THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION OF BONDS
The bonds to be formed having been chosen, the next step is to arrange
for their most economical order of formation--to arrange to have each
help the others as much as possible--to arrange for the maximum of
facilitation and the minimum of inhibition.
The principle is obvious enough and would probably be admitted in theory
by any intelligent teacher, but in practice we are still wedded to
conventional usages which arose long before the psychology of arithmetic
was studied. For example, we inherit the convention of studying addition
of integers thoroughly, and then subtraction, and then multiplication,
and then division, and many of us follow it though nobody has ever given
a proof that this is the best order for arithmetical learning. We
inherit also the opposite convention of studying in a so-called "spiral"
plan, a little addition, subtraction, multiplication, and division, and
then some more of each, and then some more, and many of us follow this
custom, with an unreasoned faith that changing about from one process to
another is _per se_ helpful.
Such conventions are very strong, illustrating our common tendency to
cherish most those customs which we cannot justify! The reductions of
denominate numbers ascending and descending were, until recently, in
most courses of study, kept until grade 4 or grade 5 was reached,
although this material is of far greater value for drills on the
multiplication and division tables than the customary problems about
apples, eggs, oranges, tablets, and penholders. By some historical
accident or for good reasons the general treatment of denominate numbers
was put late; by our naive notions of order and system we felt that any
use of denominate numbers before this time was heretical; we thus became
blind to the advantages of quarts and pints for the tables of 2s; yards
and feet for the tables of 3s; gallons and quarts for the tables of 4s;
nickels and cents for the 5s; weeks and days for the 7s; pecks and
quarts for the 8s; and square yards and square feet for the 9s.
Problems like 5 yards = __ feet or 15 feet = __ yards have not only the
advantages of brevity, clearness, practical use, real reference, and
ready variation, but also the very great advantage that part of the data
have to be _thought of_ in a useful way instead of _read off_ from the
page. In life, when a person has twenty cents with which to buy tablets
of a certain sort, he _thinks of_ the price in making his purchase,
asking it of the clerk only in case he does not know it, and in planning
his purchases beforehand he _thinks of_ prices as a rule. In spite of
these and other advantages, not one textbook in ten up to 1900 made
early use of these exercises with denominate numbers. So strong is mere
use and wont.
Besides these conventional customs, there has been, in those responsible
for arithmetical instruction, an admiration for an arrangement of topics
that is easy for a person, after he knows the subject, to use in
thinking of its constituent parts and their relations. Such arrangements
are often called 'logical' arrangements of subject matter, though they
are often far from logical in any useful sense. Now the easiest order in
which to think of a hierarchy of habits after you have formed them all
may be an extremely difficult order in which to form them. The criticism
of other orders as 'scrappy,' or 'unsystematic,' valid enough if the
course of study is thought of as an object of contemplation, may be
foolish if the course of study is regarded as a working instrument for
furthering arithmetical learning.
We must remember that all our systematizing and labeling is largely
without meaning to the pupils. They cannot at any point appreciate the
system as a progression from that point toward this and that, since they
have no knowledge of the 'this or that.' They do not as a rule think of
their work in grade 4 as an outcome of their work in grade 3 with
extensions of a to a_1, and additions of b_2 and b_3 to b and b_1, and
refinements of c and d by c_4 and d_5. They could give only the vaguest
account of what they did in grade 3, much less of why it should have
been done then. They are not much disturbed by a lack of so-called
'system' and 'logical' progression for the same reason that they are not
much helped by their presence. What they need and can use is a
_dynamically_ effective system or order, one that they can learn easily
and retain long by, regardless of how it would look in a museum of
arithmetical systems. Unless their actual arithmetical habits are
usefully related it does no good to see the so-called logical relations;
and if their habits are usefully related, it does not very much matter
whether or not they do see these; finally, they can be brought to see
them best by first acquiring the right habits in a dynamically effective
order.
DECREASING INTERFERENCE AND INCREASING FACILITATION
Psychology offers no single, easy, royal road to discovering this
dynamically best order. It can only survey the bonds, think what each
demands as prerequisite and offers as future help, recommend certain
orders for trial, and measure the efficiency of each order as a means of
attaining the ends desired. The ingenious thought and careful
experimentation of many able workers will be required for many years to
come.
Psychology can, however, even now, give solid constructive help in many
instances, either by recommending orders that seem almost certainly
better than those in vogue, or by proposing orders for trial which can
be justified or rejected by crucial tests.
Consider, for example, the situation, 'a column of one-place numbers to
be added, whose sum is over 9,' and the response 'writing down the sum.'
This bond is commonly firmly fixed before addition with two-place
numbers is undertaken. As a result the pupil has fixed a habit that he
has to break when he learns two-place addition. If _oral_ answers only
are given with such single columns until two-place addition is well
under way, the interference is avoided.
In many courses of study the order of systematic formation of the
multiplication table bonds is: 1 x 1, 2 x 1, etc., 1 x 2, 2 x 2, etc.,
1 x 3, 2 x 3, etc., 1 x 9, 2 x 9, etc. This is probably wrong in two
respects. There is abundant reason to believe that the x 5s should be
learned first, since they are easier to learn than the 1s or the 2s, and
give the idea of multiplying more emphatically and clearly. There is
also abundant reason to believe that the 1 x 5, 1 x 2, 1 x 3, etc.,
should be put very late--after at least three or four tables are
learned, since the question "What is 1 times 2?" (or 3 or 5) is
unnecessary until we come to multiplication of two- and three-place
numbers, seems a foolish question until then, and obscures the notion of
multiplication if put early. Also the facts are best learned once for
all as the habits "1 times _k_ is the same as _k_," and "_k_ times 1 is
the same as _k_."[8]
[8] The very early learning of 2 x 2, 2 x 3, 3 x 2, 2 x 4, 4 x 2,
3 x 3, and perhaps a few more multiplications is not considered
here. It is advisable. The treatment of 0 x 0, 0 x 1, 1 x 0, etc.,
is not considered here. It is probably best to defer the 'x 0'
bonds until after all the others are formed and are being used
in short multiplication, and to form them in close connection
with their use in short multiplication. The '0 x' bonds may well
be deferred until they are needed in 'long' multiplication,
0 x 0 coming last of all.
In another connection it was recommended that the divisions to 81 / 9
be learned by selective thinking or reasoning from the multiplications.
This determines the order of bonds so far as to place the formation of
the division bonds soon after the learning of the multiplications. For
other reasons it is well to make the proximity close.
One of the arbitrary systematizations of the order of formation of bonds
restricts operations at first to the numbers 1 to 10, then to numbers
under 100, then to numbers under 1000, then to numbers under 10,000.
Apart from the avoidance of unreal and pedantic problems in applied
arithmetic to which work with large numbers in low grades does somewhat
predispose a teacher, there is little merit in this restriction of the
order of formation of bonds. Its demerits are many. For example, when
the pupil is learning to 'carry' in addition he can be given better
practice by soon including tasks with sums above 100, and can get a
valuable sense of the general use of the process by being given a few
examples with three- and four-place numbers to be added. The same holds
for subtraction. Indeed, there is something to be said in favor of using
six- or seven-place numbers in subtraction, enforcing the 'borrowing'
process by having it done again and again in the same example, and
putting it under control by having the decision between 'borrowing' and
'not borrowing' made again and again in the same example. When the
multiplication tables are learned the most important use for them is not
in tedious reviews or trivial problems with answers under 100, but in
regular 'short' multiplication of two- and three- and even four-place
numbers. Just as the addition combinations function mainly in the
higher-decade modifications of them, so the multiplication combinations
function chiefly in the cases where the bond has to operate while the
added tasks of keeping one's place, adding what has been carried,
writing down the right figure in the right place, and holding the right
number for later addition, are also taken care of. It seems best to
introduce such short multiplication as soon as the x 5s, x 2s, x 3s,
and x 4s are learned and to put the x 6s, x 7s, and the rest to work
in such short multiplication as soon as each is learned.
Still surer is the need for four-, five-, and six-place numbers when
two-place numbers are used in multiplying. When the process with a
two-place multiplier is learned, multiplications by three-place numbers
should soon follow. They are not more difficult then than later. On the
contrary, if the pupil gets used to multiplying only as one does with
two-place multipliers, he will suffer more by the resulting interference
than he does from getting six- or seven-place answers whose meaning he
cannot exactly realize. They teach the rationale and the manipulations
of long multiplication with especial economy because the principles and
the procedures are used two or three times over and the contrasts
between the values which the partial products have in adding become
three instead of one.
The entire matter of long multiplication with integers and United States
money should be treated as a teaching unit and the bonds formed in close
organization, even though numbers as large as 900,000 are occasionally
involved. The reason is not that it is more logical, or less scrappy,
but that each of the bonds in question thus gets much help from, and
gives much help to, the others.
In sharp contrast to a topic like 'long multiplication' stands a topic
like denominate numbers. It most certainly should not be treated as a
large teaching unit, and all the bonds involved in adding, subtracting,
multiplying, and dividing with all the ordinary sorts of measures should
certainly not be formed in close sequence. The reductions ascending and
descending for many of the measures should be taught as drills on the
appropriate multiplication and division tables. The reduction of feet
and inches to inches, yards and feet to yards, gallons and quarts to
quarts, and the like are admirable exercises in connection with the
(_a_ x _b_) + _c_ = .... problems,--the 'Bought 3 lbs. of sugar at
7 cents and 5 cents worth of matches' problems. The reductions of
inches to feet and inches and the like are admirable exercises in
the _d_ = (.... x _b_) + _c_ or 'making change' problem, which in its
small-number forms is an excellent preparatory step for short division.
They are also of great service in early work with fractions. The
feet-mile, square-foot-square-inch, and other simple relations give a
genuine and intelligible demand for multiplication with large numbers.
Knowledge of the metric system for linear and square measure would
perhaps, as an introduction to decimal fractions, more than save the
time spent to learn it. It would even perhaps be worth while to invent a
measure (call it the _twoqua_) midway between the quart and gallon and
teach carrying in addition and borrowing in subtraction by teaching
first the addition and subtraction of 'gallon, twoqua, quart, and pint'
series! Many of the bonds which a system-made tradition huddled together
uselessly in a chapter on denominate numbers should thus be formed as
helpful preparations for and applications of other bonds all the way
from the first to the eighth half-year of instruction in arithmetic.
The bonds involved in the ability to respond correctly to the series:--
5 = .... 2s and .... remainder
5 = .... 3s and .... remainder
88 = .... 9s and .... remainder
should be formed before, not during, the training in short division.
They are admirable at that point as practice on the division tables; are
of practical service in the making-change problems of the small purchase
and the like; and simplify the otherwise intricate task of keeping one's
place, choosing the quotient figure, multiplying by it, subtracting and
holding in mind the new number to be divided, which is composed half of
the remainder and half of a figure in the written dividend. This change
of order is a good illustration of the nearly general rule that "_When
the practice or review required to perfect or hold certain bonds can, by
an inexpensive modification, be turned into a useful preparation for new
bonds, that modification should be made._"
The bonds involved in the four operations with United States money
should be formed in grades 3 and 4 along with or very soon after the
corresponding bonds with three-place and four-place integers. This
statement would have seemed preposterous to the pedagogues of fifty
years ago. "United States money," they would have said, "is an
application of decimals. How can it be learned until the essentials of
decimal fractions are known? How will the child understand when
multiplying $.75 by 3 that 3 times 5 cents is 1 dime and 5 cents, or
that 3 times 70 cents is 2 dollars and 1 dime? Why perplex the young
pupils with the difficulties of placing the decimal point? Why disturb
the learning of the four operations with integers by adding at each step
a second 'procedure with United States money'?"
The case illustrates very well the error of the older oversystematic
treatment of the order of topics and the still more important error of
confusing the logic of proof with the psychology of learning. To prove
that 3 x $.75 = $2.25 to the satisfaction of certain arithmeticians, you
may need to know the theory of decimal fractions; but to do such
multiplication all a child needs is to do just what he has been doing
with integers and then "Put a $ before the answer to show that it means
dollars and cents, and put a decimal point in the answer to show which
figures mean dollars and which figures mean cents." And this is general.
The ability to operate with integers plus the two habits of prefixing $
and separating dollars from cents in the result will enable him to
operate with United States money.
Consequently good practice came to use United States money not as a
consequence of decimal fractions, learned by their aid, but as an
introduction to decimal fractions which aids the pupil to learn them. So
it has gradually pushed work with United States money further and
further back, though somewhat timidly.
We need not be timid. The pupil will have no difficulty in adding,
subtracting, multiplying, and dividing with United States money--unless
we create it by our explanations! If we simply form the two bonds
described above and show by proper verification that the procedure
always gives the right answer, the early teaching of the four operations
with United States money will in fact actually show a learning profit!
It will save more time in the work with integers than was spent in
teaching it! For, in the first place, it will help to make work with
four-place and five-place numbers more intelligible and vital. A pupil
can understand $16.75 or $28.79 more easily than 1675 or 2879. The
former may be the prices of a suit or sewing machine or bicycle. In the
second place, it permits the use of a large stock of genuine problems
about spending, saving, sharing, and the like with advertisements and
catalogues and school enterprises. In the third place, it permits the
use of common-sense checks. A boy may find one fourth of 3000 as 7050 or
75 and not be disturbed, but he will much more easily realize that one
fourth of $30.00 is not over $70 or less than $1. Even the decimal point
of which we used to be so afraid may actually help the eye to keep its
place in adding.
INTEREST
So far, the illustrations of improvements in the order of bonds so as to
get less interference and more facilitation than the customary orders
secure have sought chiefly to improve the mechanical organization of the
bonds. Any gain in interest which the changes described effected would
be largely due to the greater achievement itself. Dewey and others have
emphasized a very different principle of improving the order of
formation of bonds--the principle of determination of the bonds to be
formed by some vital, engaging problem which arouses interest enough to
lighten the labor and which goes beyond or even against cut-and-dried
plans for sequences in order to get effective problems. For example, the
work of the first month in grade 2B might sacrifice facilitations of the
mechanical sort in order to put arithmetic to use in deciding what
dimensions a rabbit's cage should have to give him 12 square feet of
floor space, how much bread he should have per meal to get 6 ounces a
day, how long a ten-cent loaf would last, how many loaves should be
bought per week, how much it costs to feed the rabbit, how much he has
gained in weight since he was brought to the school, and so on.
Such sacrifices of the optimal order if interest were equal, in order to
get greater interest or a healthier interest, are justifiable. Vital
problems as nuclei around which to organize arithmetical learning are of
prime importance. It is even safe probably to insist that some genuine
problem-situation requiring a new process, such as addition with
carrying, multiplication by two-place numbers, or division with
decimals, be provided in every case as a part of the introduction to
that process. The sacrifice should not be too great, however; the search
for vital problems that fit an economical order of subject matter is as
much needed as the amendment of that order to fit known interests; and
the assurance that a problem helps the pupil to learn arithmetic is as
important as the assurance that arithmetic is used to help the pupil
solve his personal problems.
Much ingenuity and experimentation will be required to find the order
that is satisfactory in both quality and quantity of interest or motive
and helpfulness of the bonds one to another. The difficulty of
organizing arithmetic around attractive problems is much increased by
the fact of class instruction. For any one pupil vital, personal
problems or projects could be found to provide for many arithmetical
abilities; and any necessary knowledge and technique which these
projects did not develop could be somehow fitted in along with them. But
thirty children, half boys and half girls, varying by five years in age,
coming from different homes, with different native capacities, will not,
in September, 1920, unanimously feel a vital need to solve any one
problem, and then conveniently feel another on, say, October 15! In the
mechanical laws of learning children are much alike, and the gain we
may hope to make from reducing inhibitions and increasing facilitations
is, for ordinary class-teaching, probably greater than that to be made
from the discovery of attractive central problems. We should, however,
get as much as possible of both.
GENERAL PRINCIPLES
The reader may by now feel rather helpless before the problem of the
arrangement of arithmetical subject matter. "Sometimes you complete a
topic, sometimes you take it piecemeal months or years apart, often you
make queer twists and shifts to get a strategic advantage over the
enemy," he may think, "but are there no guiding principles, no general
rules?" There is only one that is absolutely general, to _take the order
that works best for arithmetical learning_. There are particular rules,
but there are so many and they are so limited by an 'other things being
equal' clause, that probably a general eagerness to think out the _pros_
and _cons_ for any given proposal is better than a stiff attempt to
adhere to these rules. I will state and illustrate some of them, and let
the reader judge.
_Other things being equal, one new sort of bonds should not be started
until the previous set is fairly established, and two different sets
should not be started at once._ Thus, multiplication of two- and
three-place numbers by 2, 3, 4, and 5 will first use numbers such that
no carrying is required, and no zero difficulties are encountered, then
introduce carrying, then introduce multiplicands like 206 and 320.
If other things were equal, the carrying would be split into two
steps--first drills with (4 x 6) + 2, (3 x 7) + 3, (5 x 4) + 1, and the
like, and second the actual use of these habits in the multiplication.
The objection to this separation of the double habit is that the first
part of it in isolation is too artificial--that it may be better to
suffer the extra difficulty of forming the two together than to teach so
rarely used habits as the (_a_ x _b_) + _c_ series. Experimental tests
are needed to decide this point.
_Other things being equal, bonds should be formed in such order that
none will have to be broken later._ For example, there is a strong
argument for teaching long division first, or very early, with
remainders, letting the case of zero remainder come in as one of many.
If the pupils have been familiarized with the remainder notion by the
drills recommended as preparation for short division,[9] the use of
remainders in long division will offer little difficulty. The exclusive
use of examples without remainders may form the habit of not being exact
in computation, of trusting to 'coming out even' as a sole check, and
even of writing down a number to fit the final number to be divided
instead of obtaining it by honest multiplication.
[9] See page 76.
For similar reasons additions with 2 and 3 as well as 1 to be 'carried'
have much to recommend them in the very first stages of column addition
with carrying. There is here the added advantage that a pupil will be
more likely to remember to carry if he has to think _what_ to carry. The
present common practice of using small numbers for ease in the addition
itself teaches many children to think of carrying as adding one.
_Other things being equal, arrange to have variety._ Thus it is
probably, though not surely, wise to interrupt the monotony of learning
the multiplication and division tables, by teaching the fundamentals of
'short' multiplication and perhaps of division after the 5s, 2s, 3s, and
4s are learned. This makes a break of several weeks. The facts for the
6s, 7s, 8s, and 9s can then be put to varied use as fast as learned. It
is almost certainly wise to interrupt the first half-year's work with
addition and subtraction, by teaching 2 x 2, 2 x 3, 3 x 2, 2 x 4, 4 x 2,
2 x 5, later by 2 x 10, 3 x 10, 4 x 10, 5 x 10, later by 1/2 + 1/2,
1-1/2 + 1/2, 1/2 of 2, 1/2 of 4, 1/2 of 6, and at some time by certain
profitable exercises wherein a pupil tells all he knows about certain
numbers which may be made nuclei of important facts (say, 5, 8, 10, 12,
15, and 20).
_Other things being equal, use objective aids to verify an arithmetical
process or inference after it is made, as well as to provoke it._ It is
well at times to let pupils do everything that they can with relations
abstractly conceived, testing their results by objective counting,
measuring, adding, and the like. For example, an early step in adding
should be to show three things, put them under a book, show two more,
put these under the book, and then ask how many there are under the
book, letting the objective counting come later as the test of the
correctness of the addition.
_Other things being equal, reserve all explanations of why a process
must be right until the pupils can use the process accurately, and have
verified the fact that it is right._ Except for the very gifted pupils,
the ordinary preliminary deductive explanations of what must be done are
probably useless as means of teaching the pupils what to do. They use up
much time and are of so little permanent effect that, as we have seen,
the very arithmeticians who advocate making them, admit that after a
pupil has mastered the process he may be allowed to forget the reasons
for it. I am not sure that the deductive proofs of why we place the
decimal point as we do in division by a decimal, or invert and multiply
in dividing by a fraction, and the like, are worth teaching at all. If
they are to be taught at all, the time to teach them is (except for the
very gifted) after the pupil has mastered the process and has confidence
in it. He then at least knows what process he is to prove is right, and
that it is right, and has had some chance of seeing _why_ it is right
from his experience with it.
One more principle may be mentioned without illustration. _Arrange the
order of bonds with due regard for the aims of the other studies of the
curriculum and the practical needs of the pupil outside of school._
Arithmetic is not a book or a closed system of exercises. It is the
quantitative work of the pupils in the elementary school. No narrower
view of it is adequate.