Arithmetic in Upper Elementary GradesAuthor(s): Pauline Frazier and Margaret BurmanSource: The Arithmetic Teacher, Vol. 6, No. 3 (APRIL 1959), pp. 165-166Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41184208 .

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Arithmetic in Upper Elementary Grades Pauline Frazier and Margaret Burman

Los Angeles City Schools, Calif.

MOST EFFECTIVE LEARNING рГОСе- dures emphasize meaning and under-

standing. A modern program in arithmetic aims to develop meanings so that the child perceives purpose and logic in what he is taught. When the learner can see the rea- sons for the uses or applications of the arith- metic that is being studied, he will be able to understand more clearly and more effi- ciently use the number relationships in- volved. This program makes use of chil- dren's experiences, a growing readiness for learning, a step-by-step course in arithmetic, and a continuous review of skills. It also guides the child to become increasingly aware of the place numbers have in the world in which he lives, such as in science or business, and provides him with opportuni- ties to apply his newly-learned concepts in arithmetic to these everyday problems of community living.

Learning is a slow process. From the time a child is first introduced to an arithmetic process to the time he has mastered it, a relatively long space of time will have elapsed. Arithmetic introduced and learned at each grade level will be dependent upon previous learning, and it will also serve as a basis for future learning. Basic concepts are introduced in the primary grades and re- applied in more complex form in each suc- ceeding grade. The child's readiness for new concepts is dependent upon this gradual and sequential building of experiences with meaning at each grade level from primary grades through upper grades.

Extending Lower Grade Learnings The upper grade arithmetic program of

the Los Angeles City Elementary Schools includes further practice and more complex problems in the four fundamental processes of addition, subtraction, multiplication and

division of whole numbers. It includes com- mon and decimal fractions and per cents. The modern program attempts to build an increased understanding of the orderly structure of the decimal number system, the laws by which it operates, and an appreci- ation of the simplicity and efficiency of the decimal number system as compared with other number systems such as the Roman numerals. As soon as we record numbers of 10 or larger we are dealing with a notational system which we received from the Hindus and Arabs. It is a decimal system which em- ploys the important principle of "positional notation" and this can be contrasted with the principles used with Roman numerals.

The upper grade program also embraces the teaching of useful technical terms of arithmetic to be used in expressing relation- ships and ideas. It strives to develop an un- derstanding of units of measures and skill- ful use of measurement to solve problems in real life situations. The teacher seeks to increase the pupil's ability to make de- pendable estimates and generalizations. The pupil uses practical applications of arith- metic to gain a greater understanding of the importance of arithmetic in intelligent buy- ing and selling, and in solving personal and community problems.

Although children always learn better and remember longer if they fully understand what they are doing, forgetting is normal. For this reason teaching, drill, and reteach- ing is needed. Also each child is a unique individual with his own interests, abilities, and needs. This variation in achievement and rate of learning is characteristic of any classroom group. The child's attitude toward the subject is the result of a combination of many factors such as interests, rate of learn- ing, aptitude, health, background of ex- periences, and environment.

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166 The Arithmetic Teacher

Arithmetic becomes most meaningful when it is used in solving problems within the realm of the child's experiences, or for which he has a real need to arrive at a satis- factory solution. Therefore, children should be taught sound and understandable ways in which they can solve these problems. Home, as well as school, provides many oppor- tunities for children to develop these prob- lem-solving abilities. At home they can learn to plan personal budgets for their allowances or earnings and for their time. They can learn to save money in a bank account which will necessitate the use of deposit slips and other business forms. They may learn to cook by measurement and to figure the cost of preparing food. They might help in caring for and replanting or land- scaping your home in order to obtain a practical understanding of measurement of time used, and cost of improvements and repairs, or help in a workshop with plan- ning, measurement and construction. Happy arithmetic experiences which occur at home with parents enable the child to approach new classroom experiences in arithmetic with confidence and anticipation.

Studies and experience show that usually the child who is successful in arithmetic understands the meaning behind the printed symbol. He has made practical application of his arithmetic facts, and in doing so he has perceived logic and purpose for numbers in daily living. He has some knowledge of the wise use of money, buying and selling, taxes, time, measurement, and graphs. He has ac- quired some appreciation of the contribu- tions arithmetic has made to other fields of endeavor such as science.

The arithmetic program of the Los Ange- les City Elementary Schools is one built upon basic concepts in sequential steps with more complexity at each grade level. Since forgetting is to be expected to some degree it is necessary to reteach, drill, and find many practical applications of arithmetic both at school and at home.

Meanings in Multiplication (Continued from page 151)

of the "situation structure" of a problem and per- form the necessary computations according to formula or another recognized pattern which pro- duces the correct answer? If a person understands the nature of the denominations and how the number of them may be used to produce in a "situation- structure" some other amount of a different denomi- nation, why quarrel with labels and how they appear on paper? The operating numbers merely are tagged with a label to identify their significance in the "situation-structure" and the operations are performed with the numbers and not with the labels. Many rather intelligent people and most scientists use labels to identify the "situation- structure" as an aid to thinking about the relation- ships. Is Dr. Christofferson correct in his comparison of the furor over such ideas as "the multiplier is always abstract" with the former argument over "how many demons can dance on the point of a pin"?

Differences in Arithmetic {Continued from page 153)

3. Sister Mary Adelbert, "Teaching Arithmetic Meaningfully for Permanent Retention," The Catholic Educator, XX (1950), 263.

4. Sister Anna Eugene Leaver, "A Statistical Study to Determine the Amount of Forgetting in Re- ligion and Arithmetic by Fifth Grade Pupils During a Three Months Interval," (unpublished) Master's Thesis, Boston College, Chestnut Hill, 1950.

Editor's Note. Sister Josephina has presented statistical evidence showing the amount of for- getting of fifth graders over the summer vacation period. The amount of forgetting seems alarmingly high. Is this in accord with current studies in other schools? What will we do, first, to obtain learning that is not so easily forgotten and, second, to rebuild this forgotten learning the following year? Is it wisest to start a year with a diagnosis of previous learnings and then to reteach before starting new work? Or should we proceed cautiously and rebuild as needed during the first half of the year? Some teachers start the year by using the book for the previous grade and refreshing and relearning cer- tain critical things before progressing into new work. One cannot assume "Once learned, never forgotten." But we ought to be seeking a method of learning that results in less forgetting. The editor would like to see Sister Josephina's tests applied to a group of pupils who had spent at least two previ- ous years in a learning pattern that featured visual and manipulative aids plus emphasis upon under- standing and discovery. Would such a group be less prone to forgetting? We should have this infor- mation.

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