THE CONSTITUTION OF ARITHMETICAL ABILITIES (CONTINUED): THE SELECTION OF
THE BONDS TO BE FORMED
When the analysis of the mental functions involved in arithmetical
learning is made thorough it turns into the question, 'What are the
elementary bonds or connections that constitute these functions?' and
when the problem of teaching arithmetic is regarded, as it should be in
the light of present psychology, as a problem in the development of a
hierarchy of intellectual habits, it becomes in large measure a problem
of the choice of the bonds to be formed and of the discovery of the best
order in which to form them and the best means of forming each in that
order.
THE IMPORTANCE OF HABIT-FORMATION
The importance of habit-formation or connection-making has been grossly
underestimated by the majority of teachers and writers of textbooks.
For, in the first place, mastery by deductive reasoning of such matters
as 'carrying' in addition, 'borrowing' in subtraction, the value of the
digits in the partial products in multiplication, the manipulation of
the figures in division, the placing of the decimal point after
multiplication or division with decimals, or the manipulation of the
figures in the multiplication and division of fractions, is impossible
or extremely unlikely in the case of children of the ages and experience
in question. They do not as a rule deduce the method of manipulation
from their knowledge of decimal notation. Rather they learn about
decimal notation by carrying, borrowing, writing the last figure of each
partial product under the multiplier which gives that product, etc. They
learn the method of manipulating numbers by seeing them employed, and by
more or less blindly acquiring them as associative habits.
In the second place, we, who have already formed and long used the right
habits and are thereby protected against the casual misleadings of
unfortunate mental connections, can hardly realize the force of mere
association. When a child writes sixteen as 61, or finds 428 as the sum
of
15
19
16
18
--
or gives 642 as an answer to 27 x 36, or says that 4 divided by 1/4 = 1,
we are tempted to consider him mentally perverse, forgetting or perhaps
never having understood that he goes wrong for exactly the same general
reason that we go right; namely, the general law of habit-formation. If
we study the cases of 61 for 16, we shall find them occurring in the
work of pupils who after having been drilled in writing 26, 36, 46, 62,
63, and so on, in which the order of the six in writing is the same as
it is in speech, return to writing the 'teen numbers. If our language
said onety-one for eleven and onety-six for sixteen, we should probably
never find such errors except as 'lapses' or as the results of
misperception or lack of memory. They would then be more frequent
_before_ the 20s, 30s, etc., were learned.
If pupils are given much drill on written single column addition
involving the higher decades (each time writing the two-figure sum),
they are forming a habit of writing 28 after the sum of 8, 6, 9, and 5
is reached; and it should not surprise us if the pupil still
occasionally writes the two-figure sum for the first column though a
second column is to be added also. On the contrary, unless some counter
force influences him, he is absolutely sure to make this mistake.
The last mistake quoted (4 / 1/4 = 1) is interesting because here we
have possibly one of the cases where deduction from psychology alone can
give constructive aid to teaching. Multiplication and division by
fractions have been notorious for their difficulty. The former is now
alleviated by using _of_ instead of x until the new habit is fixed. The
latter is still approached with elaborate caution and with various means
of showing why one must 'invert and multiply' or 'multiply by the
reciprocal.'
But in the author's opinion it seems clear that the difficulty in
multiplying and dividing by a fraction was not that children felt any
logical objections to canceling or inverting. I fancy that the majority
of them would cheerfully invert any fraction three times over or cancel
numbers at random in a column if they were shown how to do so. But if
you are a youngster inexperienced in numerical abstractions and if you
have had _divide_ connected with 'make smaller' three thousand times and
never once connected with 'make bigger,' you are sure to be somewhat
impelled to make the number smaller the three thousand and first time
you are asked to divide it. Some of my readers will probably confess
that even now they feel a slight irritation or doubt in saying or
writing that 16/1 / 1/8 = 128.
The habits that have been confirmed by every multiplication and division
by integers are, in this particular of '_the ratio of result to number
operated upon_,' directly opposed to the formation of the habits
required with fractions. And that is, I believe, the main cause of the
difficulty. Its treatment then becomes easy, as will be shown later.
These illustrations could be added to almost indefinitely, especially in
the case of the responses made to the so-called 'catch' problems. The
fact is that the learner rarely can, and almost never does, survey and
analyze an arithmetical situation and justify what he is going to do by
articulate deductions from principles. He usually feels the situation
more or less vaguely and responds to it as he has responded to it or
some situation like it in the past. Arithmetic is to him not a logical
doctrine which he applies to various special instances, but a set of
rather specialized habits of behavior toward certain sorts of quantities
and relations. And in so far as he does come to know the doctrine it is
chiefly by doing the will of the master. This is true even with the
clearest expositions, the wisest use of objective aids, and full
encouragement of originality on the pupil's part.
Lest the last few paragraphs be misunderstood, I hasten to add that the
psychologists of to-day do not wish to make the learning of arithmetic a
mere matter of acquiring thousands of disconnected habits, nor to
decrease by one jot the pupil's genuine comprehension of its general
truths. They wish him to reason not less than he has in the past, but
more. They find, however, that you do not secure reasoning in a pupil by
demanding it, and that his learning of a general truth without the
proper development of organized habits back of it is likely to be, not a
rational learning of that general truth, but only a mechanical
memorizing of a verbal statement of it. They have come to know that
reasoning is not a magic force working in independence of ordinary
habits of thought, but an organization and cooeperation of those very
habits on a higher level.
The older pedagogy of arithmetic stated a general law or truth or
principle, ordered the pupil to learn it, and gave him tasks to do which
he could not do profitably unless he understood the principle. It left
him to build up himself the particular habits needed to give him
understanding and mastery of the principle. The newer pedagogy is
careful to help him build up these connections or bonds ahead of and
along with the general truth or principle, so that he can understand it
better. The older pedagogy commanded the pupil to reason and let him
suffer the penalty of small profit from the work if he did not. The
newer provides instructive experiences with numbers which will stimulate
the pupil to reason so far as he has the capacity, but will still be
profitable to him in concrete knowledge and skill, even if he lacks the
ability to develop the experiences into a general understanding of the
principles of numbers. The newer pedagogy secures more reasoning in
reality by not pretending to secure so much.
The newer pedagogy of arithmetic, then, scrutinizes every element of
knowledge, every connection made in the mind of the learner, so as to
choose those which provide the most instructive experiences, those which
will grow together into an orderly, rational system of thinking about
numbers and quantitative facts. It is not enough for a problem to be a
test of understanding of a principle; it must also be helpful in and of
itself. It is not enough for an example to be a case of some rule; it
must help review and consolidate habits already acquired or lead up to
and facilitate habits to be acquired. Every detail of the pupil's work
must do the maximum service in arithmetical learning.
DESIRABLE BONDS NOW OFTEN NEGLECTED
As hitherto, I shall not try to list completely the elementary bonds
that the course of study in arithmetic should provide for. The best
means of preparing the student of this topic for sound criticism and
helpful invention is to let him examine representative cases of bonds
now often neglected which should be formed and representative cases of
useless, or even harmful, bonds now often formed at considerable waste
of time and effort.
(1) _Numbers as measures of continuous quantities._--The numbers one,
two, three, 1, 2, 3, etc., should be connected soon after the beginning
of arithmetic each with the appropriate amount of some continuous
quantity like length or volume or weight, as well as with the
appropriate sized collection of apples, counters, blocks, and the like.
Lines should be labeled 1 foot, 2 feet, 3 feet, etc.; one inch, two
inches, three inches, etc.; weights should be lifted and called one
pound, two pounds, etc.; things should be measured in glassfuls,
handfuls, pints, and quarts. Otherwise the pupil is likely to limit the
meaning of, say, _four_ to four sensibly discrete things and to have
difficulty in multiplication and division. Measuring, or counting by
insensibly marked off repetitions of a unit, binds each number name to
its meaning as ---- _times whatever 1 is_, more surely than mere
counting of the units in a collection can, and should reenforce the
latter.
(2) _Additions in the higher decades._--In the case of all save the very
gifted children, the additions with higher decades--that is, the bonds,
16 + 7 = 23, 26 + 7 = 33, 36 + 7 = 43, 14 + 8 = 22, 24 + 8 = 32, and the
like--need to be specifically practiced until the tendency becomes
generalized. 'Counting' by 2s beginning with 1, and with 2, counting by
3s beginning with 1, with 2, and with 3, counting by 4s beginning with
1, with 2, with 3, and with 4, and so on, make easy beginnings in the
formation of the decade connections. Practice with isolated bonds should
soon be added to get freer use of the bonds. The work of column addition
should be checked for accuracy so that a pupil will continually get
beneficial practice rather than 'practice in error.'
(3) _The uneven divisions._--The quotients with remainders for the
divisions of every number to 19 by 2, every number to 29 by 3, every
number to 39 by 4, and so on should be taught as well as the even
divisions. A table like the following will be found a convenient means
of making these connections:--
10 = .... 2s
10 = .... 3s and .... rem.
10 = .... 4s and .... rem.
10 = .... 5s
11 = .... 2s and .... rem.
11 = .... 3s and .... rem.
.
.
.
89 = .... 9s and .... rem.
These bonds must be formed before short division can be efficient, are
useful as a partial help toward selection of the proper quotient figures
in long division, and are the chief instruments for one of the important
problem series in applied arithmetic,--"How many _x_s can I buy for _y_
cents at _z_ cents per _x_ and how much will I have left?" That these
bonds are at present sadly neglected is shown by Kirby ['13], who found
that pupils in the last half of grade 3 and the first half of grade 4
could do only about four such examples per minute (in a ten-minute
test), and even at that rate made far from perfect records, though they
had been taught the regular division tables. Sixty minutes of practice
resulted in a gain of nearly 75 percent in number done per minute, with
an increase in accuracy as well.
(4) _The equation form._--The equation form with an unknown quantity to
be determined, or a missing number to be found, should be connected with
its meaning and with the problem attitude long before a pupil begins
algebra, and in the minds of pupils who never will study algebra.
Children who have just barely learned to add and subtract learn easily
to do such work as the following:--
Write the missing numbers:--
4 + 8 = ....
5 + .... = 14
.... + 3 = 11
.... = 5 + 2
16 = 7 + ....
12 = .... + 5
The equation form is the simplest uniform way yet devised to state a
quantitative issue. It is capable of indefinite extension if certain
easily understood conventions about parentheses and fraction signs are
learned. It should be employed widely in accounting and the treatment of
commercial problems, and would be except for outworn conventions. It is
a leading contribution of algebra to business and industrial life.
Arithmetic can make it nearly as well. It saves more time in the case of
drills on reducing fractions to higher and lower terms alone than is
required to learn its meaning and use. To rewrite a quantitative
problem as an equation and then make the easy selection of the
necessary technique to solve the equation is one of the most universally
useful intellectual devices known to man. The words 'equals,' 'equal,'
'is,' 'are,' 'makes,' 'make,' 'gives,' 'give,' and their rarer
equivalents should therefore early give way on many occasions to the '='
which so far surpasses them in ultimate convenience and simplicity.
(5) _Addition and subtraction facts in the case of fractions._--In the
case of adding and subtracting fractions, certain specific
bonds--between the situation of halves and thirds to be added and the
responses of thinking of the numbers as equal to so many sixths, between
the situation thirds and fourths to be added and thinking of them as so
many twelfths, between fourths and eighths to be added and thinking of
them as eighths, and the like--should be formed separately. The general
rule of thinking of fractions as their equivalents with some convenient
denominator should come as an organization and extension of such special
habits, not as an edict from the textbook or teacher.
(6) _Fractional equivalents._--Efficiency requires that in the end the
much used reductions should be firmly connected with the situations
where they are needed. They may as well, therefore, be so connected from
the beginning, with the gain of making the general process far easier
for the dull pupils to master. We shall see later that, for all save the
very gifted pupils, the economical way to get an understanding of
arithmetical principles is not, usually, to learn a rule and then apply
it, but to perform instructive operations and, in the course of
performing them, to get insight into the principles.
(7) _Protective habits in multiplying and dividing with fractions._--In
multiplying and dividing with fractions special bonds should be formed
to counteract the now harmful influence of the 'multiply = get a larger
number' and 'divide = get a smaller number' bonds which all work with
integers has been reenforcing.
For example, at the beginning of the systematic work with multiplication
by a fraction, let the following be printed clearly at the top of every
relevant page of the textbook and displayed on the blackboard:--
_When you multiply a number by anything more than 1 the result is larger
than the number._
_When you multiply a number by 1 the result is the same as the number._
_When you multiply a number by anything less than 1 the result is
smaller than the number._
Let the pupils establish the new habit by many such exercises as:--
18 x 4 = .... 9 x 2 = ....
4 x 4 = .... 6 x 2 = ....
2 x 4 = .... 3 x 2 = ....
1 x 4 = .... 1 x 2 = ....
1/2 x 4 = .... 1/3 x 2 = ....
1/4 x 4 = .... 1/6 x 2 = ....
1/8 x 4 = .... 1/9 x 2 = ....
In the case of division by a fraction the old harmful habit should be
counteracted and refined by similar rules and exercises as follows:--
_When you divide a number by anything more than 1 the result is smaller
than the number._
_When you divide a number by 1 the result is the same as the number._
_When you divide a number by anything less than 1 the result is larger
than the number._
State the missing numbers:--
8 = .... 4s 12 = .... 6s 9 = .... 9s
8 = .... 2s 12 = .... 4s 9 = .... 3s
8 = .... 1s 12 = .... 3s 9 = .... 1s
8 = .... 1/2s 12 = .... 2s 9 = .... 1/3s
8 = .... 1/4s 12 = .... 1s 9 = .... 1/9s
8 = .... 1/8s 12 = .... 1/2s
12 = .... 1/3s
12 = .... 1/4s
16 / 16 = 9 / 9 = 10 / 10 = 12 / 6 =
16 / 8 = 9 / 3 = 10 / 5 = 12 / 4 =
16 / 4 = 9 / 1 = 10 / 1 = 12 / 3 =
16 / 2 = 9 / 1/3 = 10 / 1/5 = 12 / 2 =
16 / 1 = 9 / 1/9 = 10 / 1/10 = 12 / 1 =
16 / 1/2 = 12 / 1/2 =
16 / 1/4 = 12 / 1/3 =
16 / 1/8 = 12 / 1/4 =
12 / 1/6 =
(8) _'% of' means 'hundredths times'._--In the case of percentage a
series of bonds like the following should be formed:--
5 percent of = .05 times
20 " " " = .20 "
6 " " " = .06 "
25% " = .25 x
12% " = .12 x
3% " = .03 x
Four five-minute drills on such connections between '_x_ percent of' and
'its decimal equivalent times' are worth an hour's study of verbal
definitions of the meaning of percent as per hundred or the like. The
only use of the study of such definitions is to facilitate the later
formation of the bonds, and, with all save the brighter pupils, the
bonds are more needed for an understanding of the definitions than the
definitions are needed for the formation of the bonds.
(9) _Habits of verifying results._--Bonds should early be formed between
certain manipulations of numbers and certain means of checking, or
verifying the correctness of, the manipulation in question. The
additions to 9 + 9 and the subtractions to 18 - 9 should be verified by
objective addition and subtraction and counting until the pupil has sure
command; the multiplications to 9 x 9 should be verified by objective
multiplication and counting of the result (in piles of tens and a pile
of ones) eight or ten times,[4] and by addition eight or ten times;[4]
the divisions to 81 / 9 should be verified by multiplication and
occasionally objectively until the pupil has sure command; column
addition should be checked by adding the columns separately and adding
the sums so obtained, and by making two shorter tasks of the given task
and adding the two sums; 'short' multiplication should be verified eight
or ten times by addition; 'long' multiplication should be checked by
reversing multiplier and multiplicand and in other ways; 'short' and
'long' division should be verified by multiplication.
[4] Eight or ten times _in all_, not eight or ten times for each fact
of the tables.
These habits of testing an obtained result are of threefold value. They
enable the pupil to find his own errors, and to maintain a standard of
accuracy by himself. They give him a sense of the relations of the
processes and the reasons why the right ways of adding, subtracting,
multiplying, and dividing are right, such as only the very bright pupils
can get from verbal explanations. They put his acquisition of a certain
power, say multiplication, to a real and intelligible use, in checking
the results of his practice of a new power, and so instill a respect for
arithmetical power and skill in general. The time spent in such
verification produces these results at little cost; for the practice in
adding to verify multiplications, in multiplying to verify divisions,
and the like is nearly as good for general drill and review of the
addition and multiplication themselves as practice devised for that
special purpose.
Early work in adding, subtracting, and reducing fractions should be
verified by objective aids in the shape of lines and areas divided
in suitable fractional parts. Early work with decimal fractions
should be verified by the use of the equivalent common fractions
for .25, .75, .125, .375, and the like. Multiplication and division
with fractions, both common and decimal, should in the early stages
be verified by objective aids. The placing of the decimal point in
multiplication and division with decimal fractions should be verified
by such exercises as:--
20 It cannot be 200; for 200 x 1.23 is much more than 24.6.
______ It cannot be 2; for 2 x 1.23 is much less than 24.6.
1.23 )24.60
246
----
The establishment of habits of verifying results and their use is very
greatly needed. The percentage of wrong answers in arithmetical work in
schools is now so high that the pupils are often being practiced in
error. In many cases they can feel no genuine and effective confidence
in the processes, since their own use of the processes brings wrong
answers as often as right. In solving problems they often cannot decide
whether they have done the right thing or the wrong, since even if they
have done the right thing, they may have done it inaccurately. A wrong
answer to a problem is therefore too often ambiguous and uninstructive
to them.[5]
[5] The facts concerning the present inaccuracy of school work in
arithmetic will be found on pages 102 to 105.
These illustrations of the last few pages are samples of the procedures
recommended by a consideration of all the bonds that one might form and
of the contribution that each would make toward the abilities that the
study of arithmetic should develop and improve. It is by doing more or
less at haphazard what psychology teaches us to do deliberately and
systematically in this respect that many of the past advances in the
teaching of arithmetic have been made.
WASTEFUL AND HARMFUL BONDS
A scrutiny of the bonds now formed in the teaching of arithmetic with
questions concerning the exact service of each, results in a list of
bonds of small value or even no value, so far as a psychologist can
determine. I present here samples of such psychologically unjustifiable
bonds with some of the reasons for their deficiencies.
(1) _Arbitrary units._--In drills intended to improve the ability to see
and use the meanings of numbers as names for ratios or relative
magnitudes, it is unwise to employ entirely arbitrary units. The
procedure in II (on page 84) is better than that in I. Inches,
half-inches, feet, and centimeters are better as units of length than
arbitrary As. Square inches, square centimeters, and square feet are
better for areas. Ounces and pounds should be lifted rather than
arbitrary weights. Pints, quarts, glassfuls, cupfuls, handfuls, and
cubic inches are better for volume.
All the real merit in the drills on relative magnitude advocated by
Speer, McLellan and Dewey, and others can be secured without spending
time in relating magnitudes for the sake of relative magnitude alone.
The use of units of measure in drills which will never be used in _bona
fide_ measuring is like the use of fractions like sevenths, elevenths,
and thirteenths. A very little of it is perhaps desirable to test the
appreciation of certain general principles, but for regular training it
should give place to the use of units of practical significance.
[Illustration: FIG. 3.
A ----------
B ------------------------------
C --------------------
D ----------------------------------------
I. If _A_ is 1 which line is 2? Which line is 4? Which line is 3?
_A_ and _C_ together equal what line? _A_ and _B_ together equal
what line? How much longer is _B_ than _A_? How much longer is _B_
than _C_? How much longer is _D_ than _A_?]
[Illustration: FIG. 4.
A ----------
B ------------------------------
C --------------------
D ----------------------------------------
II. _A_ is 1 inch long. Which line is 2 inches long? Which line is
4 inches long? Which line is 3 inches long? _A_ and _C_ together
make ... inches? _A_ and _B_ together make ... inches? _B_ is ...
... longer than _A_? _B_ is ... ... longer than _C_? _D_ is ...
... longer than _A_?]
(2) _Multiples of 11._--The multiplications of 2 to 12 by 11 and 12 as
single connections should be left for the pupil to acquire by himself as
he needs them. These connections interfere with the process of learning
two-place multiplication. The manipulations of numbers there required
can be learned much more easily if 11 and 12 are used as multipliers in
just the same way that 78 or 96 would be. Later the 12 x 2, 12 x 3,
etc., may be taught. There is less reason for knowing the multiples
of 11 than for knowing the multiples of 15, 16, or 25.
(3) _Abstract and concrete numbers._--The elaborate emphasis of the
supposed fact that we cannot multiply 726 by 8 dollars and the still
more elaborate explanations of why nevertheless we find the cost of 726
articles at $8 each by multiplying 726 by 8 and calling the answer
dollars are wasteful. The same holds of the corresponding pedantry about
division. These imaginary difficulties should not be raised at all. The
pupil should not think of multiplying or dividing men or dollars, but
simply of the necessary equation and of the sort of thing that the
missing number represents. "8 x 726 = .... Answer is dollars," or
"8, 726, multiply. Answer is dollars," is all that he needs to think,
and is in the best form for his thought. Concerning the distinction
between abstract and concrete numbers, both logic and common sense as
well as psychology support the contention of McDougle ['14, p. 206f.],
who writes:--
"The most elementary counting, even that stage when the counts were not
carried in the mind, but merely in notches on a stick or by DeMorgan's
stones in a pot, requires some thought; and the most advanced counting
implies memory of things. The terms, therefore, abstract and concrete
number, have long since ceased to be used by thinking people.
"Recently the writer visited an arithmetic class in a State Normal
School and saw a group of practically adult students confused about this
very question concerning abstract and concrete numbers, according to
their previous training in the conventionalities of the textbook. Their
teacher diverted the work of the hour and she and the class spent almost
the whole period in reestablishing the requirements 'that the product
must always be the same kind of unit as the multiplicand,' and 'addends
must all be alike to be added.' This is not an exceptional case.
Throughout the whole range of teaching arithmetic in the public schools
pupils are obfuscated by the philosophical encumbrances which have been
imposed upon the simplest processes of numerical work. The time is
surely ripe, now that we are readjusting our ideas of the subject of
arithmetic, to revise some of these wasteful and disheartening
practices. Algebra historically grew out of arithmetic, yet it has not
been laden with this distinction. No pupil in algebra lets _x_ equal the
horses; he lets _x_ equal the _number_ of horses, and proceeds to drop
the idea of horses out of his consideration. He multiplies, divides, and
extracts the root of the _number_, sometimes handling fractions in the
process, and finally interprets the result according to the conditions
of his problem. Of course, in the early number work there have been the
sense-objects from which number has been perceived, but the mind
retreats naturally from objectivity to the pure conception of number,
and then to the number symbol. The following is taken from the appendix
to Horn's thesis, where a seventh grade girl gets the population of the
United States in 1820:--
7,862,166 whites
233,634 free negroes
1,538,022 slaves
---------
9,633,822
In this problem three different kinds of addends are combined, if we
accept the usual distinction. Some may say that this is a mistake,--that
the pupil transformed the 'whites,' 'free negroes,' and 'slaves' into a
common unit, such as 'people' of 'population' and then added these
common units. But this 'explanation' is entirely gratuitous, as one will
find if he questions the pupil about the process. It will be found that
the child simply added the figures as numbers only and then interpreted
the result, according to the statement of the problem, without so much
mental gymnastics. The writer has questioned hundreds of students in
Normal School work on this point, and he believes that the ordinary
mind-movement is correctly set forth here, no matter how well one may
maintain as an academic proposition that this is not logical. Many
classes in the Eastern Kentucky State Normal have been given this
problem to solve, and they invariably get the same result:--
'In a garden on the Summit are as many cabbage-heads as the total number
of ladies and gentlemen in this class. How many cabbage-heads in the
garden?'
And the blackboard solution looks like this each time:--
29 ladies
15 gentlemen
--
44 cabbage-heads
So, also, one may say: I have 6 times as many sheep as you have cows. If
you have 5 cows, how many sheep have I? Here we would multiply the
number of cows, which is 5, by 6 and call the result 30, which must be
linked with the idea of sheep because the conditions imposed by the
problem demand it. The mind naturally in this work separates the pure
number from its situation, as in algebra, handles it according to the
laws governing arithmetical combinations, and labels the result as the
statement of the problem demands. This is expressed in the following,
which is tacitly accepted in algebra, and should be accepted equally in
arithmetic:
'In all computations and operations in arithmetic, all numbers are
essentially abstract and should be so treated. They are concrete only in
the thought process that attends the operation and interprets the
result.'"
(4) _Least common multiple._--The whole set of bonds involved in
learning 'least common multiple' should be left out. In adding and
subtracting fractions the pupil should _not_ find the least common
multiple of their denominators but should find any common multiple that
he can find quickly and correctly. No intelligent person would ever
waste time in searching for the least common multiple of sixths, thirds,
and halves except for the unfortunate traditions of an oversystematized
arithmetic, but would think of their equivalents in sixths or twelfths
or twenty-fourths or _any other convenient common multiple_. The process
of finding the least common multiple is of such exceedingly rare
application in science or business or life generally that the textbooks
have to resort to purely fantastic problems to give drill in its use.
(5) _Greatest common divisor._--The whole set of bonds involved in
learning 'greatest common divisor' should also be left out. In reducing
fractions to lowest terms the pupil should divide by anything that he
sees that he can divide by, favoring large divisors, and continue doing
so until he gets the fraction in terms suitable for the purpose in hand.
The reader probably never has had occasion to compute a greatest common
divisor since he left school. If he has computed any, the chances are
that he would have saved time by solving the problem in some other way!
The following problems are taken at random from those given by one of
the best of the textbooks that make the attempt to apply the facts of
Greatest Common Divisor and Least Common Multiple to problems.[6] Most
of these problems are fantastic. The others are trivial, or are better
solved by trial and adaptation.
1. A certain school consists of 132 pupils in the high school, 154
in the grammar, and 198 in the primary grades. If each group is
divided into sections of the same number containing as many pupils
as possible, how many pupils will there be in each section?
2. A farmer has 240 bu. of wheat and 920 bu. of oats, which he
desires to put into the least number of boxes of the same capacity,
without mixing the two kinds of grain. Find how many bushels each
box must hold.
3. Four bells toll at intervals of 3, 7, 12, and 14 seconds
respectively, and begin to toll at the same instant. When will
they next toll together?
4. A, B, C, and D start together, and travel the same way around an
island which is 600 mi. in circuit. A goes 20 mi. per day, B 30,
C 25, and D 40. How long must their journeying continue, in order
that they may all come together again?
5. The periods of three planets which move uniformly in circular
orbits round the sun, are respectively 200, 250, and 300 da.
Supposing their positions relatively to each other and the sun
to be given at any moment, determine how many da. must elapse
before they again have exactly the same relative positions.
[6] McLellan and Ames, _Public School Arithmetic_ [1900].
(6) _Rare and unimportant words._--The bonds between rare or unimportant
words and their meanings should not be formed for the mere sake of
verbal variety in the problems of the textbook. A pupil should not be
expected to solve a problem that he cannot read. He should not be
expected in grades 2 and 3, or even in grade 4, to read words that he
has rarely or never seen before. He should not be given elaborate drill
in reading during the time devoted to the treatment of quantitative
facts and relations.
All this is so obvious that it may seem needless to relate. It is not.
With many textbooks it is now necessary to give definite drill in
reading the words in the printed problems intended for grades 2, 3, and
4, or to replace them by oral statements, or to leave the pupils in
confusion concerning what the problems are that they are to solve. Many
good teachers make a regular reading-lesson out of every page of
problems before having them solved. There should be no such necessity.
To define _rare_ and _unimportant_ concretely, I will say that for
pupils up to the middle of grade 3, such words as the following are rare
and unimportant (though each of them occurs in the very first fifty
pages of some well-known beginner's book in arithmetic).
absentees
account
Adele
admitted
Agnes
agreed
Albany
Allen
allowed
alternate
Andrew
Arkansas
arrived
assembly
automobile
baking powder
balance
barley
beggar
Bertie
Bessie
bin
Boston
bouquet
bronze
buckwheat
Byron
camphor
Carl
Carrie
Cecil
Charlotte
charity
Chicago
cinnamon
Clara
clothespins
collect
comma
committee
concert
confectioner
cranberries
crane
currants
dairyman
Daniel
David
dealer
debt
delivered
Denver
department
deposited
dictation
discharged
discover
discovery
dish-water
drug
due
Edgar
Eddie
Edwin
election
electric
Ella
Emily
enrolled
entertainment
envelope
Esther
Ethel
exceeds
explanation
expression
generally
gentlemen
Gilbert
Grace
grading
Graham
grammar
Harold
hatchet
Heralds
hesitation
Horace Mann
impossible
income
indicated
inmost
inserts
installments
instantly
insurance
Iowa
Jack
Jennie
Johnny
Joseph
journey
Julia
Katherine
lettuce-plant
library
Lottie
Lula
margin
Martha
Matthew
Maud
meadow
mentally
mercury
mineral
Missouri
molasses
Morton
movements
muslin
Nellie
nieces
Oakland
observing
obtained
offered
office
onions
opposite
original
package
packet
palm
Patrick
Paul
payments
peep
Peter
perch
phaeton
photograph
piano
pigeons
Pilgrims
preserving
proprietor
purchased
Rachel
Ralph
rapidity
rather
readily
receipts
register
remanded
respectively
Robert
Roger
Ruth
rye
Samuel
San Francisco
seldom
sheared
shingles
skyrockets
sloop
solve
speckled
sponges
sprout
stack
Stephen
strap
successfully
suggested
sunny
supply
Susan
Susie's
syllable
talcum
term
test
thermometer
Thomas
torpedoes
trader
transaction
treasury
tricycle
tube
two-seated
united
usually
vacant
various
vase
velocipede
votes
walnuts
Walter
Washington
watched
whistle
woodland
worsted
(7) _Misleading facts and procedures._--Bonds should not be formed
between articles of commerce and grossly inaccurate prices therefor,
between events and grossly improbable consequences, or causes or
accompaniments thereof, nor between things, qualities, and events which
have no important connections one with another in the real world. In
general, things should not be put together in the pupil's mind that do
not belong together.
If the reader doubts the need of this warning let him examine problems 1
to 5, all from reputable books that are in common use, or have been
within a few years, and consider how addition, subtraction, and the
habits belonging with each are confused by exercise 6.
1. If a duck flying 3/5 as fast as a hawk flies 90 miles in an hour,
how fast does the hawk fly?
2. At 5/8 of a cent apiece how many eggs can I buy for $60?
3. At $.68 a pair how many pairs of overshoes can you buy for $816?
4. At $.13 a dozen how many dozen bananas can you buy for $3.12?
5. How many pecks of beans can be put into a box that will hold just
21 bushels?
6. Write answers:
537 Beginning at the bottom say 11, 18, and 2 (writing it in
365 its place) are 20. 5, 11, 14, and 6 (writing it) are 20,
? 5, 10. The number, omitted, is 62.
36
----
1000
_a._ 581 _b._ 625 _c._ 752 _d._ 314 _e._ ?
97 ? 414 429 845
364 90 130 ? 223
? 417 ? 76 95
---- ---- ---- ---- ----
1758 2050 2460 1000 2367
(8) _Trivialities and absurdities._--Bonds should not be formed between
insignificant or foolish questions and the labor of answering them,
nor between the general arithmetical work of the school and such
insignificant or foolish questions. The following are samples from
recent textbooks of excellent standing:--
On one side of George's slate there are 32 words, and on the other
side 26 words. If he erases 6 words from one side, and 8 from the
other, how many words remain on his slate?
A certain school has 14 rooms, and an average of 40 children in a
room. If every one in the school should make 500 straight marks on
each side of his slate, how many would be made in all?
8 times the number of stripes in our flag is the number of years
from 1800 until Roosevelt was elected President. In what year was
he elected President?
From the Declaration of Independence to the World's Fair in Chicago
was 9 times as many years as there are stripes in the flag. How
many years was it?
(9) _Useless methods._--Bonds should not be formed between a described
situation and a method of treating the situation which would not be a
useful one to follow in the case of the real situation. For example, "If
I set 96 trees in rows, sixteen trees in a row, how many rows will I
have?" forms the habit of treating by division a problem that in reality
would be solved by counting the rows. So also "I wish to give 25 cents
to each of a group of boys and find that it will require $2.75. How many
boys are in the group?" forms the habit of answering a question by
division whose answer must already have been present to give the data of
the problem.
(10) _Problems whose answers would, in real life, be already
known._--The custom of giving problems in textbooks which could not
occur in reality because the answer has to be known to frame the problem
is a natural result of the lazy author's tendency to work out a problem
to fit a certain process and a certain answer. Such bogus problems are
very, very common. In a random sampling of a dozen pages of "General
Review" problems in one of the most widely used of recent textbooks, I
find that about 6 percent of the problems are of this sort. Among the
problems extemporized by teachers these bogus problems are probably
still more frequent. Such are:--
A clerk in an office addressed letters according to a given list.
After she had addressed 2500, 4/9 of the names on the list had not
been used; how many names were in the entire list?
The Canadian power canal at Sault Ste. Marie furnished 20,000
horse power. The canal on the Michigan side furnished 2-1/2
times as much. How many horse power does the latter furnish?
It may be asserted that the ideal of giving as described problems only
problems that might occur and demand the same sort of process for
solution with a real situation, is too exacting. If a problem is
comprehensible and serves to illustrate a principle or give useful
drill, that is enough, teachers may say. For really scientific teaching
it is not enough. Moreover, if problems are given merely as tests of
knowledge of a principle or as means to make some fact or principle
clear or emphatic, and are not expected to be of direct service in the
quantitative work of life, it is better to let the fact be known. For
example, "I am thinking of a number. Half of this number is twice six.
What is the number?" is better than "A man left his wife a certain sum
of money. Half of what he left her was twice as much as he left to his
son, who receives $6000. How much did he leave his wife?" The former is
better because it makes no false pretenses.
(11) _Needless linguistic difficulties._--It should be unnecessary to
add that bonds should not be formed between the pupil's general attitude
toward arithmetic and needless, useless difficulty in language or
needless, useless, wrong reasoning. Our teaching is, however, still
tainted by both of these unfortunate connections, which dispose the
pupil to think of arithmetic as a mystery and folly.
Consider, for example, the profitless linguistic difficulty of problems
1-6, whose quantitative difficulties are simply those of:--
1. 5 + 8 + 3 + 7
2. 64 / 8, and knowledge that 1 peck = 8 quarts
3. 12 / 4
4. 6 / 2
5. 3 x 2
6. 4 x 4
1. What amount should you obtain by putting together 5 cents, 8
cents, 3 cents, and 7 cents? Did you find this result by adding or
multiplying?
2. How many times must you empty a peck measure to fill a basket
holding 64 quarts of beans?
3. If a girl commits to memory 4 pages of history in one day, in
how many days will she commit to memory 12 pages?
4. If Fred had 6 chickens how many times could he give away 2
chickens to his companions?
5. If a croquet-player drove a ball through 2 arches at each
stroke, through how many arches will he drive it by 3 strokes?
6. If mamma cut the pie into 4 pieces and gave each person a piece,
how many persons did she have for dinner if she used 4 whole pies
for dessert?
Arithmetically this work belongs in the first or second years of
learning. But children of grades 2 and 3, save a few, would be utterly
at a loss to understand the language.
We are not yet free from the follies illustrated in the lessons of pages
96 to 99, which mystified our parents.
LESSON I
[Illustration: FIG. 5.]
1. In this picture, how many girls are in the swing?
2. How many girls are pulling the swing?
3. If you count both girls together, how many are they?
_One_ girl and _one_ other girl are how many?
4. How many kittens do you see on the stump?
5. How many on the ground?
6. How many kittens are in the picture? One kitten and one other
kitten are how many?
7. If you should ask me how many girls are in the swing, or how
many kittens are on the stump, I could answer aloud, _One_; or I
could write _One_; or thus, _1_.
8. If I write _One_, this is called the _word One_.
9. This, _1_, is named a _figure One_, because it means the same as
the word _One_, and stands for _One_.
10. Write 1. What is this named? Why?
11. A figure 1 may stand for _one_ girl, _one_ kitten, or _one_
anything.
12. When children first attend school, what do they begin to learn?
_Ans._ Letters and words.
13. Could you read or write before you had learned either letters
or words?
14. If we have all the _letters_ together, they are named the
Alphabet.
15. If we write or speak _words_, they are named Language.
16. You are commencing to study Arithmetic; and you can read and
write in Arithmetic only as you learn the Alphabet and Language
of Arithmetic. But little time will be required for this purpose.
LESSON II
[Illustration: FIG. 6.]
1. If we speak or write words, what do we name them, when taken
together?
2. What are you commencing to study? _Ans._ Arithmetic.
3. What Language must you now learn?
4. What do we name this, 1? Why?
5. This figure, 1, is part of the Language of Arithmetic.
6. If I should write something to stand for _Two_--_two_ girls,
_two_ kittens, or _two_ things of any kind--what do you think we
would name it?
7. A _figure Two_ is written thus: _2._ Make a _figure two_.
8. Why do we name this a _figure two_?
9. This figure two (2) is part of the Language of Arithmetic.
10. In this picture one boy is sitting, playing a flageolet. What
is the other boy doing? If the boy standing should sit down by the
other, how many boys would be sitting together? One boy and one
other boy are how many boys?
11. You see a flageolet and a violin. They are musical instruments.
One musical instrument and one other musical instrument are how
many?
12. I will write thus: 1 1 2. We say that 1 boy and 1 other boy,
counted together, are 2 boys; or are equal to 2 boys. We will now
write something to show that the first 1 and the other 1 are to be
counted together.
13. We name a line drawn thus, -, a _horizontal line_. Draw such a
line. Name it.
14. A line drawn thus, |, we name a _vertical line_. Draw such a
line. Name it.
15. Now I will put two such lines together; thus, +. What kind of a
line do we name the first (-)? And what do we name the last? (|)?
Are these lines long or short? Where do they cross each other?
16. Each of you write thus: -, |, +.
17. This, +, is named _Plus_. _Plus_ means _more_; and + also means
_more_.
18. I will write.
_One and One More Equal Two._
19. Now I will write part of this in the Language of Arithmetic.
I write the first _One_ thus, 1; then the other _One_ thus, 1.
Afterward I write, for the word _More_, thus, +, placing
the + between 1 and 1, so that the whole stands thus: 1 + 1.
As I write, I say, _One and One more_.
20. Each of you write 1 + 1. Read what you have written.
21. This +, when written between the 1s, shows that they are to be
put together, or counted together, so as to make 2.
22. Because + shows what is to be done, it is called a _Sign_. If
we take its name, _Plus_, and the word _Sign_, and put both words
together, we have _Sign Plus_, or _Plus Sign_. In speaking of this
we may call it _Sign Plus_, or _Plus Sign_, or _Plus_.
23. 1, 2, +, are part of the Language of Arithmetic.
_Write the following in the Language of Arithmetic_:
24. One and one more.
25. One and two more.
26. Two and one more.
(12) _Ambiguities and falsities._--Consider the ambiguities and false
reasoning of these problems.
1. If you can earn 4 cents a day, how much can you earn in 6 weeks?
(Are Sundays counted? Should a child who earns 4 cents some day
expect to repeat the feat daily?)
2. How many lines must you make to draw ten triangles and five
squares? (I can do this with 8 lines, though the answer the book
requires is 50.)
3. A runner ran twice around an 1/8 mile track in two minutes. What
distance did he run in 2/3 of a minute? (I do not know, but I do
know that, save by chance, he did not run exactly 2/3 of 1/8 mile.)
4. John earned $4.35 in a week, and Henry earned $1.93. They put
their money together and bought a gun. What did it cost? (Maybe $5,
maybe $10. Did they pay for the whole of it? Did they use all their
earnings, or less, or more?)
5. Richard has 12 nickels in his purse. How much more than 50 cents
would you give him for them? (Would a wise child give 60 cents to a
boy who wanted to swap 12 nickels therefor, or would he suspect a
trick and hold on to his own coins?)
6. If a horse trots 10 miles in one hour how far will he travel in
9 hours?
7. If a girl can pick 3 quarts of berries in 1 hour how many quarts
can she pick in 3 hours?
(These last two, with a teacher insisting on the 90 and 9, might
well deprive a matter-of-fact boy of respect for arithmetic for
weeks thereafter.)
The economics and physics of the next four problems speak for
themselves.
8. I lost $15 by selling a horse for $85. What was the value of the
horse?
9. If floating ice has 7 times as much of it under the surface of
the water as above it, what part is above water? If an iceberg is
50 ft. above water, what is the entire height of the iceberg? How
high above water would an iceberg 300 ft. high have to be?
10. A man's salary is $1000 a year and his expenses $625. How many
years will elapse before he is worth $10,000 if he is worth $2500
at the present time?
11. Sound travels 1120 ft. a second. How long after a cannon is
fired in New York will the report be heard in Philadelphia, a
distance of 90 miles?
GUIDING PRINCIPLES
The reader may be wearied of these special details concerning bonds now
neglected that should be formed and useless or harmful bonds formed for
no valid reason. Any one of them by itself is perhaps a minor matter,
but when we have cured all our faults in this respect and found all the
possibilities for wiser selection of bonds, we shall have enormously
improved the teaching of arithmetic. The ideal is such choice of bonds
(and, as will be shown later, such arrangement of them) as will most
improve the functions in question at the least cost of time and effort.
The guiding principles may be kept in mind in the form of seven simple
but golden rules:--
1. Consider the situation the pupil faces.
2. Consider the response you wish to connect with it.
3. Form the bond; do not expect it to come by a miracle.
4. Other things being equal, form no bond that will have to be broken.
5. Other things being equal, do not form two or three bonds when one
will serve.
6. Other things being equal, form bonds in the way that they are
required later to act.
7. Favor, therefore, the situations which life itself will offer, and
the responses which life itself will demand.