Froebel's Gifts

by Kate Douglas Smith Wiggin; Nora Archibald Smith

Edward Wiebe says in regard to the relation of the seventh gift to geometry and general mathematical instruction: "Who can doubt that the contemplation of these figures and the occupations with them must tend to facilitate the understanding of geometrical axioms in the future, and who can doubt that all mathematical instruction by means of Froebel's system must needs be facilitated and better results obtained? That such instruction will be rendered fruitful in practical life is a fact which will be obvious to all who simply glance at the sequence of figures even without a thorough explanation, for they contain demonstratively the larger number of those axioms in elementary geometry which relate to the conditions of the plane in regular figures."

As the tablets are used in the kindergarten, they are intended only "to increase the sum of general experience in regard to the qualities of things," but they may be made the medium of really advanced instruction in mathematics, such as would be suitable for a connecting-class or a primary school. All this training, too, may be given in the concrete, and so lay the foundation for future mathematical work on the rock of practical observation.

The kindergarten child is expected only to know the different kinds of triangles from each other, and to be familiar with their simple names, to recognize the standard angles, and to know practically that all right angles are equally large, obtuse angles greater, and acute less than right angles. All this he will learn by means of play with the tablets, by dictations and inventions, and by constant comparison and use of the various forms.

How and when Tablets should be introduced.

As to the introduction of the tablets, the square is first of all of course given to the child. A small cube of the third gift may be taken and surrounded on all its faces by square tablets, and then each one "peeled off," disclosing, as it were, the hidden solid. We may also mould cubes of clay and have the children slice off one of the square faces, as both processes show conclusively the relation the square plane bears to the cube whose faces are squares. If the first tablets introduced are of pasteboard, as probably will be the case, the new material should be noted and some idea given of the manufacture of paper.

There is a vast difference in opinion concerning the introduction of this seventh gift, and it is used by the child in the various kindergartens at all times, from the beginning of his ball plays up to his laying aside of the fifth gift. It seems very clear, however, that he should not use the square plane until after he has received some impression of the three dimensions as they are shown in solid bodies, and this Mr. Hailmann tells us he has no proper means of gaining, save through the fourth gift.[63]

As to the triangular tablets, it is evident enough they should not be dealt with until after the child has seen the triangular plane on the solid forms of the fifth gift. Mr. Hailmann says that a clear idea of the extension of solids in three dimensions can only come from a familiarity with the bricks, and again that the abstractions of the tablet should not be obtruded on the child's notice until he has that clear idea.

Though the six tablets which surround the cube may be given to the child at the first exercise, it is better to dictate simple positions of one or two squares first, and let him use the six in dictation and many more in invention.

Order of introducing Triangles.

The first triangle given is the right isosceles, showing the angle of forty-five degrees, and formed by bisecting the square with a diagonal line. The child should be given a square of paper and scissors and allowed to discover the new form for himself, letting him experiment until the desired triangle is obtained. He should then study the new form, its edges and angles, and then join his two right-angled triangles into a square, a larger triangle, etc. Then let him observe how many positions these triangles may assume by moving one round the other. He will find them acting according to the law of opposites already familiar to him, and if not comprehended,[64] yet furnishing him with an infallible criterion for his inventive work.