of a fraction. Half of a stick is still a stick to the child, just as
half of a stone is still called a stone in common parlance. The half
should be radically different from the unit; hence an object resembling
a sphere or a circle is best adapted for the first lessons in fractions.
In teaching decimals the square or rectangle is better than the circle.
It is difficult to divide a circumference into ten equal parts. On the
contrary, the square is easily divided into tenths by vertical lines,
and then into hundredths by horizontal lines, thus furnishing also a
convenient device for the first lessons in percentage.
[Sidenote: Basal concepts.]
[Sidenote: John Fiske on symbolic conceptions.]
It is one of the aims of the training-class and the normal school to
point out the best methods of developing the different basal concepts
which lie at the foundation of the branches to be taught. Many of
these are complex, and require great skill on the part of the teacher.
The difficulty is well stated in John Fiske’s discussion of Symbolic
Conceptions. He says, “Of any simple object which can be grasped in a
single act of perception, such as a knife or a book, an egg or an orange,
a circle or a triangle, you can frame a conception which almost, or
quite exactly, _represents_ the object. The picture, or visual image,
in your mind when the orange is present to the senses is almost exactly
reproduced when it is absent. The distinction between the two lies
chiefly in the relative faintness of the latter. But as the objects of
thought increase in size and in complexity of detail, the case soon comes
to be very different. You cannot frame a truly representative conception
of the town in which you live, however familiar you may be with its
streets and houses, its parks and trees, and the looks and demeanor of
the townsmen; it is impossible to embrace so many details in a single
mental picture. The mind must range to and fro among the phenomena, in
order to represent the town in a series of conceptions. But practically,
what you have in mind when you speak of the town is a fragmentary
conception in which some portion of the object is represented, while you
are well aware that with sufficient pains a series of mental pictures
could be formed which would approximately correspond to the object. To
some extent the conception is representative, but to a great degree it
is symbolic. With a further increase in the size and complexity of the
objects of thought, our conceptions gradually lose their representative
character, and at length become purely symbolic. No one can form a mental
picture that answers even approximately to the earth. Even a homogeneous
ball eight thousand miles in diameter is too vast an object to be
conceived otherwise than symbolically, and much more is this true of the
ball upon which we live, with all its endless multiformity of detail. We
imagine a globe, and clothe it with a few terrestrial attributes, and in
our minds this fragmentary notion does duty as a symbol of the earth.
“The case becomes still more striking when we have to deal with
conceptions of the universe, of cosmic forces such as light and heat,
or of the stupendous secular changes which modern science calls us to
contemplate. Here our conceptions cannot even pretend to represent
the objects; they are as purely symbolic as the algebraic equations
whereby the geometer expresses the shapes of curves. Yet so long as
there are means of verification at our command we can reason as safely
with these symbolic conceptions as if they were truly representative.
The geometer can at any moment translate his equation into an actual
curve, and thereby test the results of his reasoning; and the case is
similar with the undulatory theory of light, the chemist’s conception of
atomicity, and other vast stretches of thought which in recent times have
revolutionized our knowledge of nature. The danger in the use of symbolic
conceptions is the danger of framing illegitimate symbols that answer to
nothing in heaven or earth, as has happened first and last with so many
short-lived theories in science and in metaphysics.”
The word conception as used in this quotation is synonymous with concept,
but elsewhere it is also used in two other senses,—namely, to signify
the mind’s _power_ to conceive objects, their relations and classes, and
to name the activity by which the concept is produced. Hence the term
concept is preferred in this discussion.
[Sidenote: Concepts of distance.]
[Sidenote: Large cities.]
To give a full account of the development of the basal concepts in the
different branches of study would require a treatise on the methods of
teaching these branches. All that can be attempted is to draw attention
to some of the typical methods and devices adopted by eminent teachers
in the development of the concepts which Mr. Fiske calls symbolic
conceptions. Distance is one of the concepts at the basis of geography
and astronomy. To say that the circumference of the earth is twenty-five
thousand miles, that the distance of the moon from the earth is two
hundred and forty thousand miles, and that the distance of the sun is
ninety-two and one-half millions of miles may mean very little to the
human mind, especially to the mind of a child. Supposing, however, that
a boy finds a mile by actual measurement, and that he finds he can walk
four miles an hour, he can gradually rise to the thought of walking forty
miles in a day of ten hours, or two hundred and forty miles in the six
working days of a week. In one hundred and four weeks, or two years, he
could walk around the globe. To walk to the moon would require a thousand
weeks, or about twenty years. It is by the method of gradual approach
that concepts of great distance, of immense magnitudes, of the infinitely
large and the infinitely small, must be developed. To this category
belong large cities like New York and London, quantities denoting the
size of the earth and its distance from the sun and the fixed stars, the
fraction of a second in which a snap-shot is taken, or an electric flash
is photographed; such quantities are apt to remain as mere figures or
symbols in the mind of the learner unless the method of gradual approach
is adopted. Starting with a town or a ward with which the pupil is
familiar, several may be joined in idea until the concept of a city of
fifty or sixty thousand population is reached. It takes about twenty of
these to make a city like Philadelphia, and five cities like Philadelphia
to make a city like London. A lesson on how London is fed will add much
to the formation of an adequate idea of such a large city.[5]
[Sidenote: Shape of the earth.]
An adequate idea of the shape of the earth can be formed only by gradual
development. The three kinds of roundness (dollar, pillar, ball) must
be taught; then the various easily intelligible reasons for believing
it to be round like a ball may follow in the elementary grade. As the
pupil advances he may be told of the dispute between Newton and the
French, the former affirming it to be round like an orange,—that is,
flattened at the poles,—the latter asserting that it resembled a lemon
with the polar axis longer than the equatorial diameter; and how, by
measuring degrees of latitude and finding that their length increases
as we approach the poles, the French mathematicians, in spite of their
wishes to the contrary, proved Newton’s view to be correct. The same
lesson might be taught by starting with the rotation of the earth,
showing by experiment the tendency of revolving bodies to bulge out at
the equator, and then drawing the inference that the degrees of latitude
are shortest where the curvature is greatest, and that they are longest
where the curvature is least. Either method is strictly logical; but the
method which follows the order of discovery, whenever it is feasible, is
calculated to arouse the greater interest in minds of average capacity.
The teacher who is a master of his art will supplement the historical
lesson by a lesson passing from cause to consequence, so as to fix and
clarify the concept formed by passing from the ground of knowledge to the
necessary inference. Finally, by drawing attention to the fact that the
equatorial diameters are not all of the same length, he will build up in
the pupil’s mind a concept of the real shape of the earth,—a shape unlike
any mathematical figure treated of in the text-books on geometry. The
attempt to give a complete idea of the shape of the earth in the first
lessons on geography would have ended in confusion of thought; the wise
teacher develops complex concepts gradually and not more rapidly than
the learner is able to advance. This process may be called enriching the
concept. The successive concepts, although only partial representations
of what is to be known, are adequate for the thinking required at a given
stage of development; the number of complete or exhaustive concepts in
any department of knowledge is small indeed.
[Sidenote: The order of discovery and of instruction.]
Instructive as it often is to follow the order of discovery, it must not
be inferred that this is invariably the best order of instruction. What
teacher of astronomy would be so foolish as to lead a student through the
nineteen imaginary paths which Kepler tried before he discovered that an
elliptical orbit fitted the recorded observations of Tycho Brahe![6]
Much may be learned from the methods pursued by eminent teachers. It
will abundantly pay any teacher of science to study Faraday’s lectures
on the chemistry of a candle,—a series which for models of developing
the fundamental concepts of chemistry is unsurpassed. The devices used
by such teachers are often very suggestive. For instance, in teaching
the concept of the new geography that the earth revolves not like a body
with a liquid interior, but like a body with an interior as rigid as
glass, Lord Kelvin suggests a comparison of the spinning of a hard-boiled
egg and of an egg not boiled at all,—an experiment easily made in every
school-room.
[Sidenote: Ideas of great distances.]
A few quotations from the astronomer Young will show how concepts of
great distances can be developed so as to be more than a numeral with a
row of ciphers annexed:
“If one were to try to walk such a distance, supposing that he
could walk four miles an hour, and keep it up for ten hours
every day, it would take sixty-eight and one-half years to make
a single million of miles, and more than sixty-three hundred
years to traverse the whole. If some celestial railway could
be imagined, the journey to the sun, even if our trains ran
sixty miles an hour, day and night, without a stop, would
require over one hundred and seventy-five years. To borrow
the curious illustration of Professor Mendenhall, if we could
imagine an infant’s arm long enough to enable him to touch
the sun and burn himself, he would die of old age before the
pain could reach him, since, according to the experiments of
Helmholtz and others, a nervous shock is communicated only
at the rate of one hundred feet per second, or one thousand
six hundred and thirty-seven miles a day, and would need more
than one hundred and fifty years to make the journey. Sound
would do it in about fourteen years if it could be transmitted
through celestial space, and a cannon-ball in about nine, if
it were to move uniformly with the same speed as when it left
the muzzle of the gun. If the earth could be suddenly stopped
in her orbit, and allowed to fall unobstructed towards the sun
under the accelerating influence of his attraction, she would
reach the centre in about two months. I have said if she could
be stopped, but such is the compass of her orbit that to make
its circuit in a year she has to move nearly nineteen miles
a second, or more than fifty times faster than the swiftest
rifle-ball; and in moving twenty miles her path deviates from
perfect straightness by less than one-eighth of an inch.”[7]
Professor Young uses a very suggestive device in his astronomy for
showing the comparative sizes and distances of heavenly bodies:
“Representing the sun by a globe two feet in diameter, the
earth would be twenty-two-hundredths of an inch in diameter—the
size of a very small pea or a ‘twenty-two caliber round
pellet.’ Its distance from the sun on that scale would be just
two hundred and twenty feet, and _the nearest star_ (still on
the same scale) _would be eight thousand miles away at the
antipodes_.”[8]
Sometimes the employment of a new unit aids in realizing the idea of
very great distances. The ordinary astronomical unit is the distance
of the sun from the earth; it is not large enough to be convenient in
expressing the distances of fixed stars. Hence astronomers have found
it more satisfactory to take as a unit the distance light travels in a
year, which is about sixty-three thousand times the distance of the sun
from the earth. The tables of fixed stars give distances in terms of this
unit from 3.5 upward. A glance at these figures fills the mind with an
idea of the infinite grandeur of the universe and with feelings of awe
and sublimity akin to those which must fill the soul on approaching the
throne of Almighty God.
[Sidenote: Time of snap-shot.]
Scientists assert that the infinitely great is more easily conceived
than the infinitely small; that quantities represented by billions and
trillions are more easily grasped than fractions of a unit with a million
in the denominator; that ages of time are more easily comprehended than
fractions of a second. In a lecture delivered at the International
Electrical Exhibition, Professor Charles F. Himes employed a very
ingenious device for giving an idea of how a “snap-shot” may be made,
or a photographic impression taken of an electric spark, or a flash of
lightning. He exhibited a photograph of the sparks of a Holtz machine,
which are of shorter duration than any instantaneous drop or slide could
be made to give. “They impressed themselves upon an ordinary collodion
plate as they passed. Suppose we assume one-twenty-thousandth of a second
as the time, and we will be within bounds. That is a fraction difficult
to comprehend. Our mental dividing engine fails as we work towards
zero. The twenty-thousandth of a second is so small that it eludes
our mental grasp.... Looking at it from another point of view, let us
regard the effect as a space-effect instead of a time-effect. Light has
a velocity, in round numbers, of one hundred and ninety thousand miles
per second. That would be one hundred and ninety miles in one-thousandth
of a second, nineteen in one-ten-thousandth, or, say, ten miles in our
one-twenty-thousandth of a second. Ten miles of light drive in upon our
plate in that time; or, if we held the corpuscular theory of Newton,
a chain of these little pellets ten miles long would have delivered
themselves upon the sensitive surfaces. Ten miles is comprehensible,
one mile is, so that we could easily conceive of an effect in one-tenth
of the time allowed to our electric sparks. But let us take another
look at it. Light is not corpuscles, but undulations, tiny wavelets,
ripplets of ether, eight hundred million million in a second for violet,
a number we can easily understand, as Sir William Thomson[9] has told
us. That would make eight hundred thousand million in one-thousandth,
eight thousand million in one-ten-thousandth, or forty thousand million
impulses striking our sensitive molecules in our one-twenty-thousandth
of a second. Surely that number should produce an effect. We can readily
conceive that one thousand million wavelets would produce an appreciable
effect. They would represent one-eight-hundred-thousandth of a second,
say one-millionth of a second. That would seem, then, to be ample time to
produce a photographic effect.”[10]
[Sidenote: Idea of total depravity.]
Many teachers of science spend all their spare time in reading scientific
literature and in posting themselves upon the latest achievements in
their specialty. It might be to them a less delightful occupation if they
traversed fields of investigation already well explored for the purpose
of seeing how the student can be led over these most expeditiously and
with minimum expenditure of time and effort. Thought bestowed upon
the best way of imparting the elements of science would have a most
beneficial effect upon their methods of instruction, and would greatly
increase their skill in teaching. Many of the most abstruse and complex
ideas can be resolved by analysis into their elements, and thereby be
made intelligible to people of ordinary training. An eminent teacher of
theology felt called upon to impart to a promiscuous audience an idea
of the doctrine of total depravity as taught by the Church. He started
by referring first to the popular mistake that the doctrine teaches the
utter depravity of the human race, then to the ancient heresy that the
depravity of human nature resides in the body, and not in the soul, and,
finally, to the meaning of total as signifying not that man is as bad
as he can become, but that he is depraved, or has a tendency towards
sin not merely in his physical body, but in the totality of his being.
Analysis prepared them to see that by total depravity is not meant
that men are as bad as they can be, nor that they do not have in their
natural condition certain amiable qualities or certain laudable virtues;
that the doctrine means that depravity, or the sinful condition of man,
infects the whole man,—intellect, feeling, heart, and will,—and that in
each unrenewed person some lower affection, and not the love of God,
is supreme. Such analysis of a complex concept into its elements, the
explicit setting forth what it is and what it is not, followed by the
synthesis of the parts into a thought-unit, is the plan pursued by the
best teachers in teaching difficult subjects. By analysis we resolve
complex concepts into their elements, which may be simple percepts or
their relations. Things are separated in thought which go together in
time, space, motion, force, or substance. Every essential attribute or
constituent can then be viewed by itself until the mind has gone around
it with the bounding line of thought, grasped its nature and essence,
and explored it in its different aspects and relations. In this way the
most abstruse subjects are shorn of their difficulties, the most complex
problems are solved and elucidated.
[Sidenote: Value of analysis.]
The bearing of all this upon the art of teaching is easily shown. A
teacher of geometry, whose mind was quite logical, failed, through
lack of power, to make things plain. If the class did not grasp the
demonstration of a theorem, he invariably started at the beginning,
tried to throw light upon every link in the chain of proof, and by the
time he reached the point of difficulty the members of the class were
thinking of something else. A younger colleague pursued a different plan.
Starting some pupil upon the demonstration, he detected the difficulty,
and by a few words of explanation, or by a well-framed question, he
focussed attention upon the simple elements, into which he resolved the
difficulty, and frequently surprised the class by showing the simplicity
of what had puzzled their minds. Under the clarifying light of analysis
half the difficulties and half the sophistries of human thinking vanish
like dew and mist before the morning sun.
[Sidenote: The moral nature.]
For the purpose of making an impression upon the moral nature
word-painting is sometimes very helpful. All the text-books on physiology
and hygiene intended for use in the public schools seek to teach the
evils of strong drink by showing the effect of alcoholic stimulants upon
different parts of the human system. Yet the most exhaustive lessons on
how whiskey is made, and what are its exhilarating and its pernicious
effects, cannot equal the effects of the word painting of Robert
Ingersoll and the paraphrase by Dr. Buckley. In making a gift to a friend
the former penned the following eulogy on whiskey:
“I send you some of the most wonderful whiskey that ever drove
the skeleton from the feast or painted landscapes in the brain
of man. It is the mingled souls of wheat and corn. In it you
will find the sunshine and the shadow that chased each other
over the billowy fields, the breath of June, the carol of the
lark, the dew of night, the wealth of summer, and autumn’s rich
content, all golden with imprisoned light. Drink it, and you
will hear the voice of men and maidens singing the ‘Harvest
Home,’ mingled with the laughter of children. Drink it, and
you will feel within your blood the starlit dawns, the dreamy,
tawny dusks of perfect days. For forty years this liquid joy
has been within the staves of oak, longing to touch the lips of
man.”
This was Dr. Buckley’s statement of the other side:
“I send you some of the most wonderful whiskey that ever
brought a skeleton into the closet, or painted scenes of lust
and bloodshed in the brain of man. It is the ghosts of wheat
and corn, crazed by the loss of their natural bodies. In it
you will find a transient sunshine chased by a shadow as cold
as an Arctic midnight, in which the breath of June grows icy
and the carol of the lark gives place to the foreboding cry
of the raven. Drink it, and you shall have ‘woe,’ ‘sorrow,’
‘babbling,’ and ‘wounds without cause.’ Your eyes shall
behold strange women, and ‘your heart shall utter perverse
things.’ Drink it deep, and you shall hear the voices of demons
shrieking, women wailing, and worse than orphaned children
mourning the loss of a father who yet lives. Drink it deep and
long, and serpents will hiss in your ears, coil themselves
about your neck, and seize you with their fangs; for at the
last it biteth like a serpent and stingeth like an adder. For
forty years this liquid death has been within staves of oak,
harmless there as purest water. I send it to you that you may
put an enemy in your mouth to steal away your brains, and yet I
call myself your friend.”
[Sidenote: The languages.]
There comes a stage of development of the learner at which the word
itself becomes the object of thought. Words are then classified as
parts of speech, and their function in sentences is studied. Their
properties and endings must be learned and compared. There is abundant
room for thought in the eleven hundred variations of the Greek verb.
The variations of words by declension and conjugation can be made the
material for thought, and as these are always at hand in the text-book,
no excursions to the field being needed to secure specimens, and no
preparation of difficult experiments being required on the part of
the teacher, the ancient languages have held their own in the schools
with most wonderful tenacity. The study of language has not merely the
advantage of supplying material for thought in the words, grammatical
forms, and sentences which are always at hand in the text, but through
the classics it brings the learner into intellectual contact with the
best thoughts of the best men in ancient and modern times. To translate
an author like Virgil or Demosthenes is to think the thoughts of a
master mind, to weigh words as in a most nicely adjusted balance, and
finally to arrange them in sentences that shall adequately convey the
meaning of the original text.
[Sidenote: Science.]
Science is, of course, a product of the human mind, quite as much as
the so-called humanities, and answers the same purpose when studied
as literature; but then it ceases to have the value of training the
intellect in the rigid methods of original research and scientific
investigation. Whilst it is the function of the laboratory to initiate
the student into the mysteries of the methods by which new discoveries
are made and verified, and thus to enable him to avail himself of the
labors of others through their publications, it does not bring the
student into living contact with human hopes, emotions, and aspirations
as do the poems of Goethe, Schiller, and Shakespeare.
[Sidenote: History.]