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C.U.P.M. Recommendations for ElementaryTeachers: Are They Realistic?

William A. Miller Central Michigan University,Mt. Pleasant, Mich. and George KaprelianWisconsin State University, Whitewater, Wis.

The motivation for this article stems from the following factors:(1) The writers5 experience at a C.U.P.M. conference in the fall

of 1964;(2) The number of recent articles which appear to equate good

elementary teacher-training programs with a requirement of 12hours in mathematics;

(3) The writers^ experience in helping to evaluate and restructurethe educational programs at Wisconsin State University-Whitewater;

(4) The writers^ experience with elementary teachers in summerand in-service mathematics institutes which indicate to them thatthe pre-service and in-service programs in mathematics cannot bedivorced;

(5) Recognition of the many areas in which the elementary teachersmust be competent: teaching mathematics, science, reading, socialstudies, language arts, healthpossibly music, physical educationetc.;

(6) The writers belief that the elementary school pupil will bebetter served if some members of the elementary school staff havemuch stronger training in mathematics than C.U.P.M/s Level Irecommendations.The C.U.P.M. report1 recommends that the pre-service preparation

of elementary school teachers include, as a minimum in mathematicscontent, 12 semester hours (four courses). These recommendationsinclude (1) a two-course sequence devoted to the structure of thereal number system (six semester hours), (2) basic concepts of algebra(three semester hours) and (3) informal geometry (three semesterhours).

Fisher2 states that "in almost every state, resolutions were passedin support of the Level I recommendations, and the delegates to theconferences agreed that the present requirements in mathematicsfor elementary school teachers were inadequate/5 This certainlyhappened at the state conference which the writers attended. How-ever, when they pointed out the possible problems involved in

1 Committee on the Undergraduate Program in Mathematics, "Recommendations of the Mathematical Asso-ciation of America for the Training of Teachers of Mathematics." American Mathematical Monthly, LXVII,December 1960.

2 John J. Fisher, "Extent of Implementation of CUPM Level I Recommendations." Arithmetic Teacher, XIV,March 1967, pp. 194-197.

10 School Science and Mathematics

implementing the program and suggested that the recommendationbe considered jointly with postgraduate training, the chairman of thepanel replied that the duty of the committee was limited to theconsideration of undergraduate programs. It is needless to add thatalternatives were not considered, non-mathematically oriented in-dividuals concerned with teacher training were not consulted, andthe programs as recommended were passed almost unanimously.At our university (which we think is typical) the recommended

programs become unrealistic and unworkable when the requiredliberal arts and education courses are considered. Our universitycurrently requires 128 to 130 semester hours of credit for graduation.Included is a minimum of 46 hours in the liberal arts for all student.For the future elementary teacher, the curriculum design includes anadditional 33 hours of service courses (e.g. conservation, naturestudy, science, music, art, etc.), the majority of which are liberal artscourses taught by faculty in the appropriate departments. In addition,30 hours are required in professional education, eight to ten of whichare in the clinical student intern program. Admittedly, the numberof required courses in the above areas is open to question; however,the required liberal arts courses crept into the curriculum 25 yearsago as a reaction to the old normal school programs. The presentprogram for elementary teachers, which requires three semesterhours in mathematics (certainly not adequate for teaching) and twosemester hours in the methods of teaching mathematics leaves thestudent a total of 19 to 20 elective credits. In a democratic society,a very good case can be made for increasing the number of electivehours. As one studies the program, about the only area which appearsas if it can be reduced is the liberal arts program, which we believewas accepted as a bad bargain in another period of history.We are opposed to a five-year program for an undergraduate degree

in teacher education. Our work with experienced teachers indicatesthat mathematics content and methods courses are of much morevalue, if taken at a time when a teacher realizes and understands theproblems which children have in learning mathematical conceptsand skills. Moreover, at this period in the professional training of theelementary teacher, behavioral objectives and mathematics conceptswill have real maening to a teacher close to the actual teachingsituation. Unlike Creswell3 we have observed a far greater interestexpressed in the application of concepts to the teaching situation.This is evidenced by comparing the number and variety of teachingaids and materials developed as well as the mathematical achieve-ment of undergraduates and graduates in comparable courses.

a John F. Creswell, "How Effective are Modem Mathematic Workshops?" Arithmetic Teacher, XIV, March1967,pp. 205-209.

C.C/.P.M. Recommendations for Elementary Teachers 11

The writers are in agreement with C.U.P.M. recommendationsregarding the competence required of a teacher of mathematics ingrades 4 through 6. In fact, even though these are minimum recom-mendations, we believe the recommendation for geometry are totallyinadequate. One must also recognize that the nature of the elemen-tary teachers duties are changing and will probably continue tochange in the future. Team teaching, middle school structures andsemi-departmentalized organizations are becoming more common.As a constructive alternative to the C.U.P.M. recommendations

for the elementary curriculum, we suggest that the following pro-grams be considered. One, that no more than six semester hours inmathematics be required in the undergraduate program. Preferably,one course should be in the structure of the rational numbers andone course in elementary geometry, if the teacher elects a minor ofapproximately 21 hours in some specialty field. These minors may bein mathematics, foreign language, English, science, etc. Second, haveavailable additional courses, designed with the special needs ofteachers in mind, which may be taken as electives. These coursesshould carry both undergraduate and graduate credit, and if takenas graduate courses, apply toward a graduate degree. Third, designgraduate programs which have special emphasis in mathematics forexperienced teachers who have decided that they wish to specializein mathematics. Of course, it is axiomatic that these courses betaught by individuals who are competent in mathematics, interestedin the problems of teaching elementary mathematics, and haveempathy toward the elementary teachers^ problems.

It is recommended that the following courses be designed andconsidered for such programs. Courses from number three on shouldall be available as either undergraduate or graduate courses.

(1) One course devoted to the structure of the rational numbers.(2) One geometry course devoted to a study of space, plane, line,

simple closed curves, polygons, separation properties, etc. developedintuitively; concepts of linear and angle measurement as well asmeasurement in the plane and space should be considered, with abrief introduction to axiomatics and coordinate geometry.

(3) A course which extends the rational numbers, develops the realnumbers, with a brief introduction to the complex numbers.

(4) The foundations of geometry, formal investigation of thepossible approaches to congruence and similarity, consider distancepreserving motions, reflections, rotations and translations; elemen-tary measure theory. Non-euclidean and finite geometries shouldalso be considered.

(5) Elementary Number Theory; properties of integers, euclideanalgorithm, unique factorization, linear diophantine equations, ele-

12 School Science and Mathematics

mentary properties of modular congruence, Euler-Fermat theorems,primitive roots, decimal expansion of rational numbers, basal systems,prime and composite numbers, perfect numbers.

(6) Elementary Probability and StatisticsIntroduction to pro-bability theory from a set-theoretic point of view. Applications ofstatistics, combinations, permutations, the binomial theorem, finitesample spaces, conditional probability, Bayes^ theorem, randomevents, Chebyshevs theorem, joint and continuous distributions,binomial, normal, hypergeometric and poisson distributions, mea-sures of variability.

(7) Elementary Logic, sentential connectives, truth tables andtautologies, common rules of inference, valid and invalid patterns,quantification, class logic. Introduction to mathematical systemsand m-valued logics.

(8) Analytic Geometry; coordination of the line, plane and space,polar coordinates, distance, vector, conic sections, elementary matricesand linear transformations.

(9) Abstract Algebra. One course devoted to algebraic structuressuch as group, rings, integral domains, fields, etc. The approachshould be somewhat intuitive, building from the familiar structuresto the abstract axiomatic.

(10) Linear Algebra. Finite dimensional vectors, spaces, matrices,linear transformations, applications, linear programming and games,solutions of linear equations.

(11) History of MathematicsIt should be noticed that the above courses bypass the usual cal-

culus sequence. If a three to six hour sequence can be developed andtaught from an intuitive approach, it should probably be added tothe above list.

In summary, we are definitely opposed to requiring more coursesin any area without a corresponding reduction in the current list ofrequired courses. On the other hand, we do not feel that the currentmathematics program is adequate and we do not think the solutionto the problem is to require 12 credits in mathematics for everyundergraduate elementary major. Such a program will be a waste ofboth student and instructor manpower at the college level, sincemany future teachers will be working in other specialized areas. Wethink a solution may be to have a special minor available at both theundergraduate and graduate level which bypasses the traditionalsequence.