Journal of Industrial Teacher Education
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 University
Career 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 handson 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, stepbystep 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 functionalbased 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 quasiexperimental design used in this study was a pretestposttest 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 (67102), median (3466), and low (033) (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 1Sample Selected for the Study
CTE Program Area 
Experimental
Group 
Control
Group 
Business
Computer Operations 
15 
7 
Family and Consumer Sciences
Child Care 
12 
5 
Health Occupations
Dental Assistant 
10 
12 
Industrial
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 (KuderRichardson [KR21]), which indicates that the instrument has high internal consistency and is measuring a uniform or single construct of generalizable mathematics skills. The testretest reliability of the total performance assessment was r = .65 (Pearson ProductMoment 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 subskills 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 15minute 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 selfreport inventory. During the sixweek 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 sixweek intervention was completed, the posttest was administered to the experimental and control groups.
Qualitative information was collected using a followup 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 ChiSquare values that indicate the homogeneity of variances of the GSPA on the on the pretest. The ChiSquare value for the Total Performance Assessment scale (ChiSquare = 12.22, p = .09) suggests acceptance of the null hypothesis and implies that, overall, the variances for the mathematics skills were generally equal.
Table 2Homogeneity of Variance of the GMSI on the Pretest
Performance  
Scale  ChiSquare  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 ShapiroWilk 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 3Normal 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 .22 
P: 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 Ftest 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 crosscase approach (Patton, 1990). The crosscase 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 (selfreport) 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 fourpoint 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 4Comparison of Mathematics Backgrounds between the Experimental and Control
Moderator  Mathematics  Experimental  Control  FValue  
Variables  Background  n  M  S.D.  n  M  S.D.  
Program 
All

MCN
MGPA 
48

2.88
2.63 
1.27
.62 
36

2.69
2.63 
1.37
.70 
<1
<1 
Bus

MCN
MGPA 
15

2.73
2.53 
.80
.61 
7

2.71
2.64 
.76
.48 
<1
<1 

FCS

MCN
MGPA 
12

3.83
2.86 
1.53
.43 
5

4.40
2.80 
1.14
.27 
<1
<1 

HO

MCN
MGPA 
10

2.20
2.50 
.79
.71 
12

2.00
2.25 
.85
.72 
<1
<1 

Ind

MCN
MGPA 
11

2.64
2.59 
1.36
.74 
12

2.67
2.93 
1.61
.79 
<1
1.05 

Gender 
Male

MCN
MGPA 
22

2.67
2.62 
1.15
.65 
18

2.72
2.87 
1.41
.68 
<1
1.16 
Female

MCN
MGPA 
26

3.04
2.63 
1.34
.61 
18

2.67
2.39 
1.37
.65 
1
1.26 

Math
Level 
High

MCN
MGPA 
42

2.95
2.67 
1.31
.60 
26

2.92
2.75 
1.20
.45 
<1
<1 
Median

MCN
MGPA 
6

2.33
2.33 
.82
.75 
10

2.10
2.30 
1.66
1.09 
<1
<1 
Table 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 5Comparison of the Pretest Means Between the Experimental and Control Groups
Dependent  Experimental  Control  
Moderate Variables  Variables  n  M  S.D.  n  M  S.D.  FValue  
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  
Gender 
Male  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
(67102) 
Subtotal

42

86.43

9.04

26

84.81

10.32

<1

Median
(3466) 
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 Ftests 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 6Pretest 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 7Analysis 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 8Comparison 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
35.08 
6.39
0.78 

31.78
32.57 
8.27
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 
<1 
FCS
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 
<1 
Ind
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 M 
26

84.77 82.35 
12.48 1.82 
18

72.89 76.38 
15.08 2.20 
4.22* 
Level of Math Skills  
High
Total M Adjusted M 
42

89.70 88.74 
5.89 1.06 
26

84.54 84.73 
8.39 1.35 
5.43* 
Median
Total M Adjusted M 
6

60.75 74.12 
6.55 5.20 
10 
58.00 64.63 
7.13 4.02 
4.08 
Note: * 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 9Comparison of the Pretest and Posttest Means for the Experimental Group
Moderator
Variables 
n 
Dependent
Variables 
Pretest

Posttest

F value 

M  SD  M  SD  
All
Programs 
48

Whole #
Fractions Decimals Total 
23.25
26.17 33.69 83.10 
1.63
5.45 7.29 12.43 
23.56
28.06 35.06 87.29 
1.27
4.14 6.39 10.00 
1.64
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
Math 
42

M

86.43

9.04

89.05

7.17

4.03*

Median
Math 
6

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 10Comparison of the Pretest and Posttest Means for Students in the Control Group
Moderator
Variables 
n 
Dependent
Variables 
Pretest

Posttest

F value 

M  SD  M  SD  
All

36

Whole #
Fractions Decimals Total 
23.03
23.03 31.42 77.47 
2.29
6.57 9.00 15.34 
22.39
24.47 31.78 78.64 
2.02
5.27 8.27 13.77 
3.09
3.10 <1 <1 
Bus  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.
Followup Interviews
Qualitative data were collected from posthoc 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 crosscase 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 (YuenYee & 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 followup 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 'nonacademic' students.
Problemsolving 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 (YuenYee & 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 nonmathematical, 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 learningbydoing 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 quasiexperimental 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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Greenan is Professor in the Department of Curriculum and Instruction at Purdue University, West Lafayette, Indiana. Greenan can be reached at jgreenan@purdue.edu.