Mathematics Achievement: The Role of Homework and Self-Efficacy

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Volume 22 ✤ Number 2 ✤ Winter 2011 ✤ pp. 310–339

a

Mathematics Achievement:

The Role of Homework and Self-Efficacy

Beliefs

Anastasia KitsantasJehanzeb CheemaHerbert W. Ware

George Mason University

According to Cooper, aAccording to Cooper, a homework involves tasks assigned to stu-dents by schoolteachers that are meant to be carried out dur-ing noninstructional time (Bembenutty, 2011). Homework is assigned for a variety of reasons such as to supplement learning activities and to practice concepts (Cooper, Robinson, & Patall, 2006). Given that learners conduct homework during noninstruc-tional time with little direction from the instructor and a less constricted timeline to complete it, researchers view homework as a tool to help students develop self-regulatory skills and self-efficacy to pursue academic tasks (Bembenutty, 2009; Kitsantas & Zimmerman, 2009; Zimmerman, Bonner, & Kovach, 1996). Zimmerman (1998) defined self-regulation as “self-generated thoughts, feelings, and actions for attaining academic goals” (p. 73). Self-efficacy refers to one’s beliefs in his or her ability to Self-efficacy refers to one’s beliefs in his or her ability to Self-efficacyperform at a designated level (Bandura, 1997). However, past research has challenged the benefits of homework with the view

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kitsantas, A., Cheema, J., & Ware, H. W. (2011). Mathematics achievement: The role of homework and self-efficacy beliefs. Journal of Advanced Academics, 22, 310–339.

The present study used the u.S. portion of the Program for International

Student Assessment (PISA) to examine how homework resources, math-

ematics self-efficacy, and time spent on homework impacted mathemat-

ics achievement across gender and ethnicity. The findings showed that

achievement gaps diminished with the increase in availability of home-

work resources and the increase in mathematics self-efficacy. Increased

proportions of homework time spent on mathematics homework were

associated with a decrease in mathematics achievement. These find-

ings suggest that educators should attempt to provide the resources for

students to complete their homework and structure homework assign-

ments accordingly. Interestingly, the findings also suggest that educators

need to focus on enhancing self-efficacy with respect to mathematics

for all students.

312 Journal of Advanced Academics

HoMEWoRk AND MATHEMATICS ACHIEVEMENT

that the use of homework expands the achievement differences between high and low socioeconomic status (SES) students, where students from higher SES backgrounds have more resources and their parents are better prepared to assist them than students from lower SES backgrounds (Balli, Wedman, & Demo, 1997).

Using the U.S. portion of the Program for International Student Assessment (PISA), the objective of the current study was to further examine how homework based on the number of hours spent outside the classroom and homework support resources such as having a quiet place to study and books are related to students’ self-efficacy and achievement in mathematics across race and gender.

Homework and Academic Achievement

Past studies examined the relationship between academic achievement and homework using variables such as the amount of homework assigned, time spent on homework, and the amount of homework actually completed (Cooper, Lindsay, Nye, & Greathouse, 1998; Trautwein, Köller, Schmitz, & Baumert, 2002; Zimmerman & Kitsantas, 2005). Generally, research using these variables remained inconclusive because most studies found that homework is not related with academic achievement in ele-mentary school. However, for the high school student population, some studies did show positive correlations between homework and achievement (Cooper, 2009). For example, Cooper et al. (1998) found that the most potent factor affecting achievement was the amount of homework the student actually completed as opposed to the amount of homework that was assigned. Although this pattern was consistent across most students, the proportion of homework completed was found to especially impact the aca-demic achievement of upper elementary and high school students as opposed to younger elementary school students.

Furthermore, Trautwein et al. (2002) analyzed a series of surveys administered to 1,976 middle school students and found that although the frequency of mathematics homework did posi-

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Kitsantas, Cheema, and Ware

tively impact mathematics achievement, the amount of homework and the length of time it took to complete the homework had no effect on achievement. This finding suggests that students who are regularly assigned mathematics homework in their classes gain more understanding in mathematics. However, homework assignments that require an extended amount of time to complete have no impact on mathematics achievement. The authors specu-lated that when students are given large amounts of homework, their motivation toward the topic declines. Additional analyses revealed that homework is more beneficial to the achievement of low-achieving students than to the achievement of high-achieving students. Specifically, the results showed that as teach-ers assigned more homework in their classes, the achievement gap between high- and low-achieving students became less evi-dent. Other researchers (e.g., Keith, Diamond-Hallam, & Fine, 2004) examined the longitudinal effects of completing homework either in school or out of school using the National Education Longitudinal Study (NELS) dataset. Keith et al.’s (2004) results revealed that time spent completing homework in school had a relatively large effect on student achievement whereas time spent completing homework outside of school had an insignificant effect on such achievement.

Numerous studies have also documented homework achieve-ment relationships between and within ethnic groups by immi-grant status, SES, and gender (Aldous, 2006; Carpenter & Ramirez, 2008; Trautwein et al., 2002). The findings show that time spent on homework differed between and within ethnic groups (e.g., Hispanic, Black, and White) and that it affected academic retention patterns (Carpenter & Ramirez, 2008). In fact, time spent on homework predicted dropout for Hispanic and White students, but not for Black students. Other research-ers (Plunkett, Behnke, Sands, & Choi, 2009) also examined how parental engagement in academic studies differed in terms of gender and ethnic background in a sample of immigrant minor-ity adolescents. Their results showed that across ethnicity and gender, student perceptions of their mothers’ engagement in their academic school work had a stronger impact on their achieve-



ment than the fathers’ engagement. Limited research however has examined how homework support resources influence stu-dent academic achievement and self-efficacy beliefs in mathemat-ics, particularly among different race groups. Homework support resources are defined as the different available resources that stu-dents can use while doing homework (e.g., having a desk and/or books).

Mathematics Self-Efficacy, Homework, and Achievement Across Race and Gender

Past research has associated completion of homework with student reports of high self-efficacy beliefs and use of self-regula-tory processes with the latter variable playing an especially impor-tant role in academic achievement (Bembenutty & Zimmerman, 2003; Kitsantas & Zimmerman, 2009). Specifically, research findings show that academic self-efficacy beliefs are positively related to homework completion (Pintrich & De Groot, 1990) with girls reporting higher self-efficacy in timely homework completion than boys (Pajares, 2001). Findings also show that homework helps children develop responsibility (Corno & Xu, 2004) and family homework help is associated with the use of management strategies regardless of the helper’s educational level (Xu & Corno, 2003, 2006).

Research findings with regard to race and gender differ-ences in self-efficacy in mathematics are ambiguous (Schunk & Meece, 2005). Generally, White students exhibited higher self-efficacy for mathematics than African American students (Pajares & Kranzler, 1995), and girls reported lower mathemat-ics self-efficacy beliefs even when they performed at similar or higher levels than boys (Pajares, 1996). Researchers also exam-ined how students form their efficacy beliefs based on information they receive from sources such as mastery experiences, vicarious experiences, social persuasions, and emotional and physiological states. For example, Usher and Pajares (2006) found that only mastery and persuasion experiences predicted African American



middle school students’ self-efficacy beliefs, whereas all sources of self-efficacy contributed to White students’ self-efficacy beliefs. Students’ level of self-efficacy beliefs may also depend on a vari-ety of other contextual factors such as social capital (Schunk & Meece, 2005). Specifically, the more social capital that families have, the more able they are to provide substantive resources to support student learning and motivation. Therefore, it is impor-tant to examine how such predictors of motivation and achieve-ment differ as a function of race.

Purpose of the Study

The present study attempted to identify the extent to which mathematics self-efficacy beliefs, time spent on math-ematics homework, and homework support resources predicted high school student mathematics achievement. This study also attempted to explore the role of race and gender in these relationships.

Method

Data Source

The data come from the 2003 Program for International Student Assessment (PISA) student and school questionnaires (National Center for Educational Statistics [NCES], 2003). The PISA assessed reading literacy, mathematics literacy, and sci-ence literacy skills of 15-year-olds in the U.S. This survey was administered by the NCES. Of the three subjects, reading, math-ematics, and science, only one subject is surveyed in depth on a rotational basis each year while the other two are given rela-tively less attention. In 2003, the primary subject was mathemat-ics (Organization for Economic Cooperation and Development [OECD], 2005). The target population for this study was the entire 15-year-old high school student population of the United



States. The sample comprised of 5,456 students from 274 par-ticipating schools chosen through multi-stage stratified random sampling (OECD, 2005).

Participants

After listwise deletion, the U.S. sample size was reduced to 3,776 students and 221 schools. However, a preliminary missing value analysis indicated that these values were missing at random. Therefore, missing values were imputed using all possible con-tinuous and categorical predictors. Missing values on categorical predictors such as gender and race were not imputed. Because missing values are missing at random and not missing completely at random, the maximum likelihood imputation method utilizing the expectation-maximization algorithm was employed.

The final sample size for analysis was 5,200 students (boys, n = 2,603; girls, n = 2,597) and the ethnic breakdown was 3,097 Caucasian, 799 African American, 883 Hispanic, 169 Asian, and 252 of mixed or other ethnicity. Student ages (M = 15.83, SD = 0.29) ranged from 15.25 to 16.33 years. Because the PISA student survey was designed to assess mathematics literacy of 15-year-old students, the reported final sample size reflects stu-dents from grades 9 (n = 1,618), 10 (n = 3,249), and 11 (n = 333) only. A very small number of students were found in grade 7 (n = 8) and grade 8 (n = 91) and as a result were not included in our analyses.

Of the 5,200 students in our sample, 98.4% (n = 5,115) pro-vided information on their place of birth (USA = 91.1%; other = 7.3%); 97.2% (n = 5,053) reported the language they spoke at home (English = 88.4%; other = 8.8%); 99.2% (n = 5,160) reported their family structure (single parent family = 29.2%; nuclear family = 54.9%; mixed family = 10.6%; other = 4.6%); and 98.0% (n = 5,098) reported their parental education level (less than high school = 5.5%; high school = 45.6%; more than high school = 46.9%).



Measures

Mathematics achievement. This variable was based on 85 test items and was reported on a continuous scale as a set of five plausible values for each student. The plausible values were ran-dom elements from the set of scores that could be attributed to each student and their variation helped capture the measurement error at the individual student level (Wu, 2005). A detailed tech-nical discussion of plausible values can be found in Mislevy (1991) and Mislevy, Beaton, Kaplan, and Sheehan (1992). A sample item included,

In a pizza restaurant, you can get a basic pizza with two toppings: cheese and tomato. You can also make up your own pizza with extra toppings. You can choose from four different extra toppings: olives, ham, mushrooms and salami. Ross wants to order a pizza with two different extra toppings. How many different combinations can Ross choose from?

Each response was coded as either correct or incorrect with no partial credit awarded (OECD, 2005). For the U.S. sample, on a scale of 0 to 1,000, the plausible values for mathematics had a mean of 483 (for OECD, M = 500). The reliability estimate for the five plausible values was α = 0.98. For each student, the mean plausible value was used as a measure of mathematics achievement.

Mathematics self-efficacy. This scale included eight ques-tions that measured a student’s confidence in performing various mathematical calculations ranging from solving elementary linear equations in one variable to calculating a percentage. A sample question included, “How confident do you feel about having to do the following calculation(s)? Solving an equation like 3x + 5 = 17.” The response scale range was from 1–4, 1 (very confident), 2 (confi-dent), 3 (not very confident), and 4 (not at all confident). Responses were reverse coded and scaled in such a way that higher scores are indicative of higher levels of this variable (OECD, 2005). The



reliability of this scale was (α = 0.86). Item means ranged from 1.45 to 2.21, item variances ranged from 0.50 to 0.78, and cor-rected item-total correlations ranged from .54 to .66.

Relative time spent on mathematics homework. This vari-able represented the ratio of actual number of self-reported hours spent by a student solely on mathematics homework to actual number of hours spent on all homework. Prior research suggests that the effect of absolute time spent on homework may dif-fer across grade level (Cooper, 2009). Using the ratio instead of solely its numerator is more meaningful because it allows some standardization across students belonging to different schools and grades. For instance, the use of the number of hours spent on mathematics homework would not let us properly compare two students who spend an equal amount of time on mathemat-ics homework but their percentage times differ. The rationale is similar to using the relative amount of school funding (number of dollars spent per student within a school) rather than the absolute figure (total number of dollars for the school). Relative time spent on mathematics homework ranged from 0 to 1 (M = .20, SD = .17) where 1 indicates that the student had allocated all of the homework time to mathematics and 0 indicates that the student did not spend any time on mathematics homework.

Homework support resources. This variable was based on eight questions and measured homework-related SES character-istics. The homework resources variable included: a desk to study at, a room of their own for the student, a quiet place to study, a computer for use with school work, a link to the Internet, their own calculator, books to help with their homework, and diction-aries. Each question required respondents to report whether or not they had a specific item or service at home. For instance, one question asked whether the student had “a quiet place to study” at home. Another inquired whether the student had “a desk for study” at home. The response choice was either “Yes” or “No.” Item means ranged from 1.07 to 1.47, item variances ranged from 0.06 to 0.25, and corrected item-total correlations ranged from .25 to .49. Homework support resources represented the proportion of homework support available to a student and ranged from 0 to



1. Thus, the eight items related to this measure were a random sample from the population of all possible items that support homework activities at home. A student who answered “Yes” to all eight questions had a value of 1 and a student who answered “No” to all questions had a value of 0. The reliability of this scale was α = 0.70.

The measures presented in this section were based on survey items, which were validated by subject matter experts who took into consideration the cultural and educational contexts in which these items were expected to be used. The work of subject mat-ter experts was supported by a pilot study and a field trial that included a large number of test items and was designed to assess both the qualitative and quantitative aspects of item responses. Only those items that survived scrutiny of the experts and the field trial were retained and became the basis for measures used in this study.

Data Collection Approach

The PISA questionnaire standards required recruiting up to 35 students at random within each school from a list of all students born in 1987. They were first given a 20–30-minute long back-ground questionnaire and then a timed 2-hour paper-and-pencil test including both multiple choice and constructed response items on mathematics, science, and reading. Approximately two thirds of the questions were related to mathematics. In the U.S., all materials were written and administered in English. Students that were unable to read or speak the language of the test were excluded from the sample (Lemke et al., 2004).

Data Analytical Approach

Descriptive statistics were reported for all variables. Four weighted least squares (WLS) multiple linear regression models were fitted. Dummy variables were created for both race and gender. The base model included only demographic variables, race and gender, to which homework-related predictors, self-effi-



cacy, and interaction effects were added sequentially in blocks. Statistical analyses were performed after taking into consider-ation the appropriate sampling weights. All continuous variables, including mathematics achievement, were standardized to have a mean of 0 and standard deviation of 1 before inclusion in a regres-sion model. In order to evaluate the robustness of the multiple regression results, a repeated measures WLS MANOVA was fit-ted with the five plausible values specified as dependent variables. Results from the multiple regression and MANOVA procedures were then compared for consistency.

The residuals from each fitted model were analyzed to ver-ify that model assumptions were not violated. In order to check for heteroscedasticity, studentized deleted residuals were plot-ted against standardized predicted values, and in order to detect departures from normality, the histogram and probability plot of residuals were analyzed. No evidence of a severe violation of either the homoscedasticity or normality assumption was found. Multicollinearity diagnostics included computation of variance inflation factors (VIF) for the four models, which ranged between 1.00 and 2.45 lying well below the VIF ≥ 10 rule-of-thumb threshold for severe multicollinearity. Condition index for the four models ranged between 3.00 and 3.58, which is well below the corresponding rule-of-thumb cut-off values of 15 and 30 for moderate and severe multicollinearity.

Results

Descriptive and Correlational Findings

Summary statistics for mathematics achievement and its pre-dictors are presented in Table 1. Descriptive statistics showed that on average students spent about a fifth of their homework time on mathematics homework (M = 0.20, SD = 0.17). In terms of relative time spent on mathematics homework, 5.3% reported spending no time, 69.0% reported spending more than 0% but less than or equal to 20%, 17.1% reported spending more than



20% but less than or equal to 40%, 2.3% reported spending more than 40% but less than or equal to 60%, 5.1% reported spending more than 60% but less than or equal to 80%, and 1.2% reported spending more than 80% of their total homework time on math-ematics. In terms of total hours spent on mathematics homework per week (M = 3.69, SD = 2.31), 5.3% of the students reported not spending any time on mathematics homework, 80.7% reported spending more than 0 but less than or equal to 5 hours, 13.3% reported spending more than 5 but less than or equal to 10 hours, and 0.7% reported spending more than 10 hours.

Some significant correlations also emerged among the vari-ables. For example, mathematics self-efficacy and mathematics achievement were highly correlated (r = .54, p < .001). Homework support resources had a moderately positive correlation with achievement (r = .32, p < .001) and with mathematics self-efficacy (r = .23, p < .001), but negatively associated with relative amount

Table 1

Means, Standard Deviations, Cronbach’s Alphas, and Correlations for Mathematics Achievement and Its Predictors

r

M SD α† 1 2 31. Average mathematics achievement‡

484.19 89.14 -- --

2. Mathematics self-efficacy 0.00 1.00 .86 0.54* --

3. Relative time spent on mathematics homework‡

0.20 0.17 -- -0.17* -0.10* --

4. Homework support resources‡ 0.85 0.20 .70 0.32* 0.23* -0.06*

Note. N = 5,200.†Cronbach’s alpha is not applicable for mathematics achievement, which is based on plausible values, and relative time spent on mathematics homework, which is a ratio. ‡These variables were standardized before inclusion in the regression models. *Correlation is significant at the .001 level (2-tailed).



of time spent on mathematics homework (r = –.06, p < .001). Finally, although the relationship between relative time spent on mathematics homework and mathematics self-efficacy was weak, it was statistically significant (r = –.10, p < .001).

The results of WLS ANOVA on the relative amount of time spent on mathematics homework by gender showed a significant difference between males (M = 0.20, SD = 0.17) and females (M = 0.21, SD = 0.18), F(1, 5198) = 5.50, p = .019, with females on average spending approximately 5% more time on mathematics homework as compared to males. A similar analysis of relative amount of time spent on mathematics homework by race revealed a significant race effect, F(4, 5198) = 14.99, p < .001. After apply-ing the Tukey post hoc adjustment, mean relative mathematics homework time significantly differed between Blacks (M = 0.23, SD = 0.19) and Whites (M = 0.19, SD = 0.16), t = 6.33, p < .001, with Black students on average spending approximately 21% more time on mathematics homework than their White counterparts. A significant difference was also found between Hispanics (M = 0.22, SD = 0.19) and Whites, t = 5.52, p < .001, with Hispanic students on average spending approximately 16% more time on mathematics homework than their White counterparts.

For the second homework variable, homework support resources, only about 0.7% of the students reported having no access to any of the critical homework support items or services at home, 1.9% reported access to more than 0% but less than or equal to 25% of support items, 5.8% reported access to more than 25% but less than or equal to 50% of support items, 22.1% reported access to more than 50% but less than or equal to 75% of support items, and 69.5% reported access to more than 75% of such items. When analyzed across gender categories, a significant mean difference was detected on homework support resources between males (M = 0.84, SD = 0.20) and females (M = 0.86, SD = 0.19), F(1, 5198) = 12.80, p < .001, with the mean homework support for females being slightly (2.4%) but significantly higher than that for males. When analyzed across race, significant mean differences in homework support resources emerged, F(4, 5195) = 77.67, p < .001. After applying the Tukey post hoc adjustment,



significant differences in mean homework support resources were found between Blacks (M = 0.80, SD = 0.23) and Whites (M = 0.88, SD = 0.17), t = 11.41, p < .001; Hispanics (M = 0.78, SD = 0.24) and Whites, t = 15.28, p < .001; multiple/other races (M = 0.85, SD = 0.18) and Whites, t = 3.33, p = .008; Asians (M = 0.89, SD = 0.17) and Blacks, t = 5.71, p < .001; multiple/other races and Blacks, t = 3.46, p = .005; Asians and Hispanics, t = 7.31, p < .001; and multiple/other races and Hispanics, t = 5.31, p < .001. Results of these comparisons can be summarized as follows: On average, White students had 10% more homework support resources available than Black students, 13% more homework support than Hispanic students, and 4% more homework support than students who identified themselves as belonging to multiple/other races. On average, Asian students had about 11% more homework support than Black students and 14% more homework support than Hispanic students. Finally, the mean homework support level of multiple/other race students was approximately 6% higher than that of Black students and 9% higher than that of Hispanic students.

Although several of the differences in mean mathematics homework time and homework support resources are small when gender and race categories are considered independently, such differences were controlled for in the analyses that follow. Full factorial two-way WLS ANOVA models were also estimated for both the amount of time spent on mathematics homework and homework support resources. However, no change in the pattern of significant effects was revealed. The gender by race interaction was not significant for either variable. These two-way ANOVA models were reestimated without the interaction term. However, again no change in the pattern of significant effects was observed.

Results of Multiple Regression Analysis

Model 1. The first multiple regression equation predicted mathematics achievement from gender and race. Results showed a difference of 0.10 standard deviations (p < .001) in mean math-ematics achievement between males and females after controlling



for race with mean mathematics achievement of males exceeding that of females. Controlling for gender, the mean mathematics achievement of White students was 1.03 standard deviations (p < .001) higher than that of Black students, 0.75 standard deviations (p < .001) higher than that of Hispanic students, 0.28 standard deviations (p < .001) higher than that of multiple/other race stu-dents, and not significantly different from that of Asian students (p = .487). Gender and race cumulatively accounted for about 17% of the total variation in mathematics achievement (see Table 2).

Model 2. The base model was augmented by adding two homework-related predictors of mathematics achievement: the relative amount of time spent on mathematics homework and homework support resources. This model attempted to deter-mine the proportion of variation in mathematics achievement that could be explained by the two homework-related predictors over and above that explained by race and gender. The model accounted for approximately 24% (ΔR2 = .08, p < .001) of the total variation in mathematics achievement. Thus, the two homework-related predictors accounted for an additional 8% of the total variation in mathematics achievement over and above that pre-dicted by race and gender.

Model 3. In the third model, mathematics self-efficacy was added as a predictor of mathematics achievement in addition to gender, race, relative time spent on mathematics homework, and homework support. The primary function of this model was to determine the proportion of variation in mathematics achieve-ment that could be explained by mathematics self-efficacy over and above that explained by race, gender, and the homework-related predictors. This model accounted for approximately 44% (ΔR2 = .20, p < .001) of the total variation in mathematics achievement. Thus, mathematics self-efficacy accounted for an additional 20% of the total variation in mathematics achievement over and above that predicted by race, gender, and homework-related variables. Estimation results showed that the significant difference in mean mathematics achievement between males and females observed in the previous two models disappeared with the introduction of mathematics self-efficacy (β = 0.01, p =



Table 2

Regression Analysis Summary for Predicting Mathematics Achievement

Model 1 Model 2

Predictors B SE B β † t ‡ P B SE B β † t ‡ P

Black -1.03 0.04 -0.36 -27.61 .001 -0.88 0.04 -0.31 -24.51 .001

Hispanic -0.75 0.04 -0.28 -21.20 .001 -0.58 0.04 -0.22 -16.81 .001

Asian -0.05 0.07 -0.01 -0.70 .487 -0.05 0.07 -0.01 -0.71 .477

Multiple/other -0.28 0.06 -0.06 -4.67 .001 -0.21 0.06 -0.05 -3.63 .001

gender 0.10 0.03 0.05 3.98 .001 0.12 0.03 0.06 4.76 .001

Relative mathematics homework time

-0.12 0.01 -0.12 -9.83 .001

Homework support

0.26 0.01 0.25 20.16 .001

R2 0.166 207.41 .001 0.243 261.11 .001

Model 3 Model 4

Predictors B SE B β † t ‡ P B SE B β † t ‡ P

Black -0.82 0.03 -0.29 -26.41 .001 -0.85 0.03 -0.30 -27.10 .001

Hispanic -0.50 0.03 -0.19 -16.76 .001 -0.50 0.03 -0.18 -16.24 .001

Asian -0.15 0.06 -0.03 -2.60 .009 -0.16 0.06 -0.03 -2.53 .011

Multiple/other -0.25 0.05 -0.05 -5.04 .001 -0.24 0.05 -0.05 -4.81 .001

gender 0.01 0.02 ≈0.00 0.40 .687 0.01 0.02 ≈0.00 0.38 .703


-0.08 0.01 -0.08 -7.89 .001 -0.07 0.02 -0.07 -4.87 .001

Homework support

0.15 0.01 0.15 13.74 .001 0.14 0.02 0.14 8.44 .001

Mathematics self-efficacy

0.47 0.01 0.46 42.67 .001 0.51 0.01 0.50 37.28 .001

Mathematics self-efficacy x Black

-0.21 0.03 -0.07 -6.14 .001

Mathematics self-efficacy x Hispanic

-0.09 0.03 -0.04 -2.98 .003

Mathematics self-efficacy x Asian

0.02 0.06 ≈0.00 0.27 .786



Mathematics self-efficacy x Multiple/other

-0.05 0.05 -0.01 -1.10 .272

Relative mathematics homework time x Black

0.04 0.03 0.02 1.50 .133

Relative mathematics homework time x Hispanic

-0.07 0.03 -0.03 -2.53 .011

Relative mathematics homework time x Asian

≈0.00 0.06 ≈0.00 -0.02 .984

Relative mathematics homework time x Multiple/other -0.07 0.05 -0.02 -1.47 .141

Homework support x Black

0.01 0.03 0.01 0.36 .722

Homework support x Hispanic

0.03 0.03 0.02 1.25 .210

Homework support x Asian

-0.06 0.07 -0.01 -0.79 .432

Homework support x Multiple/other

0.16 0.05 0.03 2.99 .003

R2 0.439 1820.85 .001 0.447 5.87 .001

Note. N = 5,200. R2 = proportion of explained variance; in the last row for each model, reported signifi-cance is for F test of Δ R2. Female and White are reference categories for gender and race. †β = Standardized slope coefficient. ‡F value reported for the Δ R2 test.

Table 2, continued



.687). The pattern of significant differences in mean mathemat-ics achievement between different racial groups remained largely the same with the introduction of mathematics self-efficacy as a predictor. The only exception was the White-Asian gap that was now significant (β = –0.15, p = .009). For a one standard deviation increase in each of the individual predictors with all remaining predictors held constant, the increase in mathematics achieve-ment was –0.08 standard deviations (p < .001) for relative time spent on mathematics homework, 0.15 standard deviations (p < .001) for homework support, and 0.47 standard deviations for mathematics self-efficacy (p < .001).

Model 4. In order to analyze interaction effects, a final model was fitted that contained the following interactions: race by home-work support, race by relative time spent on mathematics home-work, and race by mathematics self-efficacy. Regression results did not reveal any change in the significance pattern for predic-tors that were included in Model 3 with estimates of regression coefficients being very similar to those obtained for Model 3. The analysis of two by two interactions involving homework-related variables, self-efficacy, and race revealed a number of significant effects. First, it was observed that the effect of homework support on mathematics achievement was significantly different between White and multiple/other race students. A one standard deviation increase in homework support caused the mathematics achieve-ment of White students to increase by 0.14 standard deviations (p < .001), and that of multiple/other race students to increase by 0.30 standard deviations (p = .003). Second, the effect of relative mathematics homework time on mathematics achievement was significantly different between White and Hispanic students. A one standard deviation increase in relative amount of time spent on mathematics homework caused the mathematics achievement of White students to decrease by 0.07 standard deviations (p < .001), and that of Hispanic students to decrease by 0.14 standard deviations (p = .011). Third, the effect of mathematics self-efficacy on mathematics achievement was significantly different between White and Black students, and between White and Hispanic students. A one standard deviation increase in mathematics self-



efficacy raised mathematics achievement of White students by 0.51 standard deviations (p < .001), that of Black students by 0.30 standard deviations (p < .001), and that of Hispanic students by 0.42 standard deviations (p < .001). Model 4 accounted for approximately 45% (ΔR2 = 0.01, p < .001) of the total variation in mathematics achievement. Although the inclusion of interactions adds little to the proportion of explained variance in mathematics achievement, such addition is nevertheless significant and reveals important differences in the effects of continuous predictors on mathematics achievement across race categories.

Results of MANOVA Analysis

A full factorial repeated measures WLS MANOVA model was fitted with all possible two-way interactions for factors and covariates. Nonsignificant interactions obtained in this prelimi-nary model were dropped and the model reestimated. The final MANOVA results are summarized in Table 3. By comparing the pattern of significance of the MANOVA model with that of the multiple regression Model 4, it becomes clear that the two approaches generate very similar results. The only predictor that is not significant in the MANOVA Model is gender (p = .780). The partial effect size estimates, η2, for the MANOVA model range from approximately 0 to .104. These estimates provide an indi-cation of the variance in dependent variable uniquely accounted for by each predictor. For instance, the partial effect size of .068 for race suggests that about 6.8% of the variation in mathematics achievement was due to racial differences among students.

Discussion

This study sought to determine the role of homework support resources, self-efficacy, and proportion of time spent on math-ematics homework in improving the mathematics achievement of 15-year-olds as measured in PISA 2003. A secondary interest was in how homework resources may be useful in closing the achieve-



Table 3

MANOVA Analysis Summary for Predicting Mathematics Achievement

MANoVA

Source SS† df MS F p ηp2

gender 0.20 1 0.20 0.08 .780 ≈0.00

Race 972.62 4 243.16 94.52 .001 .068

Homework support 200.27 1 200.27 77.86 .001 .015


93.02 1 93.02 36.14 .001 .007

Mathematics self-efficacy 1551.38 1 1551.38 603.01 .001 .104

Race x Homework support 25.10 4 6.28 2.44 .045 .002

Race x Relative mathematics homework time

26.83 4 6.71 2.61 .034 .002

Race x Mathematics self-efficacy

95.41 4 23.85 9.27 .001 .007

Error 13323.94 5179 2.57

Marginal Mean Comparisons for gender and Race

Estimates Pairwise Comparisons

Factor M SE Difference (D) MD SE t p 95% CI d‡

gender Female 474.29 1.84 Female – Male 0.52 1.88 0.28 .780 (-3.16, 4.21)

Male 473.76 1.80 White – Black 75.46 2.77 27.23 .001 (67.67, 83.24) 0.94

Race White 503.81 1.23 White – Hispanic 40.62 2.68 15.14 .001 (33.09, 48.16) 0.49

Black 428.35 2.48 White – Asian 10.67 5.54 1.92 .544 (-4.90, 26.23) 0.13

Hispanic 463.19 2.38White – Multiple/other

22.18 4.44 5.00 .001 (9.72, 34.64) 0.27

Asian 493.14 5.40 Black – Hispanic -34.83 3.44 -10.12 .001 (-44.50, -25.17) -0.42

Multiple/other

481.63 4.26 Black – Asian -64.79 5.95 -10.89 .001 (-81.49, -48.09) -0.83

Black – Multiple/other

-53.28 4.93 -10.80 .001 (-67.13, -39.42) -0.66

Hispanic – Asian -29.96 5.90 -5.07 .001 (-46.54, -13.37) -0.34

Hispanic – Multiple/other

-18.44 4.88 -3.78 .002 (-32.16, -4.73) -0.21

Asian – Multiple/other

11.51 6.88 1.67 .945 (-7.82, 30.84) 0.13

Note. N = 5,200. Adjustment for multiple comparisons: Bonferroni. Female and White are reference categories for gender and race. †Type III sum of squares. ‡ Square root of the weighted average of the variances in the groups forming the con-trast used as denominator for Cohen’s d.



ment gap between minority and White students. Achievement gaps diminished with the introduction of two variables: (a) the availability of resources for homework completion and (b) the development of higher levels of mathematics self-efficacy. In gen-eral, increased proportions of homework time spent on math-ematics homework decreased mathematics scores. Specifically, our multiple regression Model 3 with its rather large contribution of explained variance (44%), offers guidance to teachers, parents, siblings, tutors, and mentors who provide homework assistance to 15-year-olds. For Blacks, Hispanics, and Asians, the achieve-ment gap with Whites shrank when mathematics self-efficacy was introduced in Model 3, suggesting that educators need to focus on enhancing self-efficacy with respect to mathematics for all students.

Homework Support Resources

The greater the proportion of homework support resources that were available to the students, the higher their mathemat-ics score. For Blacks, Hispanics, and Asians, the achievement gap with Whites shrank when homework support resources were introduced to the regression equation. Thus, if one were to use this model to improve mathematics scores as measured in PISA, one would seek ways to increase student access to the resources identified with the variables in this analysis. Given that these resources constitute a form of social capital (Schunk & Meece, 2005), they also have the potential to enhance an individual’s self-efficacy beliefs. Although some of these resources seem to be characteristic only of a home site, increasing the number of these available at any site where homework support is provided should be viewed as important.

The availability of these resources or the lack thereof has other implications. Brock, Lapp, Flood, Fisher, and Han (2007) found that teachers in urban school systems cognizant of the lack of resources available to some students prompted teachers to avoid assigning homework requiring resources that the students did not have. Alternatively, the teachers would attempt to provide



the resources for students to complete their homework. Viewed in the context of the impact of homework support resources on mathematics achievement, this illustrates the importance of the teacher’s awareness of the resources available to students and the need to structure homework assignments accordingly. The risk here is that students with fewer resources could be offered assign-ments below their true potential.

Mathematics Self-Efficacy

For Blacks, Hispanics, and Asians, the achievement gap with Whites was reduced when mathematics self-efficacy was intro-duced in Model 3. Our Model 4, in which we examined the interaction of ethnic group with mathematics efficacy, produced significant interactions in this regard, but only a slight increase in explained variance. This seems consistent with the ambiguous findings of others regarding ethnic-efficacy interactions (Britner & Pajares, 2001; Pajares & Kranzler, 1995). Although the ques-tions used to assess mathematics self-efficacy in this survey seem very specific, they are within the realm of the mathematics expo-sure of American 15-year-olds. The implication is not that these specific content points should be addressed with the students, but that the students should be helped to the point that they feel effi-cacious in handling the mathematics content to which they have been exposed. Based on empirical findings, different methods can be used to enhance students’ efficacy beliefs. For example, Usher and Pajares (2006) showed that mastery experiences and social persuasion were the most potent predictors of academic self-efficacy for girls while mastery and vicarious experiences were the most potent predictors of academic self-efficacy for boys. In terms of ethnic differences, mastery experiences and physiological states were the most potent predictors of academic self-efficacy for White students while mastery experiences and social persua-sion were the most potent predictors of self-efficacy for African American students. Additional evidence of gender differences comes from Usher (2009), where interviews with teachers and parents of middle school students about their child’s performance



in mathematics suggested that parents and teachers were more likely to attribute girls’ performance to hard work while express-ing their surprise that boys could perform at such high levels regardless of their poor work habits. Usher suspected that the messages that parents and teachers are giving to their students may be implicitly impacting and informing their self-efficacy beliefs. In terms of ethnic differences, Usher found that African American boys tended to rely more heavily on social persuasion than White students.

Proportion of Homework Time Devoted to Mathematics

Increased proportions of homework time spent on mathe-matics actually decreased mathematics achievement. Although this was a surprising finding, a lack of understanding of a sub-ject can lead to inefficient and disproportionate effort as well as diminished motivation. Trautwein et al. (2002) found that while the frequency of homework positively affected mathemat-ics achievement, the amount and length of time it took middle school children to complete homework did not have an effect on achievement in mathematics. When coupled with the work of Keith et al. (2004), completing homework at school may have a greater benefit than leaving it for completion outside of school.

Implications for Practice

The results of the present study were based upon a representa-tive sample of public school 15-year-olds attending 9th, 10th, and 11th grades in U.S. high schools surveyed relative to mathematics in 2003. Because no allowance was made for differences among high schools or for other subjects, results are best interpreted for the general population of 15-year-olds and not for the population of a particular high school or for implications for other subject areas. Nevertheless, the findings of the present study have sig-nificant implications for practice. First, the results revealed that for teachers, parents, siblings, and tutors who provide homework



assistance to 15-year-olds—whether it be at home, in a school lab, at a community center study hall, or at the workplace of a mentor, it is important to provide the site with as many of the homework resources as possible, particularly for Black and Hispanic students. This may include creating a library within the classroom from which the student might check out a calculator, necessary books, dictionaries, or drawing instruments. Within the school, the cost of acquiring these resources might come from the library, textbook, or equipment funds allocated in the school budget. Alternatively, partnerships with local businesses could be potential sources of funding for these items, either within the school or for sites outside the school. Public recognition of such partnerships and contributions could serve as an inducement for participation. The teacher can become the source of the simplest of the items that the student requires, either from his or her own resources, from those of the school, or from contributions sought from school patrons. Creating a quiet study environment and providing a computer with Internet access for homework help would be desirable enhancements to the study site.

Second, based on the findings of our study, teachers, parents, siblings, and tutors should make sure that the students feel effica-cious in handling the mathematics content to which they have been exposed by creating mastery experiences where students can feel successful with their work. The object is to create class-room and homework assignment settings that facilitate a progres-sion from the easy to the more difficult while increasing student beliefs in their mathematics efficacy with solution demands rang-ing from those requiring simple recall to those demanding analy-sis, synthesis, or evaluation. Within the classroom, the teacher might structure the homework in such a way that there is dif-ferentiation for the skill and understanding levels of the students. For example, success for the mathematically challenged student could be generated by assigning for those students, and verify-ing that success within the class, a cluster of problems sufficient for them to demonstrate that they can solve the less complex problems. Adding to that, cluster a problem requiring applica-tion of the technique can serve to increase the students’ belief



that they can apply the technique. For the more able student, reducing the number of problems offered requiring the lower level of understanding and substituting problems requiring more sophisticated application or analysis serves to challenge the more capable student. In both instances, the efficacy research related to mastery experiences suggests that verifying that students can solve the problems, before sending them off to pursue homework, likely will reduce the proportion of time spent on homework and increase mathematics achievement. Of course, these strategies require that the teacher know which students are challenged and which ones are more capable.

Third, our findings show that increasing the amount of time spent on mathematics homework does not lead to higher mathe-matics achievement scores. That is, students who spend a relatively high proportion of time on mathematics homework are likely to have low achievement scores, low self-efficacy beliefs, and low values on homework support resources. This observation fits the notion that students who have low mathematics scores and spend more time on mathematics homework do it precisely because of low self-efficacy and fewer homework support resources. Having students complete homework at school may have greater benefits than having them complete it outside of school. Furthermore, because mathematics self-efficacy beliefs and homework support resources are related to the time spent on mathematics home-work, educators should pay particular attention to addressing the resources and self-efficacy issues.

Providing homework support to the student at sites outside the mathematics classroom has implications for training those who help the student. Although it may be unreasonable to expect every teacher, parent, mentor, or supervisor of a study site to attend training sessions related to aiding students with math-ematics homework, it is not unreasonable to offer those oppor-tunities—either at the school or at neighborhood work sites. Familiarizing those individuals with the resources likely to be helpful for mathematics homework completion is a starting point. Communicating to them the resources needed for a specific unit (as simple as a ruler or straightedge, compass, and protractor for



geometry activities) constitutes a beginning. Finally, emphasizing mastery over extensive time spent on homework and acknowl-edging the differences in the ways students learn is important. For example, girls and African Americans learn from mastery and social experiences and boys learn from mastery and vicari-ous experiences (Usher & Pajares, 2006). Girls can also learn from hard work and boys can perform at high levels in spite of seemingly poor work habits (Usher, 2009). Emphasizing, again, mastery over time spent on the homework is important. It also is reasonable to point out that we expect those who help the student to be facilitators of the homework process. We do not expect them to be masters of mathematics.

Limitations

This study has both statistical and practical limitations. First, as with every secondary data set, we were limited to the items that the participants were asked. Some of the items could have been better worded to elicit more accurate responses. In addition, given that we pooled all of the students into one group for a linear regression, we made no adjustment for a second level of analysis in which characteristics of their schools were considered. Students are not only nested within schools but are also nested within classrooms with the classrooms in turn nested within schools. Although such a three-level hierarchical design can be potentially more informative, unfortunately our source data do not provide any classroom level information. For the PISA 2003 survey, stu-dents were sampled randomly within each school from a list of all students who were born in the 1987 calendar year without any regard to which classroom they came from. Thus, even though a student can be matched with the correct school, there is no way to tell which classroom the student originated from within that school. Furthermore, for the schools included in our sample, the number of students sampled from an individual school ranged between 3 and 29 (M = 19.70, SD = 5.47), which means that even if classroom information were available, there would be cases in



our sample where 3 or fewer cases belonged to a classroom. Such a small number of cases per school or classroom thus rules out any effective application of two-level or three-level hierarchical modeling in our study. Future studies using a multilevel analysis with an alternative data set, given the availability of such data, in which the school is the level-2 unit could further inform practice.

From the practical perspective, our inability to characterize specific school settings limits our ability to prescribe practice in a specific school. The availability of homework resources can vary from neighborhood to neighborhood. The proportion of Black and Hispanic students varies from school to school. We do not know how mathematics self-efficacy is addressed in dif-ferent schools. We have no knowledge of the specific homework practices employed in each school. We lack knowledge, in this data set, of the forms of homework support provided in each school neighborhood. The absence of this data, however, offers fruitful ground for a school’s self-assessment of its mathematics homework policies and practices.

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Mathematics Achievement: The Role of Homework and Self-Efficacy