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THE

PSYCHOLOGY OF NUMBER

AND ITS APPLICATIONS TO
METHODS OF TEACHING ARITHMETIC

BY

JAMES A. MCLELLAN, A. M., LL. D.
PRINCIPAL OF THE ONTARIO SCHOOL OF PEDAGOGY, TORONTO

AND
JOHN DEWEY, Ph. D.

HEAD PROFESSOR OF PHILOSOPHY IN THE UNIVERSITY OF CHICAGO

“The art of measuring brings the world into subjection to man ; the art
of writing prevents his knowledge from perishing along with himself: to-
gether they make man-what Nature has not made him--all-powerful and
eternal.”_MOMMSEN.

NEW YORK
D. APPLETON AND COMPANY

COPYRIGAT, 1895,
By D. APPLETON AND COMPANY.

ELECTROTYPED AND PRINTED
AT THE APPLETON PRESS, U. S. A.

hat. Bill 7-6-39 38966

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In presenting this book on the Psychology of Number it is believed that a special want is supplied. There is no subject taught in the elementary schools that taxes the teacher's resources as to methods and devices to a greater extent than arithmetic. There is no subject taught that is more dangerous to the pupil in the way of deadening his mind and arresting its development, if bad methods are used. The mechanical side of training must be joined to the intellectual in such a form as to prevent the fixing of the mind in thoughtless habits. While the mere processes become mechanical, the mind should by ever-deepening insight continually increase its power to grasp details in more extensive combinations.

Methods must be chosen and justified, if they can be justified at all, on psychological grounds. The concept of number will at first be grasped by the pupil imperfectly. He will see some phases of it and neglect others. Later on he will arrive at operations which demand a view of all that number implies. Each and every number is an implied ratio, but it does not express the ratio as simple number. The German language is fortunate in having terms that express the two aspects of numerical quantity. Anzahl expresses the multiplicity and Einheit the unity. Any number, say six, for example, has these two aspects : it is a manifold of units; the constituent unit whatever it is, is repeated six times. It is a unity of these, and as such may be a constituent unit of a larger number, five times six, for instance, wherein the five represents the multiplicity (Anzahl) and the six the constituent unity (Einheit).*

Number is one of the developments of quantity. Its multiplicity and unity correspond to the two more general aspects of quantity in general, namely, to discreteness and continuity.

There is such a thing as qualitative unity, or individuality. Quantitative unity, unlike individuality, is always divisible into constituent units. All quantity is a unity of units. It is composed of constituent units, and it is itself a constituent unit of a real or possible larger unity. Every pound contains within it ounces; every pound is a constituent unit of some hundredweight or ton.

The simple number implies both phases, the multiplicity and the unity, but does not express them adequately. The child's thought likewise possesses the same inadequacy; it implies more than it explicitly states or holds in consciousness.

This twofoldness of number becomes explicit in multiplication and division, wherein one number is the unit and the other expresses the multiplicity—the times the unit is taken. Fractions form a more adequate expression of this ratio, and require a higher consciousness of the nature of quantity than simple numbers do. Hence the difficulty of teaching this subject in the ele

* Hegel, Logik, Bd. I, 1st Th., S. 225.

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