The mathematics education of elementary school teachers: pre-service and in-serviceAuthor(s): J. Fred WeaverSource: The Arithmetic Teacher, Vol. 12, No. 1 (JANUARY 1965), pp. 71-75Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185092 .

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Focal point J. Fred Weaver

The mathematics education of elementary school teachers: pre-service and in-service

Xhe mathematics education of elemen- tary school teachers is not a new problem, nor has it been a problem of limited or superficial concern. A few years ago in The Arithmetic Teacher Jack N. Sparks summarized concisely a number of attacks upon this problem,1 citing refer- ences from 1949 through 1959.* Concern over the problem was voiced by several writers prior to 1949. Now, within the past several years, intensified attention has been given to the mathematics education of elementary school teachers.

Pre-service education

Suggestions pertaining to pre-service mathematics education have been pre- sented by Phillips,2 by Davis,3 by Shryock,4 by Ruddell, Dutton, and Reckzeh,5 and by others. Undoubtedly, the most potentially influential recommendations have been those prepared by the Panel on Teacher Training of the Committee on the Under- graduate Program in Mathematics (CUPM) of the Mathematical Association of America (MAA),6 covering five levels of instruction from elementary school mathe- matics teaching through college mathe- matics teaching. Relevant recommenda- tions covering "Level I" - teachers of elementary school mathematics - were re- printed in The Arithmetic Teacher,7 as was a subsequent progress report on the

♦ Footnotes appear at the end of the article.

implementation of the CUPM recommen- dations.8 The gist of these Level I recom- mendations follows:

"As a prerequisite for the college train- ing of elementary school teachers, we recommend at least two years of college preparatory mathematics, consisting of a year of algebra and a year of geometry, or the same material in integrated courses. It must also be assured that these teachers are competent in the basic techniques of arithmetic. The exact length of the train- ing program will depend on the strength of their preparation. For their college training, we recommend the equivalent of the following courses :

(a) A two-course sequence devoted to the structure of the real number sys- tem and its subsystems.

(ò) A course devoted to the basic con- cepts of algebra.

(c) A course in informal geometry. "The material in these courses might, in

a sense, duplicate material studied in high school by the prospective teacher, but we urge that this material be covered again, this time from a more sophisticated, col- lege-level point of view.

"Whether the material suggested in (a) above can be covered in one or two courses will clearly depend upon the previ- ous preparation of the student."9

Course Guides to implement these recommendations have been prepared

January 1965 71

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under the direction of the Panel on Teacher Training of the CUPM.10 Text- books have been written with the express purpose of reflecting these recommenda- tions, and more such texts are certain to follow. Progress has been made and will continue to be made. In general, however, at the present time the pre-service mathe- matical preparation of elementary school teachers falls short of the recommenda- tions of the Panel on Teacher Training of the CUPM.

In-service education

Extensive and diverse efforts have been devoted to the in-service mathematical preparation of elementary school teachers. The direction taken is very similar to that taken at the pre-service level. However, for understandable reasons, a wider vari- ety of approaches has been utilized.

During the past several years The Arithmetic Teacher has reported many in-service efforts - too many to identify at this time. Special attention is directed to the fact, however, that films, kinescopes, and the like, with accompanying text ma- terials, occupy a prominent place in the exploration of effective ways to cope with the truly enormous problem of in-service mathematics education for elementary school teachers. The following are illustra- tive of such projects: Patterns in Arith- metic, directed by Henry Van Engen;11 Foundations of Mathematics for Elementary School Teachers, directed by E. Glenadine Gibb;12 Sets and Systems, directed by M. Virginia Biggy;13 the "Elementary Film Project" of the School Mathe- matics Study Group;14 and In-Service Films in Mathematics for Elementary School Teachers, directed by Harry D. Ruderman for the National Council of Teachers of Mathematics.16 Also of signifi- cance in this connection is the newly or- ganized "Project for the Improved Use of Newer Educational Media in Elementary School Mathematics/' sponsored by the National Council of Teachers of Mathe- matics and directed by James H. Zant.18

Materials on pre- and in-service training of elementary school teachers

Within the past five years many profes- sional books have been published that per- tain to the mathematics education of ele- mentary school teachers. Some of these books deal exclusively or very predomi- nantly with mathematical content at vari- ous levels of mathematical sophistication. Other books include an extensive amount of material pertaining to methods and materials of instruction, and vary in the extent of their consideration of mathe- matical content per se. Needless to say, not all are of the same quality, nor are they comparable in scope.

There follows a bibliography of 42 books published during the five-year pe- riod, 1960-1964, that should at least be known by persons concerned with the pre- and in-service preparation of elementary school teachers. They differ in the ways just indicated, and in other ways as well. But they do represent available materials from which appropriate and judicious selections may be made.

Bibliography ♦1 Anderson, Concepts of Informal Ge-

ometry. School Mathematics Study Group, Studies in Mathematics, Vol. V. New Haven, Conn. : Yale Univer- sity Press, 1960. See review by John D. Hancock in The Arithmetic Teacher, IX (October, 1962), 344.

2 Banks, Learning and Teaching Arith- metic, 2nd ed. Boston: Allyn and Bacon, 1964.

*3 Bell, Chinn, McDermott, Pieters, and Willerding, A Brief Course in Mathe- matics for Elementary School Teachers. School Mathematics Study Group, Studies in Mathematics, Vol. IX. Stanford, Calif.: University Press, 1963.

*4 Bell, Hammond and Herrara, Funda- mentals of Arithmetic for Teachers. New York: John Wiley and Sons,

* Deals exclusively or predominantly with mathematical oontent.

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1962. See review by Clarence H. Heinke in The Arithmetic Teacher, X (January, 1963), 44-45.

*5 Brumfiel, Eicholz, Shanks, O'Daffer, Principles of Arithmetic. Reading, Mass.: Addison- Wesley Press, 1963.

6 Corle, Teaching Mathematics in the Elementary School. New York: Ronald Press, 1964.

*7 Crouch and Baldwin, Mathematics for Elementary Teachers. New York: John Wiley and Sons, 1964.

8 DeVault (ed.), Improving Mathe- matics Programs. Columbus, Ohio: Merrill Books, 1961. See review by George S. Cunningham in The Arithmetic Teacher, IX (March, 1962), 164.

9 Dutton and Adams, Arithmetic for Teachers. Englewood Cliffs, N.J.: Prentice-Hall, 1961.

*10 Educational Research Council of Greater Cleveland, Key Topics in Mathematics for the Primary Teacher. Chicago : Science Research Associates, 1962.

*11 Evenson, Modern Mathematics: In- troductory Concepts and Their Im- plications. Chicago: Scott, Foresman and Company, 1962. See review by James R. Smart in The Arithmetic Teacher, X (March, 1963), 162.

12 Grossnickle and Brueckner, Discover- ing Meanings in Elementary School Mathematics. New York: Holt, Rine- hart and Winston, 1963. See review by Ann C. Peters in The Arithmetic Teacher, XI (March, 1964), 211-213.

*13 Haag, Structure of Algebra. Reading, Mass.: Addison- Wesley Press, 1964.

*14 Haag, Structure of Elementary Al- gebra. School Mathematics Study Group, Studies in Mathematics, Vol. III. New Haven, Conn.: Yale Uni- versity Press, 1960.

*15 Hacker, Barnes, and Long, Funda- mental Concepts of Arithmetic. Engle- wood Cliffs, N.J. : Prentice-Hall, 1963.

*16 Hamilton and Landin, Set Theory and

the Structure of Arithmetic. Boston: Allyn and Bacon, 1961.

17 Härtung, Van Engen, Knowles, and Gibb, Charting the Course for Arith- metic. Chicago: Scott, Foresman and Company, 1960.

*18 Henkin, Smith, Varineau, and Walsh, Retracing Elementary Mathematics. New York: Macmillan Company, 1962.

19 Hollister and Gunderson, Teaching Arithmetic in the Primary Grades. Boston: D.C. Heath and Company, 1964.

20 Howard and Dumas, Basic Proce- dures in Teaching Arithmetic. Boston: D.C. Heath and Company, 1963. See review by Mary Folsom in The Arithmetic Teacher, XI (January, 1964), 46.

*21 Keedy, A Modern Introduction to Basic Mathematics. Reading, Mass.: Addison- Wesley Press, 1963.

*22 Larsen and Ludlow, Arithmetic for Colleges, 3rd ed. New York: Mac- millan Company, 1963.

*23 Moise, Elementary Geometry from an Advanced Standpoint. Reading, Mass. : Addison-Wesley Press, 1963.

*24 Mueller, Arithmetic: Its Structure and Concepts, 2nd ed. Englewood Cliffs, N.J.: Prentice-Hall, 1964.

25 National Council of Teachers of Math- ematics, Enrichment Mathematics for the Grades, 27th Yearbook. Washing- ton: National Council of Teachers of Mathematics, 1963. See review by Ann C. Peters in The Arithmetic Teacher, X (December, 1963), 518- 519.

26 National Council of Teachers of Mathematics, Instruction in Arith- metic 25th Yearbook. Washington: National Council of Teachers of Mathematics, 1960. See review by Herbert F. Miller in The Arithmetic Teacher, VII (November, 1960), 373-375.

*27 Ohmer, Aucoin, and Cortez, Ele- mentary Contemporary Mathematics.

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New York: Blaisdell Publishing Com- pany, 1964.

*28 Osborn, DeVault, Boyd, and Hous- ton, Extending Mathematics Under- standing. Columbus, Ohio: Merrill Books, 1961.

29 Overman, The Teaching of Arithmetic. Chicago: Lyons and Carnahan, 1961.

*30 Peterson and Hashisaki, Theory of Arithmetic. New York: John Wiley and Sons, 1963. See review by Paul B. Johnson in The Arithmetic Teacher, XI (May, 1964), 362-363.

*31 Schaaf, Basic Concepts of Elementary Mathematics. New York: John Wiley and Sons, 1960.

*32 School Mathematics Study Group, Concepts of Algebra. Studies in Mathe- matics, Vol. VIII. New Haven, Conn.: Yale University Press, 1961.

*33 School Mathematics Study Group, Intuitive Geometry. Studies in Mathe- matics, Vol. VII. New Haven, Conn.: Yale University Press, 1961. See review by John D. Hancock in The Arithmetic Teacher, IX (October, 1962), 345-346.

*34 School Mathematics Study Group, Number Systems. Studies in Mathe- matics, Vol. VI. New Haven, Conn. : Yale University Press, 1961.

35 Shipp and Adams, Developing Arith- metic Concepts and Skills. Englewood Cliffs, N.J.: Prentice-Hall, 1963.

36 Spitzer, The Teaching of Arithmetic, 3rd ed. Geneva, 111.: Houghton Mif- flin Company, 1962.

37 Swenson, Teaching Arithmetic to Chil- dren. New York: Macmillan Com- pany, 1964.

38 Thorpe, Teaching Elementary Arith- metic , New York: Harper and Broth- ers, 1962. See review by Herbert F. Miller in The Arithmetic Teacher, IX (December, 1962), 462-465.

*39 University of Maryland Mathe- matics Project. Mathematics for Ele- mentary School Teachers, Book I and Book II. College Park: University

of Maryland Mathematics Project, 1964.

*40 Ward and Hardgrove, Modern Ele- mentary Mathematics. Reading, Mass.: Addison- Wesley Press, 1964.

*41 Webber and Brown, Basic Concepts of Mathematics. Reading, Mass. : Ad- dison- Wesley Press, 1963.

*42 Youse, Arithmetic: A Modern Ap- proach. Englewood Cliffs, N.J. : Pren- tice-Hall, 1963.

In addition to the preceding publica- tions, booklets such as the following have been released for use by parents, or teachers, or both:

a) Barker, Curran, and Metcalf, The "New" Mathematics for Teachers and Par- ents of Elementary School Children. San Francisco: Fearon Publishers, 1964.

b) Deans, Elementary School Mathe- matics: New Directions. Washington: U. S. Government Printing Office, 1963. See review by Julia Adkins in The Arithme- tic Teacher, XI (April, 1964), 276.

c) Engler, Helping Your Child Learn the New Arithmetic. New York: Criterion Books, 1961. See review by James R. Smart in The Arithmetic Teacher, IX (May, 1962), 294.

d) May, Major Concepts of Elementary Modern Mathematics. Wilmette, 111. : John Colburn Associates, 1962. See review by Dale G. Jungst in The Arithmetic Teacher, XI (February, 1964), 127-128.

e) National Council of Teachers of Mathematics, Topics in Mathematics for Elementary School Teachers. Washington: National Council of Teachers of Mathe- matics, 1964. (Series of eight booklets, soon to be published as the 29th NCTM Yearbook.)

/) Phillips and Kluttz, Modern Mathe- matics and Your Child. Washington: U. S. Government Printing Office, 1963.

g) Traston, Insight into Modern Mathe- matics. Ideal, 1963.

The past five years have been extremely active ones in relation to the pre- and in- service mathematics education of ele-

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mentary school teachers. The next five years well may be even more active and challenging ones, particularly in light of things such as the Report of the Cambridge Conference on School Mathematics.17

Notes 1 Jack N. Sparks, "Arithmetic Understand-

ings Needed by Elementary-School Teachers," The Arithmetic Teacher, VIII (December, 1961), 395-403. 2 Clarence Phillips, "A Combined Content- Methods Course for Elementary Mathematics Teachers/' The Arithmetic Teacher, VII (March, 1960), 138-140. 3 Catherine Linn Davis, Highlights of a Sum- mer Conference," The Arithmetic Teacher, VIII (January, 1961), 14-18. 4 Jerry Shryock, "A Mathematics Course for Prospective Elementary School Teachers," The Arithmetic Teacher, X (April, 1963), 208- 211.

6 Arden K. Ruddell, Wilbur Dutton, and John Reckzeh, "Background Mathematics for Elementary Teachers," Instruction in Arith- metic (Washington, D.C.: National Council of Teachers of Mathematics, Twenty-Fifth Year- book, 1960), pp. 296-319. 6 " Recommendations of the Mathematical Association of America for the Training of Teachers of Mathematics," American Mathe- matical Monthly, LXVII (December, 1960), 982-991.

7 "The Training of Elementary-School Mathe- matics Teachers," The Arithmetic Teacher, VII (December, 1960), 421-425. 8 Clarence E. Hardgrove and Bernard Jacob- son, "CUPM Report on the Training of Teach- ers of Elementary School Mathematics," The Arithmetic Teacher, XI (February, 1964), 89-93.

0 Quoted from sources referred to in Notes 6 and 7.

10 "Course Guides for the Training of Teachers of Elementary School Mathematics," unpub- lished paper (fourth draft, July, 1964). 11 See "Patterns in arithmetic," The Arith- metic Teacher, X (April, 1963), 217-221. 12 See "Foundations of Mathematics for Ele- mentary Schools: An In-Service Project," The Arithmetic Teacher, X (October, 1963), 359- 364.

18 See "Sets and Systems," The Arithmetic Teacher, XI (December, 1964), 581-582. 14 Series of 30 half-hour films, distributed by Modern Learning Aids.

16 For information, communicate with the National Council of Teachers of Mathematics.

18 For information, communicate with the Na- tional Council of Teachers of Mathematics.

17 Goals for School Mathematics: The Report of the Cambridge Conference on School Mathe- matics, organized and administered by Educa- tional Services Incorporated. Geneva, 111.: Houghton Mifflin, 1963. (For an abridged ver- sion of the report see "The Cambridge Confer- ence on School Mathematics," The Arithmetic Teacher, XI (March, 1964), 207-210.)

A principals' conference on the "new" mathematics The changing curricular approach to elementary school mathematics requires in-service educa- tion of teachers as one of its basic tenets. Getting teachers merely to understand the fundamentals of mathematics is a sine qua non for administrators. The Philadelphia Public Schools, with over 200 elementary school or- ganizations, feel this situation quite keenly. A curriculum guide was newly issued for the Primary Unit (Years 1, 2, and 3) in September, 1963, and a concentrated effort is now being given to publication of a guide for intermediate grades during the 1964-1965 school year. The great necessity of orienting 4000 elementary school teachers to a changing curricular program easily takes on great administrative priority.

That the principal is the instructional leader of his school has long been a fundamental pre- mise in Philadelphia. In a sense, the principal is the director of curriculum construction, the director of in-service education, and the director of research for his own school. The principal's unique position and the need for teacher in-

service education led the Arithmetic Curriculum Committee to formulate a "Principals' Con- ference on The 'New' Mathematics." Such a conference would (1) unify thinking on the philosophy of arithmetic teaching in Phila- delphia; (2) sensitize principals to the need for continual arithmetic in-service education, and (3) present principals with newer material which would be applicable to most school situations in the city. This conference, it was hoped, would lead to teacher in-service education through initial in-service education of principals.

Enthusiasm is high among the collaborating teachers, the Arithmetic Curriculum Committee members, and the principals who took part in the three conferences. The Committee feels that the conference is a first step, but a necessary and valid first step. Plans are now being made for an extension of arithmetic in-service op- portunities for elementary teachers during the next school year. - Frederick R. Rabinowitz, Arithmetic Collaborating Teacher School District of Philadelphia

January I960 75

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