Part 6 (1/2)

Give each one of a cla.s.s 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 att.i.tude 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 pa.s.siveness 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 amus.e.m.e.nt, 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 amus.e.m.e.nts of his son of five years old. Pascal and Professor Pierce found out such amus.e.m.e.nts 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 amus.e.m.e.nt, 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.

CHAPTER X.

READING.

THIS art should be taught simultaneously with writing, or, more properly, printing; and I should certainly advise that it do not come till children are hard upon seven years old, if they have entered the Kindergarten at three. For it properly belongs to the second stage of education, after the Kindergarten exercises on the blocks, sticks, peas, &c., are entirely exhausted; and the children have become very expert in sewing, weaving, p.r.i.c.king, and drawing. They will then have received a certain cultivation of intellect which will make it possible to teach Reading on a philosophical method, which will make the acquisition a real cultivation of mind, instead of the distraction it now is to those whose vernacular is English, the _pot pourri_ of languages, and whose _orthography_ should be called _Kakography_, it is so lawless.

Though we repudiate phonography, so far as to deprecate its being applied to the English language, and reducing all our libraries to a dead language as it were, we are not insensible to the truth that _phonography_ is the true principle of writing; and this method of ours takes advantage of it to a certain extent, as we shall proceed to show: for if we p.r.o.nounce the vowel-characters as in the Italian language, and the letters c and g hard, it is a fact that the largest number of syllables in English will be found strictly phonographic. It was on this hint, given by a great philologist, that the ”First Nursery Reading Book” was written, which has no word in it that makes an exception to the letters so sounded. In my own Kindergarten, where I began to teach reading when I was yet ignorant of the necessity for the previous training of which I have just given account, I began to teach on this method, reading and writing at the same time; thus:--

All the children were set before the black-board, with their slates and pencils; and I said, ”What does the cat say?” The answer was immediately ready,--”mieaou.” Now this sound goes from the highest to the lowest of the Italian vowels, beginning with the consonant m.

I said, ”Now we will learn to print 'mieaou.' How does it begin?” I answered myself,--shutting my lips, and sounding m. They all imitated the sound, which, being a semi-vowel, was continuous.

I said, ”We will write m,” putting down three short perpendiculars, and joining them on top by a horizontal; and I made the letter myself, according to this direction, and they imitated with more or less success.

I then said, ”_mi_,” sounding the i as in mach_i_ne; and adding, ”Now we must write i,--and that is one little short perpendicular with a dot over it.” I did it, and they imitated.

Then I said ”mie,” sounding e as in egg, only making it long; ”and this e is made by a curve and straight line,”--at the same time making it on the black-board, which they imitated.

Then I said ”miea,” sounding a as in ah; and, as I made it on the black-board, I said, ”We will make a little egg; and over the egg we will make a dot, and that is a snake's head; and this is the body,” I continued, as I made the curve that completed the a. They imitated with indifferent success, but I did not criticise their scrawls.