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by f, the pound by 18, the dollar by 18 or by 188; where 12 refers to inches, 3 to feet, 16 to ounces, 10 to dimes, and 100 to cents. Familiarity with fractions thus defined by and connected with the ordinary scales of measurement means easy mastery of all forms of fractions as a mode of definite measurement.
3. As to the teaching of fractions, it will be enough, for the present, to note the following points :
1. In the formal treatment of fractions nothing new is involved; there is simply a conscious direction of attention to ideas and processes which, under right teaching, have been used from the first in the formation of numerical ideas, and which have been further developed in the fundamental arithmetical operations.
2. As in “integers” so in teaching fractions, the idea and process of measurement should be ever present. To begin the teaching of fractions with vague and undefined “units” obtained by breaking up equally undefined wholes--the apple, the orange, the piece of paper, the pie—may be justly termed an irrational procedure. Half a pie, e. g., is not a numeral expression at all, unless the pie is defined by weight or volume; the constituent factors of a fraction are not present; the unity of arithmetic is ignored; the process of fractions is assumed to be something different from that of number as measurement; it becomes a question—it actually has been questioned—whether a fraction is really a number; and all this in spite of the fact that from the beginning fractions are implicit in all operations; that from first to last the process of number as a psychical act is a process of fractions.
3. The primary step in the explicit teaching of frac
tions—that is, in making the habit of fractioning already formed an object of analytical attention—is to make perfectly definite the child's acquaintance with certain standard measures, their subdivisions and relations. In all fractions—because in all exact measurement—there must be a definite unit of measure. This implies two things : (a) The definition of a standard of reference (the “primary" unit) in terms of its own unit of measure; (b) the measurement of the given quantity by means of this “ derived” unit. If the foot is unit of measure, it is unmeaning in itself; it must be mastered, must be given significance by relating it to other units in the scale of length; it is 1 (yard) = 3 in one direction; or (taking the usual divisions of the scale) it is 1 (i. e., x 12) in the other direction, i. e., as measured in inches. The teaching of fractions, then, should be based on the ordinary standard scales of measurement; on the fundamental process of parting and whol. ing in measurement, and not upon the qualitative parts of an undefined unity.
4. Under proper teaching of number as measurement the pupil soon learns to identify instantly 4 inches,
foot, $ foot as expressions for the same measured quantity. He is led easily to the conscious recognition of the true meaning of fractions as a means of indicating the exact measurement of a quantity in terms of a measuring unit which is itself exactly measured.
5. Addition and subtraction of fractions involve the principle of ratio, multiplication and division the principle of proportion. In all cases the meaning of fractions as denoting definitely measured quantity should be made clear. For example, not fx4, but f foot x *;
not xf, but $1 x }, as indicating, e. g., the cost of yard of cloth at $1 a yard.
Since a fraction expresses a quantity in a form for comparison with other quantities of the same kind, the fundamental operations as applied in fractions carry out these comparisons. Addition is always (as in "whole numbers ") of homogeneous quantities—i. e., those measured in terms of some unit of length, surface, volume, time, etc.; so with subtraction. All the first examples should deal only with definite measures; after the principle is quite familiar, and only then, fractions having denominators not corresponding to any existing scale of measurement-e. g., 17, 49, 131-may be introduced for the sake of securing mechanical facility.
The same remark applies to multiplication and division of fractions-operations which involve no principles different from the corresponding operations with “whole numbers.” Multiplication of fractions is multiplication, and division is division ; they are not new processes under old names. They make explicit use of ratio (the comparison of quantities), which is implied in the operations with "integers,” by defining the measuring unit which defines a measured quantity. They put in shorthand, as it were, the complete psychical process of measurement, and thus make a severer demand on conscious attention. But if number has been from the first taught upon the psychological method, the pupil will be quite prepared to meet this demand. There will be nothing strange in reducing fractions to a common denominator, nor any mystery in a product less than the multiplicand, or in a quotient greater than the dividend; so far as the nature of the processes is con
cerned, $1 + $will be just as intelligible as $3 + $4. If, too, the nature and relation of times and measuring parts have become familiar, there will be no more mystery in 18 feet * f = 9 feet than in measuring halfway across a room 18 feet wide; the peculiar thing would be if taking a quantity only a part of a time did not give a smaller quantity.
So in division, when the mutual relation between times and parts is understood, the operation $$10, or $4 : 1t, is just as intelligible as $80 = $10, or as $80 = 10. To say that the quotient eight, the result of $4 = $10, is greater than the dividend ($4) is to talk nonsense ; is to compare incomparable things—is to confuse parts with times, quantity with number, matter with a psychical process.
ON PRIMARY NUMBER TEACHING.
The Number Instinct. We have seen that number is not something impressed upon the mind by external energies, or given in the mere perception of things, but is a product of the mind's action in the measurement of quantity—that is, in making a vague whole definite. Since this action is the fundamental psychical activity directed upon quantitative relations, the process of numbering should be attended with interest; that is, contrary to the commonly received opinion, the study of arithmetic should be as interesting to the learner as that of any other subject in the curriculum. The training of observation and perception in dealing with nature studies is said to be universally interesting. This is no doubt true, as there is a hunger of the senses--of sight, hearing, touch—which, when gratified by the presentation of sense materials, affords satisfaction to the self. But we may surely say with equal truth that the exercise of the higher energy which works upon these raw materials is attended with at least equal pleasure. The natural action of attention and judgment working upon the sense-facts must be accompanied with as deep and vivid an interest as the normal action of the observing powers through which the sense-facts are acquired.