Thinking and learning to think

by Nathan Christ Schaeffer

Different gate-ways for different ideas.

Integers.

In seeking to build in the mind of the learner the concepts which lie at the basis of a new branch of study, it is a legitimate question to ask by which of the gate-ways of knowledge the materials or elements for the new idea can best be made to enter the mind. At the basis of arithmetic lies the idea of number,—an idea that is evoked by the question of how many applied to a collection of two or more units. Taste and smell must be ruled out from the list of senses which can be utilized to advantage. Three taps on the desk are as easily recognized as three marks or strokes on the black-board. The sense of touch is helpful in passing from concrete to abstract numbers. To think a number when the corresponding collection of objects is not visible, but is suggested by tactile impressions, helps to emancipate the thinking process from the domination of the eye; in other words, it helps to sunder the thinking of number from a specific sense, and thus aids in the evolution of the idea of number apart from concrete objects.

Fractions.

As already indicated, there are some basal concepts, like that of a fraction, in the development of which only one sense can be utilized to advantage. Whilst imparting the idea of a whole number, the appeal may be to the eye, the ear, and the sense of touch; the instruction designed to impart the idea of fractions to the normal child is limited to visible objects. In the instruction of the blind the other senses are addressed from necessity. The extent to which touch can supply the function of sight is full of hints to teachers in charge of pupils possessing all the gate-ways of knowledge.

Teaching decimals.

Moreover, not all units are equally adapted for imparting the first ideas 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.

Basal concepts.

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.