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Doctor:ATE IN PEDAgogy.
* - ".
Gr.

















NEW YORK UNIVERSITY.
Sºcijool of Belagogy.
3-Z4–2 2.
THE FALSITY OF THE GRUBE METHOD
OF
TEACHING PRIMARY ARITHMETIC,
THESIS FOR THE DOCTORATE IN PEDA GOG Y.
ACCEPTED MAY 1, 1894.
BY
S.A U. L. B A D A NES.
,--~~
*
NEW YORK, 1895.
"ve
TABLE OF CONTENTS.
PAGE'ſ.
CHAPTER I. ExPosition OF THE GRUBE METHOD. . . . . . . . . . . . . . . . . . . . . . . . . 7
- Grube's principles of instruction in elementary arithmetic... 9.
The deductions he made from his principles. . . . . . . . . . . . . . 10,
His course of study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . © tº e º e º 'º a 11
The set form under which every number from 1 to 100 is to
be taught. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * * * * * * * 11
The application of Grube's Method to the teaching of the num- .
bers four and six. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
CHAPTER II. Psychological ANALYSIS OF THE CONCEPT OF NUMBER. . . . . . 16
Description and explanation of the origin of the idea of num-
ber offered by the psychology of individual minds. . . . . . . . 16
The question of growth of the idea of number of greater im- -
portance than the question of its origin. . . . . . . . . . . . . . . . . . . 17
A complete solution of the problem of the growth of the idea
of number can be found only in human history. . . . . . . . . . . 17
The three principal stages in the development of the idea of
number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
The stage of spontaneous numbers . . . . . . . . . . . . . . . . . . . . . . . . . 18
The stage of introducing images, the beginning of the induc-
tive process of reasoning in the human race. . . . . . . . . . . . . . 21
The period of formation of numerical concepts and judg-
ments, the period when the deductive process predominates 22
CHAPTER III. CRITICISM OF THE GRUBE METHOD. . . . . . . . . . . .e. e s e s e e * c e < e i s w tº 25
Grube came to his fundamental principle not by a psychologi-
cal analysis of the idea of number, but by the method of
analogy. . . . . . . . . . . . . . . . . . . . . . . . . . * * * * * * * * * * * * * * * * * * 25
(a) His method is false from the point of view of psy-
chology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
(1) We do not acquire the idea of number by the
mere process of sense-perception, except the
first three or four units. . . ... . . . . . . . . . . . . . . . . 27
(2) His method ignores the serial nature of number 27
(3) He begins with deductions instead of inductions 28
(b) The Grube Method is false from the point of view of
arithmetic as a science and as an art. . . ... . . . . . . . . . . . 29
(1) It ignores the process of counting.. . . . . . . . . . . . . . 30
(2) It makes no use of our decimal system of numer-
ation . . . . . . . . . . . . . . . . . . . . . * * * * * * * * * * e s e e s e 31
3
4
PAGE.
(3) It substitutes the individual study of each num-
ber for the study of operations. . . . . . . . . . . . . .. 32
(c) The Grube Method is false from the point of view of
pedagogy. . . . . . . . . . . . . . . . * º º ºs e tº • * * * * g. t e s a 33
(1) The child is confused by so many operations
upon each number. . . . . . . . . . . . . . . • * * g tº a ſº ſº tº º 34
(2) It taxes the memory of the pupil to the utmost... 34
(3) It destroys the spontaneity of the pupil. . . . . . . . . 34
(4) The set form of the Grube Method is a mechanism 34
(5) The separation of each number into equal and
unequal parts is a tedious process. . . . . . . . . . . 34
CHAPTER IV. METHOD of TEACHING NUMBER FROM 1 to 100... . . . . . . . . . . 35
The guide to be taken for the arrangement . . . . . . . . . . . . . . . . . 35
The three main divisions of a course of study in elementary
arithmetic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
The demands of pedagogy in teaching primary arithmetic. . . . 37
* Numbers from 1 to 10; the four operations. . . . . . . . . . . . . . . . . 39
Counting with tens from 10 to 100; the four operations. . . . . . 44
Counting with numbers composed of units and tens from 1 to
100; the four operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
BIBLIOGRAPHY.
THE following books were consulted in the preparation of this
thesis :
1. ACKERMAN, E. Pädagogische Fragen. Dresden, 1884.
2. Alternative, The. A Study in Psychology. London, 1882. (Anon-
Aymous.)
3. BAIN, A. Logic. New York, 1884.
4. CoNDORCET. Moyens d'apprendre à compter surement et avec
facilité. Paris, 1854.
5. CANTOR, M. Mathematische Beiträge zum Kulturleben der Völker.
Halle, 1863.
6. CoMTE, A. System of Positive Polity. London, 1876.
11.
12.
13.
14.
15.
. DUHAMEL. Des Méthodes dans les sciences de raisonnement. Paris,
1865.
. Encyclopædia Metropolitana.
. EGOROFF, Th. I. Metodeca Arifmetike. Moscow, 1893.
. GRUBE, A. W. Leitfaden für das Rechnen in der Elementarschule.
Berlin, 1881.
Gou, James. A Short History of Greek Mathematics. Cambridge
1884.
GALTON, F. Narrative of an Explorer in Tropical South Africa.
London, 1853.
GOLDENBURG, A. E. Metodeca Natschalnoi Arifmetike. St. Pe-
tersburg, 1892.
HARTMAN, B., Dr. Der Rechenunterricht in der deutschen Volks-
schule. Leipzig, 1893.
HAN KEL, HERMAN. Zur Geschichte der Mathematic im Alterthum
und Mittelalter. Leipzig, 1874.
5
6
16.
17.
18.
19.
20.
33.
34.
35.
36.
37.
38.
HALL, S. Contents of Children’s Minds on Entering School. New
York, 1894.
HussBRL, E. C., Dr. Philosophie der Arithmetic. Halle a. S.,
1891. & ſ
HERBART, J. F. Allgemeine Pädagogie und Pädagogische Schrift-
en. Leipzig, 1875.
HARTMAN. Analyse des kindlichen Gedankenkreises. Annaberg,
1890. -
RNILLING, R. Zur Reform des Rechmenunterrichtes in der Volks-
schulen. München, 1884.
. KRANKE. Ausführliche Anleitung zu einem Zweckmässigen Unter-
richt im Rechnen. Hannover, 1860.
. KEHR. Geschichte der Methodic. Gotha, 1882.
. LEWES. Problems of Life and Mind. New York, 1880.
. LEROY, GEORGES C. The Intelligence and Perfectibility of Ani-
mals. London, 1870.
. LINDSAY. Mind in Lower Animals. New York, 1880.
. MILL, J. S. System of Logic. New York, 1846.
. MILL, J. Analysis of the Human Mind. London, 1878.
. PREYER. Die Seele des Kindes. Berlin, 1890. -
. PESTALOzzi's Sämtliche Werke. Brandenburg a. H., 1869–1872.
. REIN, Dr., PICKEL and SHELLER. Das erste Schuljahr. Leipzig,
1892.
. ROMANES, J. Mental Evolution in Man. New York, 1889.
STERNER, MATTHAUS. Geschichte der Rechenkunst. München u.
Leipzig, 1891.
SACHSE, I. I. Der praktische, geistbildende und erzieliche Unter-
richt im Rechnen und in der Raumlehre. Osnabrück, 1887.
TAYLOR. Prehistoric Culture. New York, 1888.
TAINE. On Intelligence. London, 1871.
TANCK, W. Das Rechnen auf der Unterstufe. Meldorf, 1884.
UNGER. Die Methodic der practischen Arithmetic in historischen
Entwickelung. Leipzig, 1888.
ZILLER. Allgemeine Pädagogie und Grundlegung zur Lehre vom
erziehenden Unterricht. Leipzig, 1884.
THE FALSITY OF THE GRUBE METHOD OF
TEACHING PRIMARY ARITHMETIC,
CHAPTER I.
EXPOSITION OF THE GRUBE METHOD,
THE science of arithmetic was little developed with the ancients,
because of their imperfect methods of notation, as is testified by their
laborious processes of multiplication, and especially of division ; hence
the modern problems of pedagogy that are related to the teaching of
arithmetic could not have arisen then. The science of arithmetic re-
ceived its greatest impetus when the decimal system of notation was
introduced into Italy and France by Gerbert (Pope Sylvester II.), who
was the first Christian priest to visit Seville and Cordova, the great
centres of Mohammedan learning. With the introduction of the Hin-
doo-Arabic system the mind found great scope in being able to use large
numbers easily. It also found no further need of oral arithmetic, and
gave itself up entirely to mechanical processes, pursuing merely a prac-
tical end.
In examining all text-books in arithmetic written up to the middle
of the eighteenth century we are struck with the mechanical ways of
teaching the arithmetical operations. First, the rule was given, which
the pupil was required to memorize ; then examples followed according
to the rule. The numbers used in the examples were very large, the
proof following each example. The interest of the learner was utterly
ignored ; all his faculties were left dormant ; appeal was made merely to
7
8
memory, and consequently the latter was overtaxed. At the end of the
eighteenth century, when all human institutions, all departments Of
human thought and human activity, were criticised with the object of
placing them on a better basis, the need of popular education was felt
for the first time, and with this need arose the demand for better
methods and improved text-books suitable for the children of the
masses. As a result of these demands text-books began to appear written
with the purpose of making the subject-matter of instruction easier,
clearer, and simpler for the learner. An excellent illustration of this
period of history is to be found in the life and work of Marquis de Con-
dorcet, a man of many-sided intellect and of a rich emotional nature.
He was an indefatigable writer, was a member of the Convention that
suggested reforms, and is especially interesting to us as a distinguished
mathematician. As a member and secretary of the Academy of Sciences
he wrote a tract on elementary arithmetic under the title “ Moyens
d'apprendre & compter surement et avec facilité’—(Means for Learning
to Reckon Certainly and Easily). The contents of this little book per-
fectly justify its title. Its pages are full of observations from which
teachers even at the present day may learn something. Condorcet's
elementary arithmetic goes to prove that amid social reforms the need
was felt of basing instruction on sound principles. But the spirit of
the age was felt by many in different parts of Europe. It reached
also Switzerland, and found there many enthusiasts for reform. Among
these was John Henry Pestalozzi, who felt that he could render the
highest usefulness to his fellow men by becoming a schoolmaster. It was
he who gave its death-blow to the mechanical form of instruction that
had preceded him.
He declared that instruction should be used as a means for the com-
plete development of the power of the individual ; he also announced
the principle that we should begin our instruction by giving the child an
experience.
He said that observation, or sense-perception, is the first step of
instruction, whether in form, number, or anything else. Since the prin-
ciple of sense-perception has but a limited application to the study of
arithmetic, and as this limit was unknown to him, Pestalozzi gave a
9
Wrong turn to the study of arithmetic: this I hope to show when we
tome to criticise the method of Grube, who extended and amplified the
principle of sense-perception.
Pestalozzi was the first to give prominence to oral arithmetic. He,
however, went to the other extreme, by attaching very little value to
written work, or to questions directly applicable to life.
His pupils barely knew the advantage of our decimal system of
notation, but his name will always remain in the history of education as
that of one who tried to make elementary arithmetic the common pos-
session of all people.
At the end of his life he wrote that he was dissatisfied with his
own method of teaching elementary arithmetic.
After Pestalozzi, the man that had the most influence on the
method of teaching this subject was Grube.
In 1842 he published the first edition of his manual of arithmetic,
under the title “ Leitfaden für das Rechnen in der Elementarschule,
mach den Grundsätzen einer heuristischen Methode ; Ein pādagogischer
Versuch zur Lösung der Frage : Wie wirkt der Unterricht sittliche
Bildung?” -
In the introduction to his manual Grube justifies the title of his
book, because it deals with the moral influence of the School, and
especially with the moral influence of instruction; further, he deals
with the moral influence of instruction in elementary arithmetic.
But in this his high aim of instruction he gave nothing new to
the world; since Herbart—a mind of greater calibre—had declared,
many years before Grube, that the aim of all instruction ought to be the
formation of character.”
What, however, we wish to consider is his principles of instruction
in elementary arithmetic, and also the deductions he made from his
principles, as these have had, after Pestalozzi, the most influence on
instruction in arithmetic. -
Grube accepted the principle of Pestalozzi, that observation or sense-
perception is the basis of all instruction: he extended it to the study of
* Grube wrote his manual one year, after Herbart's death.
;
A.
10
primary arithmetic, and by giving it a new turn formulated a principle,
known since then as the monographic, or individual and all-sided, con-
sideration of Number.
Numbers, according to Grube, are comprehended and recognized in
the same manner as sensible objects; hence the method of object-teach-
ing is to be used also for teaching numbers. Further, every number, with
all its component parts, from one to one hundred is to stand clear in the
mind of the pupil; every number is to form what is called in the school
of Herbart a method whole, and is to be taught in the same way as a
plant in Botany; and as only a thorough observation of a plant leads to
a clear percept of the plant, in the same way only thorough observation
of a number leads to a clear percept of the number. Hence the con-
sideration of number ought to be all-sided ; all operations with num-
bers, and even the applied examples, are to serve merely to make clear
the idea of an abstract number. -
Grube argues for his principles of clear intuition of numbers, and
of their monographic consideration, by saying that to separate the in-
struction of elementary arithmetic, according to the so-called four opera-
tions, is the same as to give children object-lessons of different things
under the heading of one of their common qualities, such as size, form,
color, etc., or to begin Botany with the system of Linnaeus.
Just as a child never acquires the knowledge of an object by observ-
ing the same attribute in different objects, but by observing the different
qualities of the same object ; and just as it is a mistake to show a
beginner in Botany first the root of a plant, then the stalk, and later
the leaves, since he should first see the whole plant, and only after-
Wards the parts, so it is a mistake to teach the child to-day 2 + 2 = 4,
and after several weeks to turn to the subtraction of 4 – 2 = 2, etc., as
by this method the child is not led to comprehend the number 4.
To acquire a thorough knowledge of the number 4, the child must
learn, simultaneously with the knowledge of 2 + 2 = 4, that 4 – 2–2,
2 × 2 = 4, 4 + 2 = 2.
The separation of the study of numbers according to the four opera-
tions, argues Grube, weakens the perception (intuition) of number ;
hence to strengthen the intuition of a number we are to exhaust all
11
possible operations within the limits of one number before we proceed
to the next following number. Further, he argues that only the num-
bers from one to one hundred are directly intuitable ; and as all reckon-
ing with larger numbers can be performed by relating them to the first
hundred, there is good reason why we should endeavor to have each
number with all its constituent parts in the series from one to one hun-
dred stand clearly before the soul of the pupil.
Further, Grube says that out of the all-sided intuition of each in-
dividual number the four fundamental operations will follow as a direct
result.
Grube's course of study covers four years, four hours a week.
First year. The study of numbers from 1 to 10.
Second year. The study of numbers from 10 to 100.
Third year. The study of numbers from 100 to 1000 for the first
half of the year ; and for the second half of the year the study
of any number. *
Separate exercises for each operation.
Fourth year. Fractions. The first half of the year, all-sided ob-
servation of fractions; the second half of the year, practice in the
four operations of fractions.
As the best way of judging a method is by seeing it in operation,
let us present the way Grube teaches the number 4 and the number
6. Each number is to be taught under the following set form :
I. Pure number :
(a) Measuring (that is to say, separating or resolving each
number into equal and unequal parts) and comparing;
(b) Rapid work ;
(c) Combining.
II. Applied number.
The symbols of numbers and of operations Grube introduces with
the first lesson, and with the study of each number its symbol.
The study of the number 4 is called by Grube the Fourth Step.
12
FOURTH STEP.
THE FOUR.
I. The Pure Number–Measuring and Comparing. (To be de-
veloped by the question-and-answer method.)
(a) Measuring with 1:
| | | | Four lines are used as an objective illustration of number.
1 ſ 1 + 1 + 1 + 1 = 4. (1 + 1 = 2, 2 + 1 = 3, 3–H 1 = 4)
|
| 1 4 × 1 = 4
| ii i = i -i –1 = 1
| 1 U 1 + 4 = 4
(b) Measuring with 2:
| 2 (2 + 4 = 4
| | 2 2 × 2 = 4
iſ C2 = 3
| 2 + 4 = 2
(Grube writes the dividend on the right-hand side of the divisor.)
(c) Measuring with 3:
ſº tºº 1 + 3 = 4
| | | 3. 1 × 3 + 1 = 4
| | 4 — 3 = 1; 4 — 1 = 3 -
3 + 4 = 1 (1) 3 in 4, once and 1 remainder.
Name animals with 4 legs and with 2 legs.
Wagons and vehicles with 1 wheel ; 2 wheels and 4 wheels.
4 is how many more than 3 ; 2 more than 2; 3 more than 1 °
3 is 1 less than 4 ; 1 more than 2 ; 2 more than 1.
1 is 3 less than 4 ; 2 less than 3; 1 less than 2.
4 is four times 1; twice 2.
1 is the fourth part of 4; 2 one half of 4.
Of what equal and unequal numbers can we form the number 4?
RAPID WORK.
2 X 2 – 3 + 2 × 1 + 1 – 2 doubled = ?
4 – 1 – 1 –1 + 1 – 3 is how many less than 4°
13
COMBINING.
What number must I take twice in order to get 42
4 is twice what number 2
Of what number is 2 the half º
Of what number is 1 the fourth P
What number can we take twice away from 4?
What number is 3 greater than 1 ?
How much must I add to the # of 4 in order to get 4?
How many times 1 is the half of 4 less than 3 °
APPLIED NUMBERS.
Caroline had 4 tulips in her vase, which she neglected to water: one
wilted ; then another ; then another. How many had she left?
IIow many cents in a 2-cent piece?
How many cakes can you buy for 4 cents, if each costs 1 cent
When each costs 2 cents 2 -
If a top costs 2 cents, how much will two tops cost
John paid for two cakes a 3-cent and a 1-cent piece ; what was the
cost of each cake *
One quart has 2 pints; how many pints in 2 quarts 2
What part of 4 cents is 1 cent?
What part of 4 cents is 2 cents?
The above set form is to be followed in the study of each number
from one to one hundred, the number 1 not being excepted.
When Grube reaches the Sixth Step, which is a study of the number
6, he says that the pupil has made such progress by his thorough study
of the first five numbers, that he will be able to write out for himself
that part of the set form called Measuring and Comparing.
Grube continues to say that the teacher has only to ask the pupil
what he knows of the number 6, and the latter will write out the follow-
ing table after the set form already pursued :
THE SIX.
The teacher says: “Compare 6 with 1.”
14
And the pupils write
1 + 1 + 1 + 1 + 1 + 1 = 6
6 × 1 = 6
6 — 1 –1 – 1 – 1 – 1 — 1 = 1
1 —- 6 = 6
“Compare with 2 : ” -
2 + 2 + 2 = 6
3 × 2 = 6
6–2 – 2 = 2
2 + 6 = 3
“Compare with 3:”
3 + 3 = 6
2 × 3 = 6
6 — 3 = 3
3 -i- 6.- 2
“Compare with 4:”
4 + 2 = 6
1 × 4 + 2 =
6 — 4 = 2
4 + 6 = 1 (2)
“Compare with 5: ”
5 + 1 = 6
1 × 5 + 1 = 6
6 — 5 = 1
5 + 6 = 1 (1)
“Compare the number of legs of different animals of 6 legs with
quadrupeds and bipeds.” º
Further, the pupil is to write out the following table by way of
review :
6 = 5 + 1 (is one more than 5); 4 + 2 ; 3 + 3 ; 2 + 4 ; 1 + 5.
5 = 6 — 1 (is one less than 6); 4 + 1 ; 3 + 2 ; 2 + 3 ; 1 + 4.
4 = 6 — 2, 5 – 1, 3 + 1, 2 + 2, 1 + 3.
3 = 6 – 3, 5 — 2, 4 – 1, 2 + 1, 1 + 2.
1 = 6 – 5, 5 — 4, 4 — 3, 3 — 2, 2 — 1.
6 × 1 (is 6 times 1), 3 × 2, 2 × 3.
15
3 = + x 6 (is one half of 6).
2 = # X 6.
1 = 4 × 6.
“Of what three like numbers is 6 composed ?”
“Of what three unlike numbers ?”
RAPID WORK.
1 X 2 + 1 × 2 + 1 × 2 — 1 × 2 × 3 – 5 × 5 = ?
I have 6 cents, and spend 1 cent, 2 cents, and 3 cents ; how many
cents have I left 2
4 + 2 – 3 is how much less than 62
COMPINING.
What number is 3 times 2 P
What number is twice 3 P
What number can you take three times from 6 and twice from 4 °
How many times 1 is a half of 6 more than a half of 4, and how
much less than 5 P t
If I take a number twice away from 6 and have 2 left, what is the
number 2
How many times is 4 of 6 contained in 4 °
The half of 4 equals what part of 6 P
II. APPLIED NUMBER.
How many times 1 cent, 2 cents, and 3 cents is 6 cents?
How many quarts is 6 pints 2
What will 3 litres of milk cost at 2 cents a litre.”
3 tops cost 6 cents ; what is the cost of 1?
What is the cost of 3 sheets of paper at 2 cents a sheet 2
I have 6 apples in 3 pockets ; how many apples in each pocket 2
16
CHAPTER II.
A PSYCHOLOGICAL ANALYSIS OF THE CONCEPT OF NUMBER,
THE principles of the Grube Method have found many adherents
in America, some adopting the method in its entirety, some appro-
priating only special features of the method. These principles have
reached such a wide popularity that in many states we find teachers,
at their examination for license, asked to explain the Grube Method and
its advantages; and teachers in Normal Schools are found among its
stanch supporters.
Because of its wide acceptance, it is opportune to examine the
basis of the Grube Method, to note its advantages and to point out its
many disadvantages. But before we direct our criticism to the method,
we shall have to examine a question of fundamental importance; namely,
the question of the origin and nature of the idea of number.
Until recently this question has had interest only for the math-
ematician and the metaphysician, but it ought to attract the attention
of the teacher; for his view of the nature of number must have a deci-
sive influence upon the method of instruction in primary arithmetic.
What then is the origin of number?
Believing that the metaphysical explanation of number cannot
render a service to Pedagogy, we shall limit ourselves merely to a brief
description and explanation of the origin of number as offered by
the psychology of individual minds. *
According to this science, the most fundamental condition of all
our psychical states is the experiencing of some transition or change;
without change there can be no consciousness; hence the most fun-
damental property of consciousness is the experience of difference or
discrimination. Every difference in consciousness has its mark by
which it is distinguished; however, we feel the difference only in
degree or quantity, and it is not until later that we learn to localize
Our sensations.
17
Though localization is an important characteristic of our sensa-
tion, it is not a primary characteristic, but a result of slow acquisi-
tion, as testified by experimental psychology and by observation on
infants; hence, the only characteristic that is primary in the act of
discrimination has reference to that of degree or quantity.
Of two differing sensations of light, one is felt to be more in-
tense than the other ; it is the same with pleasure and pain, and
with psychical states of every description. The attribute of quantity
is equally inseparable from objective as well as subjective facts.
Further, psychology tells us that every difference has its correla-
tive resemblance; neither is possible without reflecting the other ;
no sense of difference can therefore arise without a relative resemblance
from which it is discerned. With the part, the whole is given ;
with the individual, the class is given to which it belongs. In the
same manner we discriminate — at least implicitly, the unity and
plurality of objects.
The perception of “two º' children usually reach without the
help of their parents, from whom they only receive the word “two; ”
the number “three * is reached by children in a similar way.
How higher numbers are developed will be shown later.
For the purpose in hand, the question of the growth and develop-
ment of the idea of number is of greater importance than the ques-
tion of origin. º
As the idea of number is a social product, and was developed
by the co-operation of successive generations of the human race,
it is in human history that we may hope to find exhibited all .
the stages of the growth of the idea of number. Of course we
need not adhere in our exposition strictly to the historical order
in all its details ; this would introduce here a great deal of
material unnecessary for our purpose.
Let us see, then, what the history of the human race can tell us
of the growth of the idea of number. In studying the development
of the idea of number, we shall have to divide its genesis into three
principal periods or stages — periods which illustrate a prominent
characteristic in the process of the growth of this idea, and, at the
18
same time, suggest all the important stages of the individual devel-
opment of the human race. The three periods are the following :
1. The period of spontaneous numbers.
2. The period when images were introduced which facilitate
the ' extension of our numeration beyond the spontaneous stage.
With this period the inductive process of reasoning began in the
human race.
3. The period of the formation of numerical concepts and judg-
ments. This was the period when signs were substituted for images
—an indication of the appearance of the deductive process of rea-
soning; with it, arithmetic began to develop as a science.
The spontaneous period which precedes conscious counting is
characterized by the fact that we are able to form an exact estimate
of visual or tactual objects, or an auditory estimate of tones, by
mere comparison of sense impressions.
In the study of primitive races, of very young children, and
even of animals, we can find illustrations of this period. We may
cite as illustrations the following three well-known examples, found
in the writings of F. Galton, C. Leroy, and Preyer.
Francis Galton,” in describing the obtuseness of the Damaras, tells
us the following : “In practice, whatever they may possess in their lan-
guage, they certainly use no numeral greater than 3. When they wish
to express 4 they take to their fingers. . . . They puzzled very
much after 5, because no spare hand remains to grasp and secure
the fingers that are required for “units; ” yet they seldom lose oxen.
The way in which they discover the loss of one is, not by the num-
ber of the herd being diminished, but by the absence of a face
they know.
“When bartering is going on, each sheep must be paid for sep-
arately : thus, suppose two sticks of tobacco to be the rate of ex-
change for one sheep; it would sorely puzzle a Damara to take two
sheep and give him four sticks. I have done so and seen the man
put two of his sticks apart and take a sight over them and at the
* “Narrative of an Explorer in Tropical South Africa,” p. 133.
19 º
sheep he was about to sell. Having satisfied himself that he was
honestly paid, and finding to his surprise that exactly two sticks
remained in hand to settle the account for the other sheep, he would
be afflicted with doubts ; the transaction seemed to come out too pat
to be correct, and he would refer back to the first couple of sticks;
and then his mind got hazy and confused, and wandered from one
sheep to the other; and he broke off the transaction, till two sticks
were put into his hand and one sheep driven away, and then the
other sticks given him and the second sheep driven away.”
Charles Leroy, a well-known naturalist, whose life-work was that of
a ranger at Versailles, brings, as an illustration of the experience of the
idea of number among animals, the following case:
“Among the various ideas which necessity adds to the experience of
animals, that of number must not be overlooked. They count, that is
certain ; and though, up to the present time, their arithmetic appears
weak, perhaps it may be possible to strengthen it.
“In those countries in which game is much preserved, crows are
made war upon, because they take away the eggs, and destroy the hopes
of the laying season. The nests of these destructive birds are carefully
noticed, and to destroy the voracious family at a blow they seek to kill
the mother while sitting. -
“Among the parent birds some are very suspicious, and desert their
nests as soon as any one approaches them. To lull suspicion, a carefully
covered watch-house is made at the foot of the tree in which there is a
nest, and a man conceals himself in it to await the return of the parent
bird; but he waits in vain if she has ever before been shot at in the same
manner. She knows that fire will issue from the cave into which she
saw a man enter. While maternal love keeps her eye fixed upon her
nest, fear prevents her return till night hides her from the sportsman’s
sight. t -
“To deceive this suspicious bird, the plan was hit upon of sending
two men to the watch-house, one of whom passed on, while the other
remained; but the crow counted, and kept her distance.
“The next day three went, and again she perceived that only two
returned. In fine, it was found necessary to send five or six men to the
20
watch-house to put her out in her calculation. The crow, thinking that
this number of men had but passed by, lost no time in returning.
“This phenomenon, always repeated when the attempt is made, is
to be recorded among the very commonest instances of the sagacity of
animals.”
Of course, we must not suppose that the bird counted the men by
any process of notation, but the contrast between the simultaneous sense-
perceptions each time was sufficient for the spontaneous inferences.
Preyer tells of a child who, at the age of a year and a half, could tell
at a glance whether one of her ten animals was missing or not.* (Of
Course, such a case of fine discrimination is very rare.)
We also find him telling the following story of his own child: “In
his eighteenth month, after having been accustomed to bring to his
mother two towels, which he would afterwards carry back to their place,
on one occasion had only one towel given back to him ; he, with in-
quiring look came back to get the second.”
Professor Preyer, in his paper on Arithmogenesis, read before the
International Congress of Experimental Psychology, held in London in
1892, confirms the fact that there is such a spontaneous stage of num-
bers, and further brings arguments and evidence that unconscious
numerical discriminations take their origin in our auditory sense.
In this period, we cannot carry distinct numeration beyond the first
four numbers ; it is true that we can form estimates a good deal beyond
that point by skilfully grouping objects: then, of course, form comes to
our aid ; further, students in psychological laboratories can estimate a
great deal beyond that point, as a result of an educated discrimination.
The second stage of the genesis of the idea of number began with
the discernment of the first non-spontaneously developed number; the
condition of this discernment being the notice of an addition of a unit
to the highest spontaneously developed number. This discernment was
made permanent by indicating it by an image, which represented not
only a sum or a plural quantity, but also the process of obtaining it
from the next lower number; hence, it represented a definite number.
* W. Preyer, “The Mind of the Child,” Part II., p. 8
21
When a primitive savage met with a sum or plural quantity which
he was interested to determine, it would naturally occur to him to with-
draw from the sum a group of the highest spontaneously developed
numbers and then withdraw a unit from the remainder of the sum and
add it to the group, indicating the newly formed group by an image;
he would keep on withdrawing each time a unit from the remainder and
constantly adding it to the newly formed image, until all the units of
the plural quantity were exhausted. .
Later on, we find savages using the same image for like numbers.
In this way counting was invented.
The Indians of the Tamanaca tribe count on their fingers. The
numerical word one signifies with them one finger; the numerical word
five signifies the whole hand; six, one on the other hand; seven, two on
the other hand.
The Caraibs at the Orinoco River have separate names for the num-
bers from 1 to 4; five is called four and the other ; six is called the hand
and one more; seven, the hand and two more. *
Thus the object of counting is seen to be the determination of a
plural quantity which cannot be spontaneously grasped. Its process
consists in forming numbers by the addition of a unit to the number next
below in magnitude, and thus arranging numbers in a scale of which
each higher degree exceeds by a unit the sum of the preceding degree.
With every addition of a unit in this series, we make a new infer-
ence, and as this inference is drawn in connection with our handling of
sensible objects, and concerning them, it is therefore inductive in its
nature. -
The formation of images of the successive numbers is necessary to
these inferences in order to fix our attention and give coherence to our
reflections.
For the representation of numbers, pebbles, fine nuts, beans, knots
tied on a string, and the fingers were first used. With the extension of
the process of counting, the number of images began to multiply; this
produced a confusion in the mind of the primitive races and made it
impossible for them to retain the images. To obviate this difficulty,
they hit upon the method of making a few images denote a multitude
22
of numbers. The secret of this plan lay in grouping or regarding cer-
tain numbers as individual things. For instance, having advanced in
counting up to 5 or 10, they indicated the new group of 5 or 10 by a
special image and then they continued to add to this group the images
of preceding numbers of this group.
In a similar manner, when five fives or ten tens were reached, they
denoted the new group by an image and proceeded to add to this new
group with its images preceding numbers with their images.
Thus, in this period was laid the basis of notation, which is the
necessary complement of numeration, and possibly also the basis of the
decimal system of notation, as suggested by the fingers of both hands.
But the process of forming numbers by mere addition of a unit to
the number next below in the scale could only be extended to the forma-
tion of small numbers, as each new judgment depended upon counting
concrete things; hence it was necessary to find other methods which
should help in forming large numbers, and thus shorten the process of
counting. . -
This demand was completely satisfied in the third period of the
development of numbers. In this period the mind became utterly
detached from sensible observations; by the process of reasoning new
numbers were formed from the numbers acquired in the second period,
that is to say, by passing from judgments retained in memory to new
judgments and conclusions. 4.
Knowing from the previous (second) period that 7 -- 1 + 1 + 1 +
1 + 1 = 12; also that 1 + 1 + 1 + 1 + 1 = 5, the conclusion reached
was that 7 + 5 = 12. This process of reasoning, which is a deductive
process, when fully set forth in its logical form would give the following:
Seven + 1 + 1 + 1 + 1 + 1 = 12
But 1 + 1 + 1 + 1 + 1 =
5
Hence 7 -i- 5 = 12
Of course, in the actual psychological process, the logical order
was not followed. This is required only for the purpose of proof; that
is to say, in our ordinary deductions we do not start with two antece-
dently known judgments, from which we draw a third judgment as a
23
conclusion; but the conclusion is the first judgment that distinctly pre-
sents itself to the mind.
The method of this stage, besides offering many ways of forming
numbers additively, also suggests the process of forming numbers sub-
tractively—thus: if 7+ 5 = 12, 12 — 5 = 7, and 12 – 7 = 5. Considera-
tions of such nature have very likely led to the subtractive form of IX,
XIX, etc. The number 19 was probably more readily formed from the
concept 20 than from the concept 10.
In the efforts to form numbers deductively, it became indispensable
to substitute artificial signs for the images of the previous period, as
these images embarrassed the efforts at deductive reasoning.
In these images we find a suggestion of likeness between the sign
and the fact signified, but the artificial signs offer no resemblance to the
numbers themselves; hence they facilitate the reasoning process, espe-
cially when abstraction is needed. -
The formation of numbers deductively gave rise to the beginning
of elementary calculations. These consisted of performing operations
upon signs that stood for the direct or indirect result of counting.
The first elementary calculations were limited to addition, subtrac-
tion, and to the abbreviated form of addition, i.e., multiplication; di-
vision was developed at a later period, owing to the abrupt increase of
complexity in mathematical reasoning.
To this period belongs the invention of the method whereby a
small number of signs could represent all possible numbers. This method
is an artifice and consists in indicating the value of numerical units by
the position of their digits. -
- In addition to this fundamental artifice, another improvement was
made—the invention of the multiplication table—by the help of which
all products are reducible to the first nine numbers.
With these two improvements, that facilitate greatly our written
calculations, arithmetic as a science began, and numbers began to be
formed in a great many ways, as, in every operation of arithmetic, we
form the required number by the help of the given number.
Here ends our survey of the 'growth of the idea of number; but
before dismissing this topic we wish to add the following: that in the
24
actual growth of the idea of number there is no separate demarcation
between one period and the next; but, on the contrary, one period over-
laps the other, one period is interwoven with the other. All our
psychical states and processes are synthetical in their nature, yet they
appear to be analytical while we study them.
It is especially important to bear in mind the synthetical nature of
our psychical processes, in order to understand the second stage of the
growth of the idea of number. This stage occupies an intermediate
position between the sphere of implicit and explicit mental activity par-
taking of the characteristic qualities of each.
The images of this period, whether they are sights or sounds, recall
our sensations with a diminished energy, and at the same time, like
arbitrary symbols, they represent definite numbers; but they differ from
the symbols in not being general in their nature; hence they do not
permit of combination and permutation in an infinite number of ways.
Images have a significance only for direct and immediate purposes; they
can facilitate the rise of the inductive process, but for mediate and de-
ductive processes the use of arbitrary symbols are indispensable.
25
CHAPTER III.
A CRITICISM OF THE GRUBE METHOD,
WE are now ready to enter upon the critical examination of the
Grube Method, and shall begin naturally with the consideration of its
fundamental principles. Grube came to his fundamental principle,
that numbers are comprehended and recognized in the same way as
sensible objects, and to the deductions he made from it, not by a
thorough psychological analysis of the idea of number, but by the
method of analogy, in the following way:
We cannot have a knowledge of an object by observing it from one
point of view or by perceiving one of its qualities; hence we cannot
perceive a number by learning to-day addition, and after several weeks
subtraction, and later on multiplication, etc.
Further, in object-lessons, we are to teach one object at a time, to
consider all its qualities, and to proceed to the study of another object
only after a thorough study of the first ; hence, concludes Grube, we are
to teach one number at a time. We are to have an all-sided considera-
tion of each individual number, and we are to exhaust all possible opera-
tions within the limits of one number, before we proceed to the next.
And lastly, a thorough study of an object consists of four steps:
1. Observation of the entire object.
2. Separating of its parts and comparing them.
3. Combining all the parts.
4. The use and purpose of the object.
Hence each number is to be considered under the following set
form :
I. Pure number:
(a) Measuring (that is to say, separating or resolving each
number into equal and unequal parts) and comparing;
(b) Rapid work;
(c) Combining.
26
II. Applied number.
As we have seen, Grube came to his fundamental principle by the
method of analogy; by this method, only probabilities and hypotheses
can be established; it is seldom successful in making discoveries.
Especially does this method fail when applied to the subject of
elementary arithmetic, as there is no essential resemblance between an
object and a number in the question of (origin and growth, and no re-
semblance whatever in the contents of these two ideas (the first three
or four numbers making an exception as to their growth).
We have only to attempt to compare a number with a sensible
object, in regard to the correlative psychological processes that go on
in our minds, and the wide difference will be made manifest at once.
We perceive sensible objects as soon as they make an impression
upon our mind, and we, attend to these impressions and refer them back
to the exciting cause. These sense impressions do not disappear with
the removal of the exciting cause, but leave a trace in our mind in the
form of an image, which recalls the object when that is not present be-
fore our consciousness.
In a word, all sensible objects are, using the terms of Herbert
Spencer, “presentative representative.” The case is different with
numbers. Our psychological analysis has shown that only the first three
or four numbers are presentative representative; but the perception of
the remaining numbers up to the number 10 is not very clear, and their
clearness is diminished as they increase in magnitude. Hence they can
never be definitely perceived without counting. -
But, as will be shown later, objective means still render here "a
service; hence we may regard numbers up to 10 as representative; as to
numbers above 10, they are all mainly re-representative, all being con-
ceived by relating them to other numbers that are representative in
their nature. All work with numbers above 10 is performed upon sym-
bols that offer no sensible suggestion whatever.
We may forget the name of an object and still have a picture of it;
hence the name is not the essential element in the perception of a sensi-
ble object. But it is altogether different with a number; once we forget
27
its name the number itself disappears. This fact perfectly illustrates the
re-representative process in the idea of numbers.
Further, let a child see a lion for the first time, and the perception
of this animal will be sufficiently clear and definite to recognize it if
seen again, provided there is nothing else in the field of consciousness
to hinder it. The case is different with numbers. Let a skilful reck-
oner count a series of objects, say 13; will he be able to tell by mere
sense-perception another group of 13 similar objects that are near-by
without counting 2
Indeed, if we could acquire the idea of numbers, within the limit of
1 to 100, in the same way as the first three numbers, by the mere pro-
cess of sense-perception, all arithmetic within that limit would be utterly
superfluous. What use would there be of mediate processes when
immediate ones are at hand *
By comparing a number with a sensible object, Grube neglects one
of the most important attributes of a number—one which goes to make
up its essential content ; this is the scale relation of number.
By mere sense-perception we apprehend a sum as a plural quantity;
but to ascertain the number we must count, and thus establish a defi-
nite quantitative relation of the sum to a certain member of the scale;
hence a number cannot be comprehended as a separate individual, or as
a whole, by itself, but only as a member of a scale or series.
Indeed, if we could comprehend every number as an individual by
itself, and not as a member of a series, then why should, in our instruc-
tion, number 5 follow 4, and not the reverse; why could not 7 follow 4,
or 11 follow 8, just as, in object-lessons, the study of one object may
either precede or follow the other ? -
The view that number can impress itself upon the mind as a sensi-
ble object has resulted in the too great neglect of counting; and what
is still worse, is the fact that the acquired numbers do not stand in any
relation to each other—not being derived one from another, they easily
disappear from the mind.
- Numerous examples could be cited from primary arithmetics, whose
authors consider counting a superfluous process in the first year or two tº
of instruction in numbers, but we shall have to limit ourselves to the
28
work of one author, Dr. E. E. White,” who, while telling us that he
differs in some points most widely from the Grube Method, still holds
fast to the fundamental principle of the latter.
He gives in his “Oral Lessons in Number” the following:
“Special attention is called in this connection (in connection with
teaching number the first year) to the importance of avoiding in these
primary processes the too common practice of counting by 1.
“The numbering, combining, and separating of groups of objects
by counting leads to the pernicious habit of adding and subtracting
numbers by counting—a habit that must be overcome before a pupil
can learn to add and subtract numbers as wholes.”
Further : “The twofold aim of all exercises, in the first and second
years, is to impart a clear idea of numbers from 1 to 20, and to give the
pupil the power to add and subtract the primary or digital number
without counting.”
These conclusions, that have for a long time misdirected the in-
struction in primary arithmetic, are but the logical consequences of the
application to our subject of the principle of sense-perception, without
knowing its function and import, and hence, its limitation.
We too believe in the value of objective means for teaching primary
work, in number, but we are also aware of its limitations.
We are able to make an estimate only of three or four objects with-
out conscious counting; hence, objective means are of value in teaching
number from 1 to 4; further, they can assist us in conceiving numbers
from 4 to 10, because the series of numbers from 1 to 10 consists of
sums or differences of two numbers, of which one can be developed
without conscious counting, as, for instance, 7 -- 2, 8 – 2, 5 + 3,
4 + 4, etc.; only two examples make an exception in this series:
5-H 5 = 10, and 10 — 5 = 5.
Further, the grouping of objects in certain geometrical forms facili-
tates our process of counting, such as dots on a domino, or spots on
playing-cards. Hence there is a use of certain number-images, from
4 to 10, which may serve as an intermediate step between the spontane-
ous number and the symbolic number.
* E. E. White, “Oral Lessons in Number,” New York and Cincinnati, 1884.
29
These number-images may also help us to prepare the way for the
introduction of ciphers. ſe
This grouping of objects or points facilitates the process of count-
ing, but does not make the process superfluous; this can be proven by
the fact that there are certain number-images that complicate and pro-
tract the process of counting.
To this latter belongs the grouping of points in a circular form,
with one point as a centre, in the following way:
For the same reason we are not able to tell instantly the number of
sides of a regular polygon that has more than five sides.
The unfortunate use made by Grube of the principle of sense-
perception in teaching number having been clearly shown, we can now
enter into a more detailed criticism, in order to see how the principle of
sense-perception, with the logical consequences that follow from it, has
influenced practice.
Grube gives the two following reasons why we should endeavor to
have each number, with all its constituent parts from 1 to 100, clear in
the mind of the pupil:
First. Because, only the numbers from 1 to 100 are directly intui-
table.
Second. Because all reckoning with larger numbers can only be
performed by relating them to the first hundred.
Both reasons are false—the first, from the point of view of the
psychology of number, as the first three or four numbers only are the
direct result of sense-perception ; the second, from the point of view
of the theory of arithmetic and the psychology of number; for we have
seen in the analysis of the growth of the idea of number that, at the end
of the second stage, the concept of number is assisted by the grouping
process, which consists in regarding a certain aggregate as a unit and
deriving all units of higher order from it; in a word, of forming a
system of numeration.
30
- In our decimal system of numeration all numbers are related to
I0 or to a multiple of 10. In this case 100 is to be regarded as a con-
cept derived from 10, i.e., 10 × 10 = 100.
The individual consideration of each number, for the purpose of
making it clear to the mind, cannot be justified by the psychology of
number, as this would destroy the essential characteristic of the idea of
number, namely, its serial nature.
Further, the individual consideration with a view to making each
number clear to the mind is superfluous above 10, as this is done by the
decimal system of numeration.
In fact, every system of numeration has for its object the facilita-
tion of the conception of numbers; and the more closely we adhere to
a system the easier and clearer is our concept of number.
To convince ourselves of this, it is sufficient to represent some
numbers by a system different from our own. For example, let us
represent the numbers 8, 9, 10, 11, 12, 13, and 14 according to the sep-
tenary system, and we shall get 11, 12, 13, 14, 15, and 16. At first the
concept of these numbers will be very vague, but they will become
clearer the more we get used to this mode of representation. 4
According to our decimal system of numeration, all numbers from
10 to 100 are derived from the first 10 numbers; for instance, the
number 36 shows us that its component parts are 10 + 10 + 10 + 6, or
3 × 10 + 6; with this, our concept of the number 36 is sufficiently
clear, and cannot become any clearer. As we have noticed before, the
clearness of a concept of number is diminished as it increases in
magnitude. -
Does the child have a clearer concept of 36 when he considers it
from the point of view of numeration, or when he considers it in the
following manner: 35 + 1 = 36 = 30 + 6 = 31 + 5 = 32 + 4 = 33 + 3,
etc., without having the slightest idea of our system of numeration, and
without knowing the meaning of addition and of the other following
operations P
If a child knows how to form numbers above 10 according to the
laws of our decimal system, and knows how to count up to 36, he also
knows then how to count up to 39; hence there is no reason to break

31
the series at 36 and perform the four fundamental operations from 1 to
36. Such a procedure is very unnatural, and can be justified neither
by psychology nor by the theory of arithmetic.
One who knows how to count up to 6 may not know how to count
up to 9; but one who knows how to count up to 46 knows also how to
count to 56. -
As we have already noticed, a few pages back, that the extensive
application of the principle of sense-perception results in the neglect of
counting, so we must notice now that the individual consideration of
each number above 10 results in not making any use of our decimal
system of numeration. -
The old writers of arithmetical text-books, who were mathema-
ticians but not pedagogists, went to one extreme by beginning their
instructions in arithmetic with the explanation of our system of numer-
ation; Grube goes to the other extreme by teaching number to pupils,
for two and a half years, without availing himself of our decimal system,
and without pointing out its significance to his pupils.
We must add here that in the method of teaching number which
we shall propose, every new number from 1 to 10 is, for pedagogical
reasons to be stated later, introduced to the pupil by one kind of ma-
terial, or by certain definite objects derived out of the sphere of his
knowledge, and thus the distraction of attention by the variety of
objects in the Grube Method is avoided. -
This may at first sight give to our mode of presentation the char-
acter of the individual consideration of the Grube Method; but, in
reality, the proposed method differs widely from that of Grube, as by it
the serial nature of number is preserved; further, it does not confuse
the pupil by practising the four fundamental operations upon each in-
dividual number.
Grube uses the four fundamental operations simultaneously, as a
means of strengthening the perception of number ; then, of course, it
is supposed that the pupil knows how to perform these operations and
knows their logical meaning; but where did he acquire that knowledge P.
To use the four operations with the first lesson of instruction, as a
32
means of strengthening the percept of number in children, is to make
an application of knowledge they do not possess.
The progress that a pupil makes, applying Grube's mode of
presentation, is not in learning a new process but in learning a new
object—a new number—by means of the four operations, which being
used only as a means to an end, stand near each other instead of follow-
ing each other, and being gradually developed one from another.
The knowledge of addition presupposes the knowledge of forward
counting, of which addition is an abbreviated process; the knowledge
of subtraction presupposes the knowledge of backward counting. Fur-
ther, we ought to teach, first, addition and subtraction, and only later
multiplication and division, as the last two operations involve a more
general process and can only be developed from the former two
operations.
The need for shorter processes must first be felt by the pupil
before they are brought into use; such is the historical way, and also
the natural way for the development of the process of multiplication.
The same is also true of division. It seems to me then that a child
will never clearly see the difference in the process of 2 + 2 = 4 and
2 × 2 = 4, when taught simultaneously, but will always take one for
the other. º
Our psychological analysis of the idea of number has shown that
generically every new number is formed by the successive addition of a
unit to the next lower number in the scale; that is to say, we form
numbers at first inductively; later, we form them in many ways, all of .
which are mainly deductive; whereas Grube begins his first step of
teaching numbers deductively.
The separation and resolving of numbers into their equal and un-
equal parts, which is the first step of the set form of the Grube Method,
is nothing else but a deductive application of a previous process of
combination of units with which we ought to begin.
Instead of teaching the number 6 inductively, by counting sensible
objects, Grube starts his lesson in teaching the number 6, by asking
his pupils what they known of the number 6, and they begin to separate
it into its component parts according to the set form.
33
Now this procedure is merely the following of a mechanical
process which taxes the memory to the utmost ; and at the same
time, every conclusion of this process, if once forgotten by the pupil,
cannot be reproduced again by him, as he possesses no means by
which to do it.
The separation of a number into equal and unequal parts brings
back continually the easy exercises ; this results in an unnecessary
accumulation of material; at the same time, the new and difficult
exercises are not sufficiently emphasized, being treated with the old
and easy material.
Among many pedagogical disadvantages of the Grube Method
besides those mentioned above, we may point out the following : e
(1) The form of instruction is exclusively the catechetical ; accord-
ing to this form the teacher gives the pupil the leading questions ; the
latter is merely a passive recipient, no room being left for the exer-
cise of his self-activity. (2) The constant use of the same form of exer-
cises and their endless repetition tire the pupil and weaken his interest
for the subject.
Grube gives preference to abstract numbers, and the greater part
of his exercises consists of these numbers ; he assigns to problems (i.e.
applied arithmetic) a secondary place, and they appear only after the
all-sided study of the number.
Further, the material used in these problems has no interest for the
pupil. The uniformity of his exercises trains the pupil to a sort of
mechanism. It is sufficient for a pupil to know the first line of the set-
form, and he will write out the other lines mechanically ; hence it is
difficult, by this form of instruction, to excite in the pupil a love for
the subject as claimed by Grube.
The extensive use of fractions at an early stage of instruction, as
practised by Grube, is above the comprehension of a beginner, as e.g.,
the question on page 35 of Grube's Manual, while teaching the number
6: “How many times 1 is a half of 6 more than a half of 4;” and
“how much less than 5?” Or on page 48: “3 of 12 is what part of 82°
Grube begins written work with his first lesson in number, to which
alone prominence is given ; hence he introduces ciphers and the sym-
34
bols of the four operations with his first lesson ; but really we have no
use for these signs when dealing with spontaneously developed num-
bers. Therefore ciphers ought not to be introduced before we come
to the number 5, perhaps not until later; and the symbols of the four
operations ought to be introduced gradually, and not all at once, and at
a later stage than ciphers. sº
Here ends our critical examination of the Grube Method; we have
considered it from several points of view, and we state our conclusions
as follows:
1. Grube's Method is false from the point of view of arithmetic as
a science and an art. It ignores the process of counting. It does not
teach operations, and does not avail itself of our decimal system of
numeration. It substitutes the individual study of each number for
the essential reality of the subject of arithmetic, which is the study of
operations. +.
2. Grube's Method is false from the point of view of psychology, as
we do not acquire the idea of number by the process of sense-percep-
tion (with the exception of the first three or four numbers); it ignores
the serial nature of number and begins with deductions instead of
inductions.
3. The Grube Method is false from the point of view of pedagogy,
as with so many operations upon each number it confuses the child; it
demands too great an amount of attention on the part of the pupil—an
amount which he is utterly unable to give. The method destroys all the
spontaneity of the pupil; it taxes his memory to the utmost, as one step
never leads up to the next.
4. The set form of the Grube Method is a mere mechanism ; the
material of the subject-matter is made up of abstract numbers which
cannot excite an interest in the child ; the separation of each number
into equal and unequal parts is a tedious process and aimless in its
results.
35
CHAPTER IV.
A METHOD OF TEACHING NUMBER FROM 1 TO 100.
OUR work would be only half done if we should limit ourselves to
the mere exhibition of the falsity of the Grube Method, as no criticism
can ever be effective and complete without showing how to build an-
other structure in place of the one condemned ; for this reason we shall
devote a few pages to the main outlines of a method we would employ
in teaching numbers from 1 to 100, as it is in the teaching of these
numbers that the Grube Method has done the greatest harm.
We shall regard our task accomplished when we have answered
these three questions:
1. What is to be taken as a guide for the arrangement of the
subject-matter, and what is to be this arrangement P
2. What are the demands of pedagogy in teaching primary arith-
metic in order to make the instruction educative? -
3. What is to be the concrete form of a course of teaching number
from 1 to 100, that it may harmonize with the several principles de-
rived from psychology, pedagogy, and arithmetic as a science and as
an art 2 -
To answer the first question, we must take for granted:
First. That counting is primarily the essential process in forming
numbers, and that the different ways of counting give rise to the four
fundamental operations; hence we must begin our instruction of ele-
mentary arithmetic with counting, and gradually and systematically
develop from it the fundamental operations. -
Second. That all operations with large numbers are made possible,
and are greatly simplified, by being reduced through the help of our
system of numeration to the two fundamental tables—that of addition
and multiplication; hence we are to consider each operation first within
the limits of these tables. -
36
Third. That a system of numeration expresses every number in
the form of a sum, every addend of which (the units excepted) repre-
sents a product.
Further, in order to make use of a system of numeration in the per-
formance of operations, we must know how to transfer a unit of a lower
Order to a higher order, and the reverse; hence, in order to understand
and make use of a system of numeration, some knowledge of the four
operations is necessary. (The above statements should be fully de-
veloped in every treatise that deals in a thorough manner with the
method of teaching elementary arithmetic).
The above considerations justify the arrangement of the subject-
matter of elementary arithmetic in three main divisions, each division
having certain subdivisions, which are again subdivided into many
method-wholes.
The three main divisions are:
1. Operations with numbers from 1 to 10.
2. Operations with numbers from 10 to 100.
3. Operations with numbers above 100.
In the First Division, our decimal system of numeration can render
us no service; every operation in this division is necessarily reduced to
counting. Further, the pupil ought to remember most of the results
of the operations of this the First Division of the subject ; memory drill
is not necessary, as this ought to come naturally from sufficient prac-
tice in solving problems and working examples; otherwise in our next
division, from 10 to 100, we should need again the assistance of count-
ing with objects in teaching addition and subtraction, and we should
also need to reduce every process of multiplication to an addition, and
every process of division to a subtraction.
In the Second Division, where every number consists of two groups
of units of different order, of simple units, and units of the second
order, tens,—we begin to use to some extent our decimal system; but
in order that the pupil may be able to come into complete possession
of it, while writing numbers and operating with them, he ought to be
led up to the decimal system gradually,
The best way of doing this is by a series of exercises in numbers
37
composed only of tens, without any units; the numbers in these exer-
cises should be treated simply as units of a higher order. These exer-
cises are to serve as a transition from units of the first order to numbers
consisting of units of the first and second order.
According to the above considerations, this division is to have a sub-
division consisting of exercises with numbers composed only of units of
the second order. (An almost similar subdivision of exercises composed
of units of the second order only we find in Hentschel’s “Lehrbuch des
Rechenunterrichts,” the first edition of which was published in 1842).
An exposition of the Third Division and its arrangement of the
subject-matter is not within our present purpose; still, we may remark
that when a pupil has once been led up to 100 by our order of the sub-
ject-matter, he can afterwards follow more or less the arrangement of
the existing text-books.
We shall now attempt to answer the second question—“What are
the demands of pedagogy in teaching arithmetic?”
Pedagogy demands that we should educate through instruction; i.e.,
we are to create an interest for the ideas that are to be introduced to
the youthful mind, and thus gain the coöperation of the pupil for the
reception of the material of the subject-matter.
By what means are we to gain the interest of the pupil for our
subject?
An answer to this question can be found in the principles of instruc-
tion formulated by the Herbartian School, and known by the name of
concentration.
These principles when properly understood and fully carried out
are destined to become the presiding and influencing agents in the con-
struction of our courses of study, and to infuse new light into the
method and subject-matter of instruction; and it is only by fully carry-
ing out these principles that we can ever hope to shake off the present
routine in which the art of instruction is deeply plunged. .
These principles call our attention to the fact that the historical
priority of mathematical studies resulted in their study, independently
of other branches. This is the cause of our utter lack of interest in
the abstract division of mathematics (to this division belong arithmetic
38
algebra, and calculus), which is nothing other than a method of reason-
ing, and which can never gain our highest interest unless studied in
connection with the matter to which it is applied.*
The principles of concentration, which are of late beginning to
receive due consideration, reveal the dryness of the instruction as at
present given in the subject of arithmetic and the want of interest in
it on the part of the pupil, and point out the cause of this in the
present isolation of arithmetic from other branches of study, which is
due to the fact of beginning instruction with abstract number.
Even the applied examples of most of our text-books excite very
little interest in the pupil, as the material of these problems is mostly
taken from a sphere outside the experience of the pupil.
The above-mentioned principles, together with the criticism of the
present method of teaching elementary arithmetic, point out the way of
construction, which consists in making the synthetical method of in-
struction prevail, instead of the present analytical view of the mere
mathematician.
Under the dominance of the view of the pedagogist whose function
is to harmonize the interest of the pupil with the special end of each
branch of study, the entire course of arithmetic is to be constructed
upon a graded, well-selected, and systematized series of simple problems,
as only by means of problems (not examples) can we explain the need
and meaning of operations upon numbers.
We are to begin every method whole, not with abstract examples,
but with applied problems to which we are to return again at the end
of each new step. Further, the material of the problems ought to be
taken out of the sphere of the pupils’ experience, and thus, by means
of the material itself gain the interest of the pupil for the special
scientific end of our subject.
(While writing these lines a recently published book, under the title “Number
Work in Nature Study,” by W. S. Jackman, advocates in its introduction something
of the method for which we are pleading ; but his work cannot be regarded as
an attempt at carrying out the pedagogical principles of concentration and coördina-
* The fundamental division of mathematics into abstract and concrete, with illus-
trations of their different objects and natures, is fully explained in the ‘‘ Philosophy
of Mathematics" of Auguste Comte.
39
tion of the instruction in this subject, but rather as an attempt to show how natural
science can be benefited by the introduction of the quantitative element. A book that
could satisfy our demand would have to show how to carry out the pedagogical prin-
ciples without neglecting in the least the nature of our subject or the arrangement
of the subject-matter.)
course FROM 1 To 10.
This course consists of seventeen steps or method-wholes; the first
eight steps are forward and backward counting by successive addition
and subtraction of a unit.
First step . . . . . . . . counting from 1 to 3
Second “ . . . . . . . . <& ** 1 to 4
Third “ . . . . . . . . & 6 ** 1 to 5
Fourth “ . . . . . . . . & & * * 1 to 6
Fifth “ . . . . . . . . “ * * 1 to 7
Sixth “ . . . . . . . . & & * * 1 to 8
Seventh “ . . . . . . . . “ “ 1 to 9
IEighth “ . . . . . . . . “ ** 1 to 10.
Every step is to be subdivided into three parts:
First. The introduction of each number to the pupil by a special
object out of the sphere of the pupil’s knowledge, in order to gain the
interest of the pupil for the step.
For instance, we are to begin the third step by counting the fingers
of one hand; the fifth step, by counting the leaflets of the leaf of a
horse-chestnut tree.
Second. By drawing squares and counting them successively one
after the other; then crossing off one after another while counting
them backwards, or by using cubical blocks grouped according to the
purpose of each step. .
We consider the drawing of rows of squares by the pupils as one of
the best objective means to be used in the instruction of primary arith-
metic; especially when they are used in connection with Tillich’s calcu-
lating-box—an apparatus worthy of wider adoption than it has yet
received in this country.* *
* In a work which we have projected, we purpose to present a full discussion of
A()
The squares are within the possession of the pupils; they are easily
understood. Further, by means of these squares we can best illustrate
the serial nature of numbers; we can also illustrate the complex units
of the second order, and the fundamental operations. Their use also
brings into play the motor element. -
Third, Applied examples of forward and backward counting.
The task of solving the question of the selection of apparatus is
not an easy one, when there are over a hundred kinds in Germany alone,
each one offering some advantages.
The mode of presentation of any of these steps, for instance the
fifth, may be the following:
FIFTH STEP.
Let the pupils count the leaflets of the leaf of a horse-chestnut tree.
Let them pluck off one leaflet at a time and count backwards.
Let the pupils draw a square for every leaflet of the leaf and count
the squares. -
Let them cross off one Square at a time and count backwards.
Let the pupils count the days of the week (Sunday one, Monday
two, Tuesday three, etc.).
How many flowers are on my desk?
How many flowers remain if I throw away this one?
(In the proper season of the year the pupils may be asked to count
the stamens and the parts of the calyx of the star-flower, Trientalis.)
I had seven marbles and lost one. How many marbles had I left P
Afterwards I lost one marble. How many marbles remain *
I had seven cents and spent one cent for a pencil and one cent for a
paper. How many cents were left
How old are you? Six.
How old will you be in a year from now *
Five and one are how many ?
Six and one are how many ? etc., etc., etc.
the question where and how far objective means can render service in the study of
elementary arithmetic, and also of the question of the selection of apparatus for
objective illustrations, that can be justified by psychology and by the science and
art of arithmetic.
41
All the work with numbers from 1 to 10 can be performed orally;
written work can only indicate the operations and their results, but can-
not facilitate the way of acquiring a result; hence there is no need of
haste in beginning written work with ciphers. •
The time of the introduction of ciphers is to depend upon the three
following considerations: First. Upon the organization of the school;
that is to say, there may be a need of introducing written work earlier,
in order to keep the pupils employed when the teacher has several
grades in one class. Second. Upon the general preparation of the
pupils for written work. Third. Upon the amount of arithmetical
knowledge the pupils possess, in order that the written work may help.
them in their further progress in arithmetic. -
Under no consideration can written work with ciphers be justified
if introduced before the eighth step. Ciphers may be introduced as
abbreviations of the squares of the previous steps.
The introduction of ciphers forms the Ninth Step.
The Tenth Step consists in adding to a number or in taking away
a definite group of units. (Addition and subtraction.)
The Eleventh Step.–Solving problems and working examples, in-
volving addition and subtraction of a more complex nature than the
previous step.
The exercises of this step are of the following nature:
1st. 4 + 3 + 2 =
2d. 8 – 3 – 4 =
3d. 9 – 2 + 3 – 4 =
The Twelfth Step.—Forward counting with twos. (Multiplication
by two.) *
Counting with twos is to be conducted as a natural continuation of
the preceding exercise.
This step may be introduced in the following form:
[] [] [][] [] [] [] [I] [I] [I]
[] [] [I] [T] [] [] [I] []
[...] [ ] [] [] [I] [T]
[ ] [I] [I] [I]
[T] [T]
42
2 + 2 =; 2 + 2 +2 = ; 2 + 2 + 2 + 2 = ; 2 + 2 + 2 + 2 + 2 =;
then multiplication by two ought to be developed by means of problems
which are to be solved, first, by addition, and afterwards by multiplica-
tion. These problems, besides showing the pupils the need for successive
addition of twos, ought to make them aware of the number of times two
is taken.
After the pupils have solved many problems, where two is repeated
two, three, four, and five times, the teacher is to acquaint the pupils
with the sign of multiplication, showing them that instead of 2 + 2 + 2
+ 2 = 8, we write shorter 2 × 4 = 8, and the cross (X) stands for the
word repeat. -
This step is to be completed with problems and examples involving
the knowledge of all previous steps.
The Thirteenth Step.–Backward counting with twos. (Division
by two.) .
The pupil may be led up to this step in the following form, crossing
off two squares after each subtraction:
[T][] [T] [] [T] [T], [] [T]
[…] [I] [ ] [ ] [ ] [ ]
[] [I] [I] [T]
[...] [T]
2 – 2 = ; 4 – 2 = ; 6 — 2 = ; 8 — 2 =; 10 — 2 =.
4 – 2 – 2 =; 6 – 2 – 2 – 2 =; 8 – 2 – 2 – 2 – 2 =; 10–2 – 2
– 2 – 2 – 2 = . Then by means of problems of the following nature
we are to teach the need and meaning of division.
PROBLEM.—A boy made note-books out of eight sheets of paper ; he
used two sheets for each note-book. How many note-books did he
make P -
Solution.—When a boy has made one note-book, he has 8–2 = 6
sheets left; when he has made the second note-book, he has 6 — 2 = 4
sheets left; when he has made the third note-book, he has 4 – 2 = 2
sheets left; with the last two sheets he makes the fourth note-book.
Answer, four note-books.
In solving this problem the pupil must be led to see that to get the
43
answer he had to subtract twos from 8. Further, the pupil must
notice that the number of times he had to subtract is the same as the
number of note-books. By solving similar problems the pupil will
learn that the answer in each problem depends upon the number of
times two can be subtracted from the given number, and upon the
number of times two is contained (repeats itself) in the given number.
The Fourteenth Step.–Forward and backward counting by threes.
(Multiplication and division by three.)
The Fifteenth Step.–Forward and backward counting by fours.
(Multiplication and division by four.)
The Sixteenth Step.–Forward and backward counting by fives.
(Multiplication and division by five.)
The Seventeenth Step.–Problems and examples involving the four
Operations. - -
Every step beginning with the Tenth Step is to be subdivided into
three parts: (1) Performing operations with the assistance of squares
that are to be drawn by the pupil and grouped according to the purpose
of each step; (2) Solving problems arranged for the purpose of each
step; (3) Working examples.
-During the Tenth Step, the symbols of the operations of addition
and subtraction are to be introduced.
During the Twelfth Step, the symbol of multiplication is to be in-
troduced, and during the Thirteenth Step, the symbol of division.
The work of the Tenth Step may be regarded as completed when
the pupils are able to find every one of the following sums and every
one of the following differences:
1 + 2 1 + 3 1 + 4 1 + 5 1 + 6 1 + 7 1 + 8 1 + 9
2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + 7 2 + 8
3 + 2 3 + 3 3 + 4 3 + 5 3 + 6 3 + 7
4 + 2 4 + 3 4 + 4 4 + 5 4 + 6
5 + 2 5 + 3 5 + 4 5 -- 5
6 – 2 6 + 3 6 + 4
7 –– 2 7 -- 3
8 + 2
:
44
3 – 2 4 – 3 5 – 4 6 – 5 7 – 6 8 – ? 9 – 8 10 – 9
4 – 2 5 — 3 6 — 4 7 – 5 8 – 6 9 — 7 10 – 8 $
5 — 2 6 – 3 7 — 4 8 — 5 9 – 6 10 – 7
6 — 2 7 – 3 8 — 4 9 — 5 10 – 6
7 — 2 8 – 3 9 — 4 10 — 5
8 — 2 9 — 3 10 — 4
9 – 2 10 — 3
10 — 2
The whole material of the five steps preceding the 17th is contained
in the following table:
2 × 2 4 + 2 3 × 2 6 —- 3
X 3 6 —- 2 3 X 3 9 —– 3
× 4 8 –- 2 4 × 2 8 –— 4
× 5 10 + 2 5 × 2 10 —- 5
2
The work of this course may be regarded as completed :
First. When the pupils are able to indicate the work of the four
operations.
Second. When they can apply these operations to the solutions of
simple problems. -
Third. When they can remember most of the sums, differences,
products, and quotients. (Of course there is no need to drill the pupil
in memorizing the results of the operations, as many occasions ought
to present themselves, while solving problems, to acquire the results
naturally.)
COUNTING WITH TENS FROM 10 TO 100.
The object of this division of the course is to acquaint the pupils
with the units of the second order and to lay a basis for utilizing the
system of numeration in the performance of operations. -
All the exercises in this division consist only of the units of the
second order and are arranged nearly in the same way as the exercises
with the units of the first order.
The steps of this division are the following:
First. Forward and backward counting by successive addition and
subtraction of units of the second order.
:
&
45
Second. Written indication of the units of the second order.
Third. Addition and subtraction of units that are composed only
of the second order. (All the exercises of this step can easily be reduced
to the corresponding exercise with units of the first order.)
Fourth. Forward and backward counting by 2, 3, 4, and 5 units of
the second order. (Multiplication and division.)
All the material of the Fourth Step of the Second Division is con-
tained in the following tables:
20 × 2 20 + 2 90 —- 3 40 –– 20
20 × 3 40 —– 2 40 –- 4 60 –– 20
20 × 4 60 —- 2 80 —— 4 80 ––.20
20 × 5 80 —- 2 50 –– 5 100 –– 20
30 × 2 100 —– 2 100 –– 5 60 –– 30
30 × 3 30 —- 3 90 –– 30
40 × 2 60 –– 3 80 –- 40
50 × 2 100 —- 50
The first step is subdivided into three parts:
First. Forward and backward counting with the squares grouped to
represent the complex units—tens.
The pupils ought to be led up to the first step of this division in the
following manner: +
Show them a silver dollar; speak of its form, the material that it
is made of; speak of its value—a dollar = 100 cents; a dollar = 10
dimes. I have 8 dimes, how many more will make a dollar?
Solve oral problems of the four fundamental operations containing
dimes for their material. The pupils are now ready for the first step of
this division.
Second. Solving problems.
Third. Working examples.
All the other steps consist of two parts, viz., solving problems and
working examples.
The work of the second division of this course may be regarded as
completed :
46
First. When the pupils are thoroughly acquainted with the idea of
regarding tens as complex units of counting.
Second. When the pupils are familiar with the representation of
these complex units in writing.
Third. When the pupils see clearly the analogy between the units
of the second order and the units of the first, and are able to reduce all
problems and examples with units of the second order to the operations
with units of the first order.
Counting with numbers composed of units and tens from 1 to 100.
Counting with numbers composed of units and tens is one of the
most important divisions of the whole course of elementary arithmetic.
Here the operations with numbers offer an opportunity to make
use of our system of numeration, and of utilizing certain results of the
previous division.
Further, the operations of this division contain most of the results
which form the basis of all arithmetical tables that make operations
with large numbers possible.
And last, in this division, the pupils become acquainted, while
performing the operations, with the advantage of the decimal system of
numeration. Hence their work in the third division differs from the
elementary ways of the first division.
It is true that all the operations with numbers, even in this third
division, could be reduced to the operations with numbers as in the first
division ; but the reasons why this should not be done are very obvious.
This third division consists of several subdivisions. The first is the
formation of numbers from 1 to 100, of units and tens; the object of this
subdivision is to teach the pupils to determine the number of units and
tens. The exercises must be performed, first, by the help of the abacus,
or by means of drawing rows of squares, and then without any objective
I]] (28,1] S.
The main steps of this subdivision are:
First. The successive addition of units to tens, and the successive
subtraction of units from tens.
Second. Addition of groups of units to tens, and the successive
groups of units from tens.
47
Third. Successive addition of tens to numbers composed of units
and tens; successive subtraction of tens from numbers composed of units
and tens.
The second subdivision is addition and subtraction; with the
beginning of this division there ought to be no further need of any
objective means for facilitating the processes.
This second subdivision consists of the following steps:
First. Addition and subtraction of groups of units from numbers
composed of tens only.
Second. Addition of numbers whose sum of units does not exceed
ten, and subtracting numbers, when the units of the subtrahend are less
than the units of the minuend.
Third. Addition, when the sum of the units equals ten; and sub-
traction, when the minuend consists of tens only.
Fourth. Addition, when the sum of units exceeds ten; subtraction
when the units of the minuend are less than the units of the subtrahend.
All the exercises of this and the following subdivision consist of
problems and examples.
The object of the third subdivision is to teach multiplication and
division, and consists of the following steps:
1. Multiplication and division by... . . . . . . . . . . . . . . 2
2. 6 & & & “ “ . . . . . . . . . . . . . . . . 3
3. & 6 & 6 & “ . . . . . . . . . . . . . . . . 4
4. & & & 6 <& “ . . . . . . . . . . . . . . . . 5
5. & & & 4 “ “ . . . . . . . . . . . . . . . . 6
6. & & C & & 4 “ . . . . . . . . . . . . . . . , 7
7. & 6 & 6 << “ . . . . . . . . . . . . . . . . 8
8. & 6 & & “ “ . . . . . . . . . . . . . . . . 9
Every step is to be introduced to the pupil by forward and back-
ward counting with equal groups of units.
Ninth step. Multiplication with a multiplicand of two digits.
Tenth step. Division with a divisor of two digits.
Eleventh step. Elementary ideas of fractions.
OF MICHIGAN
II.
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