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from dollars; therefore we can divide dollars only by dollars. Thus :

Divide $42 equally among 6 men. Now we cannot divide $42 by 6 men or by 6; but if we give each man a dollar, then that will require $6, and $6 can be subtracted from $42 seven times. Hence we can give each man a dollar seven times, or we can give $7 at one time.

After the operation is performed, we may call the 7, seven dollars; then the 6 will be a mere number, and thus, indirectly, we may divide $42 by 6.

Practically, however, all such operations are performed abstractly, as 42, 6, 7, taken as mere numbers.

The study and solution of examples and their discussion in the class involve the following emphasized. points:

Points to be

1. Correct reading. Not one pupil in twenty reads a new kind of problem correctly the first time.

2. Examination preparatory to solution. A celebrated mathematician said that if his life depended on solving a complicated problem within an hour, he would give the first thirty minutes to studying it before putting down a figure.

3. Analysis and solution.

4. Reviewing, to see that there are no errors.

5. General correction by the rule of Common Sense. A mistake in pointing off may make a barrel of flour. cost 70 cts. or $70, but the pupil's common sense should teach him that neither is possible.

Cautions. Keep in mind the following cautions.

1. Present single ideas, single facts and single difficulties.

2. Call up each point in the lesson frequently.

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MENTAL ARITHMETIC

3. Teach simple processes.

4. Keep the mind active.

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5. See that pupils get a clear perception of principles. 6. Fix and hold the attention.

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Mental arithmetic.

The analysis of a problem is the same in mental as in written arithmetic. The difference is that mental arithmetic is limited to problems that may be performed mentally, without recourse to written symbols. It is a fact that those pupils who have been trained carefully in mental arithmetic take up the principles of higher mathematics more readily. The language used should be sufficient to render the solution of the example clearly intelligible to a listener, yet so brief as not to retard the process of mental calculation. Mental arithmetic should both precede and accompany the written arithmetic, step by step. In fact it would be much the better way to select a text-book that contained exercises in both mental and written arithmetic. In mental arithmetic the language should be clear, and the words enunciated distinctly. No hesitancy should be permitted-pupils should pass through the solution rapidly. Pupils should be required to construct original problems, and random exercises should be given by the teacher in addition, subtraction, multiplication, and division, to teach rapidity and accuracy in computation.

The teacher should give diversified problems of a practical nature to the class.

There is a great deal of perfectly barren mathematical knowledge in this country, particularly among those who have studied, not for knowledge, but for a certificate or a diploma.

Practical application.

Not unfrequently do we meet teachers who can demonstrate problems in algebra and geometry, who at the same time cannot make the least application of them. Again, we have met teachers who have graduated at the higher institutions of learning, who have passed over the rules of arithmetic-finished the study-who would be unable to determine how many feet there are in a board 12 feet in length and 12 inches wide.

They seem to be unaware that the rules of arithmetic were ever intended for any practical purpose.

Knowledge, so confined and abstract, is of doubtful Theory and utility, even as a mental discipline. Theory practice. and practice should be united, or we perceive nothing of the beauties of mathematics. "Detached propositions and abstract mathematical principles give us no better idea of true and living science than detached words and abstract grammar would give us of poetry or rhetoric." Small acquirements in mathematics serve only to make us timid, cautious, and distrustful of our own powers-but a step or two further gives us life, confidence, and power.

Mental dis

Mathematics should not be studied merely for discipline. The object should be to understand cipline. the subject and make it useful. Those who teach with no other view than giving discipline to the minds of their pupils, never more than half teach.

Let a person undertake the study of any science with no other object than discipline and the science will come to him with difficulty. But let him begin the study. determined to understand it and avail himself of it, and the science will come to him with ease, and with it a discipline of mind, the most effective he can attain.

In the application of arithmetic there are two distinct operations, the logical and the mechan- Logic of

ical.

arithmetic.

In too many schools greater attention is given to the mechanical. To some extent this is quite necessary, and pupils should be made very familiar with elementary processes; but after they become expert in computation, greater attention should be given to calculation, -the thinking. The undisciplined direct their attention more to the doing than to the thinking, when it should be the reverse; and nearly all the efforts of the good teacher are directed to making his pupils reason correctly. If a person fails in an arithmetical problem, the failure is usually in the logic, for false logic directs to false reasoning, and true logic points out true operations.

II. FRACTIONS

Objects first.

It is well to introduce the study of fractions by objective teaching. For this purpose the best device is a series of equal spheres, of which one is whole, another is divided into halves, and the others into thirds, fourths, fifths, sixths, eighths, etc. These have been provided in what is known as Davis's Fractional Apparatus. A similar but less perfect device is a series of circles correspondingly subdivided.

Most

teachers will have to make use of apples or other objects obtained without expense. Whatever is used, the following definitions should all be made so clear that every pupil can illustrate them by the objects employed. The term UNITY in mathematical science is applied to any number or quantity regarded as a whole; the term UNIT in arithmetic, to any number that is used as the base of a collection.

Definitions.

Every number, whether integral or fractional, has the unit 1 for a primary base.

A quantity regarded as a whole, called a unit, is the primary base of every fraction.

One of equal parts of a unit called the fractional unit, is the secondary base of every fractional number.

The value of a fraction is the number of times it contains the unit 1.

The quantity or unit that is divided into equal parts is the unit of the fraction.

One of the equal parts is called a fractional unit.

In of a pound, 1 pound is the unit of the fraction, and of the pound the fractional unit.

A fractional unit or a collection of fractional units is a fraction. (Or a fraction may be considered one or more of the equal parts of a unit, these parts corresponding to fractional units.)

Two integers are required to express a fraction, one above a short horizontal line to denote the number of fractional units, called the numerator; it numbers, or expresses how many are taken. The other below the line, expresses how many fractional units it is divided into, and is called the denominator; it denominates or

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