CHAPTER XIII.

FRACTIONS.

AFTER the previous discussion on the nature of fractions (see especially Chapter VII) and their psychological relation with the fundamental operations, a brief reference to some of the points brought out is all that is needed as an introduction to the formal teaching of fractions.

Number depends upon measurement of quantity. This measurement begins with the use of inexact units —the counting of like things—and gives rise to addition and subtraction. From this first crude measurement is evolved the higher stage in which exactly defined units of measure are used, and in which multiplication, division, and fractions arise. Multiplication and division bring out more clearly the idea of number as measurement of quantity-as denoting, that is, (2) a unit of measure and (ii) times of its repetition. The fraction carries the development of the measuring idea a step further. As a mental process it constitutes a more definite measurement by consciously using a defined unit of measure; and as a notation, it gives complete expression to this more definite mental process. Fractions therefore employ more explicitly both the conceptions involved in multiplication and division-name

ly, analysis of a whole into exact units, and synthesis of these into a defined whole. The idea of fractions is present from the first, because division and multiplication are implied from the first. There is no number without measurement, nor measurement without fractions. Even in whole numbers, as has been pointed out, both "terms" of a fraction are implied in the accurate interpretation of the measured quantities.

Since there is nothing new in the process of fractions, so in the teaching of fractions there is nothing essentially different from the familiar operations with whole numbers. If the idea of number as measurement has been made the basis of method in primary work and in the fundamental operations, the fraction idea must have been constantly used, and there is absolutely no break when the pupil comes to the formal study of fractions. There is only before him the easy task of examining somewhat more attentively the nature of the processes he has long been using. The suggestions made in reference to primary teaching and formal instruction in the fundamental operations apply with equal force to the teaching of fractions. The measuring idea is to be kept prominent: avoidance of the fixed unit fallacy and its logical outgrowth, the use of the undefined qualitative unit-the pie and apple method-as the basis for developing "fractional" units of measure, and the "properties of fractions"; the essential property of the unit in measurement-the measured part of a measured whole; the logical and psychological relation between the number that defines the measuring unit and the number that defines the measured quantity; or as it is sometimes expressed, the

"relation between the size of the parts" (the measuring units) and the number of the parts composing or equalling the measured quantity; these and all kindred points that have been brought out in discussing number as measurement, and numerical operations as simply phases in the development of the measuring idea, can not be ignored in the teaching of fractions, because they can not be ignored in the teaching of whole numbers. Exact number demands definition of the unit of measure; the fraction completely satisfies this demand by stating or defining expressly the unit of measure. In all number as representing measured quantity the questions are: What is the unit of measure, and how is it defined or measured? How many units equal or constitute the quantity? These questions only number in its fractional form completely answers. It is the completion of the psychical process of number as measurement of quantity; the idea of the quantity is made definite, and it is definitely expressed.

While the following treatment of fractions is in strict line with the principles of number set forth in these pages, and has stood the test of actual experience, it is given only by way of suggestion. The principles are universal and necessary; devices for their effective application are within certain limits individual and contingent. Principles are determined by philosophy, devices by rational experience. The teacher must be loyal to principles, but the slave of no man's devices.

I. THE FUNCTION OF THE FRACTION.

1. In its primary conception a fraction may be considered as a number in which the unit of measure is

expressly defined. In the quantities 4 dimes, 5 inches, 9 ounces, the units of measure are not explicitly defined; their value is, however, implied, or else there is not a definite conception of the quantity. In $, foot,

pound, the units of measure are explicitly defined; and each of these expressions denotes four things: 1. The unity (or standard) of reference from which the actual unit of measure is derived. 2. How this unit is derived from the unity of reference. 3. The absolute number of these derived units in the quantity. 4. This number is the ratio of the given quantity to the unity of reference.

For example, in pound, the unity of reference is one pound; it is divided into sixteen equal parts, to give the direct measuring unit; the number of these units in the given quantity is nine; the ratio of the given quantity to the unity of reference is nine.

n

n

2. If properly taught, the pupil knows—if not, he must be made to know-that any quantity can be divided into 2, 3, 4, 5 ... n equal parts, and can be expressed in the forms,,, .... Familiar with the ideas of division and multiplication which become explicit in fractions, he learns in a few minutes. (has already learned, if he has been rationally instructed) that any quantity may be measured by 2 halves, or 3 thirds, or 4 fourths or n nths; that to take a half, a third, a fourth . . . an nth of any quantity, it is only necessary to divide by 2, or 3, or 4 . . . or n; that if, for example, 16 cents, or 16 feet, or 16 pounds has been divided into four parts, the counts of the units in each case are one, two, three, four, or one fourth, two fourths, three fourths, four fourths; that each of these units

fourths-is measured by other units, and can be expressed as integers, namely, 4 cents, 4 feet, 4 pounds, and so on, with kindred ideas and operations.

3. The Primary Practical Principle in Fractions. -It is clear that this complete expression for the number process is the fundamental principle employed in the treatment of fractions: if both terms of a fraction be multiplied or divided by the same number, the numerical value of the fraction will not be changed. This principle is usually "demonstrated"; it is, however, involved in the very conception of number, and seems as difficult to demonstrate as the definition of a triangle; but intuitions and illustrations to any extent may be given. Any 12-unit quantity, for example, is measured by, 4, 1, or by 3, 8, 1; the identity of the quantity remains unchanged in the changing measurements. Moreover, if half the quantity be measured, the identity of 1, 2, 3, is seen at once. The principle is, of course, that in a given measured quantity the "size" of the units varies inversely with their number. This principle is said to be beyond the comprehension of the pupil. On the contrary, if constructive exercises, such as have been described, have been practised, there comes in good time a complete recognition of the principle. When, for instance, the child measures off any 24-unit quantity by twos, threes, fours, sixes, eights, he can not help feeling the relation between the magnitude and the number of the measuring parts. This is, in fact, the process of number.

Proof of the Principle.-If the first vague awareness of the relation does not grow into a clear comprehension of it, clearly the method is at fault. In any