FORM, POSITION, AND SIZE.
_Method of instruction, geometrical song--Anecdotes--Size--Song
measure--Observations_.
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"Geometry is eminently serviceable to improve and strengthen the
intellectual faculties."--_Jones_.
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Among the novel features of the Infant School System, that of
geometrical lessons is the most peculiar. How it happened that a mode
of instruction so evidently calculated for the infant mind was so long
overlooked, I cannot imagine; and it is still more surprising that,
having been once thought of, there should be any doubt as to its
utility. Certain it is that the various forms of bodies is one of the
first items of natural education, and we cannot err when treading in
the steps of Nature. It is undeniable that geometrical knowledge is of
great service in many of the mechanic arts, and, therefore, proper
to be taught children who are likely to be employed in some of those
arts; but, independently of this, we cannot adopt a better method of
exciting and strengthening their powers of observation. I have seen a
thousand instances, moreover, in the conduct of the children, which
have assured me, that it is a very pleasing as well as useful branch
of instruction. The children, being taught the first elements of form,
and the terms used to express the various figures of bodies, find in
its application to objects around them an inexhaustible source of
amusement. Streets, houses, rooms, fields, ponds, plates, dishes,
tables; in short, every thing they see calls for observation, and
affords an opportunity for the application of their geometrical
knowledge. Let it not, then, be said that it is beyond their capacity,
for it is the simplest and most comprehensible to them of all
knowledge;--let it not be said that it is useless, since its
application to the useful arts is great and indisputable; nor is it to
be asserted that it is unpleasing to them, since it has been shewn to
add greatly to their happiness.
It is essential in this, as in every other branch of education,
to begin with the first principles, and proceed _slowly_ to their
application, and the complicated forms arising therefrom. The next
thing is to promote that application of which we have before spoken,
to the various objects around them. It is this, and this alone, which
forms the distinction between a school lesson and practical knowledge;
and so far will the children be found from being averse from this
exertion, that it makes the acquirement of knowledge a pleasure
instead of a task. With these prefatory remarks I shall introduce
a description of the method I have pursued, and a few examples of
geometrical lessons.
We will suppose that the whole of the children are seated in the
gallery, and that the teacher (provided with a brass instrument formed
for the purpose, which is merely a series of joints like those to a
counting-house candlestick, from which I borrowed the idea,[A] and
which may be altered as required, in a moment,) points to a straight
line, asking, What is this? A. A straight line. Q. Why did you not
call it a crooked line? A. Because it is not crooked, but straight. Q.
What are these? A. Curved lines. Q. What do curved lines mean? A. When
they are bent or crooked. Q. What are these? A. Parallel straight
lines. Q. What does parallel mean? A. Parallel means when they are
equally distant from each other in every part. Q. If any of you
children were reading a book. that gave an account of some town which
had twelve streets, and it is said that the streets were parallel,
would you understand what it meant? A. Yes; it would mean that the
streets were all the same way, side by side, like the lines which we
now see. Q. What are those? A. Diverging or converging straight lines.
Q. What is the difference between diverging and converging lines and
parallel lines? A. Diverging or converging lines are not at an equal
distance from each other, in every part, but parallel lines are. Q.
What does diverge mean? A. Diverge means when they go from each other,
and they diverge at one end and converge at the other.[B] Q. What does
converge mean? A. Converge means when they come towards each other.
Q. Suppose the lines were longer, what would be the consequence? A.
Please, sir, if they were longer, they would meet together at the end
they converge. Q. What would they form by meeting together? A. By
meeting together they would form an angle. Q. What kind of an angle?
A. An acute angle? Q. Would they form an angle at the other end? A.
No; they would go further from each other. Q. What is this? A. A
perpendicular line. Q. What does perpendicular mean? A. A line up
straight, like the stem of some trees. Q. If you look, you will see
that one end of the line comes on the middle of another line; what
does it form? A. The one which we now see forms two right angles. Q.
I will make a straight line, and one end of it shall lean on another
straight line, but instead of being upright like the perpendicular
line, you see that it is sloping. What does it form? A. One side of
it is an acute angle, and the other side is an obtuse angle. Q. Which
side is the obtuse angle? A. That which is the most open. Q. And which
is the acute angle? A. That which is the least open. Q. What does
acute mean? A. When the angle is sharp. Q. What does obtuse mean? A.
When the angle is less sharp than the right angle. Q. If I were to
call any one of you an acute child, would you know what I meant? A.
Yes, sir; one that looks out sharp, and tries to think, and pays
attention to what is said to him; and then you would say he was an
acute child.
[Footnote b: Mr. Chambers has been good enough to call the instrument
referred to, a gonograph; to that name I have no objection.]
[Footnote B: Desire the children to hold up two fingers, keeping them
apart, and they will perceive they diverge at top and converge at
bottom.]
_Equi-lateral Triangle_.
Q. What is this? A. An equi-lateral triangle. Q. Why is it called
equi-lateral? A. Because its sides are all equal. Q. How many sides
has it? A. Three sides. Q. How many angles has it? A. Three angles.
Q. What do you mean by angles? A. The space between two right lines,
drawn gradually nearer to each other, till they meet in a point.
Q. And what do you call the point where the two lines meet? A. The
angular point. Q. Tell me why you call it a tri-angle. A. We call it a
tri-angle because it has three angles. Q. What do you mean by equal?
A. When the three sides are of the same length. Q. Have you any thing
else to observe upon this? A. Yes, all its angles are acute.
_Isoceles Triangle_.
Q. What is this? A. An acute-angled isoceles triangle. Q. What does
acute mean? A. When the angles are sharp. Q. Why is it called an
isoceles triangle? A. Because only two of its sides are equal. Q. How
many sides has it? A. Three, the same as the other. Q. Are there any
other kind of isoceles triangles? A. Yes, there are right-angled and
obtuse-angled.
[Here the other triangles are to be shewn, and the master must explain
to the children the meaning of right-angled and obtuse-angled.]
_Scalene Triangle_.
Q. What is this? A. An acute-angled scalene triangle. Q. Why is it
called an acute-angled scalene triangle? A. Because all its angles are
acute, and its sides are not equal. Q. Why is it called scalene? A.
Because it has all its sides _unequal_. Q. Are there any other kind of
scalene triangles? A. Yes, there is a right-angled scalene triangle,
which has one right angle. Q. What else? A. An obtuse-angled scalene
triangle, which has one obtuse angle. Q. Can an acute triangle be an
equi-lateral triangle? A. Yes, it may be equilateral, isoceles, or
scalene. Q. Can a right-angled triangle, or an obtuse-angled triangle,
be an equilateral? A. No; it must be either an isoceles or a scalene
triangle.
_Square_.
Q. What is this? A. A square. Q. Why is it called a square? A. Because
all its angles are right angles, and its sides are equal. Q. How many
angles has it? A. Four angles. Q. What would it make if we draw a
line from one angle to the opposite one? A. Two right-angled isoceles
triangles. Q. What would you call the line that we drew from one angle
to the other? A. A diagonal. Q. Suppose we draw another line from the
other two angles. A. Then it would make four triangles.
_Pent-agon_.
Q. What is this? A. A regular pentagon. Q. Why is it called a
pentagon? A. Because it has five sides and five angles. Q. Why is it
called regular? A. Because its sides and angles are equal. Q. What
does pentagon mean? A. A five-sided figure. Q. Are there any other
kinds of pentagons? A. Yes, irregular pentagons? Q. What does
irregular mean? A. When the sides and angles are not equal.
_Hex-agon_.
Q. What is this? A. A hexagon. Q. Why is it called a hexagon? A.
Because it has six sides and six angles. Q. What does hexagon mean? A.
A six-sided figure. Q. Are there more than one sort of hexagons? A.
Yes, there are regular and irregular. Q. What is a regular hexagon?
A. When the sides and angles are all equal. Q. What is an irregular
hexagon? A. When the sides and angles are not equal.
_Hept-agon_.
Q. What is this? A. A regular heptagon. Q. Why is it called a
heptagon? A. Because it has seven sides and seven angles. Q. Why is it
called a regular heptagon? A. Because its sides and angles are equal.
Q. What does a heptagon mean? A. A seven-sided figure. Q. What is
an irregular heptagon? A. A seven-sided figure, whose sides are not
equal.
_Oct-agon_.
Q. What is this? A. A regular octagon. Q. Why is it called a regular
octagon? A. Because it has eight sides and eight angles, and they are
all equal. Q. What does an octagon mean? A. An eight-sided figure. Q.
What is an irregular octagon? A. An eight-sided figure, whose sides
and angles are not all equal. Q. What does an octave mean? A. Eight
notes in music.
_Non-agon_.
Q. What is this? A. A nonagon. Q. Why is it called a nonagon? A.
Because it has nine sides and nine angles. Q. What does a nonagon
mean? A. A nine-sided figure. Q. What is an irregular nonagon? A. A
nine-sided figure whose sides and angles are not equal.
_Dec-agon_.
Q. What is this? A. A regular decagon. Q. What does a decagon mean? A.
A ten-sided figure. Q. Why is it called a decagon? A. Because it has
ten sides and ten angles, and there are both regular and irregular
decagons.
_Rect-angle or Oblong_.
Q. What is this? A. A rectangle or oblong. Q. How many sides and
angles has it? A. Four, the same as a square. Q. What is the
difference between a rectangle and a square? A. A rectangle has two
long sides, and the other two are much shorter, but a square has its
sides equal.
_Rhomb_.
Q. What is this? A. A rhomb. Q. What is the difference between a rhomb
and a rectangle? A. The sides of the rhomb are equal, but the sides of
the rectangle are not all equal. Q. Is there any other difference? A.
Yes, the angles of the rectangle are equal, but the rhomb has only its
opposite angles equal.
_Rhomboid_.
Q. What is this? A. A rhomboid. Q. What is the difference between a
rhomb and a rhomboid? A. The sides of the rhomboid are not equal, nor
yet its angles, but the sides of the rhomb are equal.
_Trapezoid_.
Q. What is this. A. A trapezoid. Q. How many sides has it? A. Four
sides and four angles, it has only two of its angles equal, which are
opposite to each other.
_Tetragon_.
Q. What do we call these figures that have four sides. A. Tetragons,
_tetra_ meaning four. Q. Are they called by another name? A. Yes,
they are called quadrilaterals, or quadrangles. Q. How many regular
tetragons are among those we have mentioned? A. One, that is the
square, all the others are irregular tetragons, because their sides
and angles are not all equal. Q. By what name would you call the whole
of the figures on this board? A. Polygons; those that have their sides
and angles equal we would call regular polygons. Q. What would you
call those angles whose sides were not equal? A. Irregular polygons,
and the smallest number of sides a polygon can have is three, and the
number of corners are always equal to the number of sides.
_Ellipse or Oval_.
Q. What is this? A. An ellipse or an oval. Q. What shape is the top or
crown of my bat? A. Circular. Q. What shape is that part which comes
on my forehead and the back part of my head? A. Oval.
The other polygons are taught the children in rotation, in the same
simple manner, all tending to please and edify them.
The following is sung:--
Horizontal, perpendicular,
Horizontal, perpendicular,
Parallel, parallel,
Parallel, lines,
Diverging, converging, diverging lines,
Diverging, converging, diverging lines.
Spreading wider, or expansion,
Drawing nearer, or contraction,
Falling, rising,
Slanting, crossing,
Convex, concave, curved lines,
Convex, concave, curved lines.
Here's a wave line, there's an angle,
Here's a wave line, there's an angle;
An ellipsis,
Or an oval,
A semicircle half way round,
Then a circle wheeling round.
Some amusing circumstances have occured from the knowledge of form
thus acquired.
"D'ye ken, Mr. Wilderspin," said a child at Glasgow one day, "that we
have an oblong table: it's made o' deal; four sides, four corners, twa
lang sides, and twa short anes; corners mean angles, and angles mean
corners. My brother ga'ed himsel sic a clink o' the eye against ane at
hame; but ye ken there was nane that could tell the shape o' the thing
that did it!"
A little boy was watching his mother making pan-cakes and wishing
they were all done; when, after various observations as to their
comparative goodness with and without sugar, he exclaimed, "I wonder
which are best, _elliptical_ pan-cakes or _circular_ ones!" As this
was Greek to the mother she turned round with "What d'ye say?" When
the child repeated the observation. "Bless the child!" said the
astonished parent, "what odd things ye are always saying; what can you
mean by liptical pancakes? Why, you little fool, don't you know they
are made of flour and eggs, and did you not see me put the milk into
the large pan and stir all up together?" "Yes," said the little
fellow, "I know what they are made of, and I know what bread is
made of, but that is'nt the shape; indeed, indeed, mother, they are
_elliptical pan-cakes_, because they are made in an _elliptical
frying-pan_." An old soldier who lodged in the house, was now called
down by the mother, and he decided that the child was right, and far
from being what, in her surprize and alarm, she took him to be.
On another occasion a little girl had been taken to market by her
mother, where she was struck by the sight of the carcasses of six
sheep recently killed, and said, "Mother, what are these?" The reply
was, "Dead sheep, dead sheep, don't bother." "They are suspended,
perpendicular, and parallels," rejoined the child. "What? What?" was
then the question. "Why, mother," was the child's answer, "don't you
see they hang up, that's suspended; they are straight up, that's
perpendicular; and they are at equal distances, that's parallel."
On another occasion a child came crying to school, at having been
beaten for contradicting his father, and begged of me to go to his
father and explain; which I did. The man received me kindly, and told
me that he had beaten the child for insisting that the table which
he pointed out was not _round_, which he repeated was against all
evidence of the senses; that the child told him that if it was round,
nothing would stand upon it, which so enraged him, that he thrashed
him, as he deserved, and sent him off to school, adding, to be thus
contradicted by a child so young, was too bad. The poor little fellow
stood between us looking the picture of innocence combined with
oppression, which his countenance fully developed, but said not a
word. Under the said table there happened to be a ball left by a
younger child. I took it up and kindly asked the man the shape of it?
he instantly replied, "_Round_." "Then," said I, "is that table the
same shape as the ball?" The man thought for a minute, and then said,
"It is _round-flat_." I then explained the difference to him between
the one and the other, more accurately, of course, than the infant
could; and told him, as he himself saw a distinction, it was evident
they were not both alike, and told him that the table was circular.
"Ah!" said be, "that is just what the little one said! but I did not
understand what circular meant; but now I see he is right." The
little fellow was so pleased, that he ran to his father directly with
delight. The other could not resist the parental impulse, but seized
the boy and kissed him heartily.
The idea of _size_ is necessary to a correct apprehension of objects.
To talk of yards, feet, or inches, to a child, unless they are shown,
is just as intelligible as miles, leagues, or degrees. Let there then
be two five-feet rods, a black foot and a white foot alternately, the
bottom foot marked in inches, and let there be a horizontal piece to
slide up and down to make various heights. Thus, when the height of a
lion, or elephant, &c. &c., is mentioned, it may be shown by the rod;
while the girth may be exhibited by a piece of _cord_, which should
always be ready. Long measure is taught as follows:
Take barley-corns of mod'rate length,
And three you'll find will make an inch;
Twelve inches make a foot;--if strength
Permit; I'll leap it and not flinch.
Three feet's a yard, as understood
By those possess'd with sense and soul;
Five feet and half will make a rood,
And also make a perch or pole.
Oh how pretty, wond'rously pretty,
Every rule
We learn at school
Is wondrously pretty.
Forty such poles a furlong make,
And eight such furlongs make a mile,
O'er hedge, or ditch, or seas, or lake;
O'er railing, fence, or gate, or stile.
Three miles a league, by sea or land,
And twenty leagues are one degree;
Just four times ninety degrees a band
Will make to girt the earth and sea.
Oh how pretty, &c.
But what's the girth of hell or heaven?
(No natural thought or eye can see,)
To neither girth or length is given;
'Tis without space--Immensity.
Still shall the good and truly wise,
The seat of heaven with safety find;
Because 'tis seen with inward eyes,
The first resides within their mind.
Oh how pretty, &c.
Whatever can be shewn by the rod should be, and I entreat teachers not
to neglect this part of their duty. If the tables be merely learnt,
the children will be no wiser than before.
Another anecdote may be added here, to shew that children even under
punishment may think of their position with advantage. Doctor J., of
Manchester, sent two of his children to an infant school, for the
upper classes, and one of his little daughters had broken some rule in
conjunction with two other little ladies in the same school; two of
the little folks were placed, one in each corner of the room, and Miss
J. was placed in the centre, when the child came home in the evening,
Doctor J. enquired, "Well, Mary, how have you got on at school to
day?" the reply was "Oh, papa, little Miss ---- and Fanny ----, and I,
were put out, they were put in the corners and I in the middle of the
room, and there we all stood, papa, a complete _triangle_ of dunces."
The worthy doctor took great pleasure in mentioning this anecdote
in company, as shewing the effect of a judicious cultivation of the
thinking faculties.
In my peregrinations by sea and land, with infants, we have had some
odd and amusing scenes. I sometimes have had infants at sea for
several days and nights to the great amusement of the sailors: I have
seen some of these fine fellows at times in fits of laughter at the
odd words, as they called them, which the children used; at other
times I have seen some of them in tears, at the want of knowledge,
they saw in themselves; and when they heard the infants sing on deck,
and explain the odd words by things in the ship, the sailors were
delighted to have the youngsters in their berths, and no nurse could
take better care of them than these noble fellows did.
I could relate anecdote after anecdote to prove the utility of this
part of our system, but as it is now more generally in the training
juvenile schools, and becoming better known, it may not be necessary,
especially as the prejudice against it is giving way, and the public
mind is better informed than it was on the subject, and moreover it
must be given more in detail in the larger work on Juvenile Training
or National Education.