School-room Methods.

ARITHMETICAL ANALYSIS.

D. S. WRIGHT, State Normal School, Cedar Falls.

A few years ago Arithmetical Analysis was every teachers' hobby. The pupils in the public schools were treated to a long course of oral analysis, first through the "Primary Arithmetic," then through the "Intellectual Arithmetic," before he was allowed, armed with slate and pencil, to begin the more pretentious work of ciphering through the higher book of the series. With patient pains-taking on the part of both teacher and pupil, all the problems of these text-books, page by page, were analyzed in accordance with the author's inflexible models, the process too often degenerating into a mere memoritor process, fruitless of good results. Finally from this long series of oral lessons the student graduated through a three page course of mental gymnastics in solving fox, frog and squirrel puzzles, into Written Arithmetic.

In the inevitable reaction that has followed, the teachers of this generation have gone to the opposite extreme. From most of our schools the Mental Arithmetic has disappeared; oral analyses are seldom or never required; and the so-called explanations given by the pupils in the written arithmetic class are a mere statement of the successive steps involved in the solution, as mechanical and formal as the solution itself. Only the how, not the why, is required or given.

Both extremes are to be avoided. The one makes parrots of the pupils; the other, ciphering machines. Middle ground is sate ground. I venture to make a plea for the re-establishment of the Mental Arithmetic Class, for a further trial of oral analysis both in mental and written work. Two principles are evident. The pupil should know the reason for every step in the process of a solution. He should be able to convey that knowledge to his teacher through the medium of speech. The language used should be the pupil's own. Models may be used as suggestive helps, but only sparingly employed for any purpose. Such an exercise properly conducted not only disciplines the reasoning faculty, but is in itself a language culture of the highest value.

I endeavor to present below what I shall call for want of a better the

name,

TWO-STEP ANALYSIS.

The complete analysis of any problem involving concrete numbers, consists of two distinct parts. 1. Reasoning from many to one. 2. Reasoning from one to many. For example, giyen the cost of seven pounds of meat to find the cost of five pounds, there must be found: 1. The cost of one pound. 2. The cost of five pounds.

Given the work accomplished by six men in a given time, to find the work accomplished by three men, there must be found: 1. The work done by one man. 2. The work done by three men.

I can best present the idea by means of a series of models, beginning with the simplest problems, and passing to the more complex. These models are not given as inflexible forms, but can be used as mere guides to the pupil in formulating his analysis

The reader will observe the substitutions of since for if. If, does not of necessity imply a reason; but if a pupil can be pursuaded to introduce his sentence with since, if he constructs a good English sentence, he will give a reason before he has done.

Ist step.

If 8 lbs. of tea cost

Since 8 lbs. of tea cost $4.80, 1 lb. of tea will cost

of $4.80 $.60.

2d step. 15 lbs. will cost $.60X 15-$9.00. Ans.

2. Prob. If 2 men do a piece of work in 18 da., in what time will 9 men do it?

Ist step.

Since 2 men do the work in 18 da., 1 man will do the work in 18 da. × 2=36 da.

2d step. 9 men will do the work in 36 da.÷9-4 da. Ans.

3. Fraction-Prob. $18 is 23 the cost of a suit of clothes; find the cost of the suit.

Ist step.

18=$9.

Since 2/3 of the cost $18, 3 of the cost is 1⁄2 of

2nd step. of the cost is $9x3=$27, Ans.

4. Prob.

A horse cost 23 times as much as a wagon,

both cost $300, find the cost of each.

and

of the cost of the wagon = the cost of the wagon

Ist step.

5.

Prob.

of the cost of the wagon =

wagon.

of $300-$30.

of the cost of the wagon = $30X7=$210.

horse.

A can do a piece of work in 3 days, and B in 5 days; in what time can both do it working together?

1st step.

Since A can do the work in 3 da., in 1 da. he can do 3 of it; since B can do the work in 5 da., in one da. he can do } and both will do in one day (} + ) or of the work.

2d step.

And to do the whole work will require as many days as is contained times in 1 or 1%, the number of days.

6. Measurement-Prob.

by 5 ft?

What is the area of a rectangle 7 ft.

1st step. A rectangle 1 ft. by. I ft. contains I sq. foot.

7. Prob. Find the volume of a solid, 8 ft. long, 5 ft. wide, and 3 ft. thick.

Ist step. 40% or

2d step.

The adaptation of percentage is simple.

II

of the number =12.

of the number =6.

of the number =30. Ans.

these models to the various applications of One model must suffice.

Prob. A man sold a watch for $80. and gained thereby 25%; find the cost of the watch.

The models for analysis of Problems in Interest in all standard text-books, conform closely to the two-step method.

12. Compound proportion-Prob. If 2 men in 3 da., working 8 hours per day, build 40 rds. of fence, how many rds. of fence will 7 men build in 12 da., working 10 hours per day?

1st step. Since 2 men in 3 da. working 8 hrs. per day build 40 rds

Like models might be given for every form of concrete arithmetical problems, but these are deemed sufficient for the purpose of this article-to hint to the intelligent teacher how his class-room work may be made more practical and efficient.

QUESTION BOX.

Ques. Will you please analyze the following sentences? "The north wind is full of courage." "The brightness and freedom of the New Learning seemed incarnate in the young and scholarly Sir Thomas More."

Ans. "The north wind is full of courage." A simple declarative sentence. "The north wind" is the logical subject. "Wind," grammatical subject, modified by "the" and "north," simple adjective elements. "Is" is the

copula. "Full of courage," the logical predicate. "Full" the grammatical predicate, modified by "of courage," a simple adverbial element of "the second class, being a prepositional phrase.

"The brightness and freedom of the New Learning seemed incarnate in the young and scholarly Sir Thomas More." Simple declarative sentence. "The brightness and freedom of the New Learning" is the logical subject. "Brightness and freedom," the compound, grammatical subject, modified by "the," and "of the New Learning," the former, a simple adjective element, the latter an adjective element of "the second class," consisting of a prepositional phrase, of which "Learning," the basis is modified by "New," an adjective element. "Seemed" is the copula. "Incarnate in the young and scholarly Sir Thomas More," is the logical predicate. “Incarnate" is the grammatical predicate, modified by "in the young and scholarly Sir Thomas More," an adjective element, consisting of the prepositional phrase, “in Sir Thomas More," of which the basis, "Sir, &c." is modified by "the," "young" and "scholarly," three simple adjective elements. "And" is a connective, the first one uniting the two subjects, the second, uniting the adjectives "young" and "scholarly."

Ques. Please define “high seas," giving authority. Books do not agree. Ans. The "high seas" are no nation's property. The term is used to designate those portions of large bodies of water, over which no nation has any control, or which can not become the territory or empire of any particular nation or state. The territory of a state or nation includes all the lands and inland waters within its boundaries, the mouths of rivers, bays and estuaries and the sea to the distance of a marine league along the coast. All portions of bodies of water not included in the territory of any state thus defined are called "high seas." This is the generally accepted definition of "high seas" by the best writers on international law. Authority: Wheaton, Halleck, Woolsey and Hall.

Ques. We say "Mr. President" in addressing the chairman or president of a society, who is a gentleman. What should we say in addressing a maiden lady chairman or president?

Ans. Say "Miss Chairman" or "Miss President."

Ques. What is the size and weight of the brain of some great person? Of an average brain and of that of an idiot?

Ans. The brains of Daniel Webster and Agassiz each weighed fifty-three and one-half ounces. An average brain weighs about three pounds. That of an idiot often does not weigh more than thirty ounces.

THE next meeting of the National Educational Association will be held in the city of Chicago next July. The location and accommodation of the city guarantee a grand meeting, both in point of numbers and interest. The Hawkeye state should turn out "one thousand teachers strong," to this meeting in July 1887.