Arithmetic on the March

Arithmetic on the MarchAuthor(s): Laura K. EadsSource: The Arithmetic Teacher, Vol. 1, No. 3 (OCTOBER 1954), pp. 10-14Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41183730 .Accessed: 12/06/2014 14:33

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 91.229.248.152 on Thu, 12 Jun 2014 14:33:19 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nctmhttp://www.jstor.org/stable/41183730?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp

10 The Arithmetic Teacher

from the very beginning may actually encourage the habituation of immature procedures and seriously impede neces- sary growth.

"Intermediate steps such as the use of the 'crutch' in subtraction, aid the learner both to understand the process and to compute accurately. With proper guidance, these temporary reactions may be

expected to give way to more direct responses in later stages of learning. "

Should we use crutches? Yes! Choose crutches which are appropriate to the arithmetic maturity level of the learner. And equally important, discontinue them when further use limits or arrests growth in arithmetic.

Arithmetic on the March Laura K. Eads

Bureau of Curriculum Research , New York City Public Schools

A RiTHMETic is growing in importance XJL at all school levels. This was quite apparent at the April 1954 meeting of the National Council of Teachers of Mathe- matics in Cincinnati. The theme of the meeting was ' 'Mathematics on the March" but arithmetic was given serious considera- tion by speakers and panelists represent- ing schools, colleges, and industry.

The program included a demonstration lesson on television plus more than a dozen serious discussions of topics dealing with teaching and learning arithmetic. The following highlights of the program are presented so that teachers and supervisors who were unable to attend may share some of the values of the Cincinnati meeting.

Emerging Practices in Mathematics Education1

This is the title of the 1954 yearbook of the National Council of Teachers of Mathematics (Twenty-Second Yearbook, Washington, D.C., 434 pages). Practices are reported by more than 50 persons from various parts of the country, from a variety of types of school situations, and on all school levels. The yearbook is or- ganized in six parts as follows:

Part One - Various Provisions for Differ- entiated Mathematics Curriculums.

Part Two - Laboratory Teaching in Math- ematics.

Part Three - Teacher Education. Part Four - New Emphasis in Subject

Matter. Part Five - The Evaluation of Mathemati-

cal Learning. Part Six- Bibliography of "What Is Going

On In Your Schools?"- 1950-1953.

Learning in a Good School2

In a good school, teaching is defined as experimenting with learning, and learning is defined as discovery. In such a school the major purpose of education is to develop persons who find security and power in being able to confront life situations adequately and in being accepted by their fellow men. Systems of grades, diplomas, special awards, etc., are recognized as interfering with the realization of this purpose.

In a good school, distinctions are not made between content and method, gener- al education and specialization, theoretical studies and practical studies, individual learning and social learning. All of these are considered important in a program of effective teaching and learning.


http://www.jstor.org/page/info/about/policies/terms.jsp

October, 1954 11

Learning in Arithmetic'

Differences among pupils in rates of learning arithmetic might be compared with the performance of cars of widely- differing powers as they proceed along smooth open roads, and as they proceed along roads with many obstructions, such as heavy traffic. When obstructions to learning are removed, and all pupils have an opportunity to learn at a rate best for them, differences among pupils become wider and wider as they progress through school.

Teachers are reluctant to accept this principle of individual differences as it affects rates of learning. Many teachers, particularly those at upper grade levels, feel that their particular classes are "un- usual" and that their pupils would not differ so widely if they had had a better background in arithmetic. Such teachers may make every effort to lessen differences among their pupils. They try to bring slow learners up to a "grade norm" and, thus, they may keep rapid learners from reaching their potential levels.

Good teaching increases differences among the learners in a class. Differences are decreased in classes where: individual attention is at a minimum, all pupils are given the same assignments, teaching aids do not challenge pupil thinking, all pupils use the same teaching aids, there are few instructional materials, arithmetic is nar- rowly conceived in terms of drill and computation and stereotyped problem solving, arithmetic experiences are isolated from other school or community experiences, teachers have inadequate knowledge and understanding of mathematical concepts, principles, and generalizations. In classes such as these, rapid learners are retarded and slow learners carry much heavier loads than they should.

Goals in Arithmetic4

The aim of arithmetic instruction should be to help pupils learn arithmetic in such

a way that it may be genuinely functional for them. Mental computation and oral responses should, therefore, receive con- siderable emphasis.

Ways in which teachers foster discovery and learning on the part of their pupils are often as important as the content they teach. Lists of specific unrelated items to be mastered are of little value. Goals such as the following can be very helpful to teachers who see the significance of arithmetic as a system of relationships :

1. Concepts and vocabulary (not word- calling, however) of number and quantity; of common fractions and decimals; of processes and procedures.

2. Understandings , principles, and generalizations of mathematical situations which range from simple direct visual comparisons to the use of analysis, judgment, and computation when two or more variables are involved; of the number system; of number relationships; of geometric forms; of the relationship of variables.

3. Factual knowledge and information of how things are done and used, of important facts and definitions, of basic number combinations and processes.

4. Skills and inferences of arithmetical processes, of when and where to use processes, of judgments and esti- mates, of interpretations and inferences.

5. Interests , attitudes , and appreciations of mathematical organization, of assurance in place of doubt and fear, of the individuality of learning and of mathematical situations, of mathematics in hobby and vocational interests.

Content and goals should be inter- preted for individual children or for groups of children in terms of: age, maturity, experience, and environment of the children; as well as the inherent nature,




structure, and difficulty of the learning materials.

Arithmetic in One School System5 The goals of the elementary and the

junior high school program in arithmetic in New York City are to foster growth in : concept development, mathematical thinking, the solving of problems from the pupils' own experiences, skill in computation - with and without the use of paper and pencil.

Teaching and learning in arithmetic develop from the concrete to the abstract through steps called: Developmental Levels of Learning. Arithmetic is de- veloped systematically from the first grade on. After its introduction, a topic is reinforced year after year on higher levels of concept development. Thus, growth in understanding is provided for as pupils become more mature and as they proceed through the grades. Grade place- ment in arithmetic thus takes on new meaning.

Articulation between the Elementary School and the Junior

High School6 The philosophy of education and the

goals in teaching mathematics should be the same for the elementary school and the junior high school. This requires that curriculum planning and the development of the mathematics curriculum be conceived as a Kindergarten through Grade 12 program.

Differences in arithmetic ability are very wide for pupils entering junior high school. Differences are widest where pupils are helped to work at maximum levels of ability in the elementary school. Some seventh grade pupils are able to deal with simple whole numbers only; others use complicated fractions, decimals, and per cents with efficiency and understanding.

Where possible, homogeneous grouping for mathematics teaching is recommended at the junior high school level. In many

junior high schools it is still a new idea to divide a class into groups for mathematics teaching and learning.

The quality of homework at the later elementary and the junior high school levels needs to be carefully evaluated. Class assignments and materials used by pupils also require more careful considera- tion in many schools.

Some bright, more average, and many slow pupils at upper grade levels prefer to learn by rote methods, rather than by discovery. They say that this is the method they are "used to." Teachers should rethink and appraise: their methods of teaching pupils at varying levels of ability, their aims or goals for mathematics teach- 'ing, their own understanding of mathematical concepts and principles.

Junior high school teachers, generally, know less about the needs of the individual pupils in their classes than elementary school teachers. On the other hand, junior high school mathematics teachers, generally, have had a greater opportunity than elementary school teachers to learn the concepts and principles underlying mathematics. Rather than deplore in- adequacies in their pupils' understanding, junior high school teachers should start their mathematics teaching where their pupils are, not where they think they ought to be. They should consider it their responsibility to develop basic concepts and understandings in elementary arithmetic where these are needed. Teachers should recognize that nothing succeeds like success and that all too many pupils ac- cumulate failure in mathematics as the weeks, months, and years pass.

Arithmetic should be a unifying thread for all mathematics teaching throughout the junior high school. An understanding of concepts in arithmetic is basic to understanding concepts in algebra. Further- more, a socially competent person is one who is able "to think and act intelligently in the many arithmetical situations that arise in social, cultural, and economic activities."4



October, 1954 13

Supervision of Mathematics Teaching7 Among the procedures discussed were:

meetings and conferences with new teachers, individual teachers, small groups of teachers, departments, etc. ; workshops for the in-service education of teachers at which participants determine problems to be studied, evaluate the workshop periodically, and study problems in ad- vance of sessions; participation of teachers in the preparation of curriculum materials, and in the preparation, analysis, and discussion of the results of tests; demonstrations by supervisors, individual teachers, and groups of teachers in class- rooms, at meetings, on radio broadcasts, on television programs; pilot classes and schools for developing procedures and materials; meetings of local and regional mathematics associations and clubs planned to meet the needs of teachers.

Research in Mathematics Education8 The National Council Committee on

Research reported its work and planned for the presentation of some recent research and research now under way.

"Mathematics Education Research Studies - 1952" is reported in: Aids for Mathematics Education, Circular No. 377, July 1953; U.S. Department of Health, Education, and Welfare; Office of Education, Washington 25, D.C. (25 p. Free) Of the 57 studies reported, 28% deal with arithmetic, from the primary grade level through high school and college levels.

Some findings of a study of the extent to which arithmetic is taught with meaning (rationalized) in one state (New Jersey) follow: the newer elementary arithmetic textbooks tend to emphasize pupil discovery, teachers tend to follow textbook procedures closely, teachers in Grades 3 and 4 are more likely to develop meaning than teachers in Grades 5 and 6, addition and subtraction are more likely to be taught with meaning than multiplication or division with whole numbers or with fractions.

Studies of specific procedures in arithmetic teaching are often inconclusive. This may be due to the brief periods usually allotted for such investigations, procedures used for evaluating learning, the nature of the problems studied, the experimental conditions set up for study.

Types of research difficult to plan and interpret but very much needed are studies in: concept formation in mathematics, appreciation in mathematics, the place of mathematics in various types of curriculum programs, sequences in learning mathematics.

Some Contributors and Topics

Thirty-Second Annual Meeting of the National Council of Teachers of Mathematics April 20-24, 19&4

1 Presentation of the Twenty-Second Yearbook. John R. Clark, Chairman, Teachers College, Co- lumbia University, New York, N. Y.; Philip Peak, Indiana University, Bloomington, Ind.; John Kinsella, New York University, New York N. Y.; Joy Mahachek, State Teachers College, Indiana, Pa.; Veryl Schult, Wilson Teachers College, Washington, D. C. 2 Teaching and Learning in a Good School. Donald P. Cottrell, Dean, College of Education, Ohio State University, Columbus, Ohio. 3 Rate of Progress in Learning Arithmetic. Esther J. Swenson, University of Alabama, Uni- versity, Ala. 4 Content and Goals in Arithmetic Education in Elementary School (with emphasis on grades five and six). Ben A. Sueltz, State Teachers Col- lege, Cortland, N. Y.

6 Developmental Mathematics in the New York City Schools. Laura K. Eads, Bureau of Curricu- lum Research, Board of Education, New York, N. Y. (A brief description of the program is given in the Twenty-Second Yearbook of the National Council of Teachers of Mathematics.)

6 Closer Articulation Between the Elementary and the Junior High School. Herschel E. Grime, Directing Supervisor of Mathematics, Public Schools, Cleveland, Ohio.

To be Effective, Guidance and Counseling Pro- grams in Mathematics Must be Continuous and Comprehensive Through the Grades. C. L. Thiele, Director of Exact Sciences, Public Schools, De- troit, Mich.

Helping Junior High School Pupils Explore Mathematics. Donald W. Lentz, Parma Schaaf Junior High School, Parma, Ohio.

Concepts From Arithmetic Which May Be Used in Teaching Fractions in Algebra. Edith Treuen- fels, Putney School, Putney, Vt.




Practical Mathematics Is Challenging to Stu- dents. Frank G. Lankford, Jr. University of Vir- ginia, Charlottesville, Va.

Determining Algebraic Ability Through Eighth Grade Mathematics. Lela Cobb, Junior-Senior High School, Wellington, Kan. 7 Better Supervision of Mathematics Teaching. Mary A. Potter, Chairman, Consultant in Math- ematics, Racine, Wis.; Herschel Grime, Mathe- matics Supervisor, Cleveland, O.; Graldine Kauffman, Supervisor of Mathematics and Sci- ence, East Chicago, Ind.; Eugene F. Peckman, Senior Supervisor, Science and Mathematics, Pittsburgh, Pa.; Louis F. Scholl, Supervisor of Mathematics, Buffalo, N. Y.; Veryl Schult, Di- rector of Mathematics, Washington, D. C.

A Junior High School Workshop for Mathe- matics Teachers. Mary A. Potter, Consultant in Mathematics, Public Schools, Racine, Wis.

Taking the Drudgery out of Problem Solving. Alice M. Hach, Slauson School, Ann Arbor, Mich.

8 Research in Mathematics Education. How- ard F. Fehr, Teachers College, Columbia Uni- versity, New York, N. Y.; Henry Van Engen, Iowa State Teachers College, Cedar Falls, Iowa; Kenneth E. Brown, Specialist for Mathematics, U. S. Office of Education, Washington, D. C; George McMeen, State Teachers College, New- ark, N. J. ; Lyman C. Peck, Iowa State Teachers

College, Cedar Falls, Iowa; Maurice Hrtung University of Chicago, Chicago, 111.; Nathan Lazar, Ohio State University, Columbus, Ohio; John Kinsella, New York University, New York, N. Y.; Myron Rosskopf, Teachers Col- lege, Columbia University, New York, N. Y. ; F. Lynwood Wren, George Peabody College for Teachers, Nashville, Tenn.

Textbooks Received The Teaching of Arithmetic, second edition,

Herbert F. Spitzer, Houghton Mifflin Company, 1954. 416 pages, $3.50. This is a revised edition of Professor Spitzer's

earlier text (1948) which is well known to many teachers and students. The new edition arrived too late for competent review in this issue. A later issue of The Arithmetic Teacher will carry a review.

Arithmetic for Today, Thomas J. Durell, Adaline P. Hagaman, and James H. Smith, Charles E. Merrill Books, 1954. 316 pages, 99j net, grades 3-6, also grades 7-8. Teacher's Manuals available, 37 net.

0S& HELPING BOYS AND GIRLS

(

Documents

Arithmetic on the March