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Topics in Mathematics for Elementary School TeachersRandall C. Hicks, Valdosta (Ga.) State CollegeandAlex F. Perrodin, University of GeorgiaAthens, Ga. 30601The advent of ^modern mathematics" in the elementary schoolcurriculum has been the stimulus for a multitude of books andcourses for elementary school teachers. Much thought, but relativelylittle research, has been involved in the selection of topics for study.The purpose of this research was to provide a sounder base for theselection of topics appropriate for the pre-service and in-serviceeducation in mathematics of elementary school teachers.PROCEDURESFour types of sources were intensively reviewed to provide informa-tion to assist educators in determining the most needed content forimproving the preparation of elementary school teachers in mathe-matics. The first of these involved a review of forty-six selected re-search studies pointing out the mathematical competencies or weak-nesses of elementary school teachers. The second source of informa-tion was provided by a review of thirty-two sets of recommendationsof mathematics educators and nationally recognized advisory groupsor organizations. The third approach was a page-by-page analysis ofsixteen recent textbooks designed for college courses in mathematicsfor elementary school teachers. Finally, as a fourth source of data,eleven arithmetic series or teachers guides for grades K-7 publishedsince 1962 were analyzed.A composite list of the mathematical topics obtained from thesefour sources was compiled and a system of rating the topics was de-vised. Topics which appeared at least once (the composite list) werecategorized as Level I. To be classified as a Level II topic, an item hadto meet one of the following criteria for selection:a. appear in at least three of the research studiesb. appear in at least five of the recomendations by mathematics educators oradvisory groupsc. appear in at least eight of the sixteen college textbooks in mathematics forelementary school teachersd. appear in at least six of the eleven arithmetic series or teachers guides forgrades K-7.Finally, to be classified as a Level III topic, an item had to meet atleast two of the four criteria listed for Level II topics.739740 School Science and MathematicsFINDINGSA total of ninety-eight topics was located in the four types ofsources. Some of the topics were actually groups of topics such as"Basic Operations with Whole Numbers." There were nineteendifferent topics identified by the research studies as exhibited needsin mathematics of elementary school teachers, (Hicks, p. 126).Thirteen of these topics were classified as Level II. Examples of theresearch studies reviewed are Allman (1952), Stoneking and Welch(1961), Nelson and Worth (1961), Cresswell (1963), ODonnell(1958), Brewer (1951), Jones (1962), Carroll (1961), Fulkerson(1960), Phillips (1959), Dutton and Cheney (1964), Nafziger (1961),Shryock (1962), Grossnickle (1962), Harper (1964), and Bean (1958).Fifty-four topics were located in the recommendations of mathe-matics educators and advisory groups, (Hicks, p. 194-196). Thirty-six topics from this source qualified as Level II. Among the educatorsand advisory groups consulted were Mayor (1960), Sueltz (1958),Jones (1960), Rehage (1960), Commission on Mathematics (1959),National Association of State Directors of Teacher Education andCertificationAmerican Association for the Advancement of Science(1960), National Council of Teachers of Mathematics (1960), Cam-bridge Conference on School Mathematics (1963), and Committee onthe Undergraduate Program in Mathematics (1961).A total of eighty-four topics was noted in the sixteen recent text-books designed for college courses in mathematics for elementaryschool teachers, (Hicks, p. 209-212). Only four of the items wereincluded in all sixteen textbooks. These four items were: basic opera-tions with whole numbers; order, ordinality, and cardinality; basicoperations with different number bases; and one-to-one correspon-dence. Thirty-four of these topics were included in at least eight of theteacher textbooks which were surveyed. The results of this examina-tion agreed with an earlier study made by Shryock (1962) in that thesame areas tended to have the most agreement as well as disagree-ment by textbook authors.There were seventy-nine topics included in the analysis of elevenarithmetic series or teachers7 guides for grades 1-7 published since1962, (Hicks, p. 282-231). It was noted that nineteen topics werecommon to all textbook series analyzed, and thirty-seven topics wereincluded in at least half of the series examined.In addition to listing the topics obtained from the four types ofsources, the topics were rated as Level I, II, or III according to thecriteria previously listed. Table I shows that fifty-one topics werecategorized as Level II and thirty-five were categorized as Level III.The Level III topics are indicated by an asterisk preceding the topictitle.Mathematics for Elementary Teachers 741TABLE I. SUGGESTED TOPICS FOR THE MATHEMATICAL STUDY OFELEMENTARY SCHOOL TEACHERSStatis-Topic ticalStudy*Set Terminology*Set OperationsDefinitions and AssumptionsNature of Proof*Relations and Functions*Whole Number Operations XUse of Zero X*Counting and 1-1 Correspon-dence*0rder and Cardinality*Field PropertiesHistory of Numbers*Different Numeration Systemsand Place Value X*Ancient Numeration Systems*Roman Numeration X*Primes and Composites*Factors and Multiples*Exponents and ExponentialNotationFundamental Theorem of Arith-metic*Divisibility RulesFinite Mathematical Systems*The Number LineSigned Numbers*Common Fractions X^Decimal Fractions X*Percentage XInterest on Money*Ratio and Proportion X*Real Numbers*Square Root*Measurement X*Precision and Error*Formulae and SubstitutionCo-ordinate Geometry*Basic Concepts of Geometry*Geometric Figures*Metric System and ConversionIndirect MeasurementScale DrawingsBasic Constructions*Equations and Symbols X*Inequations*Central Tendency XDispersion^Statistical Graphs XProbability*Problem SolvingRecom-menda-tionStudyXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXTeacherTextsXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXSchoolTextsXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXTotal3311241323142333312131444142243313331114231312742School Science and MathematicsTable I. (Continued)TopicQ .Recom-bLatls-menda- Teacher School^, ,tlcaldon Texts Texts rotalstud>rStudy*MakingEstimationsX X 2*Rationalizing AlgorithmsX X X 3*CheckingProceduresX X X 3Logarithms, Slide Rules,andCalculatorsX 1Trigonometric FunctionsX 1* Indicates topic meets criteria for Level III.It must be noted that the table can be somewhat deceptive withregard to certain topics. For example, problem solving and rationali-zation of algorithms are certainly advocated by mathematics educa-tors and advisory groups, but these topics were typically camou-flaged with discussion of more specific topics which are noted in thetable. Such ambiguity did not in any case prevent a topic from beingclassified as Level III.CONCLUSIONS AND RECOMMENDATIONSThe four types of sources used in this study reveal the wide varia-tion and lack of unanimity as to the specific mathematics backgroundneeded by elementary teachers. However, assuming that the con-sensus of researchers, mathematics educators, advisory groups,teacher textbook writers, and pupil textbook writers has value, itappears that at least the thirty-five Level III topics should be in-cluded in some manner in the mathematics preparation of elementaryschool teachers. Many of these topics have not been tested experimen-tally for their appropriateness for the mathematics curriculum forelementary school teachers, and for many topics there is no ^con-crete" evidence to support the position that elementary teachersneed to study these topics. Certainly there is little evidence extantwith regard to the most efficient sequencing of topics in the mathe-matics preparation for elementary school teachers. It is essential thatmathematics educators continue to seek well-defined evidence tosupport their views on the introduction and sequencing of topics inthe mathematics curriculum for preparing elementary schoolteachers.It is believed that the educators responsible for pre-service andin-service preparation of elementary school teachers in mathematicscan utilize Table I to appraise present teacher-preparation courses inmathematics or for the planning of such courses. For example,Mathematics for Elementary Teachers 743Hicks (p. 237-326) has developed a course outline including all butthree of the topics in Table I. Hicks outline includes seventeen otherLevel I topics not listed in Table I which would facilitate the devel-opment of the Level III topics listed.SUMMARYThis study was designed to place the development of programs ofstudy in mathematics for elementary school teachers upon a founda-tion of topics which met certain selected criteria within the confinesof a descriptive research design. There is a continuing need to examinethese topics more analytically, to develop mathematics curricula forteachers based upon the findings of this study, and to evaluate suchprograms to determine if their products (both elementary schoolteachers directly and their pupils indirectly) can react in a more posi-tive manner in mathematical situations.SELECTED REFERENCESALLMAN, REVA WHITE. "A Study of Selected Competencies of ProspectiveTeachers in Alabama," University of Michigan, Dissertation Abstracts, XII,(January, 1952), p. 143-144.BEAN, JOHN ELLIS. "The Arithmetical Understandings of Elementary SchoolTeachers," Stanford University, Dissertation Abstracts, XIX, (October, 1958),p. 708.BREWER, MADISON. ^Elementary Education Majors Understanding and Knowl-edge of Common Fractions," Colorado State College, Colorado State College ofEducation Abstracts of Field Studies, XIII, (1951), p. 6-9.CARROLL, EMMA C. "A Study of the Mathematical Understandings Possessed byUndergraduate Students Majoring in Elementary Education," Wayne StateUniversity, Dissertation Abstracts, XXII, (August, 1961), p. 494-495.CRESSWELL, JOHN L. "An Analysis of the Relationships of Selected Factors toMathematics and Arithmetic Competency of Prospective Elementary Teachersin Georgia." Unpublished Doctoral Dissertation, University of Georgia,Athens, 1963.DUTTON, WILBUR H. AND AUGUSTINE P. CHENEY. "Pre-Service and In-ServiceEducation of Elementary School Teachers in Arithmetic," The ArithmeticTeacher, XI, (March, 1964) p. 192-198.EDUCATIONAL SERVICES INCORPORATED. Goals for School MathematicsA Reportof the Cambridge Conference on School Mathematics. Boston: HoughtonMifflin Company, 1963.FULKERSON, ELBERT. ^How Well Do 158 Prospective Elementary TeachersKnow Arithmetic?" The Arithmetic Teacher, VII, (March, 1960), p. 141-146.GROSSNICKLE, FOSTER E. "Growth in Mathematical Ability Among ProspectiveTeachers of Arithmetic," The Arithmetic Teacher, IX, (May, 1962), p. 278-279.HARPER, E. HAROLD. "Elementary Teachers Knowledge of Basic ArithmeticConcepts and Symbols," The Arithmetic Teacher, XI, (December, 1964), p.543-546.HICKS, RANDALL C. "A Program of Study in Mathematics for Elementary SchoolTeachers Based Upon Exhibited and Derived Needs." Unpublished DoctoralDissertation, University of Georgia, Athens, 1966.JONES, GEORGE LUCAS. "A Study to Determine Which Basic Mathematical Con-cepts Commonly Presented in Grades Four Through Eight Are Least Under-744 School Science and Mathematicsstood by Certain Elementary Education Majors." Unpublished DoctoralDissertation, Colorado State College, Greeley, 1962.JONES, PHILLIP S. "Mathematics," Recent Research and Developments and TheirImplications for Teacher Education, p. 97-102. Thirteenth Yearbook of theAmerican Association of Colleges for Teacher Education. Washington, D. C.:American Association of Colleges for Teacher Education, 1960.MATHEMATICAL ASSOCIATION or AMERICA. Recommendations for the Training ofTeachers of Mathematics, A Summary. A Report Prepared by the Committeeon the Undergraduate Program in Mathematics. Berkeley, California: Com-mittee on the Undergraduate Program in Mathematics, January, 1961.MAYOR, JOHN R. "Science Teacher Certification," Science, (June 17, 1960), p.1779.NAFZIGER, MARY KATHERINE. "A Study of Selected Arithmetic Understandingsof Undergraduate Students in the Elementary Teacher Preparation Programat Goshen College." Unpublished Doctoral Dissertation, Northwestern Uni-versity, Evanston, Illinois, 1961.NATIONAL ASSOCIATION or STATE DIRECTORS OF TEACHER EDUCATION AND CERTI-FICATIONAmerican Association for the Advancement of Sciences. ImprovingScience and Mathematics Programs in American Schools. "Guidelines for Scienceand Mathematics in the Preparation Program of Elementary School Teachers,p. 3-10. Washington, D. C.: The Associations, 1960.NATIONAL COUNCIL or TEACHERS OF MATHEMATICS. Instruction in Arithmetic.Twenty-Fifth Yearbook of the National Council of Teachers of Mathematics.Washington, D. C.: National Council of Teachers of Mathematics, 1960.NELSON, L. DOYAL AND WALTER H. WORTH. "Mathematical Competence of Pro-spective Elementary Teachers in Canada and in the United States," TheArithmetic Teacher, VIII, (April, 1961), p. 147-151.(VDoNNELL, JOHN ROBERT. "Levels^ of Arithmetical Achievement, AttitudesToward Arithmetic, and Problem-Solving Behavior Shown By ProspectiveElementary Teachers," Pennsylvania State University, Dissertation Abstracts,XIX, (December, 1958), p. 1300.PHILLIPS, CLARENCE ALOIS. "The Relationships Between Achievement in Ele-mentary Arithmetic and Vocabulary Knowledge of Elementary Mathematicsas Possessed by Prospective Elementary Teachers," University of Illinois,Dissertation Abstracts, XX, (November, 1959), p. 1687.Program for College Preparatory Mathematics. A Report of the Commission onMathematics, "Implementation: The Vital Role of Teacher Education," p.48-50. New York: College Entrance Examination Board, 1959.REHAGE, KENNETH J. "Teachers for the Emerging Elementary School," Frontiersof Elementary Education, VI, (1960), p. 8-15.SHRYOCK, ALFRED JERRY. "A Study of Mathematics Courses for ElementarySchool Teachers." Unpublished Doctoral Dissertation, State University ofIowa, Iowa City, 1962.STONEKING, LEWIS W. AND RONALD C. WELCH. "Teachers7 and Students^ Under-standing of Arithmetic," Indiana University School of Education Bulletin,XXXVII, (September, 1961), p. 1-56.SUELTZ, BEN A. "The Mathematical Training of Elementary School Teachers,"SCHOOL SCIENCE AND MATHEMATICS, LVIII, (June, 1958), p. 441-442.WATCHES SUFFER TROPICAL INFECTIONSTropical diseases plague not only humans but their wristwatches as well.Soviet microbiologists have found that the prime cause of wristwatch failure inthe tropics is a massive attack on the lubricating oil by microorganisms.Fungi and bacteria affect watch oil under conditions of high temperature andhumidity, they report. The presence of salts of nitrogen and phosphorus acceler-ates the organisms7 breakdown of the oil. The answer, they found, lies in dopingwatch oil for the tropics with organo-mercury compounds in a ratio of one partper thousand of oil.