of Arithmetic

Wm. A. Wheatley

Superintendent, Fairfield, Conn.

Instructor in Arithmetic in the Summer Session of the Danbury State Normal School

I Accurate and rapid computation of abstract numbers This means a complete mastery of addition, subtraction, multiplication and division. First, accuracy in the fundamental operations; then, rapidity. While objects should be used freely to illustrate number facts and relations, and much practice should be given in measuring with foot-rule, yard-stick, and other common measures and weights, still the number facts themselves can be gained only by much abstract drill and repetition. Use constantly the blackboard, charts, flash-cards, hektographed sheets, a great variety of seat work and other means to make the number facts automatic.

After the fundamental operations can be performed accurately and rapidly, then common fractions, decimals and percents are to be mastered. Their computation likewise should be made so familiar as to become purely automatic. Both for rapid computation and the power to solve concrete problems, mental or intellectual, arithmetic, without the use of paper and pencil, is highly beneficial and should be regularly employed. Among others, three excellent books are Hopkins' and Underwood's "New Mental Arithmetic"; Brooks's "New Mental Arithmetic"; Williams' "Mental Drill Book in Arithmetic," Educational Publishing Company.

In written arithmetic, instead of using answer-books, have the pupils prove their own work. Better, by far, do five examples independently and prove their correctness, than ten examples with a dependent spirit and compare the results, during the work or even afterwards, with those found in the answer-book.

While teachers should conduct daily drills in computing abstract numbers, they should be careful that these drills are varied, spirited and very brief. In this connection, the following topics in psychology will be found serviceable; first impressions, mental associations, representation, memory, habit and fatigue.

I believe it would be a serious omission if I did not here

call attention to a novel and excellent drill book, arranged for practice in the fundamental operations, common and decimal fractions and denominate numbers, much as a copy-book is for writing, just written by Dr. Edward L. Thorndike, of the Teachers' College, Columbia University. There are in the series five of these exercise books, arranged to supplement the work of the regular text-book and to eliminate the eye-strain and expense of time and energy, due to copying numbers and solving examples that are written in figures not clearly formed or well spaced. These books furnish an abundance of well-graded material, they are paper-covered and inexpensive.

The fundamental operations, with computation of fractions, decimals and percents should become increasingly automatic each year until the close of the grammar school course, when the pupils should be at their very best in this work.

II Important arithmetic facts to be learned

In arithmetic we find many terms whose meaning, application and spelling should be learned. Examples: plus, minus, difference, denominator, ounce, rectangle, principal, discount, etc. Here an occasional spelling lesson on the important terms met in the arithmetic will be found useful.

Our pupils must learn tables of weights and measures, as linear, square, board, liquid and dry measures, avoirdupois weight, etc.; certain axioms, as multiplying or dividing both terms of a fraction by the same number does not

as base multiplied by rate equals the percentage.

The teacher should carefully select the essential arithmetic facts, make their meaning and application perfectly clear to the pupils and then insist on the facts being thoroughly learned. Here frequent drills with flash cards could be made effectual. În arithmetic, as in all other branches, the live teacher rearranges, re-organizes the subject-matter before she attempts to present it to her pupils. This means discarding some parts of the textbook, touching others very lightly and emphasizing and amplifying from outside sources other parts. The live teacher must be superior to any text-book and to the letter of any course of study.

III The power to interpret and solve concrete problems In interpreting simple number relations or complex problems, it is often most helpful to use objects, such as, nuts, apples, disks, etc.; the foot-rule, yard-stick, pint, quart and other measures, the pound and other weights, play money, etc.; and, in the upper grades, carefully drawn diagrams. In the Stockton, Cal., schools considerable attention is given to drawing to a scale plans of rooms, yards and gardens, and then finding the areas. They devote much time to this kind of work, even spending a whole week on such plans of their school yard.

In grades four to eight, pupils should have much practice in reading problems, stating what facts are given and what facts are to be found. Then have them state what operations are to be performed and why. In many cases, it is well to read and interpret problems without actually performing the operations. Much practice in reading problems is more imperative now than ever, owing to the increasingly large proportion of foreign pupils in our schools.

After reading the problem, go one step farther, but do not perform the operations; merely indicate the work to be done. Whether we would simply strengthen our pupils in arithmetic problems or would prepare them materially for higher mathematics and applied sciences, this practice of indicating the operations is of the greatest importance.

After interpreting a problem and indicating the operations, an excellent practice is to forecast the result, determining whether this will be larger or smaller than some fact given, and then approximating what this result will be. This practice will prevent much senseless figuring and will assist considerably in arriving at the correct

result.

When solving problems on paper or blackboard, see that the pupils' work is logically, systematically and neatly arranged, that dollars, feet and other denominations are properly labeled and that partial results and the final answer are plainly designated.

After the pupil has solved his problem, often require him to explain his work in a straightforward way, using the fewest and simplest words possible. The old-fashioned explanation, or analysis, of a problem was better than none, but too formal and too bookish to benefit much our modern pupils.

As was mentioned in another paragraph of this paper, mental arithmetic, without the use of paper and pencil, is highly beneficial in developing the power to interpret and solve problems.

Have pupils make up in class, or bring from home or their father's business, common-sense problems, containing concrete numerical data which they already know or can easily comprehend. In this way our pupils can secure at first hand practical problems from the farm, the grocery, the dry-goods store, the butcher-shop, the office. the bank, the factory, etc. To make up and then solve common-sense problems, gives a mastery to problem interpretation and solution that no book problems can supply.

Have some of the best problems made up by the pupils copied carefully in their note-books. In this way and otherwise, an occasional language lesson can be profitably based on the arithmetic.

Let us remember, also, that many good problems can be readily found in connection with other school studies, notably, elementary science, hygiene, manual arts, geography and history. Indeed, the supreme value of this power to interpret and solve problems in after life comes from the fact that every department of human knowledge and experience presents numerical problems that we must correctly solve or else suffer from our inability to solve them.

Again, those problems are the most educative in which the thought-content is continuous throughout a series. In other words, it is more profitable to solve a dozen problems either on wheat-growing, rainfall, plant food, iceharvesting, or on bean-bag games, as the case may be, before turning to another field for problems. Problems, whose thought-content is miscellaneous, may be occasionally profitable, but as a regular assignment they do not build up the power of problem interpretation and solution so well as do those problems, each group or series of which has the same general thought-content.

IV Familiarity with modern business forms and practices

In this work, the pupils should learn at first hand, so far as possible, the common business forms and practices followed in the banks, stores, shops, and factories of the neighborhood.

They should be taught to draw up correctly and readily itemized bills and accounts of goods and of labor, balances, receipts, promissory notes, checks, indorsements and the simplest contracts.

They should learn to keep personal and family accounts of income and expenditures and accounts with separate parts of a farm or separate branches of a simple business. They should also be encouraged and grounded, if possible, in the habit of keeping their own personal accounts.

They should become familiar with sending money by postal, express or telegraph order, by personal check and bank draft. They should have much practice in writing short letters, in which they subscribe for a magazine, order a bill of goods, send money to pay for goods or a receipt for money supposed to be sent them. Here, again, a language lesson may profitably be blended with one from the arith

metic.

From the fifth grade, to the close of the grammar school course, much practice should be given to this phase of arithmetic, the pupils at first filling in printed or hektographed forms furnished them and later bringing into class many original forms drawn up and filled in by themselves.

Report of Unassigned Work

D

School No. 6

Caroline Birdsall®

URING the school year 1912-1913, two hundred ninety-nine pupils have been given special help in the Unassigned Class in School No. 6, Passaic, N. J. These pupils have been sent from grades 2B to 6A inclusive and for any subjects in which they were particularly weak.

Of the two hundred ninety-nine pupils helped, two hundred sixty-two have been promoted, making twentyseven retarded, fifteen of whom passed in the subject or subjects in which they were helped.

The reasons for retardation are health, late entrance, undevelopment and inability to accomplish the grade work in one term.

Opportunity has been given to pupils who have shown. ability to advance more rapidly than the regular grade by giving them special help in the major subjects for a period a day, usually thirty minutes, until they could enter the higher grade. There have been thirty-seven such pupils, and all have completed the grade in which they were placed, thus doing three terms' work this school year.

In one class where there were two grades, 2A-3B, there have been nine 2A pupils who have been specially helped and completed both 2A and 3B work this June. This number includes several older boys, who, having the incentive have attended school more regularly and have been more attentive to their work.

While the efficiency of the pupils has been raised the percent of promotion has been steadily rising also, as is shown by comparing the year 1910-1911, the last year with no Unassigned Work, with the two successive years where this work has been added:

Transitive and Intransitive

Verbs

K. E. E.

Aim To teach the meaning of transitive and intransi

tive verbs.

Teacher

Let us see what the other contestants did. Pupil Emily remained where she was. Teacher What does that verb express?

Pupil "Was" expresses being.

(b) They had also previously learned that the prefix "in" means "not" and that the suffix "ive" means "that which."

(c) The class also knew the distinction between being verbs and action verbs.

Presentation In the grammar period, the teacher began. thus: "Boys and girls, to-day we are going to learn two classes of verbs, and the use of each. I have written a story for you on the blackboard. If you read it carefully, and see these racers in imagination, you will be able to ascertain their names and uses. Read it aloud.

A RACE

One day, the principal made this announcement: "This afternoon at 3:45, two boys and two girls will compete in a hundred yard dash. I will give each contestant a small flag and the one who plants it first at the end of the track will receive a prize. You are all invited to see the race." Accordingly, at the specified time, the competitors lined up at the starting time. At the signal, they darted off all except Emily, who was inattentive. On, on, they ran! Soon Dominic stopped. He was tired. The other two ran on, abreast. Neither one nor the other gained. Presently, Joseph spurted, and Katherine fell behind. At just twelve seconds after starting time, Joseph planted his flag at the goal.

Teacher What did Joseph do with his flag?

Pupil He carried it across from the starting place to the goal.

Teacher What did he have to do in order to win the race and plant his flag at the goal?

Pupil He had to run.

Teacher What does such a verb express?

Pupil "Run" is an action verb.

Teacher Explain why Katherine did not win. Pupil Katherine, too, had been running, but she fell behind and so could not carry her flag across to the goal. While these answers are given, the teacher makes two columns on the blackboard and writes as the several children explain.

Fig. Vi

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T

Fifth Grade Geography

Cassius S. Lyman

Superintendent of Schools, Hudson, Mass. (Book rights reserved)

IX

Farms and Profitable Farm Products

HE planting season is a good time for the fifth grade

to make a study of the Farm and Farm Products. Another good time to take up this subject is during the harvest season. Then the sixth grade can review and extend the work of grade five.

In some respects agriculture is the most important of all industries, and it is also the most extensive. Children in village and rural sections are more or less familiar with farm work in their own neighborhood, while city children know many of the products of the farm and should have some

In the hay field

idea of the methods of production. The movement for school and home gardens is so widespread in the public schools, that many children, even in the cities, are getting a first-hand acquaintance with the soil and are learning how to produce a few things from it. This should, then, be one of the first lessons to take up for extensive study in industrial geography in both the fifth and the sixth grade, and the later grades should take up some phases of the subject, such as the production of some staple crop, and make a thorough study of it. In the POPULAR EDUCATOR for May, 1913, I described how children of our village and country schools took up "Study of the Soil, Farms, Products of the Farms and their Market," in their home geography work. As this work covered only our own locality, the upper grades should greatly extend and enlarge this study.

SURFACE AND SOIL

Even in the fifth grade the children can understand that the surface and soil of any region will largely control the kinds of crops which the farmers will raise. Certain crops, like wheat, which require large, complicated harvesting machinery, can only be raised profitably on level lands; therefore, such crops are found for the most part on great plains. New England, with its hilly surface, produces little grain. There are, though, in New England, two sections where the land is level and very fertile. Aroostook County in Northern Maine has been found profitable for potatoes and hay. Aroostook potatoes have become famous, and many carloads of the tubers are shipped to the cities, where starch factories use enormous quantities. The level stretches of the Connecticut Valley have been found profitable for tobacco and vegetables. The illustration in this. article shows a section of the valley north of Northampton where large areas are devoted to onions and tobacco.

The photograph was taken from Mount Sugar Loaf, and shows great stretches of level farm lands on both sides of the Connecticut River. In the foreground the outlines of the different cultivated fields show very clearly. In addition to the large tracts of tobacco and onions there are big fields of corn and many other vegetables. Seen from the mountain itself the different kinds of vegetation, indicated by the various shades of green, make a more beautiful picture than the camera records. The well-built houses,

huge barns, and well-kept yards denote prosperity and comfort. People should live happy lives in such a rich, pleasant valley.

The Connecticut River is a blessing to thousands of people in both rural and urban communities. Good roads and steam and electric railways connect the cities and country districts so that the farm products can be easily marketed. Such cities as Hartford, Springfield, Holyoke and Northampton furnish good markets for thousands of farmers.

MARKET GARDENS

Many sections of New England are densely populated. This is especially true of the southern part. Most of the rich land near the urban communities is devoted to market gardening. Each farmer has a small tract of land which he cultivates intensively, often raising two or more crops each year. Hot houses and cold frames enable them to start their plants early, and many of the green vegetables are grown the year round. Many of the towns about Boston are filled with market gardens. Our Hudson children have seen these market gardens and greenhouses, from the car windows, on their trips to Boston. Some of the Belmont greenhouses and gardens are close to the railroad track. Many men and women who live in Boston work during the warm weather in these gardens.

Farmers in Lincoln (a small town under my supervision), which is seventeen miles from Boston, are largely engaged in market gardening. The children in this town study this industry at home. Some of our boys and girls have excellent gardens of their own. Last year one of our Lincoln boys, William C. Pierce, Jr., made the best record in the State in planting, caring for, harvesting, marketing and accounting for one-eighth of an acre of potatoes. He produced 43 bushels of potatoes at a profit of $20.35. This is at the rate of 346 bushels per acre at a profit of $162.80. At this rate a small farm would be very profitable. This work was done under the guidance of the Massachusetts Agricultural College at Amherst, and the seven boys and girls who made the best records were taken on a ten days' trip to Amherst and Washington, D. C., by Professor Morton. At Washington they met boys and girls from other states who had been doing similar work. On comparing notes it was found that William had made the second best record in the United States. Such incidents as these interest the children both in agriculture and geography. They tend also to keep the boys on the farms.