GEOMETRY.
REV. DR. HILL, the present President of Harvard College, in his articles
in Dr. Barnard's "Journal of Education," has set forth the importance of
Geometry in the earliest education, giving the Science of Form
precedence to that of numbers. Of course he does not mean that logical
demonstration is to form one of the exercises of little children! but
that observation of differences and resemblances of shape, and the
combination of forms, should be inwoven with the amusements of children.
He invented a toy on the principle of the Chinese tanagram, (published
by Hickling, Swan & Co., in Boston,) to further an exercise which begins
in the cradle with the examination of the hands and feet.
The blocks are the first materials. Take the cube and ask how many faces
it has; how many corners; and whether one face is larger than another or
equal; and finally, lead the child to describe a cube as a solid figure
with six equal sides, and eight corners. Then take a solid triangle from
the box and draw out by questions that it has five sides and six
corners, that three of its sides are equal, and two others equal; that
the three larger sides are four-sided, and the two smaller sides are
three-sided; and that the corners are sharper than those of a cube.
Make analogous use of all the blocks, and of the furniture of the room,
of the sphere and its parts, the cylinder, &c. Do not require the
definition-formulas at first, but content yourself with opening the
children's eyes to the facts which the formula afterwards shall
declare.
Paper-folding can be made subservient to another step, just short of
abstraction.
Give each one of a class a square piece of paper, and proceed thus: What
is the shape of this paper? How many sides has it? Which is the longest
side? How many corners has it? Have in hand, already cut, several acute
and obtuse angled triangles, and showing them, ask if the corners of the
square are like these corners? If they are as sharp as some of them; or
as blunt as some? Spreading out the triangle before them say, which is
the sharpest corner, and which the bluntest? and let the children
compare them with the corners of the square, by laying them upon the
square. They will see that the square corners are neither blunt nor
sharp, but as they will perhaps say, _straight_. Let them look round the
room, and on the furniture and window-sashes, find these several kinds
of corner. At least they can always find right angles in the furniture.
Then tell them there is another word for corners, namely, _angles_, that
a square corner is a _right_ angle, a sharp corner a sharp angle, and a
blunt corner a blunt angle.
If the teacher chooses she can go farther and tell them that _acute_ is
another word for sharp, and _obtuse_ another word for blunt; (or these
two _Latin words_ may be deferred till by and by, one new word _angle_
being enough to begin with.)
You can then say, "Now tell me how you describe a square, supposing
somebody should ask you that did not know;" and give them more or less
help to say: "A square is a figure with four equal sides and four
straight corners (or right angles)." To prove to them that it is
necessary to mention the right angles in describing a square, you can
make a rhombus, and show them its different shape with its acute and
obtuse angles. Having thus exhausted the description of a square, let
every one double up his square, and so get an oblong. Ask if this is a
square? What is it? How does it differ from a square? Are all four sides
different from each other? Which sides are alike? How are the corners
(or angles)? In what, then, is it like a square? In what does it
differ? Bring out from the child at last the description of an oblong,
as a four-sided figure with straight corners (or right angles), and its
opposite sides equal. Contrast it with some parallelogram which is not a
rectangle, and which you must have ready. Let them now fold their
oblongs again, and crease the folds; then ask them to unfold and say
what they have, and they will find four squares. Ask them if every
square can be folded so as to make two oblongs, and then if every oblong
can be so divided as to make two squares? If they say yes to this last
question, give them a shorter oblong, which you must have ready, and
having made them notice that it is an oblong, by asking them to tell
whether its opposite sides are equal, and its angles right angles, ask
them to fold it, and see if it will make two squares. They will see that
it will not. Then ask them if all oblongs are of the same shape; and
then if all squares are of the same shape?
The above foldings will be enough for a lesson, and if the children are
small it will be enough for two lessons.
Beginning the next time, ask them what is the difference between an
oblong and square? and if they have forgotten, do not tell them in
words, but give them square papers and let them learn it over again as
before, by their own observations. Then give them again square pieces of
paper, and ask them to join the opposite corners, and crease a fold
diagonally (but do not use the word diagonally). Then ask them what
shape they have got? They will reply, a three-sided figure. Ask them how
many corners or angles it has, and then tell them that, on account of
its being three-cornered, it is called a _triangle_. Now let them
compare the angles, and they will find that there is one straight corner
(right angle) and two sharp corners (acute angles). Ask them if the
sides are equal, and they will find that two sides are equal and the
other side longer. Set up the triangle on its base, so that the equal
sides may be in the attitude of the outstretched legs of a man; call
their attention to this by a question, and then say, on account of this
shape this triangle is called equal-legged, as well as right-angled--a
right-angled equal-legged triangle. By giving them examples to compare
it with, you can demonstrate to them that all right-angled triangles are
not equal-legged, and all equal-legged triangles are not right-angled.
Show them an equal-legged right-angled triangle, an equal-legged
acute-angled triangle, and an equal-legged obtuse-angled triangle, and
this discrimination will be obvious. The word _isosceles_ can be
introduced, if the teacher thinks best; but I keep off the Greek and
Latin terms as long as possible.
Now tell the children to put together the other two corners of their
triangles, laying the sharp corners on each other, and crossing the
fold; unfolding their papers they will find four right-angled
equal-legged triangles creased upon their square paper. Are all these of
the same shape, and of the same size? Now fold the unfolded square into
oblongs, and make a crease, and they will find, on unfolding again, that
they have six isosceles triangles, two of them being twice as large as
any one of the other four. Ask, are all these triangles of equal size?
Are all of them similar in shape? leading them to discriminate the use
in geometry of the words equal and similar. Can triangles be large and
small without altering the shape? Then similar and equal mean
differently? Are all squares similar? are all squares equal? are all
triangles equal? are all triangles similar? What is the difference
between a square and oblong? What is the difference between a square and
a triangle? What is the difference between a square and a rhombus? What
kind of corners has a rhombus? In what is a square like a rhombus? How
do you describe a triangle? What is the name of the triangles you have
learnt about? They will answer right-angled, equal-legged triangles.
Then give them each a hexagon, and ask them what kind of corners it has?
Whether any one is more blunt than another? Whether any side is greater
than another? How many sides has it? And then draw out from them that a
hexagon is a figure of six equal sides, with six obtuse angles, just
equal to each other in their obtuseness. Having done this, direct the
folding till they have divided the hexagon into six triangles, meeting
at the centre. Ask them if these are right-angled triangles, and if they
hesitate, give them a square to measure with. Then ask them if they are
equal-legged (isosceles) triangles. They may say yes, in which case
reply yes, and more than equal-legged, they are equal-_sided_. All three
sides are equal, and so they have a different name,--they are called
_equilateral_. Ask, what is the difference between equilateral and
isosceles, if you have given them these names, and help them, if
necessary, to the answer, "equilateral triangles have all the sides
equal, isosceles triangles have only two sides equal." Are equilateral
triangles all similar, that is, of the same shape? Are isosceles
triangles all similar? and if they hesitate or say yes, show two
isosceles triangles, one with the third side shorter, and one with it
longer than the other two sides.
Now give to each child a square, and tell them to fold it so as to make
two equal triangles; then to unfold it, and fold it into two equal
oblongs. Unfold it again, and there will be seen, beside the triangles,
two other figures, which are neither squares, oblongs, or triangles, but
a four-sided figure of which no two sides are equal, and only two sides
are parallel, with two right angles, one obtuse and one acute angle. Let
all this be brought out of the children by questions. As there is no
common name for this figure, name it trapezoid at once. Then let them
fold the paper to make two parallelograms at right angles with the first
two, and they will have two equal squares, and four equal isosceles
triangles, which are equal to the two squares. Now fold the paper into
two triangles, and you will have eight triangles meeting in the centre
by their vertices, all of which are right-angled and equal-legged. Ask
them if they are equal-sided? so as to keep them very clear of
confounding the isosceles with the equilateral, but use the English
terms as often as the Latin and Greek, for the vernacular keeps the mind
awake, while the foreign _technical_ puts it into a passiveness more or
less sleepy. Then give all the children octagons, and bring out from
them its description by sides and angles; and then fold it so as to make
eight isosceles triangles.
Another thing that can be taught by paper-folding is to divide polygons,
regular or irregular, into triangles, and thus let them learn that every
polygon contains as many triangles as it has sides, less two.
Proportions can also be taught by letting them cut off triangles,
similar in shape to the wholes, by creases parallel to the base. Grund's
"Plane Geometry" will help a teacher to lessons on proportion, and can
be almost wholly taught by this paper-folding. Also Professor Davies's
"Descriptive Geometry," and Hay's "Symmetrical Drawing."
Of course it will take a teacher who is familiar with geometry to do all
that may be done by this amusement, to habituate the mind to consider
and compare forms, and their relations to each other. Exercises on
folding circles can be added. It would take a volume to exhaust the
subject. Enough has been said to give an idea to a capable teacher. Care
must be taken that the consideration should be always of concrete not of
abstract forms. Mr. Hill says his "First Lessons in Geometry" were the
amusements of his son of five years old. Pascal and Professor Pierce
found out such amusements for themselves, which had the high end of
preparing them for their great attainments in logical geometry.
Sometimes surprising applications of Geometry, thus practically
appreciated, will be made by very small people. A boy of eight years of
age, with whom I read over Mr. Hill's "Geometry for Beginners" for his
amusement, in two months after invented a self-moving carriage for his
sister's dolly, that would give it a ride of ten feet! A neighboring
carpenter made it from his drafted model.