMedianEstimateHighestEstimateRange Required toInclude Half ofthe Estimates3 or 13 or 23, etc. + 22515001,000,000800-5000" " 324145080,000475-5000" " 423115050,000750-5000" " 522140044,000700-5000" " 621135041,000700-4500" " 721150037,000600-4000" " 820140033,000550-4100" " 920115028,000650-45007 or 17 or 27, etc. + 22012502,000,000600-5000" " 31911001,000,000650-4900" " 418100080,000650-4900" " 517130080,000650-4400" " 616110029,000650-4500" " 715110025,000500-4500" " 813110021,000650-3800" " 910127517,000500-40003 − 3251000100,000500-40004 − 3201050500,000525-30005 − 32011002,500,000650-42006 − 310105021,000650-32507 − 322110015,000550-30508 − 321107515,000650-30009 − 321100015,000700-260010 − 320100020,000600-250011 − 320100015,000465-255012 − 318100015,000650-21007 − 710100018,000425-30008 − 715100018,000413-31009 − 71595018,000550-300010 − 71595018,000600-395011 − 71090018,000550-300012 − 71092518,000525-310013 − 71090018,000500-260014 − 71090018,000500-310015 − 71092518,000500-300016 − 71087518,000500-25009 × 71070020,000500-20007 × 91070020,000500-17508 × 61075020,000500-25006 × 8970020,000500-250063 ÷ 995004,500300-250064 ÷ 992004,000100- 70065 ÷ 982004,000100- 60066 ÷ 972004,000100- 55067 ÷ 972004,00075- 45068 ÷ 962004,00087- 57569 ÷ 962004,00087- 45070 ÷ 952004,00075- 57571 ÷ 952004,00075- 700XX405501,000,000300-2000XO2050011,500150-2000XXX1545012,000100-1000XXO2540015,000150-1000XOO154005,000100-1000XOX1040010,000100- 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 amountof 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.
23456789Total22261541621509787664512745376465137363322, etc.50606863425038263216141127898254584013434360705230221823, etc.1530515042322930785901031038481614717352542323521291627, etc.30233229242325288185112146997571736118283552462829241428, etc.5336343823362727910481112966374585719131131382514221129, etc.19172720323219182, 12, 22, etc.35027730626019017414010418013, 13, 23, etc.2742142302091761161098814067, 17, 27, etc.1481381871641411251159111098, 18, 28, etc.26618323218512613612410213549, 19, 29, etc.1361091701541201209986994Totals11649211125972753671687471
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
MinuendsSubtrahends12345678913722214311313614918941461421032055171919216413668059697181192710657556759156808735050755062481529717554744855551241331026184631001938357124911148315036413246351248775751358030133522402935281425373649321533194820161636261727201819Total excluding1 − 1,2 − 2,etc.1258755565713571558327569301
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.
MinuendsSubtrahends11121etc.21222etc.31323etc.41424etc.51525etc.61626etc.71727etc.81828etc.91929etc.1020etc.10,20,30,40, etc.112916523251730226011,21,31,41, etc.4214223212261952171012,22,32,42, etc.47975139211124191713,23,33,43, etc.7407141513191922314,24,34,44, etc.82814581316142619715, 25, 35, 45,etc.212829545115211224816, 26, 36, 46,etc.51812273569131719217, 27, 37, 47,etc.5912403254241212118, 28, 38, 48,etc.21610232236184716019, 29, 39,etc.5771013281423160Totals153286134323234329160262186108
TABLE 6
Amount of Practice: Multiplication Bonds in Another Recent Textbook (B) of Excellent Repute. Books I and II
MultipliersMultiplicands0123456789Totals1299534472271310293261178195992912235064466848045837733223823915539413280487509388318302247199227152310941863753982422032651971631599322815268359393234263243217192197114248061802842651991961911681691651061923713528327717618715815512114511817558137272292175192164158157126126179997117314012297102101100821101098Totals1906341134142287222420951836151715351073
TABLE 7
Amount Of Practice: Divisions Without Remainder In Textbook B, Parts I And II
DividendsDivisors23456789TotalsIntegral multiples of2to9in sequence;i.e.,4 ÷ 2occurred397times,6 ÷ 2occurred256times,6 ÷ 3, 224times,9 ÷ 3, 124times.3972242501309344982312592561241527928436125768318123130655019391976325898861052524342065019849762722303316451771806954914636503731382428171227131716221691615278427236Totals1753809817499275217319142
TABLE 8
Division Bonds, With And Without Remainders. Book B
All work through grade 6, except estimates of quotient figures in long division.Dividend2345Divisor12123123412345Number ofOccurrences41386271892402639766185231364353135Dividend67Divisor1234561234567Number ofOccurrences2125622468438323725538463254Dividend89Divisor12345678123456789Number ofOccurrences1731830250222839911950124492515183038Dividend1011Divisor2345678923456789Number ofOccurrences258384612019924243221163711143Dividend1213Divisor2345678923456789Number ofOccurrences19812315229939167451615117453Dividend1415Divisor2345678923456789Number ofOccurrences772013584486699816798846Dividend1617Divisor2345678923456789Number ofOccurrences180191301469983619151466123Dividend1819Divisor2345678923456789Number ofOccurrences694913628772321610534104Dividend2021Divisor34567893456789Number ofOccurrences24866511323554128543105Dividend2223Divisor34567893456789Number ofOccurrences17161581361578118632Dividend2425Divisor34567893456789Number ofOccurrences91761850561111131055653Dividend2627Divisor34567893456789Number ofOccurrences56334634681042625Dividend2829Divisor34567893456789Number ofOccurrences43683193768051123Dividend3031Divisor456789456789Number ofOccurrences2127256713431142Dividend3233Divisor456789456789Number ofOccurrences501136395877261Dividend3435Divisor456789456789Number ofOccurrences835211103152453Dividend3637Divisor456789456789Number ofOccurrences37162226191287539Dividend3839Divisor456789456789Number ofOccurrences787115437431Dividend404142Divisor567895678956789Number ofOccurrences38923426637572830103Dividend434445Divisor567895678956789Number ofOccurrences7510327645024671020Dividend464748Divisor567895678956789Number ofOccurrences33222622037174332Dividend49505152Divisor56789678967896789Number ofOccurrences472792463823125553Dividend53545556Divisor6789678967896789Number ofOccurrences43221251165342013168Dividend5758596061Divisor678967896789789789Number ofOccurrences031322312303391125Dividend626364656667Divisor789789789789789789Number ofOccurrences461175952201101214011Dividend6869707172737475Divisor789789898989898989Number ofOccurrences13206162101610753353Dividend7677787980818283848586878889Divisor898989899999999999Number ofOccurrences3230410241524120327
All work through grade 6, except estimates of quotient figures in long division.
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 laterperiods. 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 =1⁄8of), the practice may be from ten to a hundred percent more than the above.
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.