
Dropping Out of Advanced Mathematics: The Effects of Parental Involvementby Xin Ma  1999 Based on a national sample from the Longitudinal Study of American Youth (LSAY), this study examined the effects of individual characteristics and different types of parental involvement on participation in advanced mathematics from grade 8 to 12. Results of hierarchial survival analysis show that an average student is most likely to drop out in grade 12, where there is also a large gender gap in participation in favor of males. Prior achievement is most influential in the early grades of high school, whereas prior attitude is most influential in the later grades. Schools vary significantly in participation rates at each grade level, but the variation is relatively small at both the beginning and the end of high school. There is a positive contextual effect associated with school mean socioeconomic status which is substantial in grade 9, smaller in grades 10 and 11, and negligible in grade 12. As to parental involvement, volunteer work for school is the most important schoollevel variable in the early grades (8 to 10). The effect is strong in each grade, and remains similar in magnitude across grades. Home discussion is critical in the middle grades (10 and 11). The effect is strong, and remains almost constant in the two grades. Homeschool communication has a temporary, though strong, effect in grade 9. The effect of home expectation is nonsignificant across all grades. Based on a national sample from the Longitudinal Study of American Youth (LSAY), this study examined the effects of individual characteristics and different types of parental involvement on participation in advanced mathematics from grades 8 to 12. Results of hierarchical survival analysis show that an average student is most likely to drop out in grade 12, where there is also a large gender gap in participation in favor of males. Prior achievement is most influential in the early grades of high school, whereas prior attitude is most influential in the later grades. Schools vary significantly in participation rates at each grade level, but the variation is relatively small at both the beginning and the end of high school. There is a positive contextual effect associated with school mean socioeconomic status which is substantial in grade 9, smaller in grades 10 and 11, and negligible in grade 12. As to parental involvement, volunteer work for school is the most important schoollevel variable in the early grades (8 to 10). The effect is strong in each grade, and remains similar in magnitude across grades. Home discussion is critical in the middle grades (10 and 11). The effect is strong, and remains almost constant in the two grades. Homeschool communication has a temporary, though strong, effect in grade 9. The effect of home expectation is nonsignificant across all grades. PARENTAL INVOLVEMENT School and family are the two most important institutions that affect the development of children. Although there exists a consistent conflict between the two (Edwards & Young, 1992), there is an increasing emphasis on their connection (Epstein, 1985, 1990). The National Commission on Excellence in Education (1983) listed parental involvement as one of the major goals in educational reform. Parental involvement, as one of the two elements consistently associated with mathematics achievement, seems particularly critical in mathematics education (Sharp, Sharp, & Metzner, 1995). The research on parental involvement has undergone three stages. In the first stage, sociologists noticed the importance of family background in children’s academic achievement (Heyns, 1978). For example, Coleman et al. (1966) demonstrated that parent education level has an impact on children’s school performance. The second stage marks the search for processes through which family background affects schooling outcomes. Two of the processes, for example, are exposure to school materials and parental academic expectation for children (Marjoribanks, 1972; Spaeth, 1976). However, researchers tended to consider school and family as separate in function and represent parents as passive in children’s schooling (Stevenson & Baker, 1987). In the current stage, more researchers view parental involvement as a key mediator between family background and cognitive and affective outcomes of schooling. For example, Stevenson and Baker (1987) found that the extent of parental involvement mediates the relationship between parent education level and academic achievement of children. Parental involvement has positive effects on children’s learning across a wide range of populations (e.g., Chavkin, 1993; Christenson, Rounds, & Gorney, 1992; Eccles & Harold, 1993; Edwards & Young, 1992; Epstein, 1991, 1994; Ho & Willms, 1996; Keith et al., 1993; U.S. Department of Education, 1994), though some occasional disagreements do appear (mainly at the high school level; see HooverDempsey & Sandler, 1995). Ascher (1988) stated that “the more parents participate in a sustained way at every level—in advocacy, decision making and oversight roles, as fundraisers and boosters, as volunteers and paraprofessionals, and as home tutors—the better for student achievement” (p. 113). Edwards and Young (1992) summarized that “studies point to higher student achievement when parents participate in school activities, monitor children’s homework, and otherwise support the extension into the home of the work and values of the school” (p. 73). One of the weaknesses in the research of parental involvement is the tendency of researchers to use a global measure of parental involvement as a mediator rather than to assess its multidimensional effects (see Grolnick & Slowiaczek, 1994; Ho & Willms, 1996). Thus, it is unclear what components of parental involvement affect schooling outcomes of children. Using data from the National Education Longitudinal Study (NELS), Ho and Willms (1996) reported in a factor analysis that parental involvement contains four dimensions: (a) home discussion, (b) school communication, (c) home supervision, and (d) school participation. This implies that parents participate in education both at home and at school. “The effects are most comprehensive when parents are involved both at home and school” (Christenson et al., 1992, p. 192). Students from a home environment that values academic achievement and promotes intellectual activities achieve better academically (Fraser, Welch, & Walberg, 1986; Kurdek & Sinclair, 1988). Epstein (1995) depicted a vision of a successful partnership among families, schools, and communities: they collectively and separately perform practices that affect children’s schooling and development. Such a partnership would make schools more like families and families more like schools. Positive home environment, therefore, is conductive to children due to its emphasis on the importance of schooling as well as the regular supervision and assistance in children’s social and academic life. Ho and Willms (1996) concluded that “it was involvement at home, particularly in discussing school activities and helping children plan their programs, that had the strongest relationship to academic achievement” (p. 137). Home discussion about school has been associated with students’ higher academic achievement (Christenson et al., 1992; Ho & Willms, 1996; Keith, 1991; Walberg, 1986). High achieving students regularly communicate with their parents about school life (e.g., de Kanter, Ginsburg, & Milne, 1986). Parents of high achieving students have rich verbal interaction with their children, delivering verbal cues, directions, guidance, and encouragement (e.g., Christenson et al., 1992; Gonzalez & Blanco, 1991). A number of researchers have also emphasized the positive effects of parental expectation and supervision at home on a range of educational outcomes (e.g., Astone & McLanahan, 1991; Fehrmann, Keith, & Reimers, 1987). Home supervision often includes such things as parents’ structuring children’s time for homework, modeling children’s learning, encouraging children to read at home, and limiting the time children watch television (see Christenson et al., 1992). Overall, parents’ setting standards, enforcing rules, and encouraging discussion, negotiation, and independence is associated with students’ higher academic outcomes (Christenson et al., 1992). A positive homeschool connection is related to higher academic outcomes (Redding, 1991). Both schooltohome communication (teachers inform parents about school programs and children’s progress) and hometoschool communication (parents contact teachers about their children’s school life) have been considered important (e.g., Epstein, 1987; Muller, 1993). Parents of high achieving students also actively participate in parentteacher organizations (Jencks, 1972). Corner and Haynes (1991) described a pattern that is often referred to as meaningful parental participation, in which parents actively get involved at all levels of their children’s school life, from providing general support for schools’ educational goals, to participating regularly in social and academic activities, to offering suggestions to school planning and management. Epstein (1987) identified five types of parental involvement that are critical to improve children’s education. Two of them are homeschool communications about school programs and children’s progress, and involvement at school such as attending school functions and volunteering. One theoretical explanation of how parental involvement works for children is that it improves children’s cognitive skills that make them more likely to succeed in academic work. For example, Epstein (1988) argued that parental involvement makes a child realize the importance of education, which leads to more responsible efforts in school. Patterson (1986) suggested that a child who perceives parents as involved develops more competence in skills. Parental involvement also has effects on school performance through helping the child in schoolwork and providing resources for skill development. On the other hand, Grolnick, Ryan, and Deci (1991) proposed that parental involvement does not affect children through building cognitive skills, but through its affective influence on children’s attitudes and motivations. This theory sees the child as an active processor of information and a purposeful constructor of schemas about self. Grolnick and Slowiaczek (1994) suggested that parental involvement include a general definition and several specific dimensions. Parent involvement . . . is defined as the dedication of resources by the parent to the child within a given domain. Such a definition recognizes that there is a difference between parents’ overall involvement with the child and involvement in the child’s education. Because of parents’ values, time commitments, and availability of resources, they may choose to, or be forced to, devote their time and energies to domains differentially (i.e., to school, social activities, athletics). (p. 238) This definition demonstrates the multidimensional nature of parental involvement in that several resources can be considered. Grolnick and Slowiaczek (1994) theorized three categories of parental involvement. Parents’ behavioral involvement such as visiting school and participating in educational affairs provides information useful to help their child’s schooling. Parents’ personal involvement shows care about their child’s affective experiences in and out school. This helps refine the affective characteristics of the child in general and create a positive attitude toward schooling and self in particular. Parents’ intellectual involvement exposes the child to cognitively stimulating activities such as reading books and discussing current events. Grolnick et al. (1991) suggested that all three types of parental involvement, not just the intellectual component, have a positive effect on children’s school performance. MATHEMATICS PARTICIPATION Mathematics is often referred to as the “critical filter” in that students with inadequate mathematics preparation lose many career choices available to them (Sells, 1973). A Nation at Risk: The Imperative for Educational Reform exposed the large number of students who dropped out of mathematics courses, especially elective courses (National Commission on Excellence in Education, 1983). The National Assessment of Educational Progress (NAEP) indicated that only half of high school graduates enroll in mathematics courses beyond the 10th grade (Dossey, Lindquist, & Chamber, 1988). “The problem seems to be serious enough to warrant concern and further investigation” (Stefanich & Dedrick, 1985, p. 274). Both American youth and adults do not seem to compete well in recent international comparisons of quantitative skills (e.g., Beaton et al., 1996; Organisation for Economic CoOperation and Development (OECD) & Statistics Canada, 1996). Does this underperformance have anything to do with participation in mathematics? Research on mathematics achievement has traditionally emphasized psychological factors, such as students’ anxiety about learning mathematics and their attitude toward the subject. Recently, researchers have begun to explore the effects of mathematics curriculum. Relatively little attention has been paid, however, to exposure (the amount of time students receive mathematics instruction), and when they are exposed to particular mathematical concepts. The Second International Mathematics Study (SIMS) did show that the coverage of the tested topics in U.S. classrooms was at or below the international average in most content areas (McKnight et al., 1987). Some findings from the recent research on school effectiveness also suggest that exposure may be one of the most important variables. For example, Lee and Bryk (1989) found that in schools where students took more academic courses, and where there was less latitude in choosing courses, there were higher levels of achievement and more equitable distributions in achievement along the social class lines. The consequence of inadequate preparation in mathematics is more than just underperformance in mathematics. There has been a decrease in employment in lowtechnology, lowwage industries across all OECD countries (OECD, 1994). Careers that were comfortably free of mathematics in the 1960s and 1970s now depend heavily on mathematics (National Council of Teachers of Mathematics, 1991). The occupational projections are a weaker demand for lowskilled workers, but a stronger demand for moderately skilled technical and administrative workers as well as highly skilled professionals (OECD, 1995). Most new jobs will be in the hightechnology sector, and will require competence in computerized data analyses, sophisticated mathematical models, or elaborate accounting systems (National Council of Teachers of Mathematics, 1989, 1991). Therefore the critical filter may have a much stronger impact on the economic well being of individuals in decades to come; that is, inadequate preparation in mathematics will disadvantage individuals in their ability to survive economically. RESEARCH QUESTIONS Until recently, longitudinal data and appropriate statistical methods have not been available to estimate comprehensive models of mathematics participation. Crosssectional data are not suitable to exploring when and why students cease taking mathematics courses (see Willett & Singer, 1991). Consequently, most previous research has been limited to descriptive analyses that portray the demographic characteristics of students who drop out of mathematics at a particular time point. This study aims to provide insight into the developmental nature of mathematics participation with six waves of data from the Longitudinal Study of American Youth (LSAY), which between 1987 and 1993 followed a representative national sample of students from grades 7 to 12 (Miller & Hoffer, 1994). Taking into account students’ gender and family background, their prior achievement in and attitude toward mathematics, and the effects of different types of parental involvement, this study attempts to address these groups of research questions: 1. What is the likelihood that a student will drop out of advanced mathematics at each grade level, from grades 8 through 12? What individuallevel variables are most strongly related to dropping out at each grade level? These variables include students’ sex, family socioeconomic status (SES), prior mathematics achievement, and prior attitude toward mathematics. 2. To what extent do schools vary in their dropout rates at each grade level? How is this variation related to the contextual effect of average SES of students in a school? 3. Can some of the variation in dropout rates be explained by parental involvement at each grade level? Do different types of parental involvement have different effects at each grade level? Does each type of parental involvement have the same effect across grade levels? METHOD DATA AND MEASURES Data for this study were drawn from the Longitudinal Study of American Youth (LSAY), a sixyear panel study of American public middle and high schools with a focus on mathematics and science education. The LSAY data were collected from 3,116 students in 52 schools randomly selected from across the United States. Compared with other national data sets, the LSAY is particularly strong in its longitudinal coverage of variables pertaining to students’ achievement, attitudes, and family background, and of variables describing school characteristics. This analysis employed data from the student questionnaire. At the individual level, independent variables predicting participation in advanced mathematics were (a) sex (renamed as female, coded 0 for males and 1 for females), (b) SES, (c) prior mathematics achievement, and (d) prior attitude toward mathematics. SES is a continuous variable integrating household possessions and parents’ education and occupation. Attitude toward mathematics was based on a scale of 9 items, intended to measure four attitudinal components: interest, utility, ability, and anxiety (Cronbach’s alpha ranges between 0.66 and 0.76 from grades 7 to 12). The scale was constructed such that a higher value indicates a more positive attitude. The achievement test comprised 60 items covering students ability to recall and recognize mathematical concepts, and to solve routine and complex problems (Cronbach’s alpha ranges between 0.86 and 0.95 from grades 7 to 12). The dependent variable was the status of students’ enrollment in advanced mathematics, coded 0 for nonparticipation and 1 for participation. A fairly liberal definition of “advanced mathematics” was adopted that includes average and high eighth grade mathematics, geometry (including honors), prealgebra, algebra I and II (including honors), trigonometry (including honors), analytic geometry, calculus, as well as probability and statistics. The expectation is that most students who took courses meeting these criteria would have the prerequisites for entry to collegelevel mathematics courses. Operationally, “mathematics dropout” refers to the grade in which a student stopped taking mathematics courses altogether. For example, the coursetaking pattern of 10100 means that the student took mathematics in grade 8, did not in grade 9, did in grade 10, and dropped out in grade 11. For the purpose of survival analysis, this student would figure in the analysis in the same way as a student with a pattern of 11100. This analysis parallels Ho and Willms (1996) with four composite variables: (a) home discussion (on children’s social and academic life), (b) home expectation or supervision (on time to be home, how late to stay up, household duties, TV time, and TV shows), (c) homeschool communication (on students’ schoolwork), and (d) volunteer work for school (or school participation; see Table 1). Cronbach’s alpha ranges between 0.62 and 0.83 from grades 7 to 12 for home discussion, between 0.60 and 0.68 for home expectation, and between 0.44 and 0.50 for homeschool communication (from grades 7 to 11). These variables were coded such that a higher value indicates a higher level of parental involvement. The four measures of parental involvement all came from the student questionnaire, but were aggregated to the school level and standardized at the school level. The reason why they were used as schoollevel variables is that parental involvement is often considered a measure of school climate rather than a measure of student characteristics (see, for example, Ho & Willms, 1996; Willms, 1992). Moreover, while parental involvement has effects on individual students, policy oriented studies are often interested in how school policies and practices can affect parental involvement as a school climate to improve average schooling outcomes of students. PROCEDURE OF STATISTICAL ANALYSIS This analysis combines survival analysis with multilevel modeling. Survival models have been used in medical studies to discern variables responsible for the mortality of a population (Namboodiri & Suchindran, 1987). The variables predicting when an event occurs can be either “timeinvariant,” such as a person’s sex and social status, or “timevariant,” such as a person’s attitude toward treatment and their sense of well being. Multilevel models have been used in educational studies in the past decade to analyze data that are structured hierarchically, such as data on students nested within schools (Raudenbush & Willms, 1991). Studies usually examine the extent to which the relationship between schooling outcomes and family background characteristics varies among schools, and whether this variation is attributable to particular schoollevel variables describing school policies and practices (see Willms, 1992). Willett and Singer (1991) have demonstrated the aptness of survival analysis in the study of dropout and attrition. The idea underlying hierarchical survival analysis is to first construct a survival model using survival modeling techniques, predicting the likelihood that a student will drop out of mathematics on a number of studentlevel characteristics, and then fit the survival model to the data for each school, modeling the variation among schools in their average dropout rates on a number of schoollevel characteristics. The studentlevel model was developed through survival modeling techniques. One advantage of survival analysis is that it accommodates both timeinvariant and timevarying variables. The treatment for timevarying variables is that “for each unit of time that each individual is known to be at risk, a separate observational record is created” (Allison, 1984, p. 18). These observations are referred to as “personyears” in this analysis. For example, students who dropped out of mathematics in year 2 (grade 9) contribute 2 personyears; those who did not drop out (from grades 8 to 12) contribute a maximum of 5 personyears. In this study, prior mathematics achievement (from grades 7 to 11) and prior attitude toward mathematics (from grades 7 to 11) were treated as timevarying variables to examine their effects on mathematics participation in the next school year. SES was considered stable over the fiveyear period because the LSAY did not collect data on SES annually. Thus, SES and female were treated as timeinvariant variables. The time unit in this analysis is one year, which is large enough to be treated as discrete rather than continuous time (Allison, 1984). Preliminary analyses of survival rates showed that the hazard probabilities for dropping out varied significantly across grades, and could not be modeled as a constant hazard probability. This analysis therefore included a set of five dummy variables, one for each grade, and the model was fitted without an intercept. This is similar to a model with an intercept and four dummy variables; however, the nointercept model facilitates the modeling of schoollevel effects in the hierarchical framework. With this construction, the five coefficients for grades 8 to 12 are referred to as time effects (see Yamaguchi, 1991). The probability of mathematics participation can then be calculated as: Four hierarchical linear models (HLM) were tested in this study. The first was similar to a “null” model (see Bryk & Raudenbush, 1992). It contained only the five dummy variables denoting grades 8 to 12 at the student level. At the school level, the withinschool coefficients (the average dropout rates) for grades 8 to 12 were allowed to vary across schools, but no schoollevel variables were entered to model the variation. This model provided estimates of the variation in the probability of mathematics participation between schools for each of the five grades. In the second model, studentlevel variables were added to the first model to estimate their effects on the probability of mathematics participation. The third model introduced the contextual variable of mean SES at the school level to discern whether the likelihood of dropping out for a student with average background characteristics depended on the socioeconomic composition of the school. Finally, parental involvement variables were entered at the school level to determine whether certain aspects of parental involvement were related to mathematics participation. In fitting these models, simplicity was considered a desirable trait (see Miller & McGill, 1984). Therefore, in the third and fourth models, only the variables that were statistically significant were retained. RESULTS Table 2 provides descriptive information about the studentlevel variables. Overall, mathematics participants had higher SES than mathematics dropouts at every grade level (although the differences were not statistically significant in grade 10). Participants, particularly females, achieved consistently better than those who dropped out. Male participants had a more positive attitude toward mathematics than male dropouts since the early grades of high school. Attitude, however, separated female participants from dropouts only in grades 11 and 12. Descriptive statistics for the schoollevel variables are shown in Table 3. Home discussion showed, in general, an increasing pattern, although it was relatively stable in grades 9 to 11. Home expectation (supervision), in contrast, decreased consistently over grades. There were two positive jumps in homeschool communication in grades 9 and 12, and it was relatively stable otherwise. The percentage of parents doing volunteer work for school increased consistently across grades. The first hierarchical survival model included only the five dummy variables representing grades 8 to 12, but allowed their coefficients to vary at the school level. This model, thus, mainly estimated variation among schools in their average participation rates. The third column in Table 4 shows the overall statistics; that is, across the 52 schools, the proportion of students continuing in advanced mathematics drops slightly in the transition from grades 8 to 9 (from 96% to 92%), and remains reasonably high until the transition from grades 11 to 12 when it drops dramatically (from 89% to 64%). Again, these proportions are average measures across the 52 schools. The HLM model also tests whether coefficients representing grades 8 to 12 vary significantly among schools, and provides Bayesian (shrunken) estimates of the coefficients for each school (see Bryk & Raudenbush, 1992). The analysis indicated that schools varied significantly in their participation rates at each grade level (from grades 8 to 12, standard deviation = 0.91, 1.34, 1.28, 0.75, and 0.46 respectively, p < 0.01). The variation in participation rates is relatively small among schools, though statistically significant, in grades 8, 11, and 12. In grades 9 and 10, there is large variation in participation rates across schools. The second model contained studentlevel variables and thus examined the effects of those studentlevel variables on mathematics participation. The last three columns of Table 4 indicate that overall an average student is 96% likely in the eighth grade, 93% likely in the ninth grade, 94% likely in the tenth grade, 90% likely in the eleventh grade, and 66% likely in the twelfth grade to participate in advanced mathematics. The model initially contained all gradebysex interaction terms. Because interactions were not statistically significant for grades 8 through 11, the model included only one interaction term (female by grade 12). The results indicate that females were 1.08 times as likely as males (or 8% more likely than males) to participate in grades 8 to 11, after taking account of their SES, prior achievement, and prior attitude. However, in the transition from grades 11 to 12, females were only 0.74 times as likely as males to participate. This is a fairly large gender gap in participation; it is especially large given that the model includes controls for prior achievement and attitude. The model also indicates that a oneunit increase in SES is associated with a 16% increase in the likelihood of participation. This effect is small; however, the model includes prior achievement and attitude, through which most of the SES effects can be mediated. SES did not interact significantly with grade level. The effect of prior mathematics achievement is much stronger than that of SES. A one standard deviation increase in prior achievement raised the likelihood of participation by 31%. The effect did not differ significantly across grades. Prior attitude toward mathematics had a small but statistically significant effect in grades 8 to 10. A one standard deviation increase in prior attitude increased the likelihood of participation by 8%, which is a small effect similar in magnitude to sex differences in those grades. However, the effects of prior attitude were much stronger in the later grades of high school. A one standard deviation difference in attitude in grades 10 and 11 was associated with a 42% and 54% increase respectively in the likelihood of participation the following year. Given that females had similar achievement scores to males in grade 11, but considerably less positive attitude toward mathematics (see Table 2), their participation in advanced mathematics in the senior year was markedly lower than that of males. The third and fourth models examined schoollevel effects. The third model introduced the contextual effect variable, school mean SES, at the school level. The results are displayed in the first three columns of Table 5. There was a positive contextual effect associated with school mean SES, which was practically large and statistically significant in grade 9, smaller in grades 10 and 11, and negligible in grade 12. In grade 8 there was a negative effect associated with school mean SES, contrary to what one would expect. This is likely to be a spurious result, especially given the small amount of variance in participation rates among schools in such an early grade. The effect for grade 9 indicates that a student with nationally average SES and nationally average prior achievement and attitude was 57% more likely to participate in advanced mathematics in that grade if the student attended a high SES school. The final model attempted to explain variation between schools in participation rates in grades 8 to 12 with the school contextual measure and schoollevel parental involvement measures. The results are shown in the last three columns of Table 5. In grade 8, volunteer work for school was statistically significant with a large effect size. Students whose parents volunteered were more than 9 times as likely to enroll in advanced mathematics as those whose parents did not volunteer. The negative effect of school mean SES is likely to be a spurious result (the same explanation may be offered for the negative effect in grade 10 as well). In grade 9, there were significant effects associated with school mean SES and homeschool communication. A one standard deviation increase in school mean SES was associated with a 30% increase in the likelihood of participation. A onepoint increase on the eightpoint scale of homeschool communication raises the likelihood of participation by 34%. The large effect of volunteer work for school became even stronger in this grade (nearly 10 times in likelihood increase). In grade 10, volunteer work for school retains its strong effect (nearly 9 times in likelihood increase). There was a significant effect associated with home discussion. A onepoint increase on the tenpoint scale of home discussion raises the likelihood of participation by 28%. In grade 11, there was another significant effect associated with home discussion, whose effect size is similar to that in grade 10. This effect also diminished the effect associated with school mean SES. In grade 12, none of the schoollevel variables (school mean SES and parental involvement variables) significantly affected the likelihood of participation. From a longitudinal perspective, volunteer work for school is a particularly important predictor of participation in advanced mathematics in the early grades (grades 8 to 10). The effect of volunteer work for school is strong in each grade, and remains similar in magnitude across grades 8 to 10. Home discussion is particularly critical in the middle grades (grades 10 and 11). The effect of home discussion is reasonably strong, and remains almost constant in the two grades. School mean SES and homeschool communication both have a temporary, though reasonably strong, effect in grade 9. DISCUSSION SUMMARY OF RESULTS Effects of student background variables. An average student is most likely to drop out of advanced mathematics in the transition from grade 11 to 12. Females are slightly more likely than males to participate in grades 8 to 11. However, there is a large gender gap in participation in favor of males in the transition from grade 11 to 12. This gap is especially worrisome given that the model controls for prior achievement and attitude. The effect of SES is small and constant across grades. The effect of prior mathematics achievement is much stronger than that of SES, and does not differ significantly over grades. In grades 8 to 10, prior attitude toward mathematics has a small effect. However, the effect of attitude is much stronger in the later grades of high school. Thereafter, prior achievement is the most influential factor in the early grades of high school, whereas prior attitude is the most influential one in the later grades. Variation among schools in participation rates. Schools vary significantly in their participation rates at each grade level from 8 to 12. The variation is relatively small, though statistically significant, at both the beginning and the end of high school. In the middle grades (grades 9 and 10), there is large variation in participation rates across schools. Effects of the school contextual variable. There is a positive contextual effect associated with school mean SES. The effect is both practically large and statistically significant in grade 9, smaller in grades 10 and 11, and negligible in grade 12. Effects of schoollevel parental involvement variables. Volunteer work for school is the most important schoollevel variable in the early grades of high school (grades 8 to 10). The effect is strong in each grade, and remains similar in magnitude across grades. Home discussion is particularly critical in the middle grades (grades 10 and 11). The effect is reasonably strong, and remains almost constant in the two grades. Homeschool communication has a temporary, though reasonably strong, effect in grade 9. The effect of home expectation (supervision) is nonsignificant across all grades. REVISITING THE THEORIES OF PARENTAL INVOLVEMENT First of all, although more studies have indicated positive effects for parental involvement than have shown no effects, many of the studies that report positive effects have no control over students’ prior academic achievement. Without such control, these studies may spuriously overestimate the effects of parental involvement. The results of this study are important in that positive effects of parental involvement on participation in advanced mathematics remain strong in some aspects such as home discussion and volunteer work for school even after the model controls for prior mathematics achievement. This provides further support for the significance of parental involvement in children’s learning. Theoretically, parental involvement is associated with schooling outcomes of children through either building up children’s cognitive skills required for academic work in school (e.g., Epstein, 1988; Patterson, 1986) or affecting children’s attitude and motivation critical to stimulate internal effort for excellence in academic work (e.g., Grolnick et al., 1991; Grolnick & Slowiaczek, 1994). Although this study does not have direct data to either support or disprove any of these theories, it offers implications as to what can make parental involvement work for children’s mathematics participation. This analysis shows that attitude toward mathematics is the most important factor in mathematics participation, particularly in the later grades of high school when dramatic drops in participation occur. Mathematics achievement plays a role only in the early grades when there are no serious drops in participation, although it is the most important factor during that period. These indicate that efforts in improving cognitive skills in mathematics do not necessarily lead to active mathematics participation. But efforts in developing positive attitude toward mathematics make a difference during the most risky period of dropping out of advanced mathematics. The implication is that parents may want to concentrate more on or spend more time in improving their children’s attitude toward mathematics. Children with positive attitude, in turn, are more likely to participate in advanced mathematics. This study also demonstrates that different types of parental involvement have different effects, in line with other researchers (e.g., Grolnick & Slowiaczek, 1994; Ho & Willms, 1996). Thus it supports the multidimensional conceptualization of parental involvement (see Grolnick & Slowiaczek, 1994; Singh et al., 1995). For example, this analysis suggests that volunteer work for school, a measure of “school participation” as conceptualized in Ho and Willms (1996), has the most important effect on mathematics participation. On the other hand, home expectation, a close relative of “home supervision” (see Ho & Willms, 1996), has no effect at all on participation. More important, adding a longitudinal perspective, this study extends the existing understanding of the multidimensional nature of parental involvement. It shows that not only may different types of parental involvement have different effects at each time point, but also they have different effect patterns over time. For example, volunteer work for school is most influential in the early grades of high school. During the middle years, however, home discussion replaces it as the most influential factor, although it is not as strong as the former in effect. Therefore, the conceptualization of parental involvement should be not only multidimensional but also developmental. This study also implies that aspects of parental involvement may have different effect patterns in different schooling outcome areas such as mathematics achievement or mathematics participation. For example, parents’ participation in schoolrelated activities has a negligible effect on mathematics achievement in grade 8 (Ho & Willms, 1996; Singh et al., 1995). In contrast, this analysis shows that parents’ involvement at school has a strong effect on mathematics participation in the early grades of high school (grades 8 to 10). Furthermore, Ho and Willms (1996) found that the discussion of schoolrelated issues at home has a strong relationship with mathematics achievement in grade 8. This study indicates, however, that home discussion has no effect on mathematics participation in grade 8, but affects mathematics participation in grades 10 and 11. IMPLICATIONS FOR MATHEMATICS PARTICIPATION One of the important findings of this study is that there are two critical transition points at which large proportions of students drop out of advanced mathematics. One is from grade 8 to 9; the other is from grade 11 to 12. Approximately 8 percent of students were not taking advanced mathematics in grade 9. Students’ prior (grade 8) mathematics achievement played a dominant role in their career path. Schools also varied in their participation rates at this grade level. With all other factors being equal, students were more likely to pursue advanced mathematics if they attended a high SES school. The most serious drop in participation rates is in the transition from grade 11 to 12. Approximately 36 percent of students were dropping out of advanced mathematics in grade 12. Students’ prior (grade 11) attitude toward mathematics determined whether they would participate in advanced mathematics. Moreover, mathematics dropouts in this last grade of high school do not appear to be schoolspecific because schools did not vary much in their participation rates. These findings suggest that students entering their freshman year may experience one of two types of segregation. Those from less advantaged backgrounds are more likely to attend schools with lower average socioeconomic composition, which may offer few advanced courses. Other students may face some kind of early tracking, which channels those with lower prior achievement into less challenging programs. The first cause of the problem is more difficult to address, because betweenschool segregation along social class or racial/ethnic lines is deeply entrenched through residential segregation and different forms of selective schooling (see Rumberger & Willms, 1992). The second cause, however, can be addressed at the local level. Schools need to ensure that students are given the opportunity to pursue advanced mathematics as long as possible. This can be achieved, for example, by providing a core mathematics curriculum for the middle school years. Many accounts of gender differences in mathematics suggest that females “leak out” of mathematics, which implies a gradual process that occurs during the entire high school career. This analysis disproves this claim. In fact, achievement scores in mathematics and participation rates in advanced mathematics are equal between males and females, if not in favor of females, prior to the 12th grade. It is only in the final year of high school that a disproportionate number of females drop out of advanced mathematics. Students’ attitude toward mathematics played a dominant role in the last two years of high school, particularly in the transition from grades 11 to 12. A large number of students, many of whom are female, do not take advanced mathematics because of their negative attitude toward mathematics. The most important strategy, therefore, for reducing mathematics dropouts is to improve students’ attitude in the later grades of high school. Both schools and families should help students realize the role of mathematics in their future careers so that they do not prematurely close too many career doors too soon. There are a variety of ways to achieve this purpose, such as home discussions, counseling programs, publicity campaigns, presentations about future job market, field trips to hightech business, and panel discussions with parents who have benefited from the study of mathematics. Mathematics curricula will also need to address the attitudinal problems in mathematics at least by making it more relevant and informative. Classroom teaching should demonstrate and reinforce mathematics as a useful and beneficial discipline. RECOMMENDATIONS FOR FURTHER STUDIES Various models of parental involvement seem to share a common assumption—it is parents who initiate parental involvement. For example, this analysis shows that students whose parents did volunteer work for school were 9 to 10 times as likely to participate in advanced mathematics as those whose parents did not volunteer. The cognitivedevelopmental model of parental involvement would suggest that children who see their parents volunteering at school translate this information into an idea about school as important, which stimulates their internal efforts in school work. But it is also possible that when students enroll in advanced mathematics courses, they get their parents increasingly involved in schoolrelated activities. This implies an alternative direction of modeling parental involvement—it is children who initiate parental involvement. Note that this alternative may be particularly important in the examination of mathematics participation, especially participation in advanced mathematics. For example, children may ask parents for advice regarding taking advanced mathematics or encourage parents to contact school about, for example, academic placement alternatives or about taking or dropping certain mathematics courses. Some children may ask parents to help them with mathematicsrelated activities such as homework and projects; others may ask parents for more learning resources such as books, journals, computers, and software. It is certainly possible that both parents and children initiate different aspects of parental involvement. For example, parents may initiate home supervision, whereas children may initiate homeschool communication. Unfortunately, available data do not contain any information on who is the initiator of parental involvement. This is something further researchers may want to bear in mind when collecting data. In general, data about parental involvement need to be more specific. For example, this study included a measure of school participation: volunteer work for school. But there was no further information regarding what kind of volunteering parents did at school. Volunteering in classroom may have different effects from attending lunch duties. Similar limitation has been noted in other studies (e.g., Ho & Willms, 1996). With measures of parental involvement being more specific, researchers may have a better chance to pinpoint the critical aspects of parental involvement that shape its significance. There may be different types of dropouts in mathematics, particularly in advanced mathematics. Schools where students come from middleclass families usually offer more chances for enriched and rigorous academic programs (e.g., Grubb, 1984; Willms, 1986). 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