high school students' achievement of selected mathematical competencies

High School Students’ Achievement ofSelected Mathematical Competencies

Edwin Giesbrecht

Algoma University CollegeSaultSteMarie,Ontario, Canada

Perhaps one of the most frequently expressed educational concernsduring the past decade is the mathematical incompetence of our youth.Educational institutions, businesses, industries, and the general publichave been seeking, and continue to seek, explanations for the drop in thelevel of basic mathematical achievement.Much of the blame for this drop in achievement level by high school

students in the United States and Canada has been directed at the "newmathematics" curriculum. The validity of this criticism is suspect.A recent report by the National Advisory Committee on Mathematics

Education (NACOME) revealed that declines in mathematics scores wereaccompanied by decreases in all school subjects�in fact, in some casesmathematics performance resisted the declining performance in otherschool subjects. Standardized test batteries, such as the Scholastic Apti-tude Tests (SAT) and the Iowa Tests of Basic Skills (ITBS), showed a de-cline during the period from 1962 to 1975 in students’ performance inmathematics, as well as in reading and language skills [1].

Second, there was reason to. believe that the fundamental principles ofthe "new mathematics" curriculum were never widely implemented inCanadian and American schools [2].The low level of competence of average, and even below average,

American citizens in mathematics, as well as in other subject areas, be-ginning in the early 1960’s, led to the formation of the National Assess-ment of Educational Progress (NAEP) in the mid-sixties. In its first re-port of mathematical competence in 1972-73, the NAEP revealed numer-ous startling results. The following were two examples:

1. Fifty-eight percent of thirteen-year-olds and 34 percent of seventeen-year-olds couldnot find the sum of 1 /2 and 1 /3 [3].

2. Sixty-five percent of seventeen-year-olds and 61 percent of young adults could notcorrectly answer a question asking for the lowest price per ounce, given the numberof ounces and the cost (twelve ounces for forty cents; fourteen ounces for forty-fivecents; one pound, twelve ounces for eighty-five cents; two pounds for ninety-ninecents) [4].

The concern over the low level of mathematical competence of theaverage citizen in Canada did not result in a significant number of earlyattempts to determine the extent of the problem or to examine relatedfactors.

277

278 School Science and Mathematics

PURPOSE

In 1976, the writer undertook to measure Saskatchewan high schoolstudents’ achievement of selected mathematical competencies, and there-by provide descriptive measures of performance in mathematics requiredfor full participation in contemporary society by the average citizen. Theset of mathematical competencies selected was the list of forty-eight com-petencies released by the National Council of Teachers in Mathematics(NCTM) in 1972. The list was divided into the following ten competencyareas: numbers and numerals, operations and properties, mathematicalsentences, geometry, measurement, relations and functions, probabilityand statistics, graphing, mathematical reasoning, business and consumermathematics. The committee of the NCTM that drew up this list believedthat acquisition of these competencies was essential for full participationm contemporary society [5].A second purpose of this study was to determine the effects of any of

the following four factors on Saskatchewan high school students’achievement of a selected list of mathematical competencies: gradelevel�grade nine, grade ten, grade eleven, and grade twelve, mathe-matics program�algebra, algebra-geometry (trigonometry), alternatemathematics, l and general mathematics, school enrolment size�small�1 to 87 students, medium�88 to 156 students, and large�157to 1816 students, student sex�female and male.The first purpose of this research project was stated in terms of the fol-

lowing research question: To what degree have Saskatchewan highschool students achieved the forty-eight mathematical competencies out-line in the 1972 report of the NCTM?

Null hypotheses regarding the differences between grade levels (ninethrough twelve), mathematics programs (algebra, algebra-geometry(trigonometry), alternate mathematics, general mathemtics), school size(small, medium, large), and student sex (female, male) were formulatedand tested.

METHOD

Stratified random sampling techniques were employed to obtain asample representative of the population of Saskatchewan students en-rolled in secondary school mathematics programs during the 1975-76school year. A proportionate number of high school students were se-lected randomly from Saskatchewan’s six school regions, from four dif-ferent mathematics programs, from three different categories of schoolenrolment sizes, and from the two student sexes.

All high schools in Saskatchewan offering mathematics programs in

1. The alternate mathematics program was designed to meet post-secondary institution entrance requirements in areasother than mathematics, natural science, and technology.

High School Students9 Mathematics Achievement 279

grades nine through twelve were requested to participate in this study.Usable test results were received from 161 high schools; that is, resultswere received from the following number of students in each of gradesnine through twelve respectively: 858, 856, 816, and 765. These numbersrepresent approximately 5 percent of the students enrolled in each highschool grade.The Beckmann-Beal Test, based on the list of forty-eight competencies

released by the NCTM in 1972, and the VR + NA subtest of the Differ-ential Aptitude Battery were administered by local teachers and guidancecounsellors to all students in the sample. The testing was conducted dur-ing the one-month time interval from May 15 to June 15, 1976.The number of students, together with the mean, standard deviation,

and range were determined for total competencies and the ten compe-tency areas in each of the following categories: grade level, mathematicsprogram, school size, and student sex. Analysis of covariance tech-niques, with intelligence as the covariate, were used to test for significantdifferences in mean attainment of total and area competencies among theabove four categories. Scheffe’s method was used to compare means ifsignificant differences existed.

RESULTS

The mean, range, and standard deviation for total competencies forstudents in each of grades nine through twelve, in each of the four typesof mathematics programs, three school enrolment sizes, and two sexesare shown in Table I. Table II shows the mean score obtained for each ofthe ten area competencies in each of the four high school grades.The mean score 2 and standard deviation respectively for total compe-

tencies in each of the four grades were as follows: grade nine�20.8(43.3%), 9.7, grade ^�24.3 (50.5%), 9.8, grade eleven�21.8 (57.9%),9.5, grade twelve�31.1 (64.7%), 9.2. While grades ten, eleven, andtwelve students enrolled in mathematics programs in 1975-76 achieved amajority of the forty-eight "basic" mathematical competencies releasedby the NCTM in 1972, grade nine students failed to achieve a majority ofthose mathematical competencies considered essential for full participa-tion in contemporary society.

Students across all four grade levels achieved consistently the lowestscores in the areas of probability and statistics, geometry, and businessand consumer mathematics. The competency area receiving the highestscore by students at all grade levels was mathematical reasoning.One-way analysis of covariance techniques were emioyed to examine

achievement of total competencies across the four grade levels. The re-

2. A competency area was said to have been attained only if a student correctly responded to both test related questionson The Beckmann-Beal Test.

School Science and Mathematics

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TABLE ilMEAN ATTAINMENT OF THE COMPETENCIES LISTED IN THE TEN AREAS BY

STUDENTS IN GRADES 9,10,11, AND 12

Grade 9 Grade 10 Grade 11 Grade 12Competency Area Mean Mean Mean Mean

Numbers andNumerals0.480.560.64 0.732.Operations and properties3.Mathematical sentences4.Geometry5.Measurement6.Relations and functions7.Probability and statistics8.Graphing9.Mathematical reasoning

0.490.550.620.690.540.600.640.690.330.430.540.600.470.530.580.650.480.570.650.720.200.220.270.360.450.560.640.710.540.610.690.75

10. Business and consumer mathematics 0.37 0.44 0.52 0.59

suits are shown in Table III. A Scheffe test of the over-all main effects ofadjusted means provided the information in Table IV.

TABLE illONE-WAY ANALYSIS OF COVARIANCE FOR THE DEPENDENT VARIABLE TOTAL

COMPETENCIES ACROSS GRADES 9, 10, 11, AND 12’

Source of Sum of Degrees MeanVariation Squares Freedom Square F-ratio

Grade 1 759.86 3 586.62 13.73*Error 124 511.00 2 914 42.73

* p<0.05

TABLE IVSCHEFFE TEST OF OVER-ALL MAIN EFFECTS OF ADJUSTED MEANS FOR TOTAL

COMPETENCIES AMONG GRADES 9,10, 11, AND 12

Grade Completion F-value*

9-109-119-1210-12

4.027.6712.663.26

* p<0.05, F(3, 2 914), for all F-values

Since there were statistically significant (p<0.05) differences in meancompetency totals adjusted for intelligence between students in gradenine and those in grades ten, eleven, or twelve, and also between studentsin grade ten and those in grade twelve, the first hypothesis, no statistical-ly significant differences exist among high school grade levels in mathe-matical competency attainment, was rejected.


One-way analysis of covariance techniques also were used to examinethe effect of high school mathematics program at the grade nine level.The results are shown in Table V. There was no statistically significantdifference in mean competency totals adjusted for intelligence betweengrade nine students enrolled in the algebra program and those studentsenrolled in the general mathematics program. This result correspondswith earlier findings reported by Beckmann [6,7].

TABLE VONE-WAY ANALYSIS OF COVARIANCE FOR THE DEPENDENT VARIABLE TOTAL

COMPETENCIES ACROSS THE MATHEMATICS PROGRAMS IN GRADE 9


Program 96.51 1 96.51 2.62*Error 27 012.00 732 36.90

* Not significant at the 0.05 level of significance

Two-way analysis of covariance techniques were used to examine theeffect of high school mathematics program, school size, and student sex,on student achievement of total competencies across the tenth, eleventh,and twelfth grade levels.

Table VI reveals the results of a two-way analysis of covariance[programs (4) x grades (3)] on the four mathematics programs in each ofgrades ten, eleven, and twelve. The results of the Scheffe test are pro-vided in Table VII.

TABLE VITWO-WAY ANALYSIS OF COVARIANCE FOR GADES 10, 11, AND 12 AND MATHEMATICS

PROGRAMS ALGEBRA, ALGEBRA-GEOMETRY (TRIGONOMETRY), ALTERNATE MATHEMATICS,AND GENERAL MATHEMATICS


Program (A) 3 073.10 3 1 024.37 23.64*Grade (B) 75.92 2 37.96 0.88A X B 296.63 6 49.44 1.14Error 93 347.00 2 154 43.34

* p<0.05

High School Studentsf Mathematics Achievement 283

TABLE VIISCHEFFE TEST OF OVER-ALL MAIN EFFECTS OF ADJUSTED MEANS FOR TOTAL

COMPETENCIES OVER MATHEMATICS PROGRAMS ALGEBRA, ALGEBRA-GEOMETRY(TRIGONOMETRY), ALTERNATE MATHEMATICS AND GENERAL MATHEMATICS

Program Comparison F-value*

Algebra�General Mathematics6.85Algebra-Geometry (Trigonometry)�Algebra9.24Algebra-Geometry (Trigonometry)�Alternate Mathematics6.97Algebra-Geometry (Trigonometry)�General Mathematics17.52

* p<0.05, F(3,2 154), for all F-values

TABLE VIIITWO-WAY ANALYSIS OF COVARIANCE FOR GRADES 9, 10, 11, AND 12 AND SCHOOL

SIZES (SMALL, MEDIUM, AND LARGE)


Size(C)447.672223.83 5.52*Grade(B)1 758.043586.01 14.44*C x B190.21631.70 0.78Error102 270.002 52040.58

* p<0.05

Since there were statistically significant (p<0.05) differences in meancompetency totals adjusted for intelligence between students enrolled inthe algebra-geometry (trigonometry) program and students enrolled inany of the algebra, alternate mathematics, or general mathematics pro-grams at the tenth, eleventh, and twelfth grade levels, and since there al-so were statistically significant (p<0.05) differences in mean competencytotals adjusted for intelligence between students enrolled in the algebraprogram and students enrolled in the general mathematics program atthe tenth, eleventh, and twelfth grade levels, the second hypothesis, nostatistically significant differences exist among high school mathematicsprograms in mathematical competency attainment, was rejected.A two-way analysis of covariance [sizes (3) x grades (4)] was con-

ducted on the three school enrolment sizes in each of grades nine throughtwelve. The results of the analysis are presented in Table VIII. TheScheffe test was applied to the over-all main effects of the adjustedmeans and the results are shown in Table IX.

Since there were statistically significant (p<0.05) differences in meancompetency totals adjusted for intelligence between students who at-tended large enrolment size schools and those who attended either smallor medium enrolment size schools, the hypothesis, no statistically signifi-cant difference exists among school enrolment size in mathematical com-petency attainment, was rejected.


TABLE IXSCHEFFE TEST OF OVER-ALL MAIN EFFECTS OF ADJUSTED MEANS FOR TOTAL

COMPETENCIES OVER SCHOOL SIZES (SMALL, MEDIUM, LARGE) IN GRADES 9, 10, 11, AND 12

Size Comparison F-value*

Small�Large 4.11Medium�Large 3.73

* p<0.05, F(2, 2 520) for all F-values

A two-way analysis fo covariance [sexes (2) x grades (4)] was per-formed on the two sexes in each of grades nine through twelve. Theresults of the analysis are shown in Table X.

TABLE XTWO-WAY ANALYSIS OF COVARIANCE FOR GRADES 9,10, 11, AND 12 AND STUDENT

SEX (FEMALE, MALE)


Sex(D)2845.2212 845.22 68.12*Grade(B)1926.223642.07 15.37*D x B90.69330.23 0.72Error121759.002 89141.77

* p<0.05

Since male high school students attained a significantly (p<0.05) great-er number of mathematical competency totals then female high schoolstudents, after adjustments were made for intelligence, the hypothesis,no statistically significant difference exists between male and female highschool students in mathematical competency attainment, was rejected.

In summary, based upon the evidence of this study, one can concludethat, when adjustments are made for intelligence, statistically significantdifferences in high school student mean achievement of total competen-cies existed among some grades, mathematics programs, school enrol-ment sizes, and between the sexes.

EDUCATIONAL IMPLICATIONS

On the basis of the findings obtained from the analysis of the data inthis research project, and within the scope of the limitations and delimi-tations underlying such a provincial study, the following educationalimplications are to be considered.

Serious consideration should be given to the problem of determiningwhat constitutes an acceptable level of basic mathematical competencefor full participation in contemporary society by the’average North


American citizen. The requirements imposed by minimal competencytests, now practised in some twenty-seven states, have not proved to becompletely satisfactory. The competencies are too minimal to stimulatethe above-average student and too difficult for the below-average stu-dent [8]. Probably what is required, rather, is a definition emphasizingfull, as opposed to limited, participation in current, technological so-ciety. Such a definition may consist of differing levels for each of the tencompetency areas outlined by the NCTM in 1972, and also for each highschool grade level.

All Departments of Education should establish and/or maintain aminimal requirement of the equivalent of two Saskatchewan credits 3 inmathematics for receipt of a high school diploma. This recommendationis tenable in view of the 1972 NCTM report which states that facility withthe majority of the forty-eight competencies contained in the report isnecessary for full participation in current technological society. The find-ings of this study revealed that students in the sample achieved masteryof a majority of the forty-eight competencies only after studying mathe-matics at both the ninth and tenth grade levels in Saskatchewan. Thepoor performance demonstrated by grade nine students in the sample isconsistent with findings reported earlier by Beckmann [6] and Niemann[9] for the State of Nebraska.The ninth grade mathematics curriculum should be examined critically

with consideration given to placing a greater emphasis on the acquisitionof those skills prerequisite to satisfactory participation in contemporarysociety, tn Saskatchewan, ninth grade students failed to achieve a major-ity of these skills, and the gain in adjusted mean competency totals wasapproximately twice as great between the ninth and tenth grade levels asit was between the tenth and eleventh or between the eleventh and twelfthgrade levels.

Serious consideration should be given to the improvement of math-ematics curricula and instruction in the following three competencyareas: probability and statistics, geometry, business and consumermathematics. Saskatchewan students in all high school grades achievedconsistently the lowest level of mathematical competence in these threeareas. This result also was reported by Niemann [9] and Cramer [10] forninth and twelfth grade level students respectively in Nebraska.

Further research of female high school students’ low scores, relative tomales, is essential in order to determine causal factors. Statisticallysignificant (p<0.05) differences in achievement of mathematical compe-tence existed between the sexes, in favor of males, for mean competencytotals, and over most competency areas, after adjustments were made for

3. Two Saskatchewan credits requires the completion of material commonly found in a first level algebra text and/oran achievement level of at least 50 percent on The Beckmann-Beal Test.


intelligence. The superior achievement by males has been supported bynumerous studies, including Davis [11] in Michigan, Alkire [12] in SouthDakota, and the NAEP [13]. Two current theories proposing solutions tothe problem of sex differences favoring males over females in mathe-matics are social conditioning and the theory that the brains of males andfemales are not alike and are intended to perform differently [14].

REFERENCES

1. Conference Board of the Mathematical Sciences. Overview and Analysis of SchoolMathematics, Grades K�12. Washington, D.C.: Conference Board of the Mathe-matical Sciences, 1975.

2. HILL, SHIRLEY. "Issues from the NACOME Report," The Mathematics Teacher,LXIX (October, 1976), 441-446.

3. National Assessment of Educational Progress, (NAEP). Mathematics Fundamentals:Selected Results from the First National Assessment of Mathematics. Washington,D. C.:U. S. Government Printing Office, 1975.

4. _____. Consumer Math: Selected Results from the First National Assessment ofMathematics. Washington, D.C.: U. S. Government Printing Office, 1975.

5. EDWARDS, E. L., JR., Eugene D. Nichols, and Glyn H. Sharpe. "Mathematical Com-petencies and Skills Essential for Enlightened Citizens," The Mathematics Teacher,LXV (November, 1972), 671-677.

6. Beckmann, Milton W. "The Level of Mathematical Competency and Relative Gains inCompetency of Pupils Enrolled in Algebra and General Mathematics." UnpublishedDoctor’s dissertation. University of Nebraska, 1951.

7. _____. "Ninth Grade Mathemtical Competence�15 Years Ago and Now," SchoolScience and Mathematics, LXIX (April, 1969), 315-319.

8. MITZMAN, BARRY, ed. "The Problems ofMinimalcy," The Math Learning Center Re-port, Spring, 1978.

9. NIEMANN, DONALD F. "A Study of the Degree to Which Seventh, Eighth, and NinthGrade Students Have Obtained Minimum Mathematical Competencies and Skills asRecommended by the National Council of Teachers of Mathematics." UnpublishedDoctor’s dissertation. University of Nebraska, 1973.

10. CRAMER, CARL F. "A Study of Achievement Levels of Nebraska High School Seniorson a Test Designed to Measure Mathematical Competencies." Unpublished Doctor’sdissertation. University of Nebraska, 1974.

11. DAVIS, DAVID JOHN. "A Comparative Study of Achievement Levels of Twelfth GradePupils on a Test Designed to Measure Functional Competence in Mathematics,"Dissertation Abstracts, 10:37, No. 2,1950.

12. ALKIRE, G. DON. "Relation of Certain Factors Resident in the Pupil in the School, andin the Teacher, to Functional Competence in Mathematics." Unpublished Doctor’sdissertation. University of Kansas, 1953.

13. National Assessment of Education Progress. Mathematics: An Overview. Washington,D. C.:U. S. Government Printing Office, 1975.

14. LAMOTT, KENNETH. "Why Men and Women Think Differently," Horizon, XIX (May,1977), 41-45.

1980 SSMA CONVENTIONINDIANAPOLIS

high school students' achievement of selected mathematical competencies

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