Topics in Mathematics for Elementary School Teachers

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  • Topics in Mathematics for Elementary School TeachersRandall C. Hicks, Valdosta (Ga.) State College

    and

    Alex F. Perrodin, University of GeorgiaAthens, Ga. 30601

    The 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 these

    four 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 or

    advisory groupsc. appear in at least eight of the sixteen college textbooks in mathematics for

    elementary school teachersd. appear in at least six of the eleven arithmetic series or teachers guides for

    grades 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.

    739

  • 740 School Science and Mathematics

    FINDINGSA total of ninety-eight topics was located in the four types of

    sources. 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 eleven

    arithmetic 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 741

    TABLE I. SUGGESTED TOPICS FOR THE MATHEMATICAL STUDY OFELEMENTARY SCHOOL TEACHERS

    Statis-Topic tical

    Study

    *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 Exponential

    NotationFundamental 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 Solving

    Recom-menda-tionStudy

    XXXXXX

    X

    XX

    X

    XX

    X

    X

    XXX

    XXXXXXXXXXX

    X

    XXXX

    TeacherTexts

    XX

    X

    XXX

    XXXXX

    X

    XX

    XXXXX

    XXXXXX

    XXX

    XX

    X

    SchoolTexts

    XX

    XX

    XXX

    XXXXX

    X

    XXX

    XXXXX

    XXX

    XXX

    XXXXX

    X

    X

    Total

    3311241

    3231

    42333

    3

    12131444142243313331114231312

  • 742School Science and Mathematics

    Table I. (Continued)

    TopicQ .Recom-bLatls-menda- Teacher School

    ^, ,tlcaldon Texts Texts rotal

    stud>rStudy*MakingEstimationsX X 2*Rationalizing AlgorithmsX X X 3*CheckingProceduresX X X 3Logarithms, Slide Rules,andCalculatorsX 1

    Trigonometric 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 743

    Hicks (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 of

    study 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 S