THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS
An inventory of the bonds to be formed in learning arithmetic should be
accompanied by a statement of how strong each bond is to be made and
kept year by year. Since, however, the inventory itself has been
presented here only in samples, the detailed statement of desired
strength for each bond cannot be made. Only certain general facts will
be noted here.
THE NEED OF STRONGER ELEMENTARY BONDS
The constituent bonds involved in the fundamental operations with
numbers need to be much stronger than they now are. Inaccuracy in these
operations means weakness of the constituent bonds. Inaccuracy exists,
and to a degree that deprives the subject of much of its possible
disciplinary value, makes the pupil's achievements of slight value for
use in business or industry, and prevents the pupil from verifying his
work with new processes by some previously acquired process.
The inaccuracy that exists may be seen in the measurements made by the
many investigators who have used arithmetical tasks as tests of fatigue,
practice, individual differences and the like, and in the special
studies of arithmetical achievements for their own sake made by Courtis
and others.
Burgerstein ['91], using such examples as
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, or 3/5 of an error per example. The children were 9-1/2
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 11-1/2 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 1-1/2
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:--
SPEED PERCENT OF
GRADE. ATTEMPTS. CORRECT ANSWERS.
Addition 8 12 80
7 11 80
6 10 70
5 9 70
4 8 70
Subtraction 8 12 90
7 11 90
6 10 90
5 9 80
4 7 80
Multiplication 8 11 80
7 10 80
6 9 80
5 7 70
4 6 60
Division 8 11 90
7 10 90
6 8 80
5 6 70
4 4 60
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 3
_B_ and 4 _A_ got 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.
3 5 6 2 3 8 9 7 4 9
7 9 6 5 5 6 4 5 8 2
3 4 7 8 7 3 7 9 3 7
8 8 4 8 2 6 8 2 9 8
2 2 4 7 6 9 8 5 6 2
6 9 5 7 8 5 2 3 2 4
9 6 4 2 7 2 9 4 4 5
3 3 7 9 9 9 2 8 9 7
6 8 9 6 4 7 7 9 2 4
8 4 6 9 9 2 6 9 8 9
-- -- -- -- -- -- -- -- -- --
20 = .... 5s
56 = .... 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!
EARLY MASTERY
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 to temporary 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 x 9 are used in examples like
729
8
---
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 x 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 x 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.
[7] These concern allowances for two errors occurring in the same
example and for the same wrong answer being obtained in both
original work and check work.
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 |APPROXIMATE |APPROXIMATE |APPROXIMATE |APPROXIMATE
THE |NUMBER OF |NUMBER OF |NUMBER OF |NUMBER OF
ELEMENTARY |WRONG ANSWERS|AGREEING |AGREEING |CHECKINGS
ADDITIONS |IN SUMS OF 5 |ANSWERS, |ANSWERS, |REQUIRED (OVER
TIMES RIGHT |THREE-PLACE |AFTER ONE |AFTER A |AND ABOVE THE
IN 1000 |NUMBERS PER |CHECKING, |CHECKING OF |FIRST GENERAL
|1000 |PER 1000 |THE FIRST |CHECKING OF
| | |DISCREPANCIES|THE 1000 SUMS)
| | | |TO SECURE TWO
| | | |AGREEING
| | | |RESULTS
-------------+-------------+-------------+-------------+--------------
960 | 700 | 90 | 216 | 4500
980 | 380 | 384 | 676 | 1200
990 | 190 | 656 | 906 | 470
995 | 95 | 819 | 975 | 210
996 | 76 | 854 | 984 | 165
997 | 54 | 895 | 992 | 115
998 | 38 | 925 | 996 | 80
999 | 19 | 962 | 999 | 40
-------------+-------------+-------------+-------------+--------------
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 3 or 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 against _inaccurate_ work 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 STRENGTH OF BONDS FOR TEMPORARY SERVICE
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 'Buy _a_ things at _b_ per thing, find
total cost' problem or a 'Buy _a_ things at _b_ per thing, what change
from _c_' problem or a 'What gain on buying for _a_ and selling for _b_'
problem or a 'How many things at _a_ each can I buy for _b_ cents'
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 they _should_ be, they _are 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 no
other 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 x in copying problems like those below:--
Add Subtract Multiply
23 79 32
61 24 3
-- -- --
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 Find the differences:--
remember that you must 39 67 78 56 45
subtract in this row. 23 44 36 26 24
-- -- -- -- --
Remember that you must Find the differences:--
subtract in this row. 85 27 96 38 78
63 14 51 45 32
-- -- -- -- --
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.
THE STRENGTH OF BONDS WITH TECHNICAL FACTS AND TERMS
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 between _cube 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 "$_X_ per M as a rate for lumber" to "Means $_X_ per 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 acquired they 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 STRENGTH OF BONDS CONCERNING THE REASONS FOR ARITHMETICAL PROCESSES
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 of _how_ you 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 do before 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 correctly _irrespective of_ the
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 lost _in their first form_, being used 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 propaedeutics to routine manipulations.
PROPAEDEUTIC BONDS
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 propaedeutic 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 cooeperates 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
propaedeutic bonds.
The bonds mentioned as illustrations are not _purely_ propaedeutic, nor
formed _only_ to 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
propaedeutic bonds may be worth while apart from their value in preparing
for other bonds. Consider, for example, exercises like those shown below
which are propaedeutic 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 x 25 3 x 15 2 x 12 4 x 11 6 x 25
3 x 25 10 x 15 2 x 15 4 x 15 6 x 15
5 x 25 4 x 15 2 x 25 4 x 12 6 x 12
10 x 25 2 x 15 2 x 11 4 x 25 6 x 11
4 x 25 7 x 15 3 x 25 5 x 11 7 x 12
6 x 25 9 x 15 3 x 15 5 x 12 7 x 15
8 x 25 5 x 15 3 x 11 5 x 15 7 x 25
7 x 25 8 x 15 3 x 12 5 x 25 7 x 11
9 x 25 6 x 15 8 x 12 9 x 12 8 x 25
State the missing numbers:--
A. 36 = .... 12s B. 44 = .... 11s C. 50 = .... 25s
60 = .... 12s 88 = .... 11s 125 = .... 25s
24 = .... 12s 77 = .... 11s 75 = .... 25s
48 = .... 12s 55 = .... 11s 200 = .... 25s
144 = .... 12s 99 = .... 11s 250 = .... 25s
108 = .... 12s 110 = .... 11s 175 = .... 25s
72 = .... 12s 33 = .... 11s 225 = .... 25s
96 = .... 12s 66 = .... 11s 150 = .... 25s
84 = .... 12s 22 = .... 11s 100 = .... 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|45 12|45 25|45 15|45 21|45 22|45
__ __ __ __ __ __
Row B. 25|55 11|55 12|55 15|55 22|55 30|55
__ __ __ __ __ __
Row C. 12|60 25|60 15|60 11|60 30|60 21|60
__ __ __ __ __ __
Row D. 12|75 11|75 15|75 25|75 30|75 35|75
___ ___ ___ ___ ___ ___
Row E. 11|100 12|100 25|100 15|100 30|100 22|100
__ __ __ __ __ __
Row F. 11|96 12|96 25|96 15|96 30|96 22|96
___ ___ ___ ___ ___ ___
Row G. 25|105 11|105 15|105 12|105 22|105 35|105
__ __ __ __ __ __
Row H. 12|64 15|64 25|64 11|64 22|64 21|64
__ __ __ __ __ __
Row I. 11|80 12|80 15|80 25|80 35|80 21|80
___ ___ ___ ___ ___ ___
Row J. 25|200 30|200 75|200 63|200 65|200 66|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 x 4) + 2,
(7 x 6) + 5, (9 x 4) + 6, given as a preparation for written
multiplication. The work of
48 68 47
3 7 9
-- -- --
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 x 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 propaedeutic or are formed _only_ for
later reconstruction. They then differ little from 'crutches.' The
typical crutch forms a habit which has actually to be broken, whereas
the purely propaedeutic 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:--
773 and 5 remainder
______
7)5416
49
--
51
49
--
26
21
--
5
The important recommendation concerning these purely propaedeutic 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 propaedeutic bonds of dubious value (with letters,
phonograms, diacritical marks, and the like) often carried to
demonstrably wasteful extremes.