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 + 1 2 - 1 1 x 1 2 / 1
1 + 2 2 - 2 2 x 1 2 / 2
1 + 3 3 x 1
1 + 4 4 x 1
1 + 5 3 - 1 5 x 1 3 / 1
1 + 6 3 - 2 6 x 1 3 / 2
1 + 7 3 - 3 7 x 1 3 / 3
1 + 8 8 x 1
1 + 9 9 x 1
4 - 1 4 / 1
4 - 2 4 / 2
11 (or 21 or 31, etc.) + 1 4 - 3 1 x 2 4 / 3
11 " + 2 4 - 4 2 x 2 4 / 4
11 " + 3 3 x 2
11 " + 4 4 x 2
11 " + 5 5 - 1 5 x 2 5 / 1
11 " + 6 5 - 2 6 x 2 5 / 2
11 " + 7 5 - 3 7 x 2 5 / 3
11 " + 8 5 - 4 8 x 2 5 / 4
11 " + 9 5 - 5 9 x 2 5 / 5
6 - 1 1 x 3 6 / 1
2 + 1 6 - 2 2 x 3 6 / 2
2 + 2 6 - 3 3 x 3 6 / 3
2 + 3 6 - 4 4 x 3 6 / 4
2 + 4 6 - 5 5 x 3 6 / 5
2 + 5 6 - 6 6 x 3 6 / 6
2 + 6 7 x 3
2 + 7 8 x 3
2 + 8 7 - 1 9 x 3 7 / 1
2 + 9 7 - 2 7 / 2
7 - 3 7 / 3
7 - 4 1 x 4 7 / 4
12 (or 22 or 32, etc.) + 1 7 - 5 2 x 4 7 / 5
12 " + 2 7 - 6 and so on 7 / 6
7 - 7 to 9 x 9 7 / 7
and so on to and so on and so on to
9 + 9 to 18 - 9 82 / 9
19 (or 29 or 39, etc.) + 9 83 / 9, etc.
If estimating for the entire series is too long a task, it will be
sufficient to use eight or ten from each, say:--
3 + 2 13, 23, etc. + 2 7 + 2 17, 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 - 3 7 - 7 9 x 7 63 / 9
4 " 8 " 7 x 9 64 "
5 " 9 " 8 x 6 65 "
6 " 10 " 6 x 8 66 "
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
======================================================================
| LOWEST | MEDIAN | HIGHEST |RANGE REQUIRED TO
ARITHMETICAL FACT |ESTIMATE|ESTIMATE|ESTIMATE | INCLUDE HALF OF
| | | | THE ESTIMATES
-----------------------+--------+--------+---------+------------------
3 or 13 or 23, etc. + 2| 25 | 1500 |1,000,000| 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 | 1250 |2,000,000| 600-5000
" " 3| 19 | 1100 |1,000,000| 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 | 1100 |2,500,000| 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 x 7 | 10 | 700 | 20,000| 500-2000
7 x 9 | 10 | 700 | 20,000| 500-1750
8 x 6 | 10 | 750 | 20,000| 500-2500
6 x 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 | 550 |1,000,000| 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 x 2 as on 9 x 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 | 9 | TOTAL
----------------+-----+-----+-----+-----+-----+-----+-----+----+------
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 | 104| 1801
3, 13, 23, etc. | 274 | 214 | 230 | 209 | 176 | 116 | 109 | 88| 1406
| | | | | | | | |
7, 17, 27, etc. | 148 | 138 | 187 | 164 | 141 | 125 | 115 | 91| 1109
8, 18, 28, etc. | 266 | 183 | 232 | 185 | 126 | 136 | 124 | 102| 1354
9, 19, 29, etc. | 136 | 109 | 170 | 154 | 120 | 120 | 99 | 86| 994
| | | | | | | | |
Totals |1164 | 921 |1125 | 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
================================================================
| SUBTRAHENDS
MINUENDS |-----------------------------------------------------
| 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.
======================================================================
| SUBTRAHENDS
|----+----+----+----+----+----+----+----+----+----
| 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
MINUENDS | 11| 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 10
| 21| 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 20
|etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.|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
| | | | | | | | | |
Totals |153 |286 |134 |323 |234 |329 |160 |262 |186 |108
=====================================================================
TABLE 6
AMOUNT OF PRACTICE: MULTIPLICATION BONDS IN ANOTHER RECENT TEXTBOOK (B)
OF EXCELLENT REPUTE. BOOKS I AND II
======================================================================
| MULTIPLICANDS
MULTIPLIERS |---------------------------------------------------------
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |Totals
------------+----+----+----+----+----+----+----+----+----+----+-------
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
| | | | | | | | | | |
Totals |1906|3411|3414|2287|2224|2095|1836|1517|1535|1073|
======================================================================
TABLE 7
AMOUNT OF PRACTICE: DIVISIONS WITHOUT REMAINDER IN TEXTBOOK B,
PARTS I AND II
======================================================================
| DIVISORS
DIVIDENDS |----------------------------------------------
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |Totals
-----------------------+----+----+----+----+----+----+----+----+------
Integral | 397| 224| 250| 130| 93| 44| 98| 23| 1259
multiples | | | | | | | | |
of 2 to 9 | 256| 124| 152| 79| 28| 43| 61| 25| 768
in sequence; | | | | | | | | |
_i.e._, 4 / 2 | 318| 123| 130| 65| 50| 19| 39| 19| 763
occurred | | | | | | | | |
397 times, | 258| 98| 86| 105| 25| 24| 34| 20| 650
6 / 2 occurred | | | | | | | | |
256 times, | 198| 49| 76| 27| 22| 30| 33| 16| 451
6 / 3, 224 times, | | | | | | | | |
9 / 3, 124 times. | 77| 54| 36| 31| 28| 27| 16| 9| 278
| 180| 91| 50| 38| 17| 13| 22| 16| 427
| 69| 46| 37| 24| 12| 17| 16| 15| 236
| | | | | | | | |
Totals |1753| 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.
Dividend 2 3 4 5
Divisor 1 2 1 2 3 1 2 3 4 1 2 3 4 5
Number of
Occurrences 41 386 27 189 240 26 397 66 185 23 136 43 53 135
Dividend 6 7
Divisor 1 2 3 4 5 6 1 2 3 4 5 6 7
Number of
Occurrences 21 256 224 68 43 83 23 72 55 38 46 32 54
Dividend 8 9
Divisor 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9
Number of
Occurrences 17 318 30 250 22 28 39 91 19 50 124 49 25 15 18 30 38
Dividend 10 11
Divisor 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
Number of
Occurrences 258 38 46 120 19 9 24 24 32 21 16 3 7 11 14 3
Dividend 12 13
Divisor 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
Number of
Occurrences 198 123 152 29 93 9 16 7 45 16 15 11 7 4 5 3
Dividend 14 15
Divisor 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
Number of
Occurrences 77 20 13 5 8 44 8 6 69 98 16 79 8 8 4 6
Dividend 16 17
Divisor 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
Number of
Occurrences 180 19 130 14 6 9 98 3 61 9 15 14 6 6 12 3
Dividend 18 19
Divisor 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
Number of
Occurrences 69 49 13 6 28 7 7 23 21 6 10 5 3 4 10 4
Dividend 20 21
Divisor 3 4 5 6 7 8 9 3 4 5 6 7 8 9
Number of
Occurrences 24 86 65 11 3 23 5 54 12 8 5 43 10 5
Dividend 22 23
Divisor 3 4 5 6 7 8 9 3 4 5 6 7 8 9
Number of
Occurrences 17 16 15 8 13 6 15 7 8 11 8 6 3 2
Dividend 24 25
Divisor 3 4 5 6 7 8 9 3 4 5 6 7 8 9
Number of
Occurrences 91 76 18 50 5 61 1 11 13 105 5 6 5 3
Dividend 26 27
Divisor 3 4 5 6 7 8 9 3 4 5 6 7 8 9
Number of
Occurrences 5 6 3 3 4 6 3 46 8 10 4 2 6 25
Dividend 28 29
Divisor 3 4 5 6 7 8 9 3 4 5 6 7 8 9
Number of
Occurrences 4 36 8 3 19 3 7 6 8 0 5 11 2 3
Dividend 30 31 32
Divisor 4 5 6 7 8 9 4 5 6 7 8 9 4 5 6 7 8 9
Number of
Occurrences 21 27 25 6 7 13 4 3 1 1 4 2 50 11 3 6 39 5
Dividend 33 34 35
Divisor 4 5 6 7 8 9 4 5 6 7 8 9 4 5 6 7 8 9
Number of
Occurrences 8 7 7 2 6 1 8 3 5 2 1 1 10 31 5 24 5 3
Dividend 36 37 38
Divisor 4 5 6 7 8 9 4 5 6 7 8 9 4 5 6 7 8 9
Number of
Occurrences 37 16 22 2 6 19 12 8 7 5 3 9 7 8 7 1 1 5
Dividend 39 40 41 42
Divisor 4 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9
Number of
Occurrences 4 3 7 4 3 1 38 9 2 34 2 6 6 3 7 5 7 28 30 10 3
Dividend 43 44 45 46
Divisor 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9
Number of
Occurrences 7 5 10 3 2 7 6 4 5 0 24 6 7 10 20 3 3 2 2 2
Dividend 47 48 49 50
Divisor 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 6 7 8 9
Number of
Occurrences 6 2 2 0 3 7 17 4 33 2 4 7 27 9 2 4 6 3 8
Dividend 51 52 53 54
Divisor 6 7 8 9 6 7 8 9 6 7 8 9 6 7 8 9
Number of
Occurrences 2 3 1 2 5 5 5 3 4 3 2 2 12 5 1 16
Dividend 55 56 57 58 59
Divisor 6 7 8 9 6 7 8 9 6 7 8 9 6 7 8 9 6 7 8 9
Number of
Occurrences 5 3 4 2 0 13 16 8 0 3 1 3 2 2 3 1 2 3 0 3
Dividend 60 61 62 63 64 65
Divisor 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9
Number of
Occurrences 3 9 1 1 2 5 4 6 1 17 5 9 5 22 0 1 10 1
Dividend 66 67 68 69 70 71
Divisor 7 8 9 7 8 9 7 8 9 7 8 9 8 9 8 9
Number of
Occurrences 2 1 4 0 1 1 1 3 2 0 6 1 6 2 1 0
Dividend 72 73 74 75 76 77 78 79
Divisor 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9
Number of
Occurrences 16 10 7 5 3 3 5 3 3 2 3 0 4 1 0 2
Dividend 80 81 82 83 84 85 86 87 88 89
Divisor 9 9 9 9 9 9 9 9 9 9
Number of
Occurrences 4 15 2 4 1 2 0 3 2 7
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 x 2,
82 is not enough for 9 x 8. If 82 is just enough for 9 x 8, 668 is too
much for 2 x 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 x 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 x 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 x 7, or 12-1/2% 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 x 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,
11-1/2
13-1/2
24
------
rather than
79-1/2
98-1/2
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, 1/2 + 1/2, 1/2 + 1/4, 1/2 of 60, 1/4 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 x 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
cooeperation 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, cooeperating 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, cooeperating 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 cooeperation 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 cooeperative 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 cooeperative 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_.