little geometry. To accomplish this we should give him the idea of
geometrical objects, and to some extent their nomenclature, and this
can be done without causing fatigue. To accomplish this he should be
taught to draw, however rudely. He can begin with straight lines, of
which he soon learns the properties; then, when he has drawn several
lines side by side, he will learn that they are parallels and will
never meet. He will learn, too, after he has drawn three intersecting
lines, that the figure within them is called a triangle, that the
figure formed by two parallel lines meeting two other parallels is a
parallelogram, and he can go on to make and learn about polygons, etc
(Fig. 3). All this nomenclature will get into his head without giving
abstract definitions, but in such a way that when he sees a
geometrical object of definite form he will recognize it at once and
give it the name that belongs to it.
[Illustration: FIG. 3.]
In the practical matter of the measurement of areas we convey
immediate comprehension as to many figures without special effort,
provided we do not present the demonstration in professional style,
limiting ourselves to making the pupil comprehend or feel things so
clearly and definitely that it shall be equivalent, as to the
satisfaction of his mind, to an absolutely rigorous demonstration. At
any rate, he will be better provided for the future than by rigorous
demonstrations that he does not understand. Taking the parallelogram,
for example, let us suppose a figure made like Fig. 4, and we saw
through it along the lines _A A'_ and _B C_. It does not need a very
great effort of attention to recognize, experimentally if need be,
that the two triangles _A A' D_ and _B B' C_ may be placed one upon
the other and are identical. If, from the figure thus formed, we take
away the right-hand triangle the parallelogram will remain; if we take
away the other triangle a rectangle will be left, or a peculiar
parallelogram, of which also we give the idea to the child as a figure
in which the angles are formed by straight lines perpendicular to one
another. Here, then, the child gains the notion of the equivalence of
a parallelogram and a rectangle of the same base and height; and this
notion, obtained by cutting up a piece of board or pasteboard, he will
carry so seriously and firmly in his head that he will never lose it.
By cutting the same parallelogram in two, along a diagonal _A C_, it
may be easily shown that the two triangles can be placed exactly one
upon the other, and that, consequently, they have equal areas. These
lessons constitute a series of classical theorems in geometry which
the child can try with his fingers and learn without even giving them
the form of theorems. I might show the same as to the area of the
trapeze and with many other theorems, but my purpose is only to
present as many examples as will make my idea understood, without
going into details.
[Illustration: FIG. 4]
Yet I can not leave this subject without showing how we can make a
very child understand some of the geometrical theorems that have
acquired a bad reputation in the world of candidates for degrees,
including even such as the _pons asinorum_ of Pythagoras; the
demonstration, that is, that if we construct the triangles _B_ and _C_
on the sides of a right-angled triangle, their sum will be equal to
the square _A_ constructed on the hypotenuse. The usual demonstration
of this theorem is not very complicated, but there is something
tiresome, artificial, and hard in it. The demonstration I propose is
almost intuitive, and the reasoning of it is both simple and rigorous.
Suppose we take two equal squares, and, making equal lengths on the
four sides of one of them, join the points so obtained as indicated in
the first of the two figures (Figs. 5 and 6) so as to form four
right-angled triangles, and then place four other squares in the
corners of the original square. These right-angled triangles are of
such sort that the sum of their sides is equal to the side of the
square. This can be demonstrated, but it strikes the eyes without
that. We see, too, that the interior figure is a square, and that it
is constructed on the hypotenuse of the triangles in question.
[Illustration: FIG. 5.]
[Illustration: FIG. 6.]
It is easy to see in the other figure, which is formed after the same
measures as its alternate, that the triangles 1, 2, 3, 4 can be
arranged so as to occupy the positions 1', 2', 3', 4' in such way as
to leave in the main square two smaller squares constructed on the
sides of one of the right-angled triangles. It follows that the square
A is equivalent to the sum of the squares _B_ and _C_. The theorem
thus becomes a kind of intuition, a thing evidently indisputable.
It is a curious fact that the origin of this demonstration is lost in
the obscurity of the past; it probably goes back to thirty or forty
centuries, at least, before the Christian era, and apparently to
India. Bhascara, in his _Bija Ganita_, after tracing a figure, a
simple combination of these two, says, "There you see it." I remark
that such a demonstration, even if dressed with geometrical terms,
assuming a character that conforms to existing ways of teaching, would
be vastly superior, even in secondary schools, to the demonstrations
of Legendre and others, which are much harder. The return to what was
done very long ago in this case constitutes a great advance upon what
we are doing now.
Having given our little one an initiation into the mysteries of
arithmetic and geometry, we introduce him to algebra, a branch which
passes in the majority of families as the hardest, most complicated,
and most abstruse that can be imagined. I do not pretend that
algebraic theories enter easily into the child's delicate brain;
rather the contrary; but I declare that some ideas in algebra can be
made comprehensible to children without fatigue. We can, for instance,
make them understand, in the way of amusement and without great
difficulty, the formula that gives the sum of the first numbers. We
take a sheet of paper ruled in squares and shade the first square of
the first line, then the first two squares of the second line, the
first three of the third, etc. (Fig. 7). The whole number of squares
shaded in this manner represents visibly the sum of the first whole
numbers up to any one we may choose--to 7 in the figure. If we give
this paper to the child and ask him to return it, he will very easily
perceive that the figures formed by the white and the black squares
are alike. The number sought for will therefore be equal to half the
sum of the squares--that is, in the present example
1 + 2 + 3 + 4 + 5 + 6 + 7 = (7 X 8) : 2 = 28,
we can prove by reasoning that if n be taken to represent the last
number we shall have for the sum
n (n + 1)
S = ----------
2
[Illustration: FIG. 7.]
I introduce this formula to define my thought better, but one can make
the child perceive the numbers that are wanted without writing down a
single character.
Somewhat similar is the method of finding the sum of the odd numbers.
For this it will be enough to take our square-ruled sheet of paper and
shade the first square on the left, then the three squares around it,
which will form with it a square (1 + 3 = 4); continuing thus we
obtain, as the figure readily shows (Fig. 8), a square formed of a
series of shaded zones, representing the series of odd numbers, the
examination of which will illustrate the property to the child.
[Illustration: FIG. 8.]
In another direction it is possible to give the child algebraic ideas
much beyond anything we would imagine. Suppose, for example, we want
to give him a conception of addition. He easily realizes that
objects--material bars, for example--can be selected so as to
represent numbers by their length. He can be readily made to
understand that if he has one bar three and another five inches long
he can obtain the sum of these lengths, in what we might call a
material way, by placing them lengthwise, one at the end of the
other--an essentially practical notion and easily carried into effect.
If we take a line and mark a starting point on it, calling it zero,
then measure off segments on it representing the bars we have been
talking about one after another, we can get the sum represented by the
length of the two segments. If, instead of measuring three plus five
inches I measure three plus two I reach another point. If, instead of
adding two and three, I wish to take one of the bars or numbers away
(3-2), or subtract, the operation will be easily performed by
measuring the two in the opposite direction. The difference will be
represented by the length that is left. If we try to form the quantity
3-5 in arithmetic we can not do it; but in proceeding in this method
and measuring back on the bar we get to a point back of the original
starting point which represents this difference--say two inches behind
where we began. Here we have in the germ the whole theory of negative
quantities, concerning which thousands and thousands of pages have
been written. Yet we find that by carefully graduating our lines we
can make it intuitive and accessible to a child who has learned that
the common operations of addition and subtraction can be represented
with material objects. The generation of negative and positive
quantities follows quite naturally.
These examples, I think, are sufficient to show that we might
considerably enlarge the field of the investigations within reach of
the child. For this purpose a small amount of very simple material,
which we can vary as we please, is needful. The first element of this
material is paper ruled in squares, a wonderful instrument, which
everybody dealing with mathematics or with science generally should
have. It is of special pedagogic use in giving children their first
ideas of form, size, and position, without which their early
instruction is only a delusion. Add to this paper dice, buttons,
beans, and match-sticks--things always easy to get--and we have all
the material we need.
There is no amusement, however puerile it may appear, not even a play
of words, that can not be utilized in teaching of this sort. For
instance, when your child has learned his addition table, if you put
him to a demonstration, assuming to prove to his comrades that six and
three make eight, his curiosity will be excited, and you may be very
sure that, once his attention has been given to this amusement, he
will never forget that six and three make nine and not eight. To make
the demonstration, we have only to group the nine match-sticks as in
the figure (Fig. 9) below. We might demonstrate in a like way that
half of twelve is seven by cutting the Roman numeral XII in two,
leaving the upper part visible. Such pleasantries have a pedagogical
value, because the paradox is precisely of a kind to attract the
attention of the child, and he will always afterward be sure not to
fall into the trap.
[Illustration: FIG. 9.]
The side of this kind of instruction on which I insist most is that,
given under the form of play, it is free from every sort of dogmatic
character. No truth should be imposed on the child; on the contrary,
he should be allowed to discover it as a fruit of his own activity. He
will be thoroughly impressed with the truths which he has thus found
out himself. They had better be few at first; the important thing is
for him to know them completely.
The instruction should also be essentially objective and free from all
abstraction. The absence of abstraction should, however, be rather
apparent than real. Abstraction is indeed one of the elements that
contribute most to give mathematical science a fearful air to
outsiders, and yet it is most usually a simplification of
matters--quite the contrary of what is generally supposed. It is, in
fact, such a simplification and so necessary that we all make it as if
by instinct, and the child makes it, not in mathematics only, but in
all the considerations of life.
Thus, when I want to give the child his first idea of the number two I
put two beans in his hand and let him contemplate them. He gets a
perfect notion of the collection two. Yet, if you look at them a
little closer and he himself looks at them closer he will find that
the two beans, whatever else they may be, are not identical, for there
exist no two objects in Nature that are not different. So when the
child introduces this idea of collection into his mind in a wholly
instinctive way, by identifying the things he sees, he begins to
perform abstraction. This abstraction delivers him from all the
complications and all the annoyances that come to him from the
contemplation of real objects. By the philosophic process of
abstraction it has been possible to construct all the sciences, and
especially the science of magnitudes.
The ideas I have been setting forth in outline are not mine, and are,
unfortunately, not recent. They may be found in somewhat different
form, but substantially the same in principle, in _l'Essai d'education
nationale_, published by Le Chalotais in 1763. The paper furnishes a
programme of studies and education which, if put into execution,
would, I believe, constitute a long advance over the present
conditions. At a later period Condorcet was occupied with the subject.
At the close of the nineteenth century the name of Jean Macé, which I
have already cited, should be held among those of men who have tried
to infuse sound and just views concerning the pedagogy of mathematics.
Another man, from whom I have borrowed a considerable part of the
examples I have cited, is Edouard Lucas, who, in his _Récréations
mathématiques_, of which one volume was published during his lifetime
and two others after his death, and in his lectures before the
Conservatoire des Arts et Métiers, strove to develop views concerning
the primary mathematical education of childhood--views which
did not differ, except in form, from those which I have
presented.--_Translated for the Popular Science Monthly from the Revue
Scientifique._
PRESENT POSITION OF SOCIOLOGY.
BY F. SPENCER BALDWIN.
The present condition of sociological thought is confused, if not
chaotic. It needs only a brief examination of the writings of
professed sociologists to discover the want of agreement among them.
There is no consensus of opinion regarding either the scope and method
of the new science, so called, or its fundamental laws and principles.
The name sociology stands for no definite body of systematic
knowledge. It is applied to an inchoate mass of speculation, often
vague and conflicting, which represents the thought of various
thinkers about social phenomena.
A few years ago a student of sociology in Chicago wrote to "all the
teachers of sociology in the United States, and to others known to be
deeply interested in the subject and entitled to express an opinion,"
asking them to answer a number of pertinent questions regarding the
nature and function of the "science."[28] About forty replied; of
these, three discreetly pleaded knowledge insufficient to entitle them
to an opinion. Comparison of the views expressed in the remaining
twenty-seven replies led the investigator to conclude that the science
is in a more or less undefined and tentative position. So little
progress toward unanimity of opinion has been made by sociologists
since the date of this census that its results may be taken as typical
of present conditions. Among the questions asked were these: "Do you
think the study is entitled to be called a science?" "In what
department does it belong?" "What is its relation to political
economy, history, political science, ethics?"
[Footnote 28: Present Condition of Sociology in the United
States. Ira W. Howarth. Annals of the American Academy,
September, 1894.]
The question whether sociology is entitled to be called a science is
answered by "fully three fourths" of the correspondents in the
affirmative. Some hedge, by affirming that it is "becoming a science."
Prof. John Bascom, of Williams College, appears to have entered into
the humor of the situation; he writes, "It will do no harm to call it
a science if we do not abate our effort to make it one."
The opinions regarding the department in which sociology belongs are
entertainingly diverse. Prof. John Dewey, of the University of
Chicago, is frank enough to admit that he doesn't "feel at all sure"
where it belongs. "It would seem well," he adds, "to have it a
separate branch, in order to make sure that it received proper
attention." This feeling of uneasiness lest the claims of sociology be
slightingly treated appears to be general among the representatives of
the new study. Most of the teachers of sociology are of the opinion
that it ought to form a department by itself. Some would place it in
the department of the social sciences, along with politics, economics,
jurisprudence, and the like. Others would change the order, making all
the social sciences divisions of sociology. On the other hand,
Professor Giddings, of Columbia University, says: "General sociology
can not be divided into special social sciences, such as economics,
law, and politics, without losing its distinctive character. It should
be looked on as the foundation or groundwork of these sciences, rather
than as their sum or as their collective name." Scattering replies
place it under psychology, moral and political science, political
economy, and anthropology. One teacher thinks it belongs under the
"humanities"; while two say it has no natural boundaries, and is
therefore not included in any one department. Altogether the
impression left by the replies to this question is that the teachers
of sociology are quite at a loss to know where to put the study in the
university curriculum. They appear to realize confusedly that they
have on their hands a pedagogical white elephant, which defies
classification.
The opinions concerning the relation of sociology to political
economy, history, political science, and ethics are almost delphic in
their vagueness. Says one, "History is its material, ethics its guide,
political economy its interpreter, and a rational system of political
science its proposed end." Says another, "Sociology is political
economy in practice, history in the making, political science as an
art, and ethics applied." After worrying over these oracular epigrams
it is refreshing to be told by another teacher that "the relation of
sociology to political economy, history, etc., is _close_."
It would be superfluous to cite further illustrations of the unsettled
state of sociological thought. The quotations that have been made show
conclusively that the accredited representatives of the new "science"
are at loggerheads upon fundamental questions. This fact the
sociologists themselves readily admit. The author of a recent treatise
on sociology speaks of the "confusion and perplexity among its
teachers, and declares that its forms are as yet varied, and perhaps
would suggest a series of pseudo-sciences instead if one genuine
science."[29] Even Professor Giddings confesses in the preface of his
Principles of Sociology that "much sociology is as yet nothing more
than careful and suggestive guesswork." Professor Small, of the
University of Chicago, in his Introduction to the Study of Society,
speaks of sociology as an "inchoate science," and remarks that "only
ignoramuses, incompetent to employ the method of any science, could
claim for sociology the merit of a completed system."
[Footnote 29: Fairbanks. Introduction to Sociology, p. 1.]
Sociologists themselves, then, confess that differences of opinion
exist among them. Let us look more carefully at the nature of these
differences. They relate to the scope, the method, the object, and the
ground-principles of the "science."
The province of sociology is defined by some very broadly, to include
the whole range of the phenomena of human association. By others the
scope of the study is limited to a narrower range of social phenomena.
Among the latter, again, there are some who would identify sociology
with the study of social origins, or the genesis of social
institutions. Others would restrict sociology to a study of the
history and function of the family. Still others understand by
sociology merely the pathology of society, devoting themselves to the
diagnosis of social diseases, as crime and pauperism.
Professor Giddings has called attention to the natural tendency on the
part of each social philosopher to create a sociology in the image of
his professional specialty. "To the economist," he says, "sociology is
a penumbral political economy--a scientific outer darkness--for
inconvenient problems and obstinate facts that will not live peaceably
with well-bred formulas. To the alienist and the criminal
anthropologist it is a social pathology. To the ethnologist it is that
subdivision of his own science which supplements the account of racial
traits by a description of social organization. To the comparative
mythologist and the student of folklore it is an account of the
evolution of culture."
The narrower conceptions of sociology, however, have been discarded
by the best-known sociologists of the present time. There is a general
tendency to adopt a broad definition of the province of sociology, to
include in the field of investigation all the phenomena of social
structure and growth.
But what is the relation of this general social science to the special
social sciences--that is, the sciences dealing with special groups of
social phenomena, as economics, politics, and jurisprudence? Is
sociology anything more than a convenient collective name for the sum
of all these? Touching this point opinions differ.[30]
[Footnote 30: See for the following: H. H. Powers. Terminology
and the Sociological Conference, in Annals of the American
Academy, March, 1895.]
At least three different conceptions of the relation of sociology to
the various special social sciences may be distinguished. Sociology
has been defined as (1) the "inclusive," as (2) the "co-ordinating,"
and as (3) the "fundamental" science of society. 1. The first
conception is that of Spencer and De Greef. Spencer defines sociology
as "the science of society," and defends his adoption of the term on
the ground that "no other name sufficiently comprehensive existed."
This implies that he conceives of sociology as an inclusive science.
De Greef, the Belgian sociologist, makes the science all
comprehensive; his scheme of classification "includes everything, from
the husbanding of corn and wine to electioneering contests in the
Institute of France."[31] 2. The second conception is that of
Professor Small, of Chicago. He defines sociology as "the synthesis of
all the particular social sciences." It does not include, it
coordinates these sciences. It concerns itself with the relations
which the various special groups of social phenomena hold to each
other and to society as a whole, leaving to special social sciences
the study of each group in minute detail. The conclusions won by these
special sciences are taken by sociology and worked over into a body of
correlated social principles. Sociology is, therefore, subsequent to
the particular social sciences and dependent upon them. 3. The third
conception is that of Professor Giddings, of Columbia University. He
defines sociology as "the science of social elements and first
principles." It is "not merely the sum of the social sciences; it is
rather their common basis." It undertakes to analyze the general
characteristics of social phenomena and to formulate the laws of
social organization and evolution. Sociology furnishes a body of
fundamental principles which make a common basis for the special
social sciences. The latter rest on sociology, which is the antecedent
and fundamental social science.
[Footnote 31: See Giddings. Principles of Sociology, p. 29.]