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SUGGESTIONS TO TEACHERS.

1. The Mental Problems should be made a thorough drill in analysis; but, since the reasoning faculty is not trained by mere logical verbiage, the solution should be concise and simple. They should also be made introductory to the written processes of which they are often a complete elucidation. Many of the written problems may also be solved mentally, thus increasing the drills in analysis.

2. All Written Problems should be solved by the pupils on slate or paper, and the solutions should be brought to the recitation for the teacher's inspection and criticism. From three to five minutes at the beginning of the recitation will suffice to ascertain the accuracy and neatness of each pupil's work. The explanations of the written processes, given by the pupil, should be both analytic and inductive. The mental problems should also be solved as written problems. This will increase the number of written problems, and make the induction of the written process from the mental an easy step.

3. The Definitions should be deduced and stated by the pupils under the guidance of the teacher, and this can usually be done in connection with the solution of the problems. See Int. Arith., p. 5, Sug. 3. When the definitions are placed before the problems, as in the applications of Percentage, they should be studied by the pupils, but their recitation may be deferred until the problems are solved, and the processes mastered.

4. The Principles should be taught inductively, when this is possible, and each should be proven or illustrated, or both, by the pupil. A thorough mastery of every principle should be made an essential condition of the pupil's progress. The recitation should secure a constant application of known principles, and a clear comprehension of all new ones.

5. The Rules should also be deduced and stated by the pupils. The true order is this: 1. A mastery of the process. 2. Recognition of the successive steps in order, and a statement of each. 3. Combination of these several statements into a general statement. 4. Comparison of this generalization with the author's rule. 5. Memorizing of the rule approved. See Int. Arith., p. 6.

6. When two or more methods or solutions are given, the one preferred should be thoroughly taught. It is well for pupils to understand different processes and explanations, but they should be made familiar with one of them.

7. Before a subject is left, the pupils should be required to make a topical analysis of the definitions, principles, and rules, and the same should be recited with accuracy and dispatch.

N. B.-See the author's "Manual of Arithmetic" for other suggestions, methods of teaching, models of analysis, illustrative solutions, etc.

(vi)

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COMPLETE ARITHMETIC.

SECTION I.

PRELIMINARY DEFINITIONS.

Art. 1. Arithmetic is the science of numbers, and the art of numerical computation.

As a science, Arithmetic treats of the relations, properties, and principles of numbers; and, as an art, it applies the science of numbers to their computation.

2. A Unit is one thing of any kind.

3. A Number is a unit or a collection of units.

4. An Integer is a number composed of whole or integral units; as, 5, 12, 20. It is also called a Whole Number.

5. Numbers are either Concrete or Abstract.

A Concrete Number is applied to a particular thing or quantity; as, 4 stars, 6 hours.

When a concrete number expresses the denominate units of currency, weight, or measure, it is called a Denominate Number. (Art. 174.) An Abstract Number is not applied to a particular thing or quantity; as, 4, 6, 20.

A concrete number is composed of concrete units, and an abstract number of abstract units.

6. A Problem is a question proposed for solution.

7.

An Example is a problem used to illustrate a process or a principle.

8. A Rule is a general description of a process.

9. An Arithmetical Sign is a character denoting an operation to be performed upon numbers, or a relation between them.

10. In the Mental Solution of a problem, the successive steps are determined mentally, and the results are held in the mind.

In the Written Solution of a problem, the results are written on a slate, paper, or other substance.

SECTION II.

NOTATION AND NUMERATION.

MENTAL EXERCISES.

1. How many hundreds, tens, and units in 368? 427? 549? 608? 724? 806? 870?

2. How many hundred-thousands, ten-thousands, and thousands in 456048? 607803? 680435? 700450?

3. Read the thousands' period in 3045; 40607; 150482; 405360; 920400; 600060.

4. Read first the thousands' period and then the units' period in 65671; 120408; 400750; 650400; 80008.

5. Read 45037406; 520600480; 138405050.

6. Read 50008140; 600650508; 805000030.

7. Read 5308008450; 35006060600; 120500408080. 8. Read 7008360004; 302000860060; 500080800008.

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