THE CONSTITUTION OF ARITHMETICAL ABILITIES
THE ELEMENTARY FUNCTIONS OF ARITHMETICAL LEARNING
It would be a useful work for some one to try to analyze arithmetical
learning into the unitary abilities which compose it, showing just what,
in detail, the mind has to do in order to be prepared to pass a thorough
test on the whole of arithmetic. These unitary abilities would make a
very long list. Examination of a well-planned textbook will show that
such an ability as multiplication is treated as a composite of the
following: knowledge of the multiplications up to 9 x 9; ability to
multiply two (or more)-place numbers by 2, 3, and 4 when 'carrying' is
not required and no zeros occur in the multiplicand; ability to multiply
by 2, 3, ... 9, with carrying; the ability to handle zeros in the
multiplicand; the ability to multiply with two-place numbers not ending
in zero; the ability to handle zero in the multiplier as last number;
the ability to multiply with three (or more)-place numbers not including
a zero; the ability to multiply with three- and four-place numbers with
zero in second or third, or second and third, as well as in last place;
the ability to save time by annexing zeros; and so on and on through a
long list of further abilities required to multiply with United States
money, decimal fractions, common fractions, mixed numbers, and
denominate numbers.
The units or 'steps' thus recognized by careful teaching would make a
long list, but it is probable that a still more careful study of
arithmetical ability as a hierarchy of mental habits or connections
would greatly increase the list. Consider, for example, ordinary column
addition. The majority of teachers probably treat this as a simple
application of the knowledge of the additions to 9 + 9, plus
understanding of 'carrying.' On the contrary there are at least seven
processes or minor functions involved in two-place column addition, each
of which is psychologically distinct and requires distinct educational
treatment.
These are:--
A. Learning to keep one's place in the column as one adds.
B. Learning to keep in mind the result of each addition until the
next number is added to it.
C. Learning to add a seen to a thought-of number.
D. Learning to neglect an empty space in the columns.
E. Learning to neglect 0s in the columns.
F. Learning the application of the combinations to higher decades
may for the less gifted pupils involve as much time and labor
as learning all the original addition tables. And even for
the most gifted child the formation of the connection
'8 and 7 = 15' probably never quite insures the presence
of the connections '38 and 7 = 45' and '18 + 7 = 25.'
G. Learning to write the figure signifying units rather than the
total sum of a column. In particular, learning to write 0 in
the cases where the sum of the column is 10, 20, etc. Learning
to 'carry' also involves in itself at least two distinct
processes, by whatever way it is taught.
We find evidence of such specialization of functions in the results with
such tests as Woody's. For example, 2 + 5 + 1 = .... surely involves
abilities in part different from
2
4
3
-
because only 77 percent of children in grade 3 do the former correctly,
whereas 95 percent of children in that grade do the latter correctly. In
grade 2 the difference is even more marked. In the case of subtraction
4
4
-
involves abilities different from those involved in
9
3,
-
being much less often solved correctly in grades 2 and 4.
6
0
-
is much harder than either of the above.
43
1 21
2 33
13 is much harder than 35.
-- --
It may be said that these differences in difficulty are due to different
amounts of practice. This is probably not true, but if it were, it would
not change the argument; if the two abilities were identical, the
practice of one would improve the other equally.
I shall not undertake here this task of listing and describing the
elementary functions which constitute arithmetical learning, partly
because what they are is not fully known, partly because in many cases a
final ability may be constituted in several different ways whose
descriptions become necessarily tedious, and partly because an adequate
statement of what is known would far outrun the space limits of this
chapter. Instead, I shall illustrate the results by some samples.
KNOWLEDGE OF THE MEANING OF A FRACTION
As a first sample, consider knowledge of the meaning of a fraction. Is
the ability in question simply to understand that a fraction is a
statement of the number of parts, each of a certain size, the upper
number or numerator telling how many parts are taken and the lower
number or denominator telling what fraction of unity each part is? And
is the educational treatment required simply to describe and illustrate
such a statement and have the pupils apply it to the recognition of
fractions and the interpretation of each of them? And is the learning
process (1) the formation of the notions of part, size of part, number
of part, (2) relating the last two to the numbers in a fraction, and, as
a necessary consequence, (3) applying these notions adequately whenever
one encounters a fraction in operation?
Precisely this was the notion a few generations ago. The nature of
fractions was taught as one principle, in one step, and the habits of
dealing with fractions were supposed to be deduced from the general law
of a fraction's nature. As a result the subject of fractions had to be
long delayed, was studied at great cost of time and effort, and, even
so, remained a mystery to all save gifted pupils. These gifted pupils
probably of their own accord built up the ability piecemeal out of
constituent insights and habits.
At all events, scientific teaching now does build up the total ability
as a fusion or organization of lesser abilities. What these are will be
seen best by examining the means taken to get them. (1) First comes the
association of 1/2 of a pie, 1/2 of a cake, 1/2 of an apple, and such
like with their concrete meanings so that a pupil can properly name a
clearly designated half of an obvious unit like an orange, pear, or
piece of chalk. The same degree of understanding of 1/4, 1/8, 1/3, 1/6,
and 1/5 is secured. The pupil is taught that 1 pie = 2 1/2s, 3 1/3s, 4
1/4s, 5 1/5s, 6 1/6s, and 8 1/8s; similarly for 1 cake, 1 apple, and the
like.
So far he understands 1/_x_ of _y_ in the sense of certain simple parts
of obviously unitary _y_s.
(2) Next comes the association with 1/2 of an inch, 1/2 of a foot, 1/2
of a glassful and other cases where _y_ is not so obviously a unitary
object whose pieces still show their derivation from it. Similarly for
1/4, 1/3, etc.
(3) Next comes the association with 1/2 of a collection of eight pieces
of candy, 1/3 of a dozen eggs, 1/5 of a squad of ten soldiers, etc.,
until 1/2, 1/3, 1/4, 1/5, 1/6, and 1/8 are understood as names of
certain parts of a collection of objects.
(4) Next comes the similar association when the nature of the collection
is left undefined, the pupil responding to
1/2 of 6 is ..., 1/4 of 8 is ..., 2 is 1/5 of ...,
1/3 of 6 is ..., 1/3 of 9 is ..., 2 is 1/3 of ..., and the like.
Each of these abilities is justified in teaching by its intrinsic
merits, irrespective of its later service in helping to constitute the
general understanding of the meaning of a fraction. The habits thus
formed in grades 3 or 4 are of constant service then and thereafter in
and out of school.
(5) With these comes the use of 1/5 of 10, 15, 20, etc., 1/6 of 12, 18,
42, etc., as a useful variety of drill on the division tables, valuable
in itself, and a means of making the notion of a unit fraction more
general by adding 1/7 and 1/9 to the scheme.
(6) Next comes the connection of 3/4, 2/5, 3/5, 4/5, 2/3, 1/6, 5/6, 3/8,
5/8, 7/8, 3/10, 7/10, and 9/10, each with its meaning as a certain part
of some conveniently divisible unit, and, (7) and (8), connections
between these fractions and their meanings as parts of certain
magnitudes (7) and collections (8) of convenient size, and (9)
connections between these fractions and their meanings when the nature
of the magnitude or collection is unstated, as in 4/5 of 15 = ...,
5/8 of 32 = ....
(10) That the relation is general is shown by using it with
numbers requiring written division and multiplication, such as
7/8 of 1736 = ..., and with United States money.
Elements (6) to (10) again are useful even if the pupil never goes
farther in arithmetic. One of the commonest uses of fractions is in
calculating the cost of fractions of yards of cloth, and fractions of
pounds of meat, cheese, etc.
The next step (11) is to understand to some extent the principle that
the value of any of these fractions is unaltered by multiplying or
dividing the numerator and denominator by the same number. The drills in
expressing fractions in lower and higher terms which accomplish this are
paralleled by (12) and (13) simple exercises in adding and subtracting
fractions to show that fractions are quantities that can be operated on
like any quantities, and by (14) simple work with mixed numbers
(addition and subtraction and reductions), and (15) improper fractions.
All that is done with improper fractions is (_a_) to have the pupil use
a few of them as he would any fractions and (_b_) to note their
equivalent mixed numbers. In (12), (13), and (14) only fractions of the
same denominators are added or subtracted, and in (12) (13), (14), and
(15) only fractions with 2, 3, 4, 5, 6, 8, or 10 in the denominator are
used. As hitherto, the work of (11) to (15) is useful in and of itself.
(16) Definitions are given of the following type:--
Numbers like 2, 3, 4, 7, 11, 20, 36, 140, 921 are called whole numbers.
Numbers like 7/8, 1/5, 2/3, 3/4, 11/8, 7/6, 1/3, 4/3, 1/8, 1/6 are
called fractions.
Numbers like 5-1/4, 7-3/8, 9-1/2, 16-4/5, 315-7/8, 1-1/3, 1-2/3 are
called mixed numbers.
(17) The terms numerator and denominator are connected with the upper
and lower numbers composing a fraction.
Building this somewhat elaborate series of minor abilities seems to be a
very roundabout way of getting knowledge of the meaning of a fraction,
and is, if we take no account of what is got along with this knowledge.
Taking account of the intrinsically useful habits that are built up, one
might retort that the pupil gets his knowledge of the meaning of a
fraction at zero cost.
KNOWLEDGE OF THE SUBTRACTION AND DIVISION TABLES
Consider next the knowledge of the subtraction and division 'Tables.'
The usual treatment presupposes that learning them consists of forming
independently the bonds:--
3 - 1 = 2 4 / 2 = 2
3 - 2 = 1 6 / 2 = 3
4 - 1 = 3 6 / 3 = 2
. .
. .
. .
. .
. .
. .
18 - 9 = 9 81 / 9 = 9
In fact, however, these 126 bonds are not formed independently. Except
perhaps in the case of the dullest twentieth of pupils, they are
somewhat facilitated by the already learned additions and
multiplications. And by proper arrangement of the learning they may be
enormously facilitated thereby. Indeed, we may replace the independent
memorizing of these facts by a set of instructive exercises wherein the
pupil derives the subtractions from the corresponding additions by
simple acts of reasoning or selective thinking. As soon as the additions
giving sums of 9 or less are learned, let the pupil attack an exercise
like the following:--
Write the missing numbers:--
A B C D
3 and ... are 5. 5 and ... are 8. 4 and ... are 5. 4 and ... are 8.
3 and ... are 9. 3 and ... are 6. 5 and ... are 6. 1 and ... are 7.
4 and ... are 7. 4 and ... are 9. 6 and ... are 9. 6 and ... are 7.
5 and ... are 7. 2 and ... = 6. 1 and ... are 8. 8 and ... are 9.
6 and ... are 8. 5 and ... = 9. 3 and ... are 7. 3 + ... are 4.
4 and ... are 6. 2 and ... = 7. 1 + ... are 3. 7 + ... are 8.
2 and ... are 5. 3 and ... = 8. 1 + ... are 5. 4 + ... are 9.
2 and ... = 8. 1 and ... = 4. 4 + ... are 8. 2 + ... are 3.
3 and ... = 6. 2 and ... = 4. 7 + ... are 9. 1 + ... are 9.
6 and ... = 9. 3 and ... = 8. 2 + ... = 4. 3 + ... = 6.
4 and ... = 6. 6 and ... = 7. 3 + ... = 8. 5 + ... = 9.
4 and ... = 7. 2 and ... = 5. 4 + ... = 5. 1 + ... = 3.
The task for reasoning is only to try, one after another, numbers that
seem promising and to select the right one when found. With a little
stimulus and direction children can thus derive the subtractions up to
those with 9 as the larger number. Let them then be taught to do the
same with the printed forms:--
Subtract
9 7 8 5 8 6
3 5 6 2 2 4 etc.
- - - - - -
and 9 - 7 = ..., 9 - 5 = ..., 7 - 5 = ..., etc.
In the case of the divisions, suppose that the pupil has learned his
first table and gained surety in such exercises as:--
4 5s = .... 6 x 5 = .... 9 nickels = .... cents.
8 5s = .... 4 x 5 = .... 6 " = .... "
3 5s = .... 2 x 5 = .... 5 " = .... "
7 5s = .... 9 x 5 = .... 7 " = .... "
If one ball costs 5 cents,
two balls cost .... cents,
three balls cost .... cents, etc.
He may then be set at once to work at the answers to exercises like the
following:--
Write the answers and the missing numbers:--
A B C D
.... 5s = 15 40 = .... 5s .... x 5 = 25 20 cents = .... nickels.
.... 5s = 20 20 = .... 5s .... x 5 = 50 30 cents = .... nickels.
.... 5s = 40 15 = .... 5s .... x 5 = 35 15 cents = .... nickels.
.... 5s = 25 45 = .... 5s .... x 5 = 10 40 cents = .... nickels.
.... 5s = 30 50 = .... 5s .... x 5 = 40
.... 5s = 35 25 = .... 5s .... x 5 = 45
E
For 5 cents you can buy 1 small loaf of bread.
For 10 cents you can buy 2 small loaves of bread.
For 25 cents you can buy .... small loaves of bread.
For 45 cents you can buy .... small loaves of bread.
For 35 cents you can buy .... small loaves of bread.
F
5 cents pays 1 car fare.
15 cents pays .... car fares.
10 cents pays .... car fares.
20 cents pays .... car fares.
G
How many 5 cent balls can you buy with 30 cents? ....
How many 5 cent balls can you buy with 35 cents? ....
How many 5 cent balls can you buy with 25 cents? ....
How many 5 cent balls can you buy with 15 cents? ....
In the case of the meaning of a fraction, the ability, and so the
learning, is much more elaborate than common practice has assumed; in
the case of the subtraction and division tables the learning is much
less so. In neither case is the learning either mere memorizing of facts
or the mere understanding of a principle _in abstracto_ followed by its
application to concrete cases. It is (and this we shall find true of
almost all efficient learning in arithmetic) the formation of
connections and their use in such an order that each helps the others to
the maximum degree, and so that each will do the maximum amount for
arithmetical abilities other than the one specially concerned, and for
the general competence of the learner.
LEARNING THE PROCESSES OF COMPUTATION
As another instructive topic in the constitution of arithmetical
abilities, we may take the case of the reasoning involved in
understanding the manipulations of figures in two (or more)-place
addition and subtraction, multiplication and division involving a two
(or more)-place number, and the manipulations of decimals in all four
operations. The psychology of these is of special interest and
importance. For there are two opposite explanations possible here,
leading to two opposite theories of teaching.
The common explanation is that these methods of manipulation, if
understood at all, are understood as deductions from the properties of
our system of decimal notation. The other is that they are understood
partly as inductions from the experience that they always give the right
answer. The first explanation leads to the common preliminary deductive
explanations of the textbooks. The other leads to explanations by
verification; _e.g._, of addition by counting, of subtraction by
addition, of multiplication by addition, of division by multiplication.
Samples of these two sorts of explanation are given below.
SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATION
MULTIPLICATION is the process of taking one number as many times as
there are units in another number.
The PRODUCT is the result of the multiplication.
The MULTIPLICAND is the number to be taken.
The MULTIPLIER is the number denoting how many times the multiplicand is
to be taken.
The multiplier and multiplicand are the FACTORS.
Multiply 623 by 3
OPERATION
_Multiplicand_ 623
_Multiplier_ 3
----
_Product_ 1869
EXPLANATION.--For convenience we write the multiplier under the
multiplicand, and begin with units to multiply. 3 times 3 units are
9 units. We write the nine units in units' place in the product. 3
times 2 tens are 6 tens. We write the 6 tens in tens' place in the
product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and 8
hundreds. The 1 thousand we write in thousands' place and the 8
hundreds in hundreds' place in the product. Therefore, the product
is 1 thousand 8 hundreds, 6 tens and 9 units, or 1869.
SHORT MULTIPLICATION WITHOUT CARRYING: INDUCTIVE EXPLANATION
1. The children of the third grade are to have a picnic. 32 are going.
How many sandwiches will they need if each of the 32 children has four
sandwiches?
_Here is a quick way to find out_:--
32 _Think "4 x 2," write 8 under the 2 in the ones column._
4 _Think "4 x 3," write 12 under the 3 in the tens column._
--
2. How many bananas will they need if each of the 32 children has
two bananas? 32 x 2 or 2 x 32 will give the answer.
3. How many little cakes will they need if each child has three
cakes? 32 x 3 or 3 x 32 will give the answer.
32 3 x 2 = .... _Where do you write the 6?_
3 3 x 3 = .... _Where do you write the 9?_
--
4. Prove that 128, 64, and 96 are right by adding four 32s, two 32s,
and three 32s.
32
32 32
32 32 32
32 32 32
-- -- --
Multiplication
You #multiply# when you find the answers to questions like
How many are 9 x 3?
How many are 3 x 32?
How many are 8 x 5?
How many are 4 x 42?
1. Read these lines. Say the right numbers where the dots are:
If you #add# 3 to 32, you have .... 35 is the #sum#.
If you #subtract# 3 from 32, the result is .... 29 is the
#difference# or #remainder#.
If you #multiply# 3 by 32 or 32 by 3, you have .... 96 is the
#product#.
Find the products. Check your answers to the first line by adding.
2. 3. 4. 5. 6. 7. 8. 9.
41 33 42 44 53 43 34 24
3 2 4 2 3 2 2 2
-- -- -- -- -- -- -- --
10. 11. 12. 13. 14. 15. 16.
43 52 32 23 41 51 14
3 3 3 3 2 4 2
-- -- -- -- -- -- --
17. 213 _Write the 9 in the ones column._
3 _Write the 3 in the tens column._
--- _Write the 6 in the hundreds column._
_Check your answer by adding._ Add
213
213
213
---
18. 19. 20. 21. 22. 23. 24.
214 312 432 231 132 314 243
2 3 2 3 3 2 2
--- --- --- --- --- --- ---
SHORT DIVISION: DEDUCTIVE EXPLANATION
Divide 1825 by 4
Divisor 4 |1825 Dividend
--------
456-1/4
Quotient
EXPLANATION.--For convenience we write the divisor at the left of
the dividend, and the quotient below it, and begin at the left to
divide. 4 is not contained in 1 thousand any thousand times,
therefore the quotient contains no unit of any order higher than
hundreds. Consequently we find how many times 4 is contained in the
hundreds of the dividend. 1 thousand and 8 hundreds are 18
hundreds. 4 is contained in 18 hundreds 4 hundred times and 2
hundreds remaining. We write the 4 hundreds in the quotient. The 2
hundreds we consider as united with the 2 tens, making 22 tens. 4
is contained in 22 tens 5 tens times, and 2 tens remaining. We
write the 5 tens in the quotient, and the remaining 2 tens we
consider as united with the 5 units, making 25 units. 4 is
contained in 25 units 6 units times and 1 unit remaining. We write
the 6 units in the quotient and indicate the division of the
remainder, 1 unit, by the divisor 4.
Therefore the quotient of 1825 divided by 4 is 456-1/4, or 456 and
1 remainder.
SHORT DIVISION: INDUCTIVE EXPLANATION
Dividing Large Numbers
1. Tom, Dick, Will, and Fred put in 2 cents each to buy an eight-cent
bag of marbles. There are 128 marbles in it. How many should each boy
have, if they divide the marbles equally among the four boys?
-----
4 |128
_Think "12 = three 4s." Write the 3 over the 2 in the tens column._
_Think "8 = two 4s." Write the 2 over the 8 in the ones column._
_32 is right, because 4 x 32 = 128._
2. Mary, Nell, and Alice are going to buy a book as a present for
their Sunday-school teacher. The present costs 69 cents. How much
should each girl pay, if they divide the cost equally among the
three girls?
----
3|69
_Think "6 = .... 3s." Write the 2 over the 6 in the tens column._
_Think "9 = .... 3s." Write the 3 over the 9 in the ones column._
_23 is right, for 3 x 23 = 69._
3. Divide the cost of a 96-cent present equally among three girls. How
much should each girl pay?
------
3|96
4. Divide the cost of an 84-cent present equally among 4 girls. How
much should each girl pay?
5. Learn this: (Read / as "_divided by_.")
12 + 4 = 16. 16 is the sum.
12 - 4 = 8. 8 is the difference or remainder.
12 x 4 = 48. 48 is the product.
12 / 4 = 3. 3 is the quotient.
6. Find the quotients. Check your answers by multiplying.
---- ---- ---- ----- ----- -----
3|99 2|86 5|155 6|246 4|168 3|219
[Uneven division is taught by the same general plan, extended.]
LONG DIVISION: DEDUCTIVE EXPLANATION
To Divide by Long Division
1. Let it be required to divide 34531 by 15.
_Operation_
Divided
Divisor 15)34531(2302-1/15 Quotient
30
--
45
45
--
31
30
--
1 Remainder
For convenience we write the divisor at the left and the quotient at the
right of the dividend, and begin to divide as in Short Division.
15 is contained in 3 ten-thousands 0 ten-thousands times; therefore,
there will be 0 ten-thousands in the quotient. Take 34 thousands; 15 is
contained in 34 thousands 2 thousands times; we write the 2 thousands in
the quotient. 15 x 2 thousands = 30 thousands, which, subtracted from 34
thousands, leaves 4 thousands = 40 hundreds. Adding the 5 hundreds, we
have 45 hundreds.
15 in 45 hundreds 3 hundreds times; we write the 3 hundreds in the
quotient. 15 x 3 hundreds = 45 hundreds, which subtracted from 45
hundreds, leaves nothing. Adding the 3 tens, we have 3 tens.
15 in 3 tens 0 tens times; we write 0 tens in the quotient. Adding to
the three tens, which equal 30 units, the 1 unit, we have 31 units.
15 in 31 units 2 units times; we write the 2 units in the quotient. 15 x
2 units = 30 units, which, subtracted from 31 units, leaves 1 unit as a
remainder. Indicating the division of the 1 unit, we annex the
fractional expression, 1/15 unit, to the integral part of the quotient.
Therefore, 34531 divided by 15 is equal to 2302-1/15.
[B. Greenleaf, _Practical Arithmetic_, '73, p. 49.]
LONG DIVISION: INDUCTIVE EXPLANATION
Dividing by Large Numbers
1. Just before Christmas Frank's father sent 360 oranges to be divided
among the children in Frank's class. There are 29 children. How
many oranges should each child receive? How many oranges will be
left over?
_Here is the best way to find out:_
12 and 12 _Think how many 29s there are in 36. 1 is right._
______ remainder _Write 1 over the 6 of 36. Multiply 29 by 1._
29 )360 _Write the 29 under the 36. Subtract 29 from 36._
29 _Write the 0 of 360 after the 7._
--- _Think how many 29s there are in 70. 2 is right._
70 _Write 2 over the 0 of 360. Multiply 29 by 2._
58 _Write the 58 under 70. Subtract 58 from 70._
-- _There is 12 remainder._
12 _Each child gets 12 oranges, and there are 12
left over. This is right, for 12 multiplied
by 29 = 348, and 348 + 12 = 360._
* * * * *
8. _In No. 8, keep on dividing by 31 until you have
________ used the 5, the 8, and the 7, and have four
31)99,587 figures in the quotient._
9. 10. 11. 12. 13.
_____ _____ _____ ____ _______
22)253 22)2895 21)8891 22)290 32)16,368
Check your results for 9, 10, 11, 12, and 13.
1. The boys and girls of the Welfare Club plan to earn money to buy
a victrola. There are 23 boys and girls. They can get a good
second-hand victrola for $5.75. How much must each earn if they
divide the cost equally?
_Here is the best way to find out_:
$.25 _Think how many 23s there are in 57. 2 is right._
----- _Write 2 over the 7 of 57. Multiply 23 by 2._
23|$5.75 _Write 46 under 57 and subtract. Write the 5 of 575
46 after the 11._
---- _Think how many 23s there are in 115. 5 is right._
115 _Write 5 over the 5 of 575. Multiply 23 by 5._
115 _Write the 115 under the 115 that is there and subtract._
---- _There is no remainder._
_Put $ and the decimal point where they belong._
_Each child must earn 25 cents. This is right, for $.25
multiplied by 23 = $5.75._
2. Divide $71.76 equally among 23 persons. How much is each person's
share?
3. Check your result for No. 2 by multiplying the quotient by the
divisor.
Find the quotients. Check each quotient by multiplying it by the
divisor.
4. 5. 6. 7. 8.
_______ _______ ________ _______ _______
23)$99.13 25)$18.50 21)$129.15 13)$29.25 32)$73.92
1 bushel = 32 qt.
9. How many bushels are there in 288 qt.?
10. In 192 qt.?
11. In 416 qt.?
Crucial experiments are lacking, but there are several lines of
well-attested evidence. First of all, there can be no doubt that the
great majority of pupils learn these manipulations at the start from the
placing of units under units, tens under tens, etc., in adding, to the
placing of the decimal point in division with decimals, by imitation and
blind following of specific instructions, and that a very large
proportion of the pupils do not to the end, that is to the fifth
school-year, understand them as necessary deductions from decimal
notation. It also seems probable that this proportion would not be much
reduced no matter how ingeniously and carefully the deductions were
explained by textbooks and teachers. Evidence of this fact will appear
abundantly to any one who will observe schoolroom life. It also appears
in the fact that after the properties of the decimal notation have been
thus used again and again; _e.g._, for deducing 'carrying' in addition,
'borrowing' in subtraction, 'carrying' in multiplication, the value of
the digits in the partial product, the value of each remainder in short
division, the value of the quotient figures in division, the addition,
subtraction, multiplication, and division of United States money, and
the placing of the decimal point in multiplication, no competent teacher
dares to rely upon the pupil, even though he now has four or more years'
experience with decimal notation, to deduce the placing of the decimal
point in division with decimals. It may be an illusion, but one seems to
sense in the better textbooks a recognition of the futility of the
attempt to secure deductive derivations of those manipulations. I refer
to the brevity of the explanations and their insertion in such a form
that they will influence the pupils' thinking as little as possible. At
any rate the fact is sure that most pupils do not learn the
manipulations by deductive reasoning, or understand them as necessary
consequences of abstract principles.
It is a common opinion that the only alternative is knowing them by
rote. This, of course, is one common alternative, but the other
explanation suggests that understanding the manipulations by inductive
reasoning from their results is another and an important alternative.
The manipulations of 'long' multiplication, for instance, learned by
imitation or mechanical drill, are found to give for 25 x _A_ a result
about twice as large as for 13 x _A_, for 38 or 39 x _A_ a result about
three times as large; for 115 x _A_ a result about ten times as large as
for 11 x _A_. With even the very dull pupils the procedure is verified
at least to the extent that it gives a result which the scientific
expert in the case--the teacher--calls right. With even the very bright
pupils, who can appreciate the relation of the procedure to decimal
notation, this relation may be used not as the sole deduction of the
procedure beforehand, but as one partial means of verifying it
afterward. Or there may be the condition of half-appreciation of the
relation in which the pupil uses knowledge of the decimal notation to
convince himself that the procedure _does_, but not that it _must_ give
the right answer, the answer being 'right' because the teacher, the
answer-list, and collateral evidence assure him of it.
I have taken the manipulation of the partial products as an illustration
because it is one of the least favored cases for the explanation I am
presenting. If we take the first case where a manipulation may be
deduced from decimal notation, known merely by rote, or verified
inductively, namely, the addition of two-place numbers, it seems sure
that the mental processes just described are almost the universal rule.
Surely in our schools at present children add the 3 of 23 to the 3 of 53
and the 2 of 23 to the 5 of 53 at the start, in nine cases out of ten
because they see the teacher do so and are told to do so. They are
protected from adding 3 + 3 + 2 + 5 not by any deduction of any sort but
because they do not know how to add 8 and 5, because they have been
taught the habit of adding figures that stand one above the other, or
with a + between them; and because they are shown or told what they are
to do. They are protected from adding 3 + 5 and 2 + 3, again, by no
deductive reasoning but for the second and third reasons just given.
In nine cases out of ten they do not even think of the possibility
of adding in any other way than the '3 + 3, 2 + 5' way, much less
do they select that way on account of the facts that 53 = 50 + 3
and 23 = 20 + 3, that 50 + 20 = 70, that 3 + 3 = 6, and that
(_a_ + _b_) + (_c_ + _d_) = (_a_ + _c_) + (_b_ + _d_)!
Just as surely all but the very dullest twentieth or so of children come
in the end to something more than rote knowledge,--to _understand_, to
_know_ that the procedure in question is right.
Whether they know _why_ 76 is right depends upon what is meant by
_why_. If it means that 76 is the result which competent people
agree upon, they do. If it means that 76 is the result which would
come from accurate counting they perhaps know why as well as they
would have, had they been given full explanations of the relation
of the procedure in two-place addition to decimal notation.
If _why_ means because 53 = 50 + 3, 23 = 20 + 3, 50 + 20 = 70, and
(_a_ + _b_) + (_c_ + _d_) = (_a_ + _c_) + (_b_ + _d_), they do not.
Nor, I am tempted to add, would most of them by any sort of teaching
whatever.
I conclude, therefore, that school children may and do reason about and
understand the manipulations of numbers in this inductive, verifying way
without being able to, or at least without, under present conditions,
finding it profitable to derive them deductively. I believe, in fact,
that pure arithmetic _as it is learned and known_ is largely an
_inductive science_. At one extreme is a minority to whom it is a series
of deductions from principles; at the other extreme is a minority to
whom it is a series of blind habits; between the two is the great
majority, representing every gradation but centering about the type of
the inductive thinker.