Saskatchewan High School Students' Attainment of Selected Mathematical Competencies 14 Years Ago and Now
By E. Giesbrecht (1991)

SSTA Research Centre Report #91-03: 14 pages, $11.


Rational for the Project

Review of the Literature


Analysis of Data



Professor E. Giesbrecht's 1976 assessment of Saskatchewan high school students' achievement of basic mathematical competence was repeated in 1990. A comparison of the results are outlined in this report.

The purpose of this longitudinal study was two-fold. First, to compare Saskatchewan high school students' level of achievement of selected mathematical competencies with the level achieved in 1976. Second, to determine again the effect, if any, of four factors on students' achievement of these competencies.

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Saskatchewan high school students' achievement of selected mathematical competencies as measured by the Beckmann-Beal Test (BBT)declined approximately 5 percent since 1976. Percentage scores achieved by Grades 9, 10, 11, and 12 were 39, 47, 51, and 61. Grade level, mathematics programme, and student gender all had statistically significant (p < 0.05) effects on the attainment of these competencies. School enrollment size did not. Statistical control for intelligence was provided by using the students' scores on the Differential Aptitude Test (DAT).

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Students' inability to perform satisfactorily in mathematics appears to be chronic.

In 1937 parents in Roslyn, Long Island demanded a return to the "three R's" because they believed that the present methods of progressive education were not as efficient as the old-fashioned methods of teaching history, spelling, reading, writing and arithmetic (New York Times, 1937:25].

In 1976 Saskatchewan students who had completed two years of high school mathematics just managed to pass (50.5 percent) the BBT, a multiple-choice test based upon forty-eight basic mathematical competencies and skills considered essential for satisfactory participation in society. The competencies did not involve advanced mathematics and were relatively simple. For example, one skill tested was the ability to list the first ten multiples of 2 through 12. (Giesbrecht, 1977:iv, 10).

In 1988 Jim Pilarski from the Marriott Hotels and Resorts stated that " ... declining educational standards eliminate some applicants from consideration. We estimate that 30 to 40% have limited mathematics and science fundamentals, and that concerns us." Ninety percent of their jobs are custodial. (Mathematical Sciences Education Board (MSEB), 1988:2).

In 1989 a Canadian survey given to 9500 Canadians found 38 percent of adults (6.6 million) unable to handle simple mathematics requiring more than addition or subtraction. "Fourteen percent (2.4 million) couldn't even consistently perform addition and subtraction . They could do little more than actually find numbers on a product label or in written material." (Sau]t Starf 1990:1).

In view of the continuing low level of mathematics achieved by Canadians, the author decided to test again Saskatchewan high school students achievement of selected mathematical skills. This study would be longitudinal in nature and would compare achievement in 1990 with that in 1976. The study would examine again the effect of mathematics programmes, grade level, school size, and student gender on mathematical achievement.

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Mathematical literacy, or numeracy, in North America has been a real concern for mathematics educators for many years. The citizenry's innumeracy (the inability to deal comfortably with the fundamental nations of number and chance) has been at an unacceptable level for most, if not all, of the twentieth century.

In 1918, H. C. Morrison, State Commissioner of Education of New Hampshire criticised secondary school mathematics by saying that the traditional courses in algebra, geometry, trigonometry and advanced algebra must be revised in order to meet the need of the adolescent, and the social purpose of the high school Mathematics must be ... presented to give immediate opportunity for functioning (Dooley, 1918:181).

When World War II broke out in 1939, many American military inductees did not possess minimum mathematical skills required for military personnel (Nimitz, 1942:88-89).

A commission consequently was established by the NCTM to address this depressing, embarrassing need. A report released in 1945 dealt with this issue and contained a list of twenty-nine mathematical concepts. The Commission stated that "If you can say 'yes' to nearly all of them then you can feel pretty secure when it comes to dealing with the problems of everyday affairs" (NCTM, 1947:318-319).

In 1975 the Canadian Chamber of Commerce (1975:i,4) stated in a report on basic educational skills that Canadian universities were finding that a significant number of freshmen lack basic mathematical skills ... and ... were functionally illiterate."

One of the biggest challenges facing mathematics educators was to determine as precisely as possible what mathematical competencies and skills were needed by the average citizen in everyday life. In 1970 the Board of Directors of the NCTM appointed The Committee of Basic Mathematical Competencies and Skills to determine a current list of mathematical competencies and skills required by the average citizen to function satisfactorily in society. In 1972, the NCTM released a list of forty-eight competencies and skills, grouped as ten competency areas (Edwards, Nichols, and Sharpe, 1972:673-674). They are shown in Table l.

In 1976 Giesbrecht (1977) conducted a study of Saskatchewan high school students' achievement of the 1972 NCTM list of mathematical competencies by administration of the BBT and the Verbal Reasoning and Numerical Ability (VR + NA) Subtest of the DAT battery in 161 schools to 742, 760, 715, and 685 grades nine through twelve students respectively. The VR + NA subtest vas administered in order that intelligence could be "factored out", and the effect of four variables, Szade level, mathematics programme, school enrollment size, and student gender, upon achievement of these mathematical competencies could be examined.

Statistically significant (p < 0.05) differences in mean competency totals, adjusted for intelligence, were found to exist between students enrolled in the following pairs of factors, in favour of the first member (underlined) in each comparison. Unadjusted mean competency totals expressed as percents are shown in parentheses.

1. Grades:

a) twelve (64.7), eleven (57.91. ten (50.5), and nine (43.3);

b) twelve and ten.

2. Programmes, (over grades ten, eleven, and twelve):

a) algebra-geometry (trigonometry and algebra,alternate mathematics, general mathematics;

b) algebra and general mathematics.

3. Size, (over all grades): small. medium, and large.

4. Gender, (over all grades): male and female.

The above significant differences generally are supported in reported research, with the exception of school size. There is no strong consensus with respect to the effect of this variable.

Saskatchewan majority of satisfactory two years of high school students were not able to achieve a the competencies considered to be essential for participation in everyday life until they had taken mathematics.

The Second International Mathematics Study (SIMS) conducted by the International Association for the Evaluation of Educational Achievement in 1981-82 found that Canadian grade twelve students placed fourteenth on the test far elementary functions and calculus, and eleventh in algebra in the fifteen participating countries, while Hang Kong and Japan placed first and second (McKnight, Travers, and Dossey, 1985:292-299).

The decline in mathematics and science scores during the period from 1970 to 1985 was accompanied by an unparalleled slump in SAT scores since 1963. The results of the fourth National Assessment of Educational Progress conducted in 1986 to determine the performance of Grades 3, 7, and 11 students found that student achievement at all age levels shows serious deficiencies" (Lindquist, 1989:160, 169).

The following two examples, with percent correct responses placed in parentheses, illustrate the types of difficulties experienced by grade eleven students in the United States.

(58%)--"Four pets are competing in a best pet contest. Eighty children vote and the pet with the most votes will vin. What is the smallest number of votes that a pet could receive and still win the contest?" (p.75). (454)--"7 1/6 - 3 1/2 = '." (p.81).

Jensen (1990:4-5) stated that one reason for the poor mathematics performance by North American students may be that the curriculum spiral is wound too tightly at the elementary level and far too much time is wasted going over material taught the year before. Higher achieving countries, such as Japarl, have a lot less repetition and, by grade eight, offer" ... an intense treatment of algebra ...

Marquis (1989:422) claimed that" ... 85 percent of high school classroom time should be spent in active instruction or controlled, monitored practice for effective teaching ... ", whereas only 12 percent is in some schools.

Many students in the United States and Canada hate mathematics and are afraid of it, consequently achieving low scores. Lincoln (1978:25-29) reported that Xavier University noted dramatic improvement of freshmen mathematical scores after implementation of a Piagetian-based strategy to help students progress from the concrete-operational level, typical of most freshmen, to the formal operational level.

Teachers probably are the most important factor in the mathematical success of their students. Chinese and Japanese elementary teachers are successful in making mathematics an exciting subject for their students (Stevenson, Lummis, Lee, and Stigler, 1990:31). Their secondary students achieved the highest mathematics scores in SIMS.

Parents also are a major player in the mathematics education of their children. Chinese and Japanese parents hold their children accountable for assigned homework. They also discourage them from holding part-time jobs (Kapoor, 1990:30). Asian children are considered as" ... clay, capable of being molded through everyday experience" into a perfect human being. All children are considered capable of learning. "Lack of achievement is attributed to a failure to work hard" (Stevenson, et al, 1990:22,32).

On the contrary, in North America it seems typical to consider children to possess very different levels of potential, the "hard work" ethic seems to be obsolete, and excellence is not generally expected of all children.

In dealing with the innumeracy problem of Saskatchewan youth, Hope (1990:281) stated that the existing curriculum should be revised because the current mathematics progrexne it is not meeting the mathematical needs of the majority of Saskatchewan's youth.

In 1989 the National Council of Supervisors of Mathematics (NCSM) assumed the task of determining what mathematics is essential for twenty-first century citizens, that is, what mathematical competencies" ... are necessary for the doors to employment and further education to remain open" (NCSM, 1989;471). A careful comparison of the competencies listed by the NCSM with the 1972 NCTM list of competencies and skills shown in Table 1 revealed that the 1972 list still is relevant, albeit slightly minimal.

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The purposes of this longitudinal study were to compare Saskatchewan high school students' level of achievement of selected mathematical competencies in 1990 with the level achieved in 1976f and to determine again the effect, if any, of four factors, grade level (9, 10, 11, 12), mathematics programme (algebra-geometry (trigonometry), algebra, alternate mathematics, general mathematics), school enrollment size (small--1 to 139, medium--162 to 264, large--269 to 1726), and student gender (female, male), on students' achievement of these competencies. The 161 schools that participated in 1976 were contacted with respect to a second test- inq. Usable test results were received from seventy-four of them.

The test instruments used were the BBT and the VR + NA Subtest of the DAT battery. The VR + NA Subtest was used to measure intelligence, the covariant in this study. The test instruments were administered by local teachers, principals, and guidance counsellors between May 15 and June 15, 1990.

Random proportionate sampling techniques were used to obtain a representative sample from the four mathematics programmes and two sexes. The sample size was set at 900 for each grade. A student was considered enrolled in the algebra- geometry (trigonometry) programme if he or she was enrolled in a geometry (trigonometry) course or had previously taken one at that grade level. Enrollment size was determined by ranking the schools according to size from smallest to largest, and then defining small schools to be the first third of the schools, medium schools the next third, and large schools the top third.

The results from both tests were entered in the Data General computer system at Algoma University College, Sault Ste. Marie, Ontario, and consequently marked and analyzed using one- and two-way ANCOVA techniques.

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Table 1 shows the number and percentage of students in each grade that attained each of the forty-eight competencies over the ten competency areas in both 1976 and 1990, as well as the percent change. The number and magnitude of the declines are substantial. They are worst at the grade eleven level, probably due in part to an increased participation rate in this grade since 1976.

Table 2 shows the mean (for 1976 also), range, and standard deviation for the forty-eight total competencies far students in each of grades nine through twelve in each of the four types of mathematics programmes, three school enrollment sizes, and two sexes.

Students scores on both the BBT (total competencies) and the DAT declined since 1976 and are shown below in percentage form.

Grade 9: BBT: 38.5, decline--4.8; DAT : 46.2, decline--5.7

Grade 10: BBT: 46.6, decline--3.9; DAT: 52.7 ,decline--3.7

Grade 11: BBT: 50.9, decline--7.0: DAT: 54.6, decline--8.7

Grade 12: BBT: 60.5, decline--4.1:, DAT: 63.5:;: decline--5.4.

Figure 1 graphically illustrates the total number of competencies achieved by grade level. Students required three years of high school mathematics in 1990 (compared with two years in 1976) to achieve at least 50 percent of these essential competencies.

Figure 2 depicts the average score achieved per competency area (Table 1) for each grade level. Grade 11 students passed in only six of the ten areas.

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Saskatchewan students' achievement of mathematical competencies considered essential for twenty-first century citizenship declined approximately 5 percent since 1976, and students required three years of high school mathematics in 1990 to achieve a simple majority as compared with two years in 1976.

Statistically significant (p < 0.05) differences in mean competency totals, adjusted for intelligence, were found to exist between students enrolled in the following pairs of factors, in favour of the first member in each comparison.

1. Grades:

a) twelve. eleven, and nine:

b) twelve. eleven, and ten.

2. Programmes, (ovez grades ten, eleven, and twelve):

a) algebra-aeometm (trigonometry), and algebra, alternate mathematics, general mathematics:

b) algebra, and general mathematics.

3. Gender, (over all grades):

Male and female.

The above significant differences resulted in the rejection of the null hypotheses concerning the effects of above three factors. There was no evidence to reject the null hypothesis concerning the effect of school size.

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1. Saskatchewan Education should define an acceptable level of basic mathematical competence for high school students.

2. Mathematics curriculum developers, department heads, and teachers at local levels in Saskatchewan high school systems should consult the 1972 NCTM list of forty-eight competencies, or the 1989 NCSM list of essential mathematics for the twenty-first century, as they plan and revise mathematics curricula in charting the course for mathematics education in Saskatchewan.

3. Consideration should be given to retaining the topics in the alternate mathematics programme in the restructuring of the mathematics curricula as suggested by Professor Hope.

4. Saskatchewan mathematics educators should address seriously students' consistent low level of achievement in the competency areas of probability and statistics, business and consumer mathematics, measurement, and operations and properties.

5. Saskatchewan Education should give full support to the recommendations for revision of the secondary school mathematics programme, as outlined by Hope in Charting the Course.

6. Because students' scores were unacceptably low in certain areas, more attention needs to be paid to the following topics: business and consumer mathematics, variation including both direct and inverse variation and proportion, use of metric units, properties of number systems, mental computation and estimation, operations on fractions and negative integers, measurement, map reading and scale drawing, elementary statistics and probability.

7. Teachers should give careful consideration to not spending too much time revisiting material from the previous year, to arranging for effective teaching and learning to be taking place during at least 85 percent of classroom time, to making a serious effort to use the mathematics laboratory approach whenever practical, and to doing their utmost to get as many students as possible to like mathematics.

8. Parents should give careful consideration to actively supporting their children by encouraging them in their school work, by holding them responsible for doing their homework, and by discouraging them from holding part-time jobs while going to school.

9. Teachers and parents collectively should think of children more as the Asians do, namely, as a piece of malleable clay capable of being molded into a state of relative perfection. They should consider adopting the mind-set that lack of achievement in mathematics by their students and children is at least partly due to them not working hard enough.

10. Every effort should be made to raise the level of mathematical achievement of girls, especially at the grade eleven level.

11. Subsequent testing with respect to students' achievement of essential mathematical skills for the twenty-first century should be conducted in five years (1995).

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