Volume 40, Number 2 • Winter 2003

## The Effects of a Generalizable Mathematics Skills Instructional Intervention on the Mathematics Achievement of Learners in Secondary CTE Programs

## Mingchang Wu

National University of Science and Technology

James P. Greenan

Purdue UniversityCareer and technical education (CTE) purports to prepare students with occupational knowledge and skills necessary to enter the workplace and live independently (Evans & Herr, 1978). Its programs are important to students and the community because (a) student populations require greater interaction with industries that provide them with work experiences and, in combination with school academic courses, increase adult employment, and (b) rapid changes in technology and the high costs of new equipment cause schools to utilize industrial sites to keep training updated and expenses within budgets (Apolloni, Feichtner, & West, 1991). Additionally, CTE programs are natural settings for functional curricula that emphasize skills and competencies that enhance opportunities for students to succeed in education and the world of work (Brown, 2000; Forman & Steen, 1999; Grubb, 1997; Hull, 2000; Keif & Stewart, 1996). Therefore, CTE programs should increasingly focus on functional skills that are necessary and transferable in educational contexts and occupational settings (Carnevale, Gainer, & Meltzer, 1990; Greenan & Tucker, 1990; Field, 1998; Secretary's Commission on Achieving Necessary Skills, 1991).

Mathematics, communications, interpersonal relations, and reasoning skills have been identified as generalizable skills. Generalizable mathematics skills have been identified as essential for success in a wide variety of CTE programs and employment settings (Greenan, 1986). The generalizable mathematics skills curriculum has the potential to integrate academic and CTE instruction, which can facilitate the transition of learners from CTE programs into continuing education and/or the world of work.

Studies have indicated that students have difficulty in conceptualizing abstract mathematical ideas and require instructional strategies that utilize concrete, real, and hands-on experiences that are relevant in their daily life. Since mathematics may appear as a hierarchically complex subject to many CTE students, effective mathematics instruction should be coherent and concentric to instructional units (Baker, 1993). To improve mathematics learning achievement, each instructional unit should be designed to meet a single and specific curriculum goal. Teaching materials should be presented in a clear, organized, step-by-step manner, from basic to advanced concepts (Yoshida, Fernandez, & Stigler, 1993). Students should practice mathematics skills through assignments until they are ready for subsequent units. Accordingly, students can effectively learn complex mathematics concepts when each phase is relevant and consistent with preceding phases.

The skills and knowledge required to live independently and succeed in employment have become increasingly emphasized in educational settings (Grubb, 1997; Hull, 2000). However, CTE and technical education programs, historically, have not assisted students to apply explicit academic knowledge to occupationally related skills and work settings (Greenan, 1986; Sarkees & West, 1990; Sitlington, 1986). Dare (2000) indicated that there is little evidence of the impact of CTE and applied academics on affecting students' academic performance and attitudes. However, the separation of CTE and academic curricula may inhibit the successful transition of youth and adults into the world of work and postsecondary education. Consequently, the integration of academic and occupational skills continues to be a major theme in educational reform (Passmore, 1994).

Generalizable mathematics skills curriculum and assessment strategies have been developed and validated for CTE, academic, and special education personnel for the purpose of revising and improving instruction (Greenan, 1984, 1985, 1986). Therefore, there is a critical need to develop functional-based instructional interventions that can assist educators to integrate generalizable mathematics skills and CTE curricula.

## Purpose and Objectives of the Study

The purpose of this study was to develop the Generalizable Mathematics Skills Instructional Intervention and determine its effects on the mathematics achievement of students in secondary CTE programs. The intervention was developed based on the generalizable mathematics skills curriculum. The objectives of this study included the following.

- To develop a generalizable mathematics skills instructional intervention for learners in secondary CTE programs; and
- To determine the effectiveness of the generalizable mathematics skills instructional intervention on the mathematics achievement of students in CTE programs.
Delimitations of the Study

The target population for this study was students enrolled in secondary CTE programs in the State of Indiana. The sample included 10

^{th}, 11^{th}, and 12^{th}grade students in four CTE program areas (Business, Family and Consumer Sciences, Health Occupations, and Industrial) and five CTE programs (Computer Operations, Dental Assistance, Child Care, Electronics, and Construction Trades) at one area career center. The results of this study, therefore, have limited generalizability to other CTE programs, teachers, and students.Assumptions of the Study

This study, like most studies, made several assumptions. The sample selected for this study was assumed to be representative of the population. It was also assumed that the teachers followed the procedures for the Generalizable Mathematics Skills Instructional Intervention according to the prescribed guidelines during the administration of the intervention. These assumptions were perceived to be reasonable and necessary.

## Methodology

The quasi-experimental design used in this study was a pretest-posttest experimental and control group design. The dependent variables in this study included generalizable mathematics skills achievement defined as those skills that include whole numbers, fractions, and decimals. The independent variables selected for this study were (a) instruction, (b) CTE program area, (c) gender, and (d) level of mathematics skills as measured by the Generalizable Mathematics Skills Performance Assessment (GMSPA): high (67-102), median (34-66), and low (0-33) (Greenan, 1985).

Population and Sample

The population for the study was students enrolled in secondary CTE programs in the State of Indiana. One urban area career center was selected to participate on the basis of program representation, student availability, and teacher interest. The programs were representative of Business, Consumer and Family Sciences, Health, and Industrial Programs. Five programs from each of the four CTE program areas were selected for both the experimental and control groups. Each program had two classes, one morning class and one afternoon class. The classes in each program were randomly assigned to the experimental or control group. Table 1 illustrates the sample of 84 students in the four CTE program areas used in this study.

Table 1

Sample Selected for the Study

CTE Program AreaExperimental

GroupControl

GroupBusiness

Computer Operations

15

7Family and Consumer Sciences

Child Care

12

5Health Occupations

Dental Assistant

10

12Industrial

Electronics

Construction Trades

11

0

____

4

8

____Total 48 36 Instrumentation

The instrument used in this study was the GMSPA developed by Greenan (1984, 1985, 1986). This instrument was administered to the experimental and control groups during the pretest and posttest administrations.

The purpose of the GMSPA is to assess how well students can perform generalizable mathematics skills (Greenan, 1984, 1985, 1986). These skills consist of 7 subscales and 150 items as follows: whole numbers (25 items), fractions (34 items), decimals (43 items), percent (9 items), mixed operations (15 items), measurement and calculations (20 items), and estimation (4 items). This study used only the whole numbers, fractions, and decimals subscales, for a total of 102 items. The total number of correct responses indicates the subscale or total scale scores.

The internal consistency reliability of the Total Performance Assessment scale was r = .97 (Kuder-Richardson [KR-21]), which indicates that the instrument has high internal consistency and is measuring a uniform or single construct of generalizable mathematics skills. The test-retest reliability of the total performance assessment was r = .65 (Pearson Product-Moment Correlation Coefficient, p < .001). The correlation indicates that the performance assessment can produce stable and similar measures over time (14 days) (Greenan, 1985, 1986).

Intervention

The Generalizable Mathematics Skills Instructional Intervention (GMSI) consists of 3 sub-skills and 22 lessons: whole numbers (4 lessons), fractions (9 lessons), and decimals (9 lessons). These three generalizable mathematics skill areas are prerequisite to learning percents, mixed operations, measurement and calculation, and estimation. Each lesson is used as a teaching unit for 15-minute durations and focuses on singular mathematics skills. Introduction to mathematics concepts, demonstration of operational procedures, problem settings, and exercises are provided in each lesson. The instructional strategies included (a) teacher demonstrations of a solution to a sample problem while verbalizing keywords for the steps used, (b) student practices on sample problems following the strategies taught in the class, and (c) student discussions with respect to the activities based on their previous knowledge and skills.

Data Collection

The pretest was administered to the experimental and control groups. Student mathematics GPA's from the previous semester and mathematics courses completed were identified using a self-report inventory. During the six-week instructional intervention, each teacher in the experimental group used the prescribed procedures and the same amount of time (15 minutes per day), and maintained the same approximate pace to teach the intervention. Teachers provided the regular CTE instruction for students in the control group. The assessment instruments did not remain with the teachers so that the teachers in the experimental group would not teach to the test. When the six-week intervention was completed, the posttest was administered to the experimental and control groups.

Qualitative information was collected using a follow-up interview questionnaire with respect to administrator, teacher, and student attitudes regarding the intervention and their teaching and learning experiences. One CTE curriculum director, four teachers, and eight students (two students from each CTE program area) in the experimental group were interviewed.

Data Analysis

The quantitative data were computed and analyzed using the Statistical Analysis System (SAS) Software version 6.08. Data were analyzed individually and collectively. Means, standard deviations, analyses of variance (ANOVA), and analyses of covariance (ANCOVA) were computed to test the hypotheses.

The assumptions underlying the use of inferential statistical methods were examined. Interval data were collected and used, and means were computed and used. In addition, homogeneity of variances was tested to identify whether or not there were significant differences between the variances for the pretest Performance Assessment scores using the Bartlett's test (Winer, Brown, & Michels, 1991). Another purpose for testing homogeneity of variances between the samples was to identify the variability of sample scores (Borg & Gall, 1989).

Table 2 illustrates the Chi-Square values that indicate the homogeneity of variances of the GSPA on the on the pretest. The Chi-Square value for the Total Performance Assessment scale (Chi-Square = 12.22, p = .09) suggests acceptance of the null hypothesis and implies that, overall, the variances for the mathematics skills were generally equal.

Table 2

Homogeneity of Variance of the GMSI on the Pretest

Performance Scale Chi-Square P Whole Numbers 18.31 .01 Fractions 11.47 .12 Decimals 12.27 .09 Total Scale 12.22 .09 P: Associated Probability Further, the GMSPA scores on the pretest were also tested through univariate analysis using the Shapiro-Wilk test to identify whether or not the data represented normal distributions (SAS Procedures Guide, 1989). Table 3 demonstrates the results of the normal distribution test on the GMSPA for the pretest. The associated probabilities indicated that the sample was not normally distributed with respect to Performance Assessment scores, W(83) = .94, p = .00. The experimental group was also not normally distributed W(47) = .93, p = .01. However, the control group was normally distributed W(35) = .96, p = .22.

The lack of normal distributions on the pretest scores for some variables was a practical limitation with regard to the results of this study. The use of inferential statistical methods was justified, since the data satisfied most of the required assumptions, including the use of interval data, means, random samples, and homogeneity of variances (Borg & Gall, 1989).

Table 3

Normal Distribution of the GMSI on the Pretest

Performance Variable W: Normal P Overall (n = 84) .94 .00 Treatment

Experimental group (n = 48)

Control group (n = 36)

.93

.o1

.96

.22P: Associated probability A 2 x 2 repeated measurement factorial analysis was conducted to identify the main effects of the treatment (GMSI) and time between the assessments (pretest and posttest), and the interaction effects of these two factors on the dependent variables for the students on each moderator variable. Analyses of variance were conducted to determine the effects of the independent variable (GMSI) in regard to the moderator variables (CTE program, gender, and level of mathematics skills) on the dependent variables (mathematics skills achievement). The Bonferroni F-test was conducted to investigate the effects of the independent variables and identify the power of the individual effects and interaction effects of the independent variables (SAS Users' Guide, 1989).

The qualitative data were analyzed using the cross-case approach (Patton, 1990). The cross-case qualitative analysis method groups the responses from different subjects according to common themes that emerge. The data from each item for each respondent were analyzed, synthesized, and summarized.

## Findings

The Student Background Inventory (self-report) was developed by personnel at the area career center to identify student mathematics GPA's from the previous year and the mathematics courses students have taken. These data were perceived to be important factors to mathematics learning. This inventory utilized a standard four-point GPA scale.

Table 4 provides the results that compare academic background (GPA's and numbers of previous mathematics courses taken) between the experimental and control groups. Overall, the data suggested that there were no significant differences (p < .05) between the experimental and control groups on GPA's, M(exp) = 2.63, M(con) = 2.63, and numbers of previous mathematics courses taken, M(exp) = 2.88, M(con) = 2.69. These data suggested that students in the experimental and control groups had equivalent mathematics backgrounds with respect to GPA's and mathematics courses previously taken.

Table 4

Comparison of Mathematics Backgrounds between the Experimental and Control

Moderator Mathematics Experimental Control F-Value Variables Background n M S.D. n M S.D.

Program

All

MCN

MGPA48

2.88

2.631.27

.6236

2.69

2.631.37

.70<1

<1Bus

MCN

MGPA15

2.73

2.53.80

.617

2.71

2.64.76

.48<1

<1FCS

MCN

MGPA12

3.83

2.861.53

.435

4.40

2.801.14

.27<1

<1HO

MCN

MGPA10

2.20

2.50.79

.7112

2.00

2.25.85

.72<1

<1Ind

MCN

MGPA11

2.64

2.591.36

.7412

2.67

2.931.61

.79<1

-1.05

Gender

Male

MCN

MGPA22

2.67

2.621.15

.6518

2.72

2.871.41

.68<1

-1.16Female

MCN

MGPA26

3.04

2.631.34

.6118

2.67

2.391.37

.651

1.26Math

Level

High

MCN

MGPA42

2.95

2.671.31

.6026

2.92

2.751.20

.45<1

<1Median

MCN

MGPA6

2.33

2.33.82

.7510

2.10

2.301.66

1.09<1

<1Table 5 illustrates the comparisons on the pretest mean scores between the experimental and control groups by CTE program, gender, and level of mathematics skills as measured by the GMSPA. Overall, the experimental group had a slightly but not significantly higher mean score than the control group on mathematics achievement, M(exp) = 83.10, M(con) = 77.47, F(1, 82) = 3.45, p = .06. The two groups had equivalent performance on the whole numbers subscale, M(exp) = 23.25, M(con) = 23.03, and the decimals subscale, M(exp) = 33.69, M(con) = 31.42. However, the experimental group had a significantly higher mean score, M(exp) = 26.17, than the control group, M(con) = 23.03, F(1, 82) = 5.72, p < .05, on the fractions subscale.

Table 5

Comparison of the Pretest Means Between the Experimental and Control Groups

Dependent Experimental Control Moderate Variables Variables n M S.D. n M S.D. F-Value

CTE

Program

All

Whole

Numbers

Fractions

Decimals

Total

48

23.25

26.17

33.69

83.10

1.63

5.45

7.29

12.43

36

23.03

23.03

31.42

77.47

2.28

6.57

9.00

15.34

<1

5.72*

1.63

3.45

Bus Subtotal 5 85.27 10.44 7 90.86 6.20 1.70 FCS Subtotal 12 84.50 13.24 5 68.80 6.72 6.19* HO Subtotal 10 71.00 9.74 12 71.58 17.54 <1 Ind Subtotal 11 89.64 9.6 12 79.17 14.66 4.02

GenderMale Subtotal 22 85.95 10.29 18 82.67 13.57 .76 Female Subtotal 26 80.69 13.72 18 72.28 15.60 3.58

Math Level

High

(67-102)Subtotal

42

86.43

9.04

26

84.81

10.32

<1

Median

(34-66)Subtotal

6

59.83

6.18

10

58.40

7.73

<1

Note: * p < .05

The pretest was also analyzed to identify the mathematics skill equivalence or difference between the two groups for each CTE program. The F-tests of mean scores indicated that there were no significant differences (p < .05) between the experimental and control groups for the Business program, M(exp) = 85.27, M(con) = 90.86, the Health Occupations program, M(exp) = 71.00, M(con) = 71.58, and the Industrial program, M(exp) = 89.64, M(con) = 79.17. However, in the Family and Consumer Sciences program, the experimental group had significantly higher generalizable mathematics achievement than the control group, M(exp) =84.50, M(con) = 68.80, F(1, 82) = 6.19, p < .05.

Both gender groups had equivalent achievement, noting no significant differences between the experimental and control groups for male students, M(exp) = 85.95, M(con) = 82.67, and female students, M(exp) = 80.69, M(con) = 72.28.

The equivalence of mathematics achievement between the experimental and control groups was also demonstrated for students with high and median generalizable mathematics skills. There were no significant differences in generalizable mathematics achievement between the experimental and control groups for students with high mathematics skill levels, M(exp) = 86.43, M(con) = 84.81, and students with median mathematics skills, M(exp) = 59.83, M(con) = 58.40.

In summary, based on the data regarding the academic backgrounds of student GPA's, mathematics courses previously taken, and the GMSPA scores on the pretest, it was concluded that the experimental and control groups generally had equivalent mathematics backgrounds, except on the fractions subscale. Therefore, the data were further analyzed using inferential statistical methods; and the research hypotheses were tested as proposed for this study.

The pretest and posttest mean scores of the experimental and control groups are presented in Table 6. A 2 x 2 factorial analysis with repeated measurement was used to identify the main effects of the intervention and time spent between administrations of the pretest and posttest, and the interaction effect between intervention and time on generalizable mathematics skills achievement.

Table 6

Pretest and Posttest Means on the GMSI for the Experimental and Control Groups

Pretest Posttest M S.D. M S.D. Experimental 83.10 12.43 87.29 10.00 Control 77.47 15.34 78.64 13.77 Overall 80.69 13.95 83.58 12.45 Table 7 illustrates that both intervention, F(1, 82) = 7.50, p = .01, and time, F(1, 82) = 6.27, p = .01, had a significant effect on generalizable mathematics skills achievement. There was no significant interaction effect between the intervention and time on mathematics achievement, F(1, 82) = 1.99, p = .16. However, the effect of the GMSI was significant on mathematics skills achievement. The power of a hypothesis test equals the probability of detecting a particular effect, that is, of rejecting a false null hypothesis. Power is the complement of the probability of a type II error. The power was p = .887 for the null hypothesis of intervention effect on mathematics skills achievement. This strongly supported the rejection of the null hypothesis and indicates the main effect of the intervention.

Table 7

Analysis of Variance of 2 x 2 Repeated Measurement for Students on the GMSI

Source DF Type III S.S Mean Square F Value Pr>F Intervention 1 2098.83 2098.83 7.50 0.01 Time 1 296.27 296.27 6.27 0.01 Int. x Time 1 93.86 93.86 1.99 0.16 Error 82 3874.16 47.25 Since there was no significant interaction effect and the pretest data indicated that the two groups had only slightly different mathematics achievement, F(1 , 82) = 3.45, p = .06, an alternative analysis was administered. The correlation between the pretest and posttest scores on the generalizable mathematics skills performance assessment was computed. The correlation between the pretest and posttest scores was 0.73, which was considered to be moderately high and strong. This correlation accounted for approximately one half of the variance between the scores. Subsequently, an ANCOVA was conducted to investigate the main effect of the intervention, using the pretest score as a covariate. The ANCOVA adjusts the posttest scores for the differences between the experimental and control groups on the corresponding pretest for the experimental and control groups with nonequivalent backgrounds. The effect of the ANCOVA adjustment compensated the experimental group for this initial disadvantage by increasing its posttest score to the level that would be predicted on the basis of the correlation between pretest and posttest scores (Borg & Gall, 1989). The data were also analyzed to test the research hypotheses proposed for this study.

The comparisons of posttest mean scores using adjusted means on the three subscales between the experimental and control groups are provided in Table 8. The effect of the intervention on generalizable mathematics skills achievement was significant for all students on the whole numbers, fractions, and decimals subscales. The posttest mean scores for the experimental group were significantly higher than the scores for the control group on the whole numbers subscale, Adjusted M(exp) = 23.53, Adjusted M(con) = 22.44, F(1, 82) = 11.14, p < .001; the fractions subscale, Adjusted M(exp) = 27.50, Adjusted M(con) = 25.33, F(1, 82) = 6.35, p < .05; and the decimals subscale, Adjusted M(exp) = 35.08, Adjusted M(con) = 32.57, F(1, 82) = 4.39, p < .05.

Table 8

Comparison of Posttest Covariance Means and Adjusted Means

Experimental Control Measure n M SD n M SD F Overall 48 36 Total

Adjusted

87.29

85.81

10.00

1.20

78.46

80.62

13.77

1.38

7.87**

Whole Numbers

Adjusted

23.56

23.53

1.27

0.21

22.39

22.44

2.02

0.25

11.41***

Fractions

Adjusted

28.06

27.50

&nbssp;4.14

0.58

24.47

25.23

5.27

0.67

6.35*

Decimals

Adjusted

35.67

35.086.39

0.78

31.78

32.578.27

0.90

4.39*CTE Program Bus

Total M

Adjusted M

15

89.27

89.78

4.10

1.02

7

90.90

88.90

6.20

1.51

<1FCS

Total M

Adjusted M

12

91.92

91.22

4.93

2.78

5

73.60

75.27

15.31

4.62

7.70*HO

Total M

Adjusted M

10

73.20

73.47

12.43

2.26

12

71.08

70.86

15.41

2.06

<1Ind

Total M

Adjusted M

11

92.36

90.63

4.70

2.19

12

81.67

83.25

10.73

2.08

5.49*Gender

Male

Total M

Adjusted M

22

90.27

89.74

4.59

1.29

18

84.39

85.03

9.65

1.43

5.39*Female

Total M

Adjusted M26

84.77

82.35

12.48

1.8218

72.89

76.38

15.08

2.20

4.22*Level of Math Skills High

Total M

Adjusted M42

89.70

88.74

5.89

1.0626

84.54

84.73

8.39

1.35

5.43*Median

Total M

Adjusted M6

60.75

74.12

6.55

5.20

10

58.00

64.63

7.13

4.02

4.08Note: * p < .05; ** p < .01; *** p < .001

The experimental group had generally higher posttest total mean scores than the control group for each CTE program. In particular, the experimental groups in the Family and Consumer Sciences program, Adjusted M(exp) = 91.22, Adjusted M(con) = 75.27, F(1, 15) = 7.70, p < .05; and the Industrial program, Adjusted M(exp) = 90.63, Adjusted M(con) = 83.25, F(1, 20) = 5.49, p < .05), had significantly higher mathematics skills achievement than those programs in the control group.

The intervention effect was significant for male and female students. Both male students, Adjusted M(exp) = 89.74, Adjusted M(con) = 85.03, F(1, 38) = 5.93, p < .05, and female students, Adjusted M(exp) = 82.35, Adjusted M(con) = 76.38, F(1, 42) =4.22, p < .05, in the experimental group had better mathematics skills than the students in the control group. The experimental group had significantly higher mean scores than the control group for students with high mathematics skills, Adjusted M(exp) = 88.74, Adjusted M(con) = 84.73, F(1, 66) = 5.43, p < .05, and for students with median mathematics skills, Adjusted M(exp) = 74.12, Adjusted M(con) = 64.63, F(1, 14) = 4.08, p < .05.

In summary, the intervention produced significant effects on generalizable mathematics skills for students. The findings indicated that the GMSI significantly improved mathematics skills achievement for male and female students, students in the Family and Consumer Sciences and Industrial programs, and students with high and median mathematics skill levels. In addition, the intervention had significant effects on the whole numbers, fractions, and decimals subscales. Therefore, the hypothesis was accepted; and it was demonstrated that the students who received the GMSI had significantly higher generalizable mathematics skills achievement than the students who did not receive the intervention, as indicated by the posttest mean scores.

The pretest and posttest mean scores for the experimental group are presented in Table 9. The posttest mean score was significantly higher than the pretest mean score on the total of the three subscales for students in the experimental group, M(pre) = 83.10, M(post) = 87.29, p < .01. The generalizable mathematics skills of students were improved on the three subscales after receiving the intervention. The posttest mean scores were significantly higher than the pretest mean scores on the fractions subscale, M(pre) = 26.17, M(post) = 28.06, p < .01, and the decimals subscale, M(pre) = 33.69, M(post) = 35.67, p < .05. The difference between the pretest and posttest mean scores was not significant for only the whole numbers subscale, M(pre) = 23.25, M(post) = 23.56.

The data were also analyzed by CTE program. The posttest mean scores were slightly, but not significantly, higher, p > .05, than the pretest mean scores for the Business program, M(pre) = 85.27, M(post) = 89.27, and the Family and Consumer Sciences program, M(pre) = 84.50, M(post) = 91.92. Additionally, there were no significant differences between the pretest and posttest mean scores for the Health Occupations program, M(pre) = 71.00, M(post) = 73.20, and the Industrial program, M(pre) = 89.64, M(post) = 92.36.

Table 9

Comparison of the Pretest and Posttest Means for the Experimental Group

Moderator

Variables

n

Dependent

Variables

Pretest

Posttest

F

valueM SD M SD All

Programs

48

Whole #

Fractions

Decimals

Total23.25

26.17

33.69

83.101.63

5.45

7.29

12.4323.56

28.06

35.06

87.291.27

4.14

6.39

10.001.64

5.84*

4.66*

7.67**Bus 15 M 85.27 10.44 89.27 4.10 3.05 FCS 12 M 84.50 13.24 91.92 4.93 3.44 HO 10 M 71.00 9.74 73.20 12.43 <1 Ind 11 M 89.64 9.60 92.36 4.70 <1 Male 22 M 85.95 10.29 90.27 4.59 4.01* Female 26 M 80.69 13.72 84.77 12.48 3.60 High

Math42

M

86.43

9.04

89.05

7.17

4.03*

Median

Math6

M

59.83

6.18

75.00

17.63

4.93*

Note: * p < .05; ** p< .01

The intervention produced different effects with respect to the generalizable mathematics skills of male and female students. The posttest mean was significantly higher than the pretest mean score for male students in the experimental group, M(pre) = 85.95, M(post) = 90.27, p < .05). However, the posttest mean score was slightly, but not significantly, higher than the pretest mean score for female students, M(pre) = 80.69, M(post) = 84.77.

The intervention yielded significant effects for students with high and median levels of mathematics skills. For students with high mathematics skills, the posttest mean was significantly higher than the pretest mean, M(pre) = 86.43, M(post) = 89.05), p < .01. In addition, the posttest mean score was significantly higher than the pretest mean score for students with median mathematics skills, M(pre) = 59.83, M(post) = 75.00, p < .05.

Since the sample size for each moderator variable was relatively small, the probability did not indicate statistical significance (Winer, Brown, & Michels, 1991). Although the experimental group increased its mathematics skills performance between the pretest and posttest, the difference was not significant.

The data suggested that the posttest mean scores were significantly higher than the pretest mean scores for students and with respect to most of the moderator variables. Therefore, the hypothesis was accepted, since the students in the experimental group had significantly higher mathematics skills achievement after receiving the GMSI intervention.

Table 10

Comparison of the Pretest and Posttest Means for Students in the Control Group

Moderator

Variables

n

Dependent

Variables

Pretest

Posttest

F

valueM SD M SD All

36

Whole #

Fractions

Decimals

Total23.03

23.03

31.42

77.472.29

6.57

9.00

15.3422.39

24.47

31.78

78.642.02

5.27

8.27

13.773.09

3.10

<1

<1Bus 7 M 90.86 6.20 90.00 5.74 <1 FCS 5 M 68.80 6.72 73.60 15.31 <1 HO 12 M 71.58 17.54 71.08 15.41 <1 Ind 12 M 79.17 14.66 81.67 9.84 <1 Male 18 M 82.67 13.57 84.39 9.65 <1 Female 18 M 72.28 15.60 72.89 15.08 <1 High Math 26 M 84.81 10.32 84.23 9.94 <1 Med Math 10 M 58.40 7.73 64.10 11.74 4.21* Note: * p < .05

The pretest and posttest mean scores for the control group were compared, and the results provided in Table 10 indicated that the students did not have significant improvement on mathematics skills achievement. There was no significant difference (p > .05) on the total scale for all students, M(pre) = 77.47, M(post) = 78.64). Also, there were no significant differences on the whole numbers subscale, M(pre) = 23.03, M(post) = 22.39; the fractions subscale, M(pre) = 23.03, M(post) = 24.47); and the decimals subscale, M(pre) = 31.42, M(post) = 31.78.

None of the four CTE programs in the control group showed significant improvement in their mathematics skills during the period in which the experimental group received the intervention. There was no significant difference (p > .05) between the pretest and posttest mean scores for the control group in the Business program, M(pre) = 90.86, M(post) = 90.00; the Family and Consumer Sciences program, M(pre) = 68.80, M(post) = 73.60; the Health Occupations program, M(pre) = 71.58, M(post) = 71.08; and the Industrial program, M(pre) = 79.17, M(post) = 81.67.

Neither male nor female students in the control group showed significant improvement in their mathematics skills. There were no significant differences (p > .05) between the pretest and posttest mean scores for male students, M(pre) = 82.67, M(post) = 84.39, and female students, M(pre) = 72.28, M(post) = 72.89.

Students with different levels of mathematics skills had different degrees of improvement between the pretest and posttest. The students with high levels of mathematics skills in the control group did not have significantly improved mathematics skills between the pretest and posttest, M(pre) = 84.81, M(post) = 84.23. However, the students with median mathematics skill levels had a significantly higher posttest than pretest mean score, M(pre) = 58.40, M(post) = 64.10, F(1 , 8) = 11.74, p < .05.

These findings indicated that most students in the control group did not possess improved mathematics skills between the pretest and posttest. Therefore, the hypothesis was accepted, since the students in the control group did not have significantly higher generalizable mathematics skills achievement. Students with median levels of mathematics skills were the only students in the control group who revealed improved mathematics skills. It is plausible to conclude that the improvement of these students' mathematics skills resulted, in part, from participating in the pretest and posttest.

Follow-up Interviews

Qualitative data were collected from post-hoc interviews with four CTE teachers, eight students from the experimental group, and a school administrator. The data were analyzed according to the posited questions using cross-case methodology. The salient findings included the following.

- CTE teachers generally believed that generalizable mathematics skills are important for student success in school and the world of work. These teachers highly valued the GMSI intervention, since it provided opportunities to complement the materials used in their programs and to explain mathematical concepts.
- Teachers believed that some students commonly have a negative attitude toward mathematics. Negative attitudes seemed to emerge from frustrations since elementary school, because students do not connect the relevance and importance of mathematics in schooling and their daily life.
- Some CTE teachers reflected that they had difficulties in mathematics or in teaching mathematics. Some teachers claimed mathematics teachers should be responsible for integrating mathematics into CTE programs beginning in the first grade.
- CTE teachers believed that applied mathematics skills that match the requirements of particular CTE programs were essential. Integrated mathematics instruction was an effective approach to teaching and learning mathematics skills in CTE programs.
- Mathematics interventions should not be taught in isolation. CTE and mathematics curricula should be integrated with cooperation and collaboration between CTE and mathematics teachers. However, a school administrator stated that this approach would likely require schedule flexibility.
- Instructor attitudes toward mathematics teaching appeared not to change as much as student attitudes toward mathematics. This could imply that teachers were not familiar with or confident in teaching mathematics within the contexts of their CTE programs. In addition, some CTE teachers assigned CTE homework to students. However, they usually did not assign mathematics homework to students. This may suggest that CTE teachers clearly value mathematics skills differently from CTE skills.
- Students in most CTE programs indicated that mathematics was important in their CTE courses and employment, and that they would like to improve their CTE competencies and mathematics skills. Students wanted mathematics to be taught and learned through applications in CTE courses and according to the needs of CTE curriculum, rather than through traditional mathematics interventions.
- Students expressed a desire that mathematics be taught while considering their present levels of mathematics skills and curriculum needs. Appropriate procedures and teaching activities used with students during generalizable mathematics skills instruction could create more effective communication.
- Most students clearly stated that they were initially reluctant to participate in the intervention. However, they accepted it and learned something helpful. The changes of students' attitudes toward mathematics were attributed to their instructors' teaching styles.
## Conclusions, Implications, and Recommendations

Conclusions

The findings indicated that there were equivalent mathematics GPA's and equivalence with respect to the numbers of mathematics courses previously taken between the experimental and control groups. Additionally, the two groups had equivalent generalizable mathematics achievement as measured by the pretest. Further, the experimental group had higher mathematics skills achievement than the control group on the posttest. The conclusions were drawn from the findings derived from testing of the hypotheses formulated for the study.

The Effects of the GMSI Intervention on Mathematics Achievement

The findings suggested that those students receiving the GMSI had significantly higher mathematics skills achievement than those students not receiving the intervention. It is plausible to conclude that the intervention had significantly positive effects on the mathematics achievement of those students who received the intervention. These findings were consistent with several previous studies claiming that effective mathematics skills instruction should (a) interpret abstract mathematics concepts using concrete objectives (Hembree, 1992), (b) integrate student interests and background with the mathematics curriculum (Yuen-Yee & Watkins, 1994), (c) increase immediate feedback through interaction between students and teachers and/or among students themselves, (d) increase opportunities for students to translate mathematics concepts and/or operations to real problems in the world (Hembree, 1992), (e) emphasize a singular conceptual unit at one time until students understand its concept and applications (Thiering, Hatherly, & McLeod, 1992), and (f) maintain student learning throughout the school year (Lamon, 1993). The GMSI also produced significant effects regarding the mathematics achievement of students in the experimental group by CTE program, gender, and level of mathematics skills.

Teacher and Student Attitudes Toward the GSMI Intervention

Based on the follow-up interviews with students, CTE teachers, and an administrator, it is reasonable to draw the following conclusions. First, CTE teachers and the CTE administrator generally perceived that generalizable mathematics skills were important for student success in CTE programs and the world of work. Second, the intervention was perceived to be effective for integrating student needs and program requirements. Students generally favored the intervention, and their attitudes toward mathematics skills were improved after they received the intervention. CTE teachers claimed that mathematics skills should be taught throughout the entire school year in order to improve students' skills. Third, some CTE teachers did not believe they could teach generalizable mathematics skills, which can, and should, be taught by mathematics teachers. Fourth, traditional mathematics teachers were perceived not to integrate mathematics with problems related to the world of work.

Implications

This study revealed that appropriate instruction could improve the achievement of CTE students who are perceived as having limited mathematics skills. The intervention was designed on the basis of the generalizable mathematics skills curriculum, functional program requirements, and student experiences. The intervention encourages students to explore and apply their experiential knowledge.

CTE students need mathematics instruction that begins with the basics and gradually progresses to advanced mathematics concepts, presenting problems in a concrete and practical context. Instruction should emphasize the integration of mathematics learning and applications in the world of work in CTE programs. Integration can facilitate the developmental processes of mathematics competencies and stimulate learner appreciation of mathematics skills (Masingila, Davidenko, & Agwu, 1994). Further, students can envision the practical value of mathematics learning through the linkage of mathematics to applications in the workplace and overcome their negative attitudes toward mathematics (Thiering, Hatherly, & McLeod, 1992).

Students were favorable towards the GMSI. Traditional mathematics instruction appears not to respond to student academic backgrounds and program requirements as well. Many students in CTE programs have had a negative attitude toward mathematics since elementary school. As a result, interest and motivation to learn mathematics often decrease due to frustration. Some students, therefore, may become stigmatized as 'non-academic' students.

Problem-solving skills assist students to effectively learn mathematics skills. To become a successful problem solver, students should be skilled in basic mathematics, especially reasoning, concepts, and translations from words to mathematics. Each teaching unit in the GMSI intervention addresses the explicit practice of a particular single mathematics skill for learners to conceptualize abstract and complicated mathematics theories (Hembree, 1992). Students cannot categorize and comprehend mathematics through extensive units in traditional CTE instruction that is based solely on CTE course needs (Lamon, 1993).

The interaction between teachers and students facilitates mathematics skills learning. When students are working on problems in class, they need clues or comments in regard to attempted solutions from teacher feedback and peer assistance. Interaction in classrooms also leads to a beneficial learning environment for effective learning (Yuen-Yee & Watkins, 1994).

The requirements and learning strategies in individual CTE programs influence student interest and motivation to relate mathematics concepts to employment goals (Stiff, 1989). This results in various positive effects on learning achievement. A student's experiences and relevant knowledge in his or her CTE program also enhances the comprehension of instructional materials. Effective mathematics skills interventions that integrate mathematics and CTE curriculum should be designed based on student experiences and needs.

Both male and female students can improve their mathematics skills through effective instructional interventions. For example, interventions have the potential to change traditional, unfavorable stigmas about female students' mathematics skills. Despite the fact that some female students have been characterized as non-mathematical, the female students in this study demonstrated that their mathematics skills could be improved when instruction related mathematics concepts to practical contexts.

Students' prior knowledge has strong effects on the acquisition of mathematics concepts. The connection between student prior knowledge and new mathematics skills may depend on his or her existing mathematics concepts and readiness for learning. This implies that it is important to consider student proficiency and experiences in the instructional design process. Stiff (1989) suggested that exemplification moves were more effective for students with low and median relevant knowledge. Therefore, teachers should supplement and vary the problems and emphasize a singular unit of mathematics skills to compensate for a student's limitations (Lamon, 1993). An instructional activity that focuses on learning-by-doing experiences with a variety of activities including real objects and models can stimulate a student's interests and assist to retain his or her learning experiences.

An important factor impeding student mathematics achievement in CTE programs is that some CTE teachers did not feel confident and/or responsible to systematically teach mathematics skills in their programs. CTE teachers were not required to take mathematics courses to become a certified CTE teacher; therefore, they were likely not comfortable teaching mathematics in their CTE programs. This suggests the need for staff development for CTE teachers in the integration of mathematics skills and CTE curricula.

Recommendations

The purpose of this study was to develop the GMSI and determine its effects on the mathematics achievement of students in secondary CTE programs. As is the case with most studies, this quasi-experimental study had several limitations. For instance, the sample used in this study was selected from one secondary area career center. Therefore, the results have limited generalizability to other CTE program areas and programs. Additionally, this study used only three subscales of the generalizable mathematics skills curriculum (i.e., whole numbers, fractions, and decimals). Therefore, the results cannot be generalized to the entire generalizable mathematics skills curriculum that also includes percents, mixed operations, measurement and calculation, and estimation subscales. Most of the students participating in this study had relatively high levels of mathematics skills. None of the students had low levels of mathematics skills. Consequently, inferences of the results to the general population of secondary CTE students are clearly limited. Further, the experimental and control groups were taught by the same teachers. This situation could introduce the threat of compensatory equalization of treatment and influence the independence of the groups. Also, the student subjects were assumed to be conscientious in regard to the GMPA. Thus, some students may not have been as attentive as other students regarding the performance assessment. Although this study had several practical limitations, they limited neither the procedures nor the impact of the results. The study revealed that a functional instructional intervention could improve the mathematics skills of students in secondary CTE programs.

Therefore, based on the findings and conclusions, several recommendations are offered for practice and future research.

- Generalizable mathematics skills instruction should be a requirement of CTE curricula.
- Generalizable mathematics skills instructional interventions in secondary CTE programs should consider a student's previous academic background and specific CTE program requirements.
- Group discussion in generalizable mathematics skills instruction should be encouraged to increase interaction among students and teachers.
- Staff development is necessary to improve the integration of mathematics skills into secondary CTE curricula.
- Future research should include more diverse samples of subjects from a variety of career and technical programs to determine the potential effects of the GMSI intervention.
- Future research should include and examine other generalizable mathematics skills and subscales in the Instructional Intervention.
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