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UC-NRLF 



HA 



LIBRARY 

OF THE 

UNIVERSITY OF CALIFORNIA. 



"Received JAN 6 f8 g 3 .. ... 189. 
Accessions No.i^Q I O . Clots No. 



How to Teach Elementary Arithmetic 



.GRUBE'S METHOD 



TEACHING ARITHMETIC EXPLAINED 



WITH A LARGE NUMBER OF 



PRACTICAL HINTS AND ILLUSTRATIONS 



BY 

F. LOUIS SOLDAN 

PRINCIPAL OF THE ST. LOUIS NORMAL SCHOOL 




THE INTERSTATE PUBLISHING COMPANY 
BOSTON: 30 FRANKLIN STREET 






COPYRIGHT, 1878, 
Bv VAILE & W1NCHELL. 



PREFACE. 



THE first of the following two essays is the same in sub- 
stance as the one read before the St. Louis Teachers' Associa- 
tion in 1870, which has been republished since extensively in 
state and city school reports and educational magazines. It 
is presented here in a somewhat changed form, because the 
practical experience in the schoolroom has shown since what 
points of the method are in such harmony with established 
views as to require no further explanation, and what details 
need full comment and amplification in order to guard against 
such mistakes as are likely to creep in. In some respects I 
was guided by many inquiries on the part of the friends of the 
method. I regret to say that I have not always been able to 
answer these questions as fully as I wished. I hope that my 
correspondents will find the desired explanation in this new 
version of the old essay. I deem it my duty, however, to say, 
in justice to Mr. Grube, that the following pages are not in 
every respect a translation from his work, as has been supposed 
by some. One gentleman has done me the credit to publish 
my essay over his own signature as a translation from Mr. 
Grube's work. It should be distinctly understood that the full 
credit for one and every idea contained herein belongs to 

3 



4 PREFACE. 

Mr. Grube, but that he is not responsible at all for the many 
imperfections in the manner in which his thoughts are stated 
here. In a few instances only, the writer has allowed himself 
to depart from Mr. Grube 's ideas. The two essays are, may 
I be allowed to repeat, not altogether a translation, but rather 
an attempt to give a condensed account of the 160 pages of 
Mr. Grube's work. 

The second essay was read before the St. Louis Normal 
School Association in 1876, when it appeared proper to supply 
the continuation of the course recommended by a method 
which had attracted the attention of many thinking educators 
of the land, from California (See San Francisco Report of 
1872) to New Hampshire (See State Report of 1876). The 
second essay contains a recapitulation and continuation of the 
first essay. It presumes as little as its predecessor to recom- 
mend, but simply submits a new and important method to the 
thoughtful consideration of those who are interested in the 
matter. If circumstances permit, this little book will be fol- 
lowed by a text-book of Primary Arithmetic, based on Grube's 
Method. 

L. S. 

ST. Louis, November, 1878. 



UNIVERSITY 




GRUBE'S METHOD 

OF 

TEACHING PRIMARY NUMBERS. 



THE old, long-established method in arithmetic is 
calculated to teach the first four processes of addition, 
subtraction, multiplication, division, in the order in 
which they are named, finishing addition with small 
and large numbers, before subtraction is begun, and so 
forth. A more recent improvement on this method 
consisted in excluding the larger numbers altogether at 
the beginning, and dividing the numbers on which the 
first four processes were taught, into classes, or so-called 
circles. The child learns each of the four processes 
with the small numbers of the first circle (i.e., from I 
to 10) before larger numbers are considered ; then the 
same processes are taught with the numbers of the 
second circle, from 10 to 100, then of the third, from 
100 to 1,000, and so forth. 

Grube, however, went beyond this principle of classi- 
fication. He discarded the use of large numbers, hun- 
dreds and thousands, at the beginning of the course, as 
others had done before him ; but instead of dividing 
the primary work in arithmetic into three or four circles 
or parts only, i.e., from I to 10, 10 to 100, etc., he con- 
sidered each number as a circle or part by itself, and 
taught it by a method that is to be set forth in the 

s 



6 G RUBE'S METHOD. 

following pages. He recommended that the child 
should learn each of the smaller numbers in succession, 
and all the operations within the range of each number, 
before proceeding to the next higher one, addition, sub- 
traction, multiplication, and division, before proceeding 
to the consideration of the next higher number. 

In order to guard against a mistake .which has been 
made rather frequently, it should be stated that such 
examples only are considered to be within the limit of 
a number, and are to be taught in connection with it, 
in which a larger number than the one that is being 
considered does not appear in any way whatsoever. 
Thus, for instance, when the number four is taught, the 
teacher should exclude at the beginning addition and 
subtraction by fours, multiplication with 4 as one of the 
factors, division with 4 as the divisor, because these 
belong to a later and more advanced part of the course, 
since they involve in the sum, minuend, product, or 
dividend numbers beyond the limit of the one that is 
being considered. But all the examples that do not 
involve a higher number than four, are illustrated and 
taught, before passing over to the next higher number, 
five. Treating, for instance, the number 2, Grube leads 
the child to perform all the operations that are possible 
within the limits of this number, i.e., all those that do 
not presuppose the knowledge of any higher number, 
no matter whether in the usual classification these oper- 
ations are called addition, subtraction, multiplication, or 
division. The child has to see and to keep in mind that 

1 + 1=2, 2X1 = 2, 2 1 = 1, 2-5-1 = 2, etc. 
The whole circle of operations up to 2 is exhausted 



TEACHING ARITHMETIC EXPLAINED. J 

before the child proceeds to the consideration of the 
number 3, which is to be treated in the same way. 

Why adhere to the abstract division of the work in 
arithmetic into addition, subtraction, etc., in the primary 
grade, where these distinctions do not help to make the 
subject any clearer to the pupil ? The first four pro- 
cesses are naturally connected, and will appear so in 
the untaught mind. If you take away i from 2, and I 
remains, the child, in knowing this, also understands 
implicitly the opposite process of adding i to I and its 
result. 

Multiplication and division are, in the same way, 
nothing but another way for adding and subtracting, so 
that we might say one operation contains all the others. 
"Every text -book of primary arithmetic professes to 
teach the numbers in some way or other," says Grube ; 
" but to know a number really means to know also its 
most simple relations to those numbers, at least, which 
are smaller than it." Any child, however, who knows 
a number and its relations, must be also able to perform 
the operations of adding, subtracting, etc., for they are 
nothing but the expression of the relation in which one 
number stands to others. Each example shows what 
must be added to or subtracted from a number to raise 
it or lower it to equality with another, or, as in multi- 
plication and division, it sets forth the multiple relation 
of two numbers. 

The four processes are the direct result of comparing, 
or "measuring," as Grube calls it, two numbers with 
each other. Only when the child can perform all these 
operations, for instance, within the limits of 2, can it 
be supposed really to have a perfect knowledge of this 
number. So Grube takes up one number after the 



8 G RUBE'S METHOD. 

other, and compares it with the preceding ones, in 
all imaginable ways, by means of addition, subtrac- 
tion, multiplication, and division. This comparing or 
"measuring" takes place always on external, visible 
objects, so that the pupil can see the objects, the num- 
bers of which he has to compare with each other. The 
adherents of this method claim for it that it is based on 
a sound philosophical theory, and that it has proved 
superior in practice to the methods in use before its 
invention. 

Some of the most important principles of this method 
of instruction are given by Grube in the following : 

"i (Language). We cannot impress too much upon the 
teacher's mind, that each lesson in arithmetic must be a lesson 
in language at the same time. This requirement is indispen- 
sable with oar method. As the pupil in the primary grade 
should be generally held to answer in complete sentences, loud, 
distinctly, and with clear articulation, so especially in arithme- 
tic, the teacher has to insist on fluency, smoothness, and neat- 
ness of expression, and should lay special stress upon the 
process of solution of each example. As long as the language 
for the number is not perfect, the idea of the number is defec- 
tive as well. An example is not finished when the result has 
been found, but when it has been solved in a proper way. 
Language is the only test by which the teacher can ascertain 
whether the pupils have perfectly mastered any step or not. 

"2 (Questions). Teachers should avoid asking too many 
questions. Such questions, moreover, as, by containing half 
the answer, prompt the scholar, should be omitted. The 
scholar must speak himself as much as possible. 

"3 (Class and Individual Recitation). In order to animate 
the lesson, answers should be given alternately by the scholars 
individually, and by the class in concert. The typical numeri- 



TEACHING ARITHMETIC EXPLAINED. 9 

cal diagrams (which, in the following, will continually re-ap- 
pear) are especially fit to be recited in concert. 

"4 (Illustrations). Every process and each example should 
be illustrated by means of objects. Fingers, lines, or any other 
objects will answer the purpose, but objects of some kind must 
always be presented to the class. 

"5 (Comparing and Measuring). The operation of each 
new stage consists in comparing or measuring each new num- 
ber with the preceding ones. Since this measuring can take 
place either in relation to difference (arithmetical ratio), or in 
relation to quotient (geometrical ratio), it will be found. to 
comprise the first four rules. A comparison of two numbers 
can only take place by means of one of the four processes. 
This comparison of the two numbers, illustrated by objects, 
should be followed by exercises in the rapid solving of prob- 
lems and a view of the numerical relations of the numbers 
just treated, in more difficult combinations. The latter offer a 
good test as to whether the results of the examination of the 
arithmetical relations of the number treated have been con- 
verted into ideas by a process of mental assimilation. In con- 
nection with this, a sufficient number of examples in applied 
numbers are given to show that applied numbers hold the same 
relation to each other that pure numbers do. 

"6 (Writing of Figures). On neatness in writing the 
figures, the requisite time must be spent. Since an invariable 
diagram for each number will re-appear in all stages of this 
course of instruction, the pupils will soon become able to 
prepare the work for each coming number by writing its 
diagram on their slates/' 



10 G RUBE'S METHOD. 

It will appear from this that Mr. Grube subjects each 
number to the following processes : 

I. Exercises on the pure number, always using objects for 

illustration. 

a. Measuring (comparing) the number with each of the 

preceding ones, commencing with i, in regard to 
addition, multiplication, subtraction, and division, 
each number being compared by all these processes 
before the next number is taken up for comparison. 
For instance, 6 is first compared with i by means of 
addition, multiplication, subtraction, and division, 

(i + i +, etc. = 6 ; 6 X i = 6 ; 

61 1, etc. = i ; 6 -j- i = 6) 

then with 2, then with 3, and so forth. 

b. Practice in solving the foregoing examples rapidly. 

c. Finding and solving combinations of the foregoing 

examples. 

II. Exercises on examples with applied numbers. 



In the following, Mr. Grube gives but the outline, 
the skeleton as it were, of his method, trusting that the 
teacher will supply the rest. The sign of division, as 
will be explained below, should be read at the begin- 
ning : " From ... I can take away ... times." By 
this way of reading, the connection between subtraction 
and division becomes evident. 



TEACHING ARITHMETIC E'XPLAINED. II 

FIRST STEP. 
THE NUMBER ONE. 

"As arithmetic consists in reciprocal ' measuring' (com- 
paring), it cannot commence with the number i, as there is 
nothing to measure it with, except itself as the absolute 
measure." 

I. The abstract (pure) number. 
One finger, one line ; one is once one. 
The scholars learn to write : 

I. 
1X1 = 1. 

II. The applied number. 

What is to be found once, in the room, at home, on the 
human body? 



SECOND STEP. 
THE NUMBER TWO. 



I. The pure number. 
a. MEASURING (comparing). 



2. 



1 + 1 = 2. 
2X1 = 2. 

2 1 = 1. 

2-5-1 = 2. (Read : From 2 I can take' 
away i twice.) 

2 is one more than i. 

1 is one less than 2. 

2 is the double of i, or twice i. 
i is one-half of 2. 



12 G RUBE'S METHOD. 

b. PRACTICE IN SOLVING EXAMPLES RAPIDLY, i + i = ? 

2 I = ? 2 -r- I = ? I + I I X 2 = ? etC. 

c. COMBINATIONS. 

What number is contained twice in 2 ? 
2 is the double of what number? 
Of what number is i one-half? 
Which number must I double to get 2 ? 
I know a number that has in it one more than one. 
Which is it? 

What number have I to add to i in order to get 2 ? 

II. Applied numbers. 

Fred had two dimes, and bought cherries for one dime. 
How many dimes had he left? 

A slate-pencil costs i cent. How much will two slate-pen- 
cils cost? 

Charles had a marble, and his sister had twice as many. 
How many did she have? 

How many one-cent stamps can you buy for 2 cents ? 



THIRD STEP. 
THE NUMBER THREE. 

I. The pure number. 
a. MEASURING. 
(i) By i. 

3- 



1 + 1 + 1 = 3. 
3x1 = 3. 

3 I 1 = 1. (Better than 3-1 1 1=0.) 

for, 3 - i = 



TEACHING ARITHMETIC EXPLAINED. 13 

This ought to be read : From 3 I can take away i 3 times, 
or, in three, i is contained three times. The ideas of "to be 
taken away " and " to be contained " must always precede the 
higher and more difficult conception of dividing. 

(2) Measuring by 2. 

2 + 1 = 3; I -f- 2 = 3- 
1X2 + 1 = 3. 

3 - 2 = i ; 3 - i = 2. 
3-5-2 = 1 (i remainder). 

(From 3, I can take away 2 once, and i will remain ; or, In 
three, 2 is contained once and one over.) 

3 is i more than 2, 3 is 2 more than i. 

2 is i less than 3, 2 is i more than i. 
i is 2 less than 3, i is i less than 2. 

3 is three times i. 

i is the third part of 3. 

i and i are equal numbers, i and 2, as well as 2 and 3 
are unequal. 

Of what equal or what unequal numbers does 3 consist, 
therefore? etc. 

b. PRACTICE IN SOLVING EXAMPLES RAPIDLY. 

How many are 3 1 1 + 2 divided by i ? 

i + i + i 2 + I + I 2 + i + i=? 

3X i 2Xi + i + i 2 + i + i = ? etc. 

The answers must be given immediately. 

No mistakes can arise as to the meaning of these exam- 
ples ; the question whether 3X1 2 means (3 x i) 2 
or 3 x (i 2) is answered jpy the fact that these examples 
represent oral work, and that it is supposed that the operation 
indicated by the first two numbers (3X1) is completed 
mentally before the next number is given. 




14 G RUBE'S METHOD. 

c. COMBINATIONS. 

From what number can you take twice i and still keep i ? 
What number is three times i ? 

I put down a number once, and again, and again once, 
and get 3 ; what number did I put down 3 times ? 

II. Applied numbers. 

How many cents must you have to buy a three-cent stamp ? 

Annie had to get a pound of tea for 2 dollars. Her mother 
gave her 3 dollars. How much money must Annie bring back ? 

Charles read one line in his primer, his sister read 2 lines 
more than he did. How many lines did she read? 

If one slate-pencil costs one cent, how much will 3 slate- 
pencils cost? 

Bertha found in her garden 3 violets, and took them to her 
parents. How can she divide them between father and mother ? 



FOURTH STEP. 
THE NUMBER FOUR. 



I. The pure number. 
a. MEASURING. 

(i) By i. 



i 

i 

i 

i 



I + I -f- I + I = 4 (i+i = a, 2 + 1 = 3). 

4X1=4- 

41 1 1 = 1. 

4-5-1=4. 



TEACHING ARITHMETIC EXPLAINED. 15 

(2) Measuring by 2. 

2 f 2 + 2 = 4. 

2X2 = 4. 

4 2 = 2. 

4-7-2 = 2. 

(3) Measuring by 3. 

3 + i = 4, i + 3 = 4. 
1X3 + 1 = 4- 

4 - 3 = i, 4 - i = 3- 

4 -j- 3 = i (i remainder). 

(In 4, 3 is contained once and i over ; or from 4 I can take 
away 3 once, and one remains.) 

Name animals with 4 legs and with 2 legs. 

Wagons and vehicles with i wheel, 2, and 4 wheels. Com- 
pare them. 

4 is i more than 3, 2 more than 2, 3 more than i. 

3 is i less than 4, i more than 2, 2 more than i. 
2 is 2 less than 4, i less than 3, i more than i. 

i is 3 less than 4, 2 less than 3, i less than 2. 

4 is 4 times i, twice 2. 

i is the fourth part of 4, 2 one-half of 4. 
Of what equal and unequal numbers can we form the num- 
ber 4 ? 

b. PROBLEMS FOR RAPID SOLUTION. 

2x2 3 + 2 x i i 2 + 2=? 

41 1 + 1 + 1 3, how many less than 4? etc. 

c. COMBINATIONS. 

What number must I double to get 4 ? 

Four is twice what number? 

Of what number is 2 one-half? 

Of what number is i the fourth part? 

\Vhat number can be taken twice from 4 ? 



1 6 G RUBE'S METHOD. 

What number is 3 more than i ? 

How much have I to add to the half of 4 to get 4 ? 

Half of 4 is how many times one less than 3 ? etc. 

II. Applied numbers. 

Caroline had 4 pinks in her flower-pot, which she neglected 
very much. For this reason, one day one of the flowers had 
withered, the second day another, and the following day one 
more. How many flowers did Caroline keep ? 

How many dollars are 2 + 2 dollars? 

Three apples and one apple ? 

4 quarts = i gallon. 

Annie bought a gallon of milk ; how many quarts did she 
have ? 

She paid i dime for the quart ; how many dimes did she pay 
for the gallon ? 



quart, 

quart, 

quart, 



dime, 
dime, 
dime. 



quart, dime. 



What part of i gallon is i quart ? 

If i quart costs 2 dimes, can you get a gallon for 4 dimes? 
A cook used a gallon of milk in 4 days. How much did she 
use each day? 



The recitations should be made interesting and animated by 
frequently varying the mode of illustration, and in this the in- 
genuity of the teacher and her inventive power can display 
themselves to their best advantage. It is, of course, superfluous 
to describe the infinite variety of objects which may be used, 



TEACHING ARITHMETIC EXPLAINED I'/ 

but a few suggestions will perhaps prove acceptable. Those 
illustrations which compel the whole class to be active* or which 
are of special interest, and arouse the attention of pupils, are 
of greater value than others. For instance : 

" Class, raise two fingers of your right hand ; two fingers of 
your left hand. How many fingers have you raised? Two 
fingers and two fingers are how many? Two and two are how 
many ? Carrie may show to the class, with her fingers, that two 
and two are four." 

This plan of illustrating should be used very frequently, as it 
requires the whole class to be active. The following illustration 
is also commendable, as it hardly ever fails to enlist the inter- 
est of the class ; every pupil likes to be allowed to illustrate a 
problem in this way : 

" From four I can take away two, how many times ? Emma 
may show that her answer is correct, by making some of the 
other girls stand." (The class know that those whom Emma 
teaches must stand until she makes them take their seats again.) 
Emma : " Four little girls are standing here. From 4 little 
girls I can take away 2 little girls once (making two of the four 
take their seats), twice (making the other two sit down). From 
4 little girls, I can take away 2 little girls twice. From 4 I can 
take away 2 twice. 4-5-2 = 2." 



i8 



GRUBE'S METHOD. 



FIFTH STEP. 
THE NUMBER FIVE. 



I. The pure number. 
a. MEASURING. 
(i) By i. 



I 
I 

I 

(2) By 2. 

2 

I 

(3) By 3. 

3 

2 

(4) By 4. 

4 

i 



1 + 1 + 1 + 1 + 1 = 5, 

5Xi = 5. 

5 1 1 1 1 = 1, 

5 -*- i = 5- 



2X2 



1=5 



5-2-2=1. 

5 -r- 2 = 2 (i remainder). 



1X3 + 2 = 5. 

5-3 = 2;5-2 = 3 . 

5-^3= i (2 remainder). 



4 + i = 5; i + 4 = 5. 

1X4 + 1 = 5- 

5 - 4 = i ; 5 - i = 4- 
5-7-4=1(1 remainder). 



The fingers are the best means of illustration here : " Hold 
up your left hand. How many fingers are you holding up ? 



TEACHING ARITHMETIC EXPLAINED. 19 

Hold the thumb away from the other fingers. How many 
fingers here? (i) ; here? (4). i finger taken from 5 fingers 
leaves how many fingers ? i from 5 = ? 4 fingers -f- i finger 
= ? 4 -h i = ? Hold your first finger and the thumb away 
from the other fingers. 5 2 = ? 3 + 2 = ? 2 + 3 = ?" 
etc. 

5 is one more than 4, 5 is 2 more than 3, 5 is 3 more than 
2, 5 is 4 more than i. (All the solutions of these examples 
are the result of observation from illustrations placed before 
the eyes of the class ; without them this kind of instruction is 
worthless.) 

4 is i less than 5 ; 4 is i more than 3, etc. 
3 is 2 less than 5, etc. 

5 = 5X1- 

i = i X 5 (i is the fifth part of five). 
5 consists of two unequal numbers, 3 + 2. 5 consists 
of two equal numbers and one unequal number, 2 + 2 + 1. 

b. PRACTICE IN THE RAPID SOLUTION OF EXAMPLES. 

(It would be a great mistake to drill on the same example 
until the pupils can remember it. Such a practice would be 
worse than valueless ; every example should be a new one to 
the pupil, and the faculty appealed to should be judgment as 
well as memory.) 

5 2 3 + 2X2, one-half of it less i, taken 5 
times = ? 

2x2 + 1 3X1X2 3 + 4=? etc. 

c. COMBINATIONS. 

What number is one fifth of 5 ? How many must I add to 
3 to get 5 ? How many must be taken away from 5 to get 3 ? 
How many times two have I added to i in order to get 5 ? I 
have taken away twice 2 from a certain number, and i remained. 
What number was it ? etc. 



20 



G RUBE'S METHOD. 



II. Applied numbers. 

How many gallons are 2 quarts ? 

Charles had 5 dimes ; he bought two copy-books, each of 
which cost two dimes. What money did he keep ? (This the 
teacher must make plain by means of lines and dots.) 



i i 

Dime. Dime. 



Copy-book. 



i i 

Dime. Dime. 

Copy-book. 



i 
Dime. 



Henry read a lesson three times, Emma read it as many 
times as he did, and two times more. How often did she read 
it ? Father had five peaches, and gave them to his 3 children. 
The youngest one received one peach ; how many did each of 
the other children receive ? etc. 



SIXTH STEP. 



THE NUMBER SIX. 

I. The pure number. 

a. MEASURING. 
aa with i 
bb with 2 
cc with 3 
d/with 4 
ee with 5 j 

ff Miscellaneous examples. 

b. RAPID SOLUTION OF PROBLEMS. 

c. COMBINATIONS OF NUMBERS. 



Each process illustrated by six lines, of 
which as many are placed in a row as are 
indicated by the number by which 6 is to 
be measured. 



II. The applied number. 



TEACHING ARITHMETIC EXPLAINED. 21 

Grube thinks that one year ought to be spent in this 
way on the numbers from I to 10. He says, " In the 
thorough way in which I want arithmetic taught, one 
year is not too long for this most important part of the 
work. In regard to extent, the scholar has not, appar- 
ently, gained very much he knows only the numbers 
from I to 10. But he knows them." 

In reference to the main principles to be observed, 
he demands, first, " that no new number shall be com- 
menced before the previous one is perfectly mastered ;" 
secondly, "that reviews should frequently and regularly 
take place ;" and lastly, "that whatever knowledge has 
been acquired and fully mastered by illustration and 
observation, must be thoroughly committed to memory." 
" In the process of measuring, pupils must_ acquire the 
utmost mechanical skill." It is essential to this method, 
that in the measuring, which forms the basis for all 
subsequent operations, the pupils have before their eyes 
a diagram illustrating the process. It matters not by 
means of what objects the pupils see the operation illus- 
trated, whether fingers, lines, or dots, but they certainly 
must see it. 

It is a feature of this method, that it teaches by the 
eye as well as by the ear, while in most other methods 
arithmetic is taught by the ear alone. If, for instance, 
the child is to measure 7 by the number 3, the illustra- 
tion to be used is : 



If lines or dots are arranged in this way, and im- 
pressed upon the child's memory as depicting the rela- 



22 G RUBE'S METHOD. 

tion between the numbers 3 and 7, it is, in fact, all 
there is to know about it. Instead of teaching all the 
variety of possible combinations between 3 and 7, it is 
sufficient to make the child keep in mind the above 
picture. The first four rules, as far as 3 and 7 are con- 
cerned, are contained in it, and will result from express- 
ing the same thing in different words, or describing the 
picture in different ways. Looking at the picture, the 
child can describe it, or read it as : 

3 + 3 + 1 = 7, or 2x3 + 1= 1, 

or 7 - 3 ~ 3 = i; 7 -*- 3 = * (0- 

The latter process to be read, From 7 I take away 3 
twice, and I remains ; or, 7 contains 3 twice and one 
more. 

Let the number to be measured be 10, and the num- 
ber by which it is to be measured be 4 ; then since the 
way to arrange the lines or dots for illustration is to 
have as many dots or lines as are indicated by the 
larger number, and as many of them in a row as are 
indicated by the smaller number, we write : 



The child will be able to see at once, by reading the 
diagram, as it were, that 

4 + 4 + 2 = 10 ; 2 X 4 + 2 = 10; 

10 -4-4= 2; 10 -=-4=2 (2), 

and to perceive at a glance a variety of other combina- 
tions. The children will, in the course of time, learn 



TEACHING ARITHMETIC EXPLAINED. 2$ 

how to draw these pictures on their slates in the proper 
way. Nor will it take long to make them understand 
that every picture of this kind is to be "read" in four 
ways, first using the word and, then times, then less, 
then, From . . . can be taken away . . . times. As 
soon as the pupils can do this, they have mastered the 
method, and can work independently all the problems, 
within the given number, which are required in measur- 
ing. 

It would be a mistake to suppose that, in teaching 
according to this method, memory is not required on 
the part of the child. Memory is as important a factor 
here as it is in all instruction. This should be empha- 
sized, because with some teachers it has become almost 
a crime to say that memory holds its place in education. 
To have a good memory, is, in their eyes, a sign of 
stupidity. Grube was too experienced a teacher to fall 
into this error. While by his method the results are 
gained in an easier and more natural way, whatever 
result is arrived at must be firmly retained by dint of 
memory assisted by frequent reviews. 

(END OF FIRST ESSAY.) 



24 G RUBE'S METHOD. 



NUMBERS ABOVE TEN. 



SECOND ESSAY. 

WHEN Grube's Method of teaching the elements of 
arithmetic was first introduced to the Teachers' Asso- 
ciation of St. Louis, in 1870, it was not presented with 
the assurance of warranted success as the only plan of 
teaching this important study, but rather as an attempt 
to demonstrate practically, in a given instance, to some 
extent at least, how far methods of teaching may be 
redeemed from the bane of vagueness, which, as long 
as it lasts, excludes them from the rank, in the science 
of Pedagogics, to which they might otherwise be en- 
titled. Grube's Method was submitted with diffidence 
to the judgment of practical teachers, without the com- 
mendation of any champion who expressed an implicit 
belief in its immediate signal success. We may speak 
disparagingly of the often frivolous distinction between 
theory and practice, which ignores the harmonious 
parallelism between the world of thought and the world 
of fact, but we" shall nevertheless insist upon practical 
usefulness as the test of any psychologically correct 
method of teaching. 

To-day Grube's Method of teaching arithmetic does 
not lack friends and supporters : it has been tried and 
adopted, not in one city alone, but has become recog- 
nized throughout the country. Long before its practi 



TEACHING ARITHMETIC EXPLAINED. 25 

cal test in the district schools of St. Louis, it was made 
part of the regular course of instruction in the schools 
of San Francisco, and many other cities have adopted it 
since. Practical experience has shown the advantages 
and disadvantages of this system as far as the part 
which was presented at that time is concerned ; namely, 
the numbers from one to ten only. 

Beyond this limit there is still disputed ground, and 
it may be allowable to say that the continuation is of- 
fered to-day in the same spirit as the beginning was 
years ago. It is simply a report on an ingenious meth- 
od which is considered worthy the notice of thoughtful 
teachers, and which seems to deserve a fair trial, con- 
tinued for a sufficient length of time to extend beyond 
the period during which a new method seems objection- 
able because it is new, and hence collides with a prac- 
tice which habit has made convenient. 

The leading idea is the same throughout Grube's 
Method. To show the principle of teaching the higher 
numbers to 100 is to recapitulate the principles that 
are to guide the teacher in his treatment of the num- 
bers from i to 10. That the four processes are taught 
with each number, before -the following one is consid- 
ered, forms, no doubt, a characteristic feature of Grube's 
Method, but it is a common mistake to suppose that it 
is the leading idea. It certainly emanates from this 
idea, but it is not the idea itself. The leading principle 
is rather that of objective illustration. 

In a very general way it may be said that in examples 
in primary arithmetic two numbers are given, and their 
relation, expressed by a third number, is to be found. 
Hence the elementary processes may be considered as 
the comparing of one number with the other, or the 



26 G RUBE'S METHOD. 

measuring of one by the other. On the basis of this 
general theory, Grube suggests a general plan of illus- 
tration, according to which the larger number of the 
two numbers given is represented by the total number 
of lines or dots placed on the blackboard. These lines 
are arranged into sets or groups, each containing as 
many lines or dots as are indicated by the smaller 
number of the two. Thus, if the numbers 6 and 2 are 
to be compared with each other, the illustration consists 
of six dots, arranged two by two. 



The measuring of 9 by 4 is illustrated by four dots and 
four dots and one dot. 



This contains the main principle of Grube's Method. 
If perception has seized this illustration, and wrought it 
into a mental picture, the solution of all the existing 
elementary relations between the two numbers has 
been grasped implicitly. For the four processes are 
simply different interpretations of this symbolic dia- 
gram. When this picture appears before the mind, it 
may be interpreted as addition or multiplication, i.e., 
our illustration may be read, 

4 + 4 + i = 9, or (2 X 4) + i = 9 ; 

and by the retrograde process, when the illustration is 
made to disappear from the blackboard, it may be in- 



TEACHING ARITHMETIC EXPLAINED. 2/ 

terpreted by subtraction or division, as 9 4 4 = I, 
or from 9 I can take away 4 twice and leave i. But 
the main point in this is, that the whole process is based 
on well selected and arranged illustrations, and is an 
object lesson on numbers. A plan of teaching which 
ignores this main point, and flatters itself to have found 
the gist of the new idea by jumbling together addition, 
subtraction, multiplication, and division, without the 
most extensive use of illustrative objects, and without 
systematic arrangement, has nothing in common with 
Grube's Method. In the latter, the clearest order and 
regularity prevail throughout. Below 10, each number 
is compared with the number i, by means of addition, 
subtraction, multiplication, and division, then with the 
number 2, then with 3, etc. The pupil will soon learn 
to perceive the regularity of this process ; and at the 
moment he has understood that part, he can by inde- 
pendent work discover the primary arithmetical rela- 
tions of a number, and prepare a synopsis or diagram 
of the same. 

A frequent and very dangerous mistake is the omis- 
sion, or neglect, of applied examples. The pure number 
as the universal expression of arithmetical truth is of 
the greatest importance, but the pupil throughout his 
school-course finds the greatest difficulty in working 
with applied numbers. Moreover, arithmetic is studied 
for life ; and in life, there are none but applied examples. 
Hence, after the universal, the pure number, has been 
mastered by means of observation, particular applica- 
tion should follow immediately, and copious examples, 
clothed in the most varied forms, should be solved. The 
training which the pupil receives from practice with 
applied problems is different in kind from that with pure 



28 G RUBE'S METHOD. 

numbers, and hence cannot be slighted in the primary 
grades without retarding the progress in the higher 
classes. Without sufficient practice in this direction, 
there is danger of mechanical and dull work, and the 
best opportunities for the pupil's display of inventive 
ingenuity are lost. 

The difficulty which the study of arithmetic presents 
in the higher grades lies not in the mechanical handling 
of numbers, in most cases the pupils succeed very well 
in that, but it lies in the fact that the words of the 
problem puzzle them. The qualitative element dis- 
turbs and conceals the quantitative. If this assertion 
is correct, a great deal of training with applied numbers 
should be given at a time of the course when the pure 
number which is considered is so small as to allow the 
scholar, after having mastered it, to concentrate his 
whole attention on the puzzle that lies in the wording, 
in the qualitative. Wherever sufficient training of this 
kind has sharpened the wit of the pupils in the lower 
grades, they will no longer consult the heading of the 
chapter as the first step in the solution of a problem, 
in order to find whether it means addition or division, 
interest or long measure, and find themselves in a help- 
less and forlorn condition when they meet an example 
which is not labeled by any heading. 

An analysis of the operation with each nifmber shows 
as the two principal elements : 

I. The number considered in its universal quantitative 
character, or pure number. The process is from objec- 
tivity to abstraction. 

II. TJie quantitative in special qualitative form, or 
applied number. Here we proceed from abstraction to 
application. 



TEACHING ARITHMETIC EXPLAINED. 29 

Under the first or pure number we have the sub- 
topics : 

a. Comparing with, or measuring by, each of the 
preceding numbers, from I to 10, considering addition, 
multiplication, subtraction, and division. 

b. Combinations of the two numbers treated of, the 
results to be within the limits of the greater one of 
the two numbers. This is a very important process, no 
doubt, but the temptation lies near to give too much 
prominence to it by forgetting that it is a part only of 
Grube's Method. The systematic comparison of num- 
bers is of greater importance, and it is an error to 
spend as much time on these combinations as if the 
method consisted of nothing but these. 

c. Sufficient practice in the rapid solution of ex- 
amples. 

In the former essay, the treatment of the numbers 
from one to five was explained. As the last step within 
the circle of numbers from one to ten, and as the 
transition to the province of larger numbers, the treat- 
ment of the number ten is of great importance. Grube 
describes it in the following way : 



30 G RUBE'S METHOD. 

TENTH STEP. 
THE NUMBER TEN. 

We have arrived at a number which is again treated as a 
unit. Hence we write it by means of the figure one ; but to 
show there is ten times as much in this as in the figure one 
which we had before, we move it one place toward the left, by 
which we mean to say, This unit means a ten. The empty 
place of the simple unit is filled out by a cipher. 

I. The pure number. 
a. MEASURING (10; i). 



I+T+I+I+I etc. = 10 



10 x i = 10 



10 i i etc. = i 



^- i = 10 



(10 ; a) 



2 + 2 etc. = 10 
5 + 2 = 10 

IO 2 2CtC.= 2 
10-^2 = 5 



(10; 3) 

3 + 3 + 3 + i = io 
3 X 3 + i = 10 
10-3-3-3 = 1 
10-5-3 = 3 (i) 



(10 ; 6) 



6 + 4=10 
1X6 +4 = 10 
10 6 = 4 
10-5- 6 = i (4) 



TEACHING ARITHMETIC EXPLAINED. 31 

Miscellaneous Measuring : 

10 consists of two equal numbers, 5+5. 

10 consists of five equal numbers, 2 + 2 + 2+ 2 + 2. 

10 consists of two equal numbers and one unequal, 
5X3 + 1. 

10 consists of four unequal numbers, 1 + 2+3 + 4. 
Review of the multiple relations within the number ten. 

A. I. i is one-half of 2, one-third of 3, one-fourth of 4, etc. 
II. 2 is one-half of 4, one-third of 6, etc. 

III. 3 is one-half of 6, one-third of 9. 

IV. 4 is one-half of 8. 
V. 5 is one-half of 10. 

B. I. 10 is 10 times i, 5 times 2, 2 times 5. 
II. 9 is 9 times i, 3 times 3. 

III. 8 is 8 times i, 4 times 2, 2 times 4. 

IV. 7 is 7 times i. 

V. 6 is 6 times i, 3 times 2, 2 times 3. 
VI. 5 is 5 times i. 

VII. 4 is 4 times i, 2 times 2. 

VIII. 3 is 3 times i. 

IX. 2 is 2 times i. 

X. i is once i. 

What numbers are contained without any remainder in 10, 
9,8? 

What numbers have no other numbers contained in them 
without remainder except the number i ? (The prime numbers 
' 3> 5. 7-) 

b. COMBINATIONS (Oral work). 

One nickel and two cents and three cents, less 6 cents, of 
this take one-half three times, and add twice two cents ; how 
many cents? 



32 G 'RUBE'S METHOD. 

(There is no better exercise for rapidity and exactness than 
this. Short combinations, slowly pronounced at first, until the 
class can solve more difficult problems, given out quickly. The 
teacher should take care not to discourage the class by ex- 
amples that can be answered by the brightest scholars only. 
No guessing should be allowed ; use illustrations.) 

(2 x 2) -f- (2 x 3) - (3 X 3) 4- (2 X 4) + i =? 
102 1 2 1 2 1=? 
1 + 2 + 3 + 4=? etc. 

c. PRACTICE IN THE RAPID SOLUTION OF EXAMPLES. 

What number is i more than twice 3 ? 

Twice five is how many more than three times three ? than 
twice four? 

A father distributed 10 apples among his children, so that 
each older child received one more than the one next below 
him in age. How many apples did each child receive? \The 
pupils know that 10 consists of 4 unequal numbers, i, 2, 3, 4, 
of which each following number is greater by one than the 
preceding. Hence the father could divide the apples so that 
the youngest received one, the next two, etc.) 

Charles had learned four words in spelling. His brother 
said, " I know twice as many as you, and 2 more." How many 
did he know? Solution : If Charles had learned 4 words, and 
his brother knew twice as many and 2 more, he knew 2X4 + 2 
words = 10 words. 

William said, " I am 5 times as old as my little sister." She 
was 2 years old. How old was William ? Solution : If the 
little sister was 2 years old, and William five times as old, he 
was 5 x 2, or 10 years old. 

II. Applied numbers. 

10 days are how many weeks and days? 
10 cents are how many dimes? nickels? 



TEACHING ARITHMETIC EXPLAINED. 33 

Fred had one dime : he bought 2 slate-pencils for one cent 
each, and a piece of candy for 5 cents. How much money did 
he spend ? How much had he left ? 

One lead-pencil costs 5 cents; how much will two lead- 
pencils cost? 

How many marbles at 2 cents apiece can you buy for 10 
cents? Solution : For 2 cents I get i marble, hence for 10 cents 
I get five marbles, since 10 cents are 5X2 cents. Or, If I 
give to the store-keeper 2 cents, he gives me i marble ; but 
if I have 10 cents, I can give him 5 times 2 cents, and so he 
gives me 5 times one marble, etc. 



With this number, says Grube, the first and most 
important step in arithmetic has been completed. If 
the subject has been taught as it should have been, one 
year is not too long a time for it. (It seems that Grube 
is speaking on the supposition that about two hours a 
week form the time given to the study of arithmetic.) 
The pupil's knowledge is not very extensive : he knows 
but the numbers from I to 10. But would he really 
possess any knowledge of arithmetic if he were able to 
count up to 100 and beyond without being able to solve 
any problem, even with the smallest number ? Learn- 
ing the names of numbers up to 100 is not the same 
as learning to count, and is simply learning by rote a 
series of words, not of much more importance for arith- 
metic than the committing to memory of a few lines of 
poetry. Special attention should be given to the prac- 
tice in the rapid solution of miscellaneous examples 
(I. c.), as these exercises are essential for a clear idea of 
number. If clear perception has preceded them, they 
will present no difficulty. Remember that no new 
number is to be taken up before the previous one has 



34 G RUBE'S METHOD. 

been thoroughly mastered, and that frequent reviews 
must help the pupil in fixing in his memory the princi- 
pal examples which have been considered and reduced 
to writing. 

Passing over to the higher numbers, Grube says, " In 
the second year the numbers from 10 to 100 are to be 
studied. The following principles must be observed in 
this work : " 

1. Fingers and lines are used for illustration, the former 
being the most natural means. 

2. The process with the numbers from 10 to 100 is the same 
as that for the smaller numbers. Multiplication and division 
form the subjects of written and oral work, while addition and 
subtraction, as a rule, need oral treatment only. Measuring 
each new number by the numbers from i to 10 is continued as 
oral, preparatory work until the pupils have acquired in it the 
greatest mechanical skill. 

3. Greatest diversity of expression and sufficient variety are 
aimed at in the selection of examples, in pure as well as in 
applied numbers, so that the pupil may free himself gradually 
from the uniformity of the elementary diagram and schedule.. 
Applied examples should not go beyond the limit of qualitative 
relations taken from daily life with which the pupil is familiar. 
This will give him an opportunity of inventing examples him- 
self, and the permission to give an example to the class may be 
made a reward for that pupil who succeeds in finding the solu- 
tion of some examples first. 

Before proceeding to describe Grube's treatment of 
some numbers of the circle from 10 to 100, it will be 
best to recall to memory the few essential points of 



TEACHING ARITHMETIC EXPLAINED. 35 

difference and agreement with the previous part of the 
course. 

1. The processes with each number remain the same, 
namely : 

1. Exercises with the pure number, by 

(a) Comparison. 

(/) Combination. 

(c) Practice in the rapid solution of examples. 

2. Exercises with applied number. 

2. Objective illustrations form the most important part of 
each exercise. Arithmetic is a series of object lessons on 
numbers. 

3. Each new number is not compared with all the numbers 
below itself, but with the numbers from i to 10 only. 

4. Comparison with these numbers by means of addition 
and subtraction forms as a rule the subject of oral work only : 
comparison by multiplication and division is practised both 
orally and in writing. 

5. In writing out these comparisons of numbers, the ex- 
amples are no longer placed side by side, but below each other : 

(n ; 2) 2 + 2=4 
4 + 2 = 6 
6 + 2=8 
8 + 2 = 10 

IO + I = II 

6. Oral comparison by addition and subtraction takes usually 
the form of, Count upward or downward by twos, threes, 
fours, etc. 




36 G RUBE'S METHOD. 

7. As the same examples occur frequently, Grube supposes 
that the pupil has acquired sufficient skill to master about two 
numbers each recitation ; he is speaking, however, of recitations 
of 60 minutes each. 

8. More time is to be given to the lower numbers from i to 

24, and especially to numbers that are of importance in applied 
examples as representing some division in compound numbers, 
such as 12 (dozen, number of months, etc.), 14 (days in 2 
weeks), 15, 1 6 (number of ounces in a pound), 18, 20, 21, 24, 

25, 28, 30, 36, 48, 56, 64, 72, etc. In connection with them 
the principal divisions of compound numbers should be taught. 

After this general explanation, an application of the 
principles set forth to a few particular numbers will 
suffice to show the process. 



TEACHING ARITHMETIC EXPLAINED. 

TWELFTH STEP. 
THE NUMBER TWELVE. 

I. a. Pure number. 
MEASURING. 



10 



12 



Oral work. Measuring. 



(12; i) (12; 2) (12; 3) 


(12; 6) 


i + i = 

2 + I = 
II + I = 


' 2 + 2 = 

4 + 2 = 

6 + 2 = 

etc. 


3 + 3 = 
6 + 3 = 
9 + 3 = 


6 + 6 = 12 


or 
i, 2, 3, 4, 
etc. 


or 
2,4,6,8, 10, 12 

. 


or 
3> 6, 9, 12 




f 12 I = 

II I = 
10 I = 


12 2 = 
10 2 = 


12-3 = 
9-3 = 


12 - 6 = 


9 c. = 








or 

12, II, 10, 

etc. 


or 12, 10, 8, 6 


or 
12, 9, 6, 3 





12X1 = 12 6X2= 12 4X3= 2X6 = 

12 -5- I = 12 12 -5- 2 = 12 -=- 3 = 1 2 -5- 6 = 



38 G RUBE'S METHOD. 

WRITTEN WORK. 

12 = 12 x i 12 = 12 x ? 12 = 11 + 1 

= 6x2 i2=6x? (i2isi more 

= 4X3 i2=4X? than n.) 

= 3X4 i2=3X? =10+2 

= 2X5 + 2 l) 12 = 12 X 

= 2x6 2)i2 = 6x etc. =9 + 3 

= * X 7 + 5 To be read : 

= 1x8 + 4 a. i can be taken away from 12, 

= 1X9 + 3 I2 times. 

= i X 10 + 2 b. i is contained or is in 12, 12 

i tenth 2 units times. 

c. i is the 1 2th part of 12, etc. 

Of what equal numbers is twelve composed ? 
Of what unequal numbers ? 

Give three numbers that make twelve, of which each follow- 
ing number is two more than the previous one. 

b. COMBINATIONS. ( Oral. ) 

(2 X 2) + (2 x 2) + (2 + 2) =? 

2 + 3 + 3 + 2 + 2-4 + 4- (4x2)=? 

Charles, Fred, and George had 1 2 apples ; they ate one-half 
of them and one more ; how many had they left ? how many did 
they eat? etc. 

c. PRACTICE IN THE RAPID SOLUTION OF EXAMPLES. 
The third part of 1 2 is what part of 8 ? 

One-half of 1 2 is how many times 3 ? 

What is the difference between one-half of 1 2 and one-half 
of 10? 

12 is three times what number? 

What number must I take from 1 2 to have 9 ? 

What number taken away from 1 2 leaves 4 ? etc. 



TEACHING ARITHMETIC EXPLAINED, 39 



II. Applied number. 

12 pieces equal a dozen. Half a dozen = ? 

12 months are called a year. (The names of the months 
are to be committed to memory.) 

What part of a dozen are six pieces ? 

What part of a year are six months ? 

3 months are a quarter (of a year). 

3 pieces are a quarter of a dozen. 

A month has about 4 weeks. Fred pays $12 a month for 
piano lessons ; how much does he pay a week ? 

Solution : One month has 4 weeks. If he pays for 4 weeks 
$12, he pays for one week the fourth part of 12, which is 3. 

A father paid $2 a month for private lessons given to his son. 
How much did he pay in a quarter ? in half a year ? 

How many slate-pencils at three cents apiece can you buy 
for 12 cents? 

Illustrate : 

000 n 

00 n 

000 n 

000 n 

Caroline learned by heart 1 2 definitions in three days, etc. 
How many each day? etc. 

The teacher should prepare collections of such examples in 
writing. 

The numbers from 10 to 100 are treated in a similar way; 
as a further illustration, the treatment of the number 30 is given 
in full. Such numbers as 17, 19, 22, 23, 26, etc., which are of 
less importance than numbers that represent some frequently 
occurring division in the denomination of number (12, 18, 24, 
36 = dozen, months, 7 = days, 10, 15 = cents, etc.), are 
treated in their relation as pure numbers only, and the pro- 
cesses taken up under II. are omitted with them. 



40 G RUBE'S METHOD. 

THIRTIETH STEP. 
THE NUMBER THIRTY. 



(3 times the fingers of two hands.) 

I. a. i . CONNECTION WITH FORMER STEPS : If I add one 
unit to 29 we have 3 tens. 
Three tens are called thirty. 

a. 2. MEASURING BY THE NUMBERS FROM i TO TEN. 
Oral. 

(3<>; i) (30; 2 ) 

Count from i to 30. 2, 4, 6, 8, 10, etc. 

Count from 30 to i. 30, 28, 26, 24, etc. 

(3; 3) (30; 4) (30; s) 

(30; 6) (30; 10) 

6, 12, 18, 24, 30 10 -f 10 = 20 

20 + 10 = 30 

30, 24, 1 8, 12, 6 30 10 = 20 

20 10 = 10 

30x1= 15x2= 5x6= 3X10 = 30 

30 -f- I = 30 -r- 2 = 30 -r- 6 = 30 -r- 10 = 3 

30 = 29 + I 

28 + 2 In counting by 2's, 3*5, etc., a pupil should 

etc. point to the illustration. The teacher should 

stop frequently in this exercise, and make the 

pupils state how many tenths and units they have counted so 



TEACHING ARITHMETIC EXPLAINED. 41 

far, and how many they have still to count up to 30. For in- 
stance : Class, i, 2, 3, 4, 5, 6, 7, "Stop." Pupil: "We are 
within the first ten, three more are necessary to complete the 
first ten, 23 units to make up 30." The same should be prac- 
tised in counting downward. 

Written Work. 













3 X 10 30 


30 = 30 X 


i 30 -;- i = 30* 


i = & X 30 ( 


= 15 x 


2 30 -f- 2 = 15 


2 = T5 X 30 


= 10 X 


3 -5-3 = 10 




= 7 X 


4 + 2 -=-4 =7 (2 


) 


= 6 X 


5 +5 = 6 


, 


= 5 x 


6 -6 = 5 




= 4 X 


7 + 2 -=-7 = 4(2 


) 



= 3 X 10 



10= 3 



30 is composed of what equal numbers? 

30 is composed of which 2, 3, 4, etc., unequal numbers? 

In these operations the 30 dots on the board should be sep- 
arated into groups of 2, 3, etc., by placing lines between them, 
i- e - (30; 3) 



If these suggestions meet with that support on the part of 
teachers which, as a rule, is most generously given to new 

* These examples are to be read by the pupil in several ways: a. From ... I can 
take away . . . times, b. In 30 . . . is contained . . . times, c. The . . . th part of 
30 is ... 



42 G RUBE'S METHOD. 

methods of value, it would be a good plan to have 10 lines or 
dots painted in a convenient place on the board in the rooms 
of the lowest grades, and 100 lines or dots arranged 10 by 10 
painted on the board in the next higher rooms. The chalk- 
dots by which the pupils divide the lines or dots into groups 
might then be wiped off, when a new relation is taken up, with- 
out erasing the painted lines. 

b. COMBINATIONS WITHIN THE LIMITS OF 30. 
i dozen -f- 4 pieces -f- 2 pieces + \ dozen = ? 

i dime -f 5 cents + i dime = how many cents ? 
(3 X 5) + (2 x 4) 4- 7 - 15 - 8 + 5 + 9 = ? 
4x6, one-half, again one-half, 5 times = ? etc. 

c. EXERCISES IN RAPID CALCULATION. 

Take 19 from 30 (19 = 10+9; 30 10 = 20; 
20 9 = ii ; 30 19 = n). 

Twice 15 (15 = i ten and 5 units, 2X1 ten = etc.). 

Compare 30 with 16 (30 = 3 tens; 16 = i ten and 6 
units ; 4 units must be added to the six to complete the second 
10, and another ten to make it 3 tens. Hence 30 is i ten and 
4 units, or 14 more than 16). 

II. Applied examples. 

(30 pieces = 2\ dozen; 30 months = 2\ years, etc.) 
A great variety of these examples should be given ; but even 
more important than this is the thoroughness with which each 
example is illustrated and worked through. Let the teacher 
move quietly in the stereotyped form of this method, so that 
the pupil becomes strong and self-active in the application 
of the familiar process. This apparently mechanical form rests 
on self-activity, and leads to self-reliance, self-confidence, and 
skill. More pupils fail in arithmetic from diffidence than from 
any other cause. In conclusion of the numbers of the second 
circle, the method of teaching the number 100 is given. 



TEACHING ARITHMETIC EXPLAINED. 



43 



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HUNDREDTH STEP. 
NUMBER ONE HUNDRED. 

Considerable time should 
be spent on the number 100. 
Besides the regular process 
which has been explained in 
^connection with the number 
30, a general review should 
take place. The multiplication 
table, of which the elements 
are known from previous in- 
struction; may be written out 
in the following well-known 
forms, and committed to 
memory thoroughly. 


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00 


ON 4^ 


to 


O 


Oo 

O 


to 
J 


to 

4h 


to 

HH 


Oo 


c^ 


to 


VO 


ON 


CO 


4k 

O 


ON 


to 


00 


to 


to 

O 


M M 

Os to 


00 


4* 


X 



II 

1 


X 

vO 

\\ 

M 




tyi 

o 


On 


O 


In 


O 


to 


O Cn 


o 


in 


ON Cn 
O 4>. 


fe 


to 


Oo |Oo 
ON| O 


to 


00 


N O\ 


g 


Oo 1 ON 





4^ |0o 
to |cn 


to 

oo 


to 


4k 


-<l 



X 



II 

1 




? 


*S 


Os 


ON 


tb 


t 


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to 


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ON 


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MD 
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8 


% 


7J 


OS 


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44 



G RUBE'S METHOD. 



The pupil will understand from this that the product remains 
the same, no matter in what order the factors are multiplied. 

MEASURING. Oral Work. 

I. a. Counting up to 100 and down by 2*s, 3% etc., to lo's, 
beginning with i, 2, or any other number. The pupil must be 
able to do this without hesitation, and correctly, before this 
part of the course can be considered finished. The written 
form for exercises of the same kind is : 



1 + 2 = 3 

3 + 2 = 5 

5 + 2 = 7 

etc. 



4 + 5=9 

9 + 5 = I 4 

14 + 5 = 19 

etc. 



or, i, 3, 5, 7> 9> etc - 
4, 9, 14, 19, etc. 



WRITTEN AND ORAL WORK. 







































10 X 10 = 


IOO. 


































IOO = IOO X 

= 50 x 
= 33 X 
= 25 x 

= 20 X 

= 16 X 


i 

2- 

3 (+ i) 

4 
5 
6 (+ 4) 


i) IOO 

2) IOO 

3) I0 
4) 100 

5) I0 
6) 100 


= IOO 

= 50* 
= 33 (0 
= 25 
= 20 
= 16 ( 4 ) 



* To be read: a. 2 is contained in 100 fifty times, b. Half of 100 is 50, etc. 



TEACHING ARITHMETIC EXPLAINED. 45 

TOO = 14 X 7 (-J- 2) 7) 100 = 14 (2) 
= 12 X 8 (+ 4) 8) 100 = 12 (4) 
= ii X 9 (-f i) 9) ioo = ii (i) 

= 10 X 10 10) IOO = 10 

= 99 + 1 

= 98 + 2 etc. 

I. b. Miscellaneous exercises in addition, subtraction, multi- 
plication, and division, with all the numbers below ioo, will be 
a test whether the pupil has the necessary mechanical skill to 
proceed to the study of numbers above ioo. 

Examples like the following should offer no difficulty. 

ADDITION: 14 + 13 + 12 + 11 

X 5 + T 7 + 1 9 + l8 

25 + 37 + 39 + J 7 



SUBTRACTION : 9016 12 11 
98 - 32 - 41 - 24 

MULTIPLICATION: 3 x 30, 4 x 22, 2 x 44, 2 x 27. 
3 X 2 5> 35 X 2 > etc - 

DIVISION: 3) 60, 3) 69. 

4) 60, 4) 72. 

12) 84, 13) 65. 

4) 53- 

A good exercise in the combination of numbers is to write a 
series of figures on the board, and to direct the pupil to add or 
multiply the first two pointed at, to subtract the next, to divide 
by the third, etc. Examples like these should present no diffi- 
culty : 

(3 X 29) - (4 X 16) + 7; 10 X 9 X 3. 

The teacher should always solve the examples mentally with 
the class. 

Grube recommends also the following exercises at this part 



46 G RUBE'S METHOD. 

of the course as a test whether the pupil has a clear or fixed 
idea of each number : 

Let the pupils count from i to 100, but instead of naming 
the numbers themselves, name two factors of which each may 
be composed. Hence, instead of counting 6, 7, 8, 9, 10, the 
pupils are to say 2 X 3, 7 X i, 2 x 4, 3 X 3, 2 x 5, etc. 
i is to be given as a factor only in case of prime numbers. 

Numbers like 52, 68, 95, etc., must be remembered as the 
product of 4 x 13,4 X 17,5 X 19, etc. ; and for this purpose 
the multiplication table up to 20 should be studied. 

I. c. Somebody had $100 and spent the fourth part of it; 
of the remainder he spent the third part. What amount did 
he keep? What part of the $100 did he keep? 

I have taken a number 3 times, and have 4 more than half a 
hundred. What number did I take three times ? 

Five times what number is 5 less than 100? 

Seventy-five is three times the fourth part of what number? 

Exercises in changing compound numbers within the limits 
of 100 to lower or higher denominations. 

One quarter of a dollar has how many cents ? Three quar- 
ters? 

Half a dollar is how many quarters ? Dimes ? 

A dollar is how many dimes ? 

How many months in 100 days? Weeks? 

A hundred months are how many years ? 

One hundred pieces are how many dozen? Pairs? 

One year and eight months are how many months ? 

One hundred ounces are how many pounds ? 

Eight pounds three ounces are how many ounces ? 

Twenty- three gallons are how many quarts ? 

One hundred quarts are how many gallons ? 

A farmer sold three mules for 99 dollars ; how much apiece 
did he get for them ? etc. 



TEACHING ARITHMETIC EXPLAINED. 47 



NUMBERS ABOVE ONE HUNDRED. 



IN teaching the numbers from looto 1,000, the tran- 
sition is made to the ordinary four processes. Instruc 
tion gradually loses the character of an object lesson, and 
appeals to memory, understanding, and reason directly. 
Not that the help of illustrations is discarded alto- 
gether, for they should be used wherever feasible ; but 
when dealing with larger numbers, the only way to il- 
lustrate is to show the analogy, with a corresponding 
example in smaller numbers, by which perception is 
enabled to help the higher powers of the mind. A few 
generalizations will be of assistance in following Grube's 
idea. 

The number 100 is the last one treated by itself. 
With it, instruction proceeds no longer from one num- 
ber to the next higher one, considering each number 
separately, but deals with the numbers from 100 to 
1,000 in general. 

Grube places the work with numbers from 100 to 
1,000 in the first half of the third year of the course. 
The first quarter is devoted almost exclusively to pure 
number, the second more to applied number. 

As the relation of the units and tens to each other 
has been considered in the previous course, the princi- 
pal part of the work at this stage is the measuring of 
hundreds by hundreds, and of hundreds by tens. 



48 G RUBE'S METHOD. 

The greater part of instruction here is oral work, or 
intellectual arithmetic ; written work is but a repetition 
of the oral. 

In the introduction to this division of his work, our 
author says, "As the future study of arithmetic is 
simply an application of the insight gained by percep- 
tion into the nature of the numbers from i to 100, the 
following part of the course has for its purpose to re- 
duce the relations of the numbers from 100-1,000 to 
those of the numbers below one hundred, or, in other 
words, to show that the relations of larger numbers 
among themselves are of the same nature as the rela- 
tions of their elements." 

By this practice the pupil arrives at the secret of ex- 
cellence in performing examples mentally, the dealing 
with numbers reduced to their smallest possible form. 

In order to arrive at a true idea of number, we must 
look upon number itself at this stage, and not yet con- 
sider the four processes as such. The latter are re- 
served for the second half of the year. Intellectual 
and written arithmetic should always be combined. 

As there is no longer any need for the isolated con- 
sideration of each number, as in the former part of the 
course, the only division of the subject-matter neces- 
sary is 

A. THE PURE NUMBER (measuring, comparing, com- 
bining}. 

B. APPLIED NUMBERS. 

Grube's six divisions of the work with pure number 
from 100 to 1,000 show the plan which he recommends ; 
and after 'having given them, nothing of the peculiar 
features of his method remains except the teaching of 
fractions. 



TEACHING ARITHMETIC EXPLAINED. 49 



FIRST STEP. 

NUMBERS FROM ONE HUNDRED TO ONE 
THOUSAND. 

Measuring by the units of the Decimal system, by units, tens, 
and hundreds. 

Illustrations should be used. Grube recommends solid 
blocks divided by lines into 10 and 100 units. Squares of 
paste-board will answer the same purpose. 

EXERCISES : 768 = 7 hundreds, 6 tens, 8 units. The 8 units 
belong to the 7th ten of the 8th hundred ; two units would 
complete the 7th ten, 3 tens more the 8th hundred, 2 hundreds 
more would complete i ,000. 

Analyze in this way 500, 704, 174, 714, 829, 999, etc. 

What number has 3 hundreds, 6 tens, 5 units ? 

How many units in 7 hundreds, 8 tens, 9 units ? 

How many units in 1,000? how many hundreds? 

Written Work. 

Hundreds Tens Units 

615 = 6 x 100 + ix 10 + 5x1 = 6 i 5 
204 = 2 x 100 + o x 10 + 4x1 = 2 o 4 
or 615 = 600 + 10 + 5 etc. 



SECOND STEP. 
HUNDREDS MEASURED BY HUNDREDS. 

A. 200 (200; 100) 

(Objective Illustration, Measuring and Comparing, 
Rapid Solution of Problems, Combinations : The same as in 
the first part of the course.) 



5O G 'RUBE'S METHOD. 

In the first part of the course the diagram under the number 
2 was : 

1 + 1 = 2 

2X1 = 2 

2 1 = 1 
2-5-1 = 2 

Hence the diagram of 200 measured by ioo is 

ioo + ioo = 200 

2 x ioo = 200 

200 ioo = ioo 

200 -f- IOO = 2 



What number is- contained twice in 200? ioo is half of 
what number? What number must I double in order to have 
200? etc. 

B. a. 300 (300; ioo) (300; 200) 

ioo -{- ioo -j- ioo = 300 
3 X ioo = 300 
300 ioo ioo = ioo 
300 -T- ioo = 3 



(300; ioo) 



(300; 200) 



20O +100 = 300 

i x 200 + ioo = 300 

3OO 2OO = IOO 

300 -f- 200 = I (lOO) 



300 is ioo more than 200, 200 more than ioo. 
200 is ioo less than 300, ioo more than ioo. 
ioo is 200 less than 300, ioo less than 200. 
300 is three times ioo, ioo is the third part of 300. 

b. 300 ioo ioo -f 200 -+ ioo = ? etc. 

c. From what number can you take twice ioo and have 
a remainder of ioo? 



TEACHING ARITHMETIC EXPLAINED. 51 

C. 400 (400; 100) (400; 200) (400; 300) 

a. i. MEASURING WITH 100. 2. MEASURING WITH 200. 



f 100 + 100 -f 100 + 100 = 400 

4 x 100 = 400 

400 100 100 100 = 100 

[ 400 -H 100 = 4 



200 -|- 200 = 400 
2 x 200 = 400 

4OO 2OO =200 
400 -r- 200 = 2 



3. MEASURING WITH 300. 4. MISCELLANEOUS MEASURING. 



!3oo + 100 = 400 
100 -f- 300 = 400 
i x 300 H- 100 = 400 
400 300 = 100 



400 is 100 more than 300 

200 more than 200 

300 more than 100 

300 is 100 less than 400 

200 is 200 less than 400 

100 is 300 less than 400 



4 is contained in 4 once. 

4 is contained in 400 a hundred times. 

2 is contained in 4 twice. 

2 is contained in 400 two hundred times. 

b. 100 + 200 -f 100 -f- 200 = ? etc. 

c. What number is twice 100 greater than 200? etc. 

D. 500, (500; 100), (500; 200), (500; 300), (500; 400), 



etc. 



E. 600, etc. 



52 G RUBE'S METHOD. 



THIRD STEP. 

MIXED HUNDREDS MEASURED BY MIXED 
HUNDREDS. 

(This step is a variation of the preceding one. It is, of 
course, neither possible nor necessary to consider every number 
which consists of hundreds and tens, since all that is required 
here is a knowledge of how to perform the operation of com- 
paring hundreds and tens with hundreds and tens. For this 
object a limited number of examples is sufficient.) 

What number is 2, 3, 4, 5 X no? 440 = 4 X no, 
= 2 X 220, 660 = 6 X ? 3 X ? 880 = 8x?4X? 2X? 
990 = 9 x ? 3 X ? Of what factors may 888 be considered 
to consist ? 999 ? 

If 333 divide 999 among themselves, how much will 
each part be ? If 3 divide 999 ? If 2 divide 888 ? 

Of what number is 120 the 3d part? the 4th? the 5th? 

What number equals the fourth part of 844 ? 

844 is four times what number? What number is contained 
4 times in 844? Half of 844 is how many more than one- 
fourth of this number? 

One-third of 333 is one-sixth of what number? 

Compare 365 with 244. (365 = 3h -f 6t -f- 5u ; 
244 = 2h -f 4t -f- 4U ; 3h 2h = ih; 6t 4t = 2t ; 
5u 4U = m; 365 244 = ih -h 2t -f lu; 365 is 121 
more than 244 ; 244 is 121 less than 365.) 

Difference between 743 and 1 20 ? 

What number is equal to the sum of 743 -f- 221 ? 

112 -j- 113 -f- 114 =? 659 222 124 =? 

in -f- 212 + 313 =? etc. 



TEACHING ARITHMETIC EXPLAINED. 53 

FOURTH STEP. 
MEASURING OF HUNDREDS BY TENS. 

I. a. Pure hundreds. 

If 100 = 10 x 10, then 

2 x roo or 200 = 2 X 10 X 10 = 20 X 10 

3 x 100 or 300 = 3 x 10 x 10 = 30 X 10 

4 X 100 or 400 = 4 X 10 x 10 = 40 X 10 
10 x 100 or 1000 = 10 x 10 x 10 = 100 x 10 

b. HUNDREDS AND TENS. 

If 100 = 10 x 10, 

no = (10 x 10) -f (i x 10) = n x 10 

120 = (lO X 10) -f- (2 X 10) = 12 X 10 

130 = (10 x 10) + (3 x 10) = 13 x 10 
990 = (90 x 10) + (9 X 10) = 99 x 10 

c. HUNDREDS, TENS, AND UNITS BY TENS. 
If 100 = 10 x 10, then 

101 = (lO X 10) + I 

109 = (10 x 10) -f 9 
906 = (90 x 10) -f 6 
814 = (81 x 10) + 4 

How many tens in 500, 900? etc. 

What number consists of 53 tens? 

What number contains 9 units more than 53 tens? 

How many times 10 in 660, 420, 870? 

10 is the 42d, 66th part of what number? 



54 G RUBE'S METHOD. 

II. Comparison of numbers. 

Compare 400 with 900 as to the number of tens they 
contain. 55 tens are how many tens less than 600? 660? 
990 ? 880 is composed of what 4 equal number of tens ? 
800 -hi 80 -}- 20 =? 210 160=? 60 tens are how 
many hundreds ? What number has 8 tens and 9 units more 
than 490? What number taken 87 times and 9 added to it is 
879? How many tens more in 73 tens than in twice 240? 
The looth part of 1,000 is contained how many times in 500? 
One-third of 630 is one-fourth of what number? The 68th 
part of 680 -f- the 24th part of 240 are how many less than 
10 X 36? 

The exercises are followed by examples which show that the 
factors in multiplication are interchangeable. 

no = ii X 10 = 10 x ii 

220 = 22 X 10 = 10 X 22 

680 = 68 X 10 = 10 X 68 

What number must I take 10 times in order to get 670? 
67 times? 

Of what number is 67 the tenth part? What is the 67th 
part of 670? 

How many times is 79 contained in 790 ? What number 
can be taken ten times from 790? 79 times? 79 times ten is 
equal to 10 times what number? 



TEACHING ARITHMETIC EXPLAINED. 55 

FIFTH STEP. 
MEASURING A NUMBER BY ITS FACTORS. 

I. a. Pure hundreds. 

100 = 2 x 50, 4 x 25, 5 x 20, etc. 
200 =2x2x50 = 4x50 

200 =2X4X25 = 8X25 

etc., etc. 

V 

b. HUNDREDS AND TENS. 

220 = 10 X 22, and since 10 = 2 x 5, 

= 2 x 5 X 22 = 2 x no, and since 22 2 x n 
= 10 X 2 x n = 10 X 22, etc. 

f. HUNDREDS, TENS, UNITS. 

426 = (10 x 42) -h 6 

= (4 X 100) -f- 26, etc. 

II. What is the difference between 980 and 377? 

The difference between 980 and 377 is three times what 
number ? 

By what number must I divide 365 to obtain five ? 

What difference between the 22d and the 3Oth part of 660? 



56 G RUBE'S METHOD. 



SIXTH STEP. 

REDUCTION OF NUMBERS FROM i TO 1,000 INTO 
THEIR ELEMENTS. 

It is immaterial in what order the numbers are considered, 
or what numbers are taken up ; the practice alone which these 
exercises afford to the pupil is important. 

A pupil who has done the work of the previous course will 
be able to separate a number into its parts quickly and accu- 
rately. The teacher gives the number, and the pupils separate 
it orally or in writing. 

360. 

300 +60 (3 x 100) + (3 X 20) 

180 4- 180 3 x 120 

200 -f- 160 10 x 36 

320 + 40 5 x 72 

336 -f 24, etc. 20 x 1 8, etc. 



TEACHING ARITHMETIC EXPLAINED. $? 



DIVISION OF THE WORK ACCORDING 
TO GRUBE. 



3d year, 2d quarter : Compound numbers, money, weights, 
measures. 

3d and 4th quarters, oral and written work : Numeration, 
Addition, Multiplication, Subtraction, and Division with any 
number according to the usual methods of analysis. 

4th year, ist term : Object lessons in fractions, on the same 
plan as the lessons with the numbers from i to 10 at the be- 
ginning of the course. 

2d term. The four processes with fractions. 



FRACTIONS. 

Leaving the work with whole numbers, after having 
considered compound and applied numbers in the second 
quarter, and passing over the four species whose treat- 
ment is about the same as can be found in any other 
arithmetic, we shall find again an original and peculiar 
application of Grube's idea in the teaching of fractions. 

The pupil is expected to take up this subject in the 
fourth year of the course after having acquired some 
knowledge of fractions by previous instruction. 

"In the same way/' says Grube, "in which the pupil 
arrived at the perception of whole numbers by measur- 



58 G RUBE'S METHOD. 

ing them by the smallest unit, fractions are now ex- 
plained to him by comparison with and reference to the 
number One, from which they have arisen. 

"While the number one has appeared so far as a 
part of other numbers, it is now considered as a whole, 
which consists of parts. The latter in relation to this 
whole are called fractions." 

As the pupils have already learned to look upon 
whole numbers as parts of larger numbers, the following 
method of teaching fractions will offer no special diffi- 
culty, since the process is the same as the one which 
has made them familiar with integers, and which con- 
sists in the perception of the manifold relations of the 
number which is being taught. 

The order in which fractions are considered is, halves, 
thirds, fourths, fifths, etc. The processes to which 
fractions are subjected are again : 

I. Pure number, and under this 

a. MEASURING. 

b. COMPARING. 

c. COMBINATIONS. 

II. Application of what has been taught with pure 
numbers, in applied examples involving the four 
processes. 

The regular illustration for fractions is the line di- 
vided into parts ; a circle divided into parts may be 
substituted for it. It is necessary to give an abundance 
of practical examples under each fraction, since the 
four processes are explained and made use of at the 
very beginning. In Division with fractions, Grube 



TEACHING ARITHMETIC EXPLAINED. 59 

urges strongly not to go here beyond the idea of 
" being contained in." It is nonsense, he says, to 
speak of 2 divided by one-half, and the like, at this 
period of instruction. That ^ is contained 4 times in 
2 will be understood by the child, because it can be 
shown to him ; but the idea of division is more difficult. 
Even examples like 4 -f- | should not be read four di- 
vided by J, but rather, 4 is twice the third part of what 
number ? or, still better, f are contained in 4 how many 
times ? 



FIRST STEP. 

HALVES. 



I. 



K 



If I divide one (a unit) into two equal parts, I obtain 2 
halves. A half is one of the 2 equal parts into which I have 
divided the whole. 

i -5- 2 = \, or \ x i = \. 

MEASURING. 

a. (Addition.) -f \ = i. 

b. (Multiplication.) i x = J. 2x^=1. 

c. (Subtraction.) i \ = J. 

d. (Division.) H- = i, i -$- } ( is contained 2 times 
in i). 



60 G RUBE'S METHOD. 

APPLICATIONS OF THESE FOUR EXAMPLES : 

i. i -*- 2 = 4 hence 2 -j- 2 = |- , 3 -7- 2 = J, 

10 -f- 2 = -ig , ioo -7- 2 = A-J-Q, etc. 



tf. 


4 + 4 = 


*i + if= 


ii + ij = 




i + 4 = 


2 i + i = 


74 + 44 = 




2 + 4 = 


124 + i = 


74 + 8 = 




3+4 = 


184 + 4 = 


74 + 8 i = 




etc. 


etc. 


etc. 


b. 


2 x | = | 


[ I X I 


1=1 x | = i J 




3X4 = 1 




1 = 2 X | = f = 




10 x 4 = J 


# = 5 3Xi 


4 = 3 x f = 1 = 



ioo X J = A f^ = 50 etc. 

6 X isi = (6 X 15) + 6 X i, etc.) 
9 X 8oJ = 
(If i X i = 4, then 4 X 6 = f = 3, -| x 9 = 4f, etc.) 

^.i 4=4 2 ij=4 2 \ i = 

2 -|= ij 6 -44= i4 64-3 = 

3 - \ = 24 9 - 34 = 3^ - 2\ = 

etc. etc. 8| 4! = 

d. i -4- J = 2 (for i = f , in | one-half is contained twice, 
hence i -s- J = 2). 

4-^4 = 8 I i_,_| = 3 6-5- ij = 

6 -5- = etc. - - = etc. 



2. ^. Compare J with i ; i = i J, i=i.-f^ 
= half of i, 1 = 2X4- 

<5. What number is equal to the difference between -J and i ? 
How many must I take from 16 to obtain 



TEACHING ARITHMETIC EXPLAINED. 6 1 

Of two numbers the smaller one is 9 J, the difference between 
it and the larger one is 6J ; what is the other number? 

Name some other two numbers that have a difference equal 
to6J. 

c. How many times must I take \ in order to have i ? 4^ 
in order to have 9? 18? 4^ is half of what number? 9 is 
twice what number? 

The quotient is 2, the divisor 4} ; what is the dividend ? 
(The quotient 2 tells that 4^ must be contained 2 times in the 
divisor, hence the divisor must be twice 4^ = 9.) I must take 
one-half of what number in order to have 4 J ? etc. 

3. a. What is meant by J dollar? dozen? (One-half dollar 
is one of the two equal parts into which a dollar may be di- 
vided.) 

b. How many half dollars in 55 cents? \ dollar -f- 5 cents, 
etc. 

c. Difference between 8 times 55 cents and 9 times 57 cents? 

(8 x 550 = 8 x $J + 8 x 50 = $ 4) 40C . 
9 x 57c = 9 x $\ r + 9*X 7C = $4% + 63C 



$5.13 $4.40 = 73C, hence the difference, etc.) 

d. The cook of a hotel buys 17^ pounds of meat + 13^ 
pounds -f- 8 pounds. This will be sufficient for how many 
persons if 8 ounces are the calculated allowance for each ? 

e. If a pound of tea costs ^ dollar, how much can be bought 
for 25 cents? 



62 G RUBE'S METHOD. 

f. If 5 yards of cloth cost 6 dollars, what is the price of loj 
yards? (i yard = fifth part of $6 = $i + fifth part of 100 
cts. = $1.20. yard = 6oc. 10 yards = $12. 10^ yards 
= $12.60.) 



3. Applied examples. 

In the treatment of the other fractions, the same 
plan is followed. Fourths, for instance, are first com- 
pared with the whole, then with halves, by addition, 
multiplication, subtraction, division, and finally with 
thirds. In the latter process, the illustration is pecul- 
iar, and consists of two parallel horizontal lines drawn 
close to each other, the upper one divided into four 
parts, the lower one into three parts, and then each 
line by light marks again into twelve parts, so that 
both show the mediating fraction of twelfths and their 
relation to fourths and thirds. 

The following is a brief abstract of the treatment of 
fourths, giving in full those details only which cannot 
be understood from what has been said in connection 
with the treatment of J-. 



TEACHING ARITHMETIC EXPLAINED. 63 

THIRD STEP. 
FOURTHS. 

A. Fourths, Halves, and Units. 

i 1 -i 



K 



1. If I divide i into 4 equal parts, each part, etc. 

i -f- 4 = J, or J x i=J. 

. | + J = ? | + J- = ? etc. (Adding by fourths.) 

b. i x J = ? 2 x i = ? etc. (Multiplying by fourths.) 

c. i - J = ? f i = ? etc. (Subtracting by fourths.) 
/ J -h J ? J -5- i = ? etc. (Dividing by fourths.) 

*>. i . Fourths as the quotient of integers : i -r- 4 = J, 
2-5-4, etc. 

2. As the product of fourths and integers : i X 3, 
i X 100 = ija. 

a. Addition (i. Mixed numbers and fourths, 4! + f; 
2. Mixed numbers 4- mixed numbers, 4| -|- 4^). 

b. Multiplication (integers X fourths and x mixed numbers, 
etc.). 

c. Subtraction. 

d. Division. 



6 4 



G RUBE'S METHOD. 



B. Fourths and Thirds. 
ILLUSTRATION : 



, 

| A 


A 


A 


I A| 


A 


A 


l 


A j i 1 


a | A | A 


B 



Or, if preferred, the circle rriay be used to illustrate the 
same principle, as follows : 




i. Fourths and thirds meet in twelfths. 



I = i - A. 4 = i + A- 
1 = f x i for j - ^ = f, 
i = i X |, for i H- J, etc. 



= 3 - 4) 



2. Compare } with f . J = ^, | - f%. 

i = I " A, = i + T 5 2- 

i = | x | (the 8th part of f taken 3 times ; see illus- 
tration), for J -5- | = | (the 8th part off (= g) is contained 
3 times (3-) ini = A - A = 3 - 8). 



TEACHING ARITHMETIC EXPLAINED. 65 

f = f X J, for | -7- \ = f . (The third part of one- 
fourth (^2) is contained 8 times in f .) 

3. Compare with f, etc. 

4. Compare halves, fourths, and thirds. 

5.. Fractions, integers, and mixed numbers. 

6. \Combinations and rapid solution of problems. 
* 

\ 
C. a Applied numbers -with fourths. 

b. Applied numbers with halves, thirds, and fourths. 

c. Examples in analysis. 

d. Miscellaneous examples. 

The other fractions are treated in a similar way. 

In giving an outline of Grube's method of teaching 
the elements of arithmetic, no attempt has been made 
to comment on any part of it, as it seemed desirable to 
submit the whole system as originally set forth to the 
judgment of practical teachers. Many points are open 
to criticism, and not a few may be obvious mistakes, 
A great number of text-books in arithmetic have been 
written in the country in which Grube's work was first 
published, which have improved the original method, 
and adapted it to the special wants of different school 
systems. It seemed better, however, to present the 
method as it was originally conceived, without giving 
expression to criticism and difference of opinion, and 
to let the well-known skill and ingenuity of the teachers 
of our common schools adapt it to our peculiar wants, 
and make such improvements and changes as may seem 
expedient. 

In regard to one point of the system, however, it 
looks as if there could be -no mistake. The thorough- 
ness with which illustrations are used is an indispen- 



66 . G RUBE'S METHOD. 

sable condition for successful work in the primary 
grades, If the introduction of the kindergarten has 
taught some lessons to all of us, the least important 
among them is certainly not the remarkable results ac- 
complished in arithmetic, when it,is taught incidentally, 
by means of the building-Sfo'ck^S!' Frcebel's "gifts." 
The writer has visited a kindergarten in which prob- 
lems like "how many twenty-sevenths in three-ninths?" 
were solved by children five or six years old without 
any perceptible difficulty. The explanation of this pro- 
ficiency lies certainly in the fact that ninths and twenty- 
sevenths are, for those children, not abstract terms, but 
names of some of the little cubes in their toy-box, and 
that ninths and twenty-sevenths are the names by which 
they know those little objects with whose comparative 
size long use has made them perfectly familiar. The 
association of arithmetical ideas with perceptible objects 
alone makes arithmetic intelligible to the child. 

There can be no doubt that many of the methods of 
instruction used in the kindergarten are excellent and 
very suggestive, and should be carried over the primary 
grades as far as the character of the schoolroom, which 
must be kept distinct from that of a kindergarten, ad- 
mits. In the common school, children learn by the 
senses of hearing and seeing ; in the kindergarten by 
seeing, hearing, and touch. The hand is a very im- 
portant means of education ; and it seems evident that 
pupils in the primary grades, who are allowed to handle 
suitable objects, in arithmetic, to count them, to arrange 
them so as to represent the problems given to the 
school, will be able to do better work than if instruc- 
tion in this important study is imparted without the 
help of objective illustrations. 




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