An Analysis of Cognitive Achievement in a NumberSystems Course for Prospective Elementary School

Teachers

Robert Sovchik

Department of Elementary EducationThe University of Akron

Akron, Ohio 44325

INTRODUCTION AND STATEMENT OF THE PROBLEM

Within the last decade a great number of mathematics courses for pro-spective elementary school teahers have been developed at colleges anduniversities across the country. These mathematics courses have largelybeen patterned after the recommendations proposed by the Committeeon the Undergraduate Program in Mathematicss (CUPM, 1960) and theensuing Course Guides (1968). The Level I recommendations for pro-spective teachers of grades K-6 were the following:

1. A two course sequence devoted to the structure of the real number system and itssubsystems

2. A course devoted to the basic concepts of algebra3. A course in informal geometry.

Very little research evidence analyzing the effectiveness of thesecourses has been made available. Investigations by Gee (1966) and Todd(1966) indicated that significant cognitive improvement did occur in thenumber systems courses they investigated. On the other hand, Reys(1968) found that a majority of prospective elementary school teacherswere scoring below the algebra norms established for eighth and ninthgrade pupils. Withnell (1967) concluded that prospective elementaryschool teachers were not increasing their understanding of mathematiicsby taking either three, six or nine semester hours of mathematics coursespatterned after the CUPM proposals.The issue of evaluating mathematics courses for prospective elemen-

tary school teachers becomes even more important when one considersthe revised recommendations offered by the CUPM in 1971. These rec-ommendations consider such topics as integral domains, exponential andlogarithmic functions, and matrix algebra. Van Engen (1972) has ques-tioned the usefulness of these topics for prospective elementary schoolteachers. Since these more ambitious recommendations have been madein the absence of a solid empirical base of evaluation of the original(1960) recommendations, it would seem logical to study the effectives ofexisting courses before revising mathematics programs for prospective el-ementary teachers along the lines of the 1971 CUPM proposals. Moody

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Achievement in a Number Systems Course 67

and Wheatley (1969) have also called for a thorough evaluation of math-ematics courses for prospective elementary school teachers. Moreover,since contemporary mathematics educators have stressed intuitive ap-proaches to the discovery of mathematics concepts, it would seem in or-der to attempt an evaluation of a mathematics course for prospective ele-mentary school teachers with an instrument designed to measure morethan recall or computation.

Thus, this investigator focused on a number systems course for pro-spective elementary school teachers patterned after the 1968 CourseGuides of the CUPM. Student cognitive change was measured by usingan investigator constructed instrument designed to measure thoughtprocesses described by Bloom (1956).

Procedure:

After a review of several number systems textbooks and the suggestedcourse guides developed by the CUPM, four general objectives were de-veloped by the investigator for Mathematics 122, a beginning numbersystems course for prospective elementary school teachers at Kent StateUniversity. Each of the four general objectives were then taxonomizedinto six detailed behavioral objectives corresponding to the six processlevels described by Bloom (1956). Multiple choice items were written foreach general objective at a particular process level described in Bloom’sTaxonomy. After several pilot tests, the final instrument developed inthis study had 60 items. Each objective had 15 items written for it in thefollowing manner: Knowledge�2 items; Comprehension�2 items; Ap-plication�2 items; Analysis�3 items; Synthesis�3 items; Evalua-tion�3 items.The sample used in this investigation consisted of four sections of a

Mathematics 122 course, a beginning number system course, at KentState University. The four participating instructors teaching those sec-tions agreed to follow the taxonomized objectives developed by the in-vestigator. The cognitive instrument developed in this study was then ad-ministered on a pretest and posttest basis to 139 students and 143 stu-dents, respectively. The study was undertaken during Fall Quarter, 1973.Also, the Mathematics Subtest score of the American College TestingProgram (ACT) was located for 116 students enrolled in the course. Theresearch hypothesis of this investigation was the following:

Statistically significant cognitive changes take place in students enrolled in mathema-tics 122.

Results

The data obtained from this investigation were analyzed in severalways. The overall reliability of the posttest instrument, calculated by the

68 School Science and Mathematics

Kuder-Richardson Formula 20, was .695. Content validity was substanti-ated by using two mathematics education professors and one mathema-tics professor as classifiers of the investigator constructed items. Theitems were classified according to the four general cognitive objectivesdeveloped for the course and process levels of Bloom’s Taxonomy.

Since the test design was essentially a six by four matrix (the six processlevels of Bloom and the four cognitive objectives yielded 24 Cells in thematrix), a situation was presented in which there were several dependentvariables. Thus, the multivariate Hotelling P statistic was used to testwhether significant change scores had occurred (Cooley & Lohnes, 1971;Morrison, 1967). It was hypothesized that the mean difference vectorwith the mean difference score for each of the four cognitive objectivesas a component would be different from the zero difference vector. Theobtained T2 value of 337.064 was significant at the .05 level of signifi-cance. This result indicated that significant cognitive improvement didoccur in Mathematics 122.

Cognitive change across the six behavior levels described in Bloom’sTaxonomy was also investigated by the T1 statistic. It was hypothesizedthat the mean difference vector with six components representing themean difference scores for the six process levels described in Bloom’sTaxonomy would be different from the zero vector. The obtained T2 val-ue (P = 502.133) was significant at the .05 level of significance. This re-sult supported the significant T2 obtained by using the mean differencescore of each of the four subtests measuring the four cognitive objectivesas components and again affirmed that significant cognitive change oc-curred in students enrolled in Mathematics 122.The change scores of the students enrolled in the sample were further

analyzed by means of a part correlation coeffecient (Glass & Stanley,1970). A Pearson Product-Moment Correlation Coeffecient was calcu-lated between the pretest and posttest of the cognitive instrument. Thecorrelation coeffecient (r = .488) was significant at the .05 level of sig-nificance. Another correlation coeffecient was calculated between theposttest of the cognitive instrument and the Mathematics Subtest of theACT. The correlation coeffecient (r = .505) was significant at the .05level of significance. Also, a correlation coeffecient was calculated be-tween the pretest of the cognitive instrument and the Mathematics Sub-test of the ACT. This correlation coeffecient (r = .487) was significant atthe .05 level of significance. Using the proceeding information, a partcorrelation coeffecient was computed between the posttest and the Math-ematics Subtest of the ACT with the effects of the pretest parted outfrom the posttest. The part correlation coeffecient (r = .312) was signifi-cant at the .05 level of significance. This indicated that cognitive im-provement in Mathematics 122 was significantly related to performance

Achievement in a Number Systems Course 69

on the Mathematics Subtest of the ACT. That is to say, those studentswith good aptitude toward mathematics, as measured by the Mathe-matics Subtest of the ACT, tended to improve their scores more than stu-dents with low aptitude.

Summary and Conclusions:

This investigation attempted to measure whether significant improve-ment in students’ change scores was evidenced during a quarter numbersystems course at Kent State University. This course was patterned afterthe recommendations of the CUP’M (1960) and subsequent CourseGuides (1968). A lack of evaluative data for these courses was indicatedand a need for research presented, a need which is especially important inlight of the 1971 CUPM revisions of proposed course content.An investigator constructed instrument based on Bloom’s Taxonomy

was administered to four sections of a Mathematics 122 course at KentState University during the Fall Quarter, 1973. (n = 139 and 143 for thepretest and posttest, respectively).The students change scores were analyzed by means of the Hotelling T2

statistic because the design indicated a situation with several dependentvariables. Significant P values were obtained at the .05 level of signifi-cance, indicating that statistically significant change scores were evi-denced by students enrolled in Mathematics 122.

Thus, cognitive improvement did occur in this number systems course.Furthermore, the significant part correlation coefficient indicated thatthe Mathematics Subtest of the ACT served as a good predictor of suc-cess in Mathematics 122. Students with high ACT scores in Mathematicstended to evidence the greatest improvement in the course.

In conclusion, this study indicated that significant cognitive improve-ment occurred in a number systems course for prospective elementaryteachers. However, the nature of attitude change in this course and therelationship between attitude change and cognitive change was not inves-tigated. The development of positive attitudes toward mathematics is animportant outgrowth of a mathematics course for prospective elementa-ry school teachers. Further investigations of these CUPM courses forprospective elementary school teachers should address themselves toboth affective and cognitive change and the relationship between thesetwo. In particular, the relationship between the categories developed byBloom (1956) and Krathwohl (1964) may be particularly enlightening. Isa student who is functioning at the organization level of the affective do-main likely to be functioning at the analysis, synthesis or evaluation lev-els of the cognitive domain?

Although this instrument tested four objectives at six process levels,the number of items for each subtest was small (15 for each objective and

70 School Science and Mathematics

8 to 12 for each process level of the cognitive domain). A useful way tostudy change scores might be to determine whether one objective wasachieved more significantly than another. The investigator examined thedata to determine whether one objective was mastered more than anoth-er; however, no statistical comparisons were made due to the small num-ber of items measuring each objective. Investigations with more itemsper cell could lead to information indicating greater accomplishment ofprocess levels or objectives in a given course. This information would beextremely valuable in planning revisions of courses servicing prospectiveelementary school teachers.

REFERENCES

BLOOM, B., (Ed.). Taxonomy of educational objectives; the classification of educationalgoals; handbook I: Cognitive domain. (New York: David McKay, 1956.

COOLEY, W. W. & LOHNES, P. R. Multivoriate data analysis. New York: John Wiley, 1971.Committee on the Undergraduate Program in Mathematics: Recommendations of the

Mathematical Association of America for the training of mathematics teachers. Ameri-can Mathematical Monthly, 1960, 67, 982-990.

Committee on the Undergraduate Program in Mathematics. Course guides for the trainingof teachers of elementary school mathematics. Berkeley: CUPM, 1968.

Committee on the Undergraduate Program in Mathematics. Recommendations on coursecontent for the training of teachers of mathematics. Berkeley: CUPM, 1971.

GEE, B. C. Attitudes toward mathematics and basic mathematical understanding of pro-spective elementary teachers at Brigham Young University. (Doctoral dissertation, Ore-gon State University) Ann Arbor, Michigan: University Microfilms, 1966. No. 66-3923.

GLASS, G. V., & STANLEY, J. C. Statistical methods in education and psychology. PrenticeHall: New Jersey, 1970.

KRATHWOHL, D. R. (Ed.). Taxonomy of educational objectives: the classification of educa-tional goals; handbook II: Affective domain. New York: David McKay, 1964.

MOODY, W. B. & WHEATLEY, G. H. Evaluating mathematics courses for prospective ele-mentary school teachers. School Science and Mathematics. 1969, 69, 703-707.

MORRISON, D. F. Multivariate statistical methods. New York: McGraw-Hill, 1967.REYS, R. E. Mathematical competencies of elementary education majors. The Journal of

Educational Research, 1968, 61, 265-266.TODD, R. M. A mathematics course for elementary teachers: does it improve understandingand attitude? Arithmetic Teacher, 1966, 13, 198-202.

VAN ENGEN, H. The mathematical training of an elementary teacher. Arithmetic Teacher,1972,19, 517-518.

WITHNELL, M. C. A comparison of the mathematical understandings of prospective ele-mentary teachers in colleges having different mathematics requirements. (Doctoral dis-sertation, University of Michigan) Ann Arbor, Michfgan: University Microfilms, 1967,No. 68-7758.