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Note. — Page references to supplementary problems will be found at appropriate places
NOTE. — The following brief statements concerning the elementary branches of mathematics, and the place of arithmetic in mathematical science, are designed for the teacher rather than for the pupil. After the completion of the book, these statements should be read aloud in class and the pupil encouraged to comment upon them and to discuss their meaning. After this has been done, the pupils may be required to commit to memory such parts as seem most important.
1. Mathematics treats of the measurement and comparison of magnitudes.
Geometry, arithmetic, and algebra are the elementary branches of mathematics.
2. Geometry treats of the measurement and comparison of certain magnitudes known as lines, angles, surfaces, and solids.
Measurement makes number necessary.
Primarily, number signifies ratio. The number six suggests that some magnitude, A, is six times some other magnitude, B; and, conversely, that the magnitude, B, is one sixth of the magnitude, A.
Number is ratio.
Secondarily, number signifies aggregation. The number six suggests the aggregation of six minor magnitudes into one group, or major magnitude.
NOTE. It is the secondary aspect of number that the child first apprehends. Thus, that which is logically secondary is pedagogically primary. To the child, the number six suggests a group of six (marbles, apples, inches). It is undoubtedly true that the mature mind ordinarily conceives of number in its aggregative aspect. It is the mathematician only who habitually thinks of number as ratio.
3. Arithmetic and Algebra treat of measured magnitudes and their relation hence of number.
Every arithmetical or algebraic problem is one of the following two simple problems, or may be resolved into such as these :
I. Given, two measured magnitudes, to find their relation.
II. Given, one measured magnitude and its relation to another magnitude, to find the measurement of the second magnitude.
Relation may be (1) by difference or (2) by quotient.
4. Every branch of mathematics is concerned with number and its expression; but it is the special province of arithmetic (1) to present the common method of numerical notation, (2) to investigate the properties and laws of number, and (3) to teach the art of computation in the simple processes of reduction, addition, subtraction, multiplication, and division.
NOTE. — For the double purpose of giving practice in simple computations and of preparing pupils for the ordinary number work of life, applications of these processes to certain classes of business transactions are presented in all arithmetical text-books.
5. It is the province of Algebra, by means of literal and figure notation combined, (1) to abridge and (2) to generalize the various processes of computation.
6. A unit is one.
7. A unit of measurement is a standard of measurement; as, a foot, a quart, an hour, a dollar, etc.
NOTE. — Counting is measuring, though sometimes inexact, owing to lack of uniformity in the magnitudes counted.
8. A unit of number is one of number.
NOTE 1. A unit of number is, in a sense, a unit of measurement. It is a standard for measuring number.
NOTE 2. - The unit of number may be simply, one or one ten, or one hundred, or one tenth, or one thousandth ; thus, we may have units of the first order, units of the second order, units of the first decimal order, etc. We may have, too, fractional units other than decimal ; as, one fourth, one fifth, one twenty-first, etc. A unit is a one, or any
group regarded for the time as one. Any ur
however great, may become a part of a major unit, and any unit, owever small, may be regarded as made up of minor units.
NOTE 3. - It is sometimes said that a unit is a standard by which we count or measure, and that a number is a unit or a collection of units. If this be true, six boys, six marbles, six books, etc., are numbers !
9. A number that is joined to the name of a standard of measurement is said to be a concrete numúer.
NOTE. - Neither four feet, six hours, nor eight apples is a concrete number; the four, six, and eight of these expressions are concrete numbers. If eight apples is a concrete number, then a red apple is a concrete color!
10. A number that is not joined to the name of a standard of measurement, as four, five, eight, ten, is said to be an abstract number.
NOTE 1. — An abstract number may suggest simply a ratio, or it may suggest both a ratio and a measured magnitude. In the expression, 8 feet = 2 feet = 4, the four suggests pure number, i.e., simply a ratio — the ratio of the magnitude 8 feet to the magnitude 2 feet. The expression 6 + 5 = 11, may suggest that 6 inches increased by 5 inches is equal to 11 inches. One may also think the 6 or the 5 or the 11 as the ratio of a certain magnitude to the standard of measurement.
NOTE 2. — Strictly speaking, number is ratio and, from its very nature, is always abstract. But the expressions, “concrete number," “denominate number," etc., have been accorded a place in the language. They are convenient terms to express certain uses of abstract number and hence are freely employed on the pages of this book.
ORDER OF PROCEDURE IN TRAINING FOR ARITH
STEP I. Since arithmetic is concerned mainly with the comparison of measured magnitudes and their numerical expression, it is of prime importance that the work should begin in the actual comparison and measurement of sentient objects.
STEP II. Since by far the greater part of the actual work in arithmetic must be done without the presence of the sense magnitudes compared, it is equally important that the pupil should early learn to image measured magnitude and to compare the images of measured magnitudes.
NOTE. — The imaging of magnitude, as the words are here used, means the mental reproduction of that which has been in the mind during an act of sense perception with special attention to quantity.
Since most of the magnitudes compared by the mathematician never have been to him objects of sense perception, and some of them never can be, it is no less important that the pupil should be constantly trained in the creation of imaginary measured magnitudes.
Note 1. – A most serious defect in ordinary arithmetical training is, that pupils image figures instead of magnitudes — that they are expected to see relation when the magnitudes compared are not present in consciousness.
NOTE 2. — The multiplication of sense-experiences in the comparison of magnitudes will not alone give the desired power. To learn to work with ideal magnitudes, one must work with ideal magnitudes. “Object teaching” must be vigorously begun and as vigorously laid aside.
NOTE 3. In the lower grades, great emphasis must be put upon the actual measurement of sense-magnitudes. In the higher grades, the chief part of the work will be in the comparison of imaginary magnitudes. But there is no grade from which Step I. can be entirely omitted, and no grade (in which formal number work should be taught) where Step I. should occupy as much time as Step II., or where Step II. should occupy as much time as Step III.