mathematical competencies of preservice elementary school teachers

Mathematical Competencies of PreserviceElementary School Teachers

Robert E. ReysUniversity of Missouri Columbia, Missouri 65201

INTRODUCTION

Inadequate mathematical preparation of elementary school-teachers has been prevalent in many educational institutions.1 Thevoluminous amount of publications critical of mathematics programsfor prospective elementary-school teachers offers evidence of thenationwide concern of educators regarding this problem. Most of theprior research has been limited to isolated courses within the mathe-matics preparatory program. Furthermore the instruments employedin these investigations were designed to measure competency in^traditional57 mathematics. Consequently little direct assistance hasbeen provided by earlier research for teacher education colleges inimproving their contemporary mathematics program for elementary-school teachers.The rapid revolution in mathematics has precipitated considerable

changes in terminology and symbolism as well as the inclusion ofmany mathematical concepts which are new to the elementary schoolcurriculum. Unfortunately the changes in the mathematics prepara-tory program for elementary teachers have not kept pace with cur-ricular changes in the elementary school mathematics program.A number of definite curricular recommendations have been pro-

posed by mathematics educators to improve the mathematicspreparatory program for the future teachers of mathematics. Inparticular the Committee on Undergraduate Programs in Mathema-tics, the major group influencing the modernization of collegiatemathematics, has recommended minimal mathematics requirementsfor elementary school-teachers.2’3 This committee has clearly statedthat a uniform mathematics preparatory program for elementaryeducation majors would not be in the best interests of all educationalinstitutions. Before adequate curricular changes can be implementedit is necessary for institutional personnel in mathematics and mathe-matics education to examine critically their preservice mathematicsprograms in light of the changing needs of their teachers.

Therefore, action research at the institutional level is essential toprovide a basis for curricular improvement in the mathematicspreparatory program for elementary-school teachers.

1 Committee on the Undergraduate Programs in Mathematics, Ten Conferences on the Training of Teachers ofElementary School Mathematics, April, 1964.

2 Committee on Undergraduate Programs in Mathematics, "Recommendation of the Mathematical Associa-tion of America for the Training of Mathematics Teachers," American Mathematical Monthly, December, 1960.

3 Johnson, Donovan A., and Robert Rahtz. The New Mathematics in Our Schools. (New York: MacmillanCompany, 1966.)

302

Mathematical Competencies of Preservice Teachers 303

PURPOSESThis investigation was undertaken to determine the present mathe-

matical status of elementary education majors at the University ofMissouri at Columbia. One of the primary purposes of this study wasto ascertain how preservice elementary-school teachers perform incontemporary mathematics.

METHODSForms W and X of the Contemporary Mathematics Test, Algebra

Level, were employed in this study. This examination contained sevensubtests including the following topics: basic concepts, algebraic ex-pressions, factors, exponents and radicals, real number system,mathematical statements, proof and nature of proof, and functionsand graphs. This instrument was selected because it proposed tomeasure mathematical competencies which seemed commensuratewith the content in the mathematics preparatory program for ele-mentary teachers at the UniversityThe 1965 spring norms were based on a nationwide standardization

of eighth and ninth grade students completing a first year course inalgebra. These statistics indicated a range on the reliability coefficientfrom .70 to .87, a mean of 26 raw score points, and a standard devia-tion of 3.5 points.4The sample of students for this study was composed of 252 ele-

mentary education majors who were enrolled in one of three coursesduring the 1965-66 academic year at the University of Missouri atColumbia: a mathematics content course, exclusively for elementaryeducation majors; an undergraduate course in methods of teachingelementary-school mathematics; and a senior college or graduatelevel methods course in problems of teaching elementary-schoolmathematics. A pre-test was administered the first week of the semes-ter and a post-test was given the final week of the semester. Onlyresults from elementary education majors who participated in boththe pre-test and post-test were analyzed.

ANALYSIS OF DATAThe scores on the pre-test and post-test were compared statistically

to determine if there was a significant gain in raw score units for thecontent and methods classes. Two t tests were performed to deter-mine if the means on the pre- and post-test did in fact differ. A com-parison was made between the means of the students in the previouslydescribed classes and the norm for eighth and ninth grade pupils.Hence a z test was employed to determine if these means were in fact

4 Contemporary Mathematics Test: Spring 1965 Norms. Monterey, California, California Test Bureau, 1965.

304 School Science and Mathematics

significantly different. In an effort to identify specific areas of weak-ness in the mathematics preparatory program for elementary schoolteachers, an index of difficulty for each item of the post-test, formW, was established.

FINDINGSPre- and Post-Test Results. The composite results of the tabulation

of the pre- and post-Algebra Level tests are reported in Table 1. Thistable shows the classes and number of students in the respectiveclasses, in addition to reporting the various statistics. These dataindicate that the elementary education majors in the mathematicscontent, undergraduate methods or graduate methods courses madesignificant gain in score points beyond the .001 level of confidence.The pre-test mean for the methods classes was more than four

points greater than the pre-test mean for the content sections. Thisdifference might be attributed to the fact that a majority of the ele-mentary education majors in the methods classes had previouslycompleted the content course in mathematics.

College Mathematics Versus No College Mathematics. It has beenpreviously reported that nearly one-fourth of the recent elementaryeducation graduates of the University of Missouri at Columbia.earned no credit in college mathematics.5 One may question whetherthe mathematical background of these students was sufficientlystrong to warrant the omission of all mathematics courses from theirprogram. Although the solution of this problem was not within thescope of this study, the scores of students in methods of teaching ele-mentary-school mathematics and problems of teaching elementary-school mathematics were analyzed to determine if students who hadearned credit in college mathematics showed approximately the samegain from the pre- to post-test as those students who had earned nocredit in college mathematics.

All of the students taking the graduate methods course constitutedone group. A second group was formed by combining students in sec-tions of the undergarduate methods course. Each of these mutuallyexclusive groups was dichotomized. One category was composed ofstudents who had earned no semester hours in college mathematics,while the other contained students who had earned credit in collegemathematics. No discrimination as to the nature or level of the par-ticular mathematics courses was made in this investigation. Theresults of these comparisons are shown in Table 2.An inspection of Table 2 shows that the mean gain from the pre- to

post-test for the two groups differed significantly only in the graduate

6 Reys, Robert E., "Are Elementary Education Graduates Satisfied With Their Mathematics Preparation?"The Arithmetic Teacher. March. 1967.


TABLE 1. COMPARISON OF PRE AND POST RESULTS OF ELEMENTARY EDUCATIONMAJORS ON ALGEBRA LEVEL OF CONTEMPORARY MATHEMATICS TEST

Class

Mathematics ContentCourse

Methods of TeachingElementaryMathematics

Problems of TeachingElementaryMathematics

N

93

141

18

Pre TestMean

16.806

20.560

21.556

Post TestMean

24.785

24.468

27.278

S&14-1-M20.598

0.436

1.158

t

13.35***

8.97***

4.94***

*** Significant beyond the .001 level.

TABLE 2. COMPARISON OF DIFFERENCES FROM PRE-TEST TO POST-TEST ONCONTEMPORARY MATHEMATICS TEST OF ELEMENTARY EDUCATION MAJORS WHOHAD COMPLETED SOME COLLEGE MATHEMATICS COURSES WITH STUDENTS WHO

HAD NOT EARNED CREDIT IN COLLEGE MATHEMATICS

Class �. No Credit ^ Creditv Mean Gain Mean Gain

Methods of TeachingElementary Mathematics

Problems of TeachingElementary Mathematics

53

6

3.842

4.833

88

12

3.948

7.500

0.441

0.972

.24

2.74*

* Significant beyond the .05 level.

methods course. The mean gain for students having earned credit inmathematics was greater than the mean gain for students earning nosemester hours in mathematics at the .05 level of confidence. A com-parison of the students in the undergraduate methods course re-vealed that the mean gain for students with credit in college mathe-matics was only slightly greater than the mean gain for students whohad not earned credit in college mathematics.

POST-TEST ANALYSIS

An analysis of the post-test scores on the Algebra Level was under-taken in order (1) to ascertain if the mathematical proficiency demon-strated on this test is adequate for elementary school-teachers, and(2) to diagnose specific areas of weakness in mathematics. Table 3shows a comparison of the post-test results in each of the respectiveclasses with the statistics supplied by the California Test Bureau.

It is evident that the post-test means for the mathematics contentand undergraduate methods courses were significantly below themeans of the eighth and ninth grade pupils completing a first yearalgebra course. The mean of the graduate methods course was above


TABLE 3. COMPARISON OF POST-TEST RESULTS OF ELEMENTARY EDUCATIONMAJORS AND NORM ON ALGEBRA LEVEL OF CONTEMPORARY MATHEMATICS TEST

^, ,7. Post Test T.T o T?Class N^^^ Norm S.E. z

Mathematics Content 93 24.785 26.000 0.363 -3.35**Methods of Teaching 141 24.468 26.000 0.295 -5.19**ElementaryMathematics

Problems of Teaching 18 27.278 26.000 0.825 1.24ElementaryMathematics

** Significant beyond the .01 level.

the norm; however, this difference was not statistically significantat the .05 level of confidence.From the analysis of the data in Tables 1 and 3 it is apparent that

although a noticeable improvement was made from the pre-test to thepost-test, the post-test means for a sizeable number of elementaryeducation majors were significantly below the means of eighth andninth grade pupils. Nearly 25 percent of the elementary educationmajors scored below the twentieth percentile. Less than 5 percentof these students scored above the ninetieth percentile.

Consequently the following analysis of the post-test results wasundertaken to aid in the diagnosis of particular areas of weakness inmathematics. An index of difficulty for each item of the post-test ofthe Algebra Level, Form W, was established. These calculations re-vealed that from a total of 54 questions, 30, 28, and 27 test items wereanswered incorrectly by 50% or more of the elementary educationmajors in mathematics content, methods of teaching elementary-school mathematics, and problems of teaching elementary-schoolmathematics, respectively. Further item analysis revealed that 21 ofthe questions had a difficulty level of less than or equal to .50 for allof the classes.In order to analyze the difficulty in specific subject areas, each test

item was placed in a content category designation as determined bythe California Test Bureau. It was found that the preservice elemen-tary-school teachers failed approximately 40, 67, and 80% of theitems from sections on the real number system, mathematical state-ments and functions and graphs, respectively. Test items on algebraicexpressions and operations were found to be answered correctly morefrequently by the preservice elementary-school teachers.

SUMMARY AND DISCUSSION

The analysis of the data showed that approximately 55% of thepreservice elementary education majors scored below the median for


eighth and ninth grade students on the Contemporary MathematicsTest, Algebra Level. This finding alone reveals a critical need for im-proving the mathematics preparation of these prospective elemen-tary-school teachers. No claim is made that the instrument em-ployed in this research assesses all the mathematical competencieswhich should be possessed by elementary education majors. Indeed,conspicious by their absence are questions related to geometry. Itwould not be unreasonable to prognosticate that similar findingswould be obtained if an instrument designed to measure competencyin elementary geometry were administered to these elementaryeducation majors.There has been considerable speculation regarding the appropriate

sequence of courses in the mathematics preparatory program. Forexample, is it pedagogically sound to offer separate courses in mathe-matics content and methods for elementary-school teachers? Theanswer to this latter question is not within the scope of this study.However, if an educational institution does offer separate contentand method courses, then a logical question would be, ^What is theproper sequence of courses?^The comparison of the mean gains of students who had earned

credit in mathematics as opposed to those students with no collegemathematics failed to produce conclusive evidence. Both groups inthe undergraduate methods course showed approximately the samegain on the test. In the graduate level methods course the mean gainwas significantly higher for those students who had earned collegecredit in mathematics.These results represent only change in performance as shown by

the Contemporary Mathematics Test for the two groups. No data wasavailable in either methods course to consider the difference in degreeof mastery of teaching techniques between the two groups. Never-theless, this evidence would lend some support to educators advocat-ing a mathematics content course prior to the methods course.

It is noteworthy that at least 27 of the 54 test items were missed byfifty percent or more of the students. When one considers that thesetest items were designed for eighth and ninth grade pupils completinga first-year course in algebra, this statistic provides further impetusfor improving the overall mathematical competency of prospectiveelementary-school teachers. Three categories on the Algebra Leveltest appeared to be the most formidable for the students: the realnumber system, mathematical statements, and functions and graphs.These particular topics play a prominent role in the elementary

school mathematics program as evidenced by a review of any con-temporary elementary school mathematics text. This vividly illus-trates the crux of the problem of adequately preparing elementary-school teachers of mathematics.


In retrospect, this investigation indicates that the mathematicsscholarship of a large percentage of elementary education majors isunsatisfactory, even though noticeable improvement was made dur-ing their mathematics preparatory program. Furthermore the ex-posure of particular areas of weakness in the mathematics prepara-tion of the students in this study gives needed direction for alleviatingthese specific problem areas in mathematics.

REFERENCESCommittee on Undergraduate Programs in Mathematics, "Recommendation of

the Mathematical Association of America for the Training of MathematicsTeachers," American Mathematical Monthly, December, 1960.

Committee on the Undergraduate Programs in Mathematics, Ten Conferences onthe Training of Teachers of Elementary School Mathematics, April, 1964.

Contemporary Mathematics Test: Spring 1965 Norms. Monterey, California,California Test Bureau, 1965.

JOHNSON, DONOVAN A., and ROBERT RAHTZ. The New Mathematics in our Schools.(New York: Macmillan Company, 1966.)

REYS, ROBERT E., "Are Elementary Education Graduates Satisfied With TheirMathematics Preparation?" The Arithmetic Teacher, March, 1967.

RADIATION SAVES SIGHT OF SIX WITH CANCERSix children with a form of eye cancer called rhabdomyosarcoma have ade-

quate to good vision as a result of radiation treatment, a New York radiologistrevealed.

Dr. Robert H. Sagerman of the Columbia-Presbyterian Medical Center saidthat when cancer in the eye socket remains localized and has not spread to otherparts of the body, intense radiation can destroy the disease without significantafter-effects. In adults the disease spreads readily, but with children it tends toremain localized.

Another report showed progress in the use of radiation to treat malignantlymph nodes. Dr. P. Ruben Kohler of the Washington University School ofMedicine, St. Louis, has found a way to block off the thoracic duct leading to thelungs, thus preventing excessive radiation in the lymph system from spilling overinto the lungs.He and his co-workers inserted a tube into the ducts of dogs used in the ex-

periment. Increased amounts of radioactivity were studied but found to have nomore effect than the lesser quantity previously given.

SPACE AGE DICTIONARYThe complex language of space is being boiled down by Oregon State Univer-

sity, Corvallis, into a 600-term dictionary that even elementary students canunderstand.More than 300 textbooks and magazine articles have been combed for common

space terms read by students in grades four through eight. Once the selectedterms are examined for scientific accuracy and reviewed by the OSU School ofEducation, the dictionary will be tested in six Oregon schools.The dictionary is being developed under contract from the National Aeronau-

tics and Space Administration, which will illustrate and publish it.

mathematical competencies of preservice elementary school teachers

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