End-of-High-School Mathematics Attainment: How Did Students Get There?

by Xiaoxia A. Newton - 2010

Background: Many studies have looked at students’ mathematics achievement in the middle and high school years and the kinds of factors that are associated with their achievement. Within this domain, however, most research utilized cross-sectional data. Cross-sectional designs have both statistical and conceptual limitations. Few studies used longitudinal data that typically included two time points, occasionally three. A pure longitudinal design is problematic as well because it does not take into account the multilevel nature of data derived from educational settings. In an attempt to account for differences in mathematics achievement, researchers have advanced different explanations, varying from affective/psychological factors (e.g., math attitude) to social factors (e.g., influences of parents, teachers, and peers). However, because of the division of psychology and sociology, subdivisions within these fields, and specialized individual research interests, a limitation of the research in this literature is that the variables are often studied in isolation rather than in concert. A promising way to resolve this problem, as Herbert Walberg argued in his psychological theory of educational productivity, is to include the chief known correlates of educational achievement derived from experimental and nonexperimental research and simultaneously analyze panel data collected on many individuals over multiple time periods on variables such as age or developmental level; ability, including prior achievement; social environment for learning; and home environment. Taken together, these studies provide a foundation for studying individual differences in secondary mathematics growth and end-of-high-school mathematics attainment and exploring various psychological and social factors that might predict such differences from a longitudinal and multilevel perspective. Building on this earlier research, the present study attempted to address these issues through longitudinal and multilevel analysis of a national probability sample of Grade 7 students who were followed until the end of high school.

Focus of Study: The present study attempted to investigate how high school seniors get to where they are in terms of end-of-high-school mathematics attainment. In addition, the study explored what factors might predict students’ attainment and their growth trajectories in mathematics during secondary school years.

Research Design: The present study is a secondary analysis of longitudinal data that tracked a national probability sample of seventh graders until they graduated from high school. Three-level hierarchical linear models were fitted to the data using a Level 1 piecewise linear growth model nested within students (Level 2) across schools (Level 3).

Conclusions: One of the important findings of this study was that on average, there was a drop in mathematics achievement during the senior year of high school for students in the sample regardless of student mathematics achievement in Grade 7. Additionally, the study found an inequitable distribution of mathematics attainment at the end of high school associated with initial differences in mathematics achievement. Several individual, school composition, and opportunities to learn variables, such as early tracking and course progress, were found to be strong predictors of students’ mathematics attainment and growth. These empirical findings point to the further directions we may take to promote student achievement in mathematics.

Recent U.S. school reform efforts have given increasing attention to public high schools (McEwen, 2006). One concern that has prompted such reform efforts centers on the inadequate preparation of many high school seniors for college-level math upon entering colleges and universities. According to a 2005 study by ACT, the college entrance exam organization, only 40% of high school seniors were ready to take the most basic college-level algebra course (ACT, 2005). Another indication of the lack of math readiness is the frequency with which colleges offer remedial courses and the pattern of student participation in remedial courses. Reading, writing, and mathematics typically are the areas of greatest need for underprepared students (Merisotis & Phipps, 2000). According to the U.S. Department of Educations (Parsad, Lewis, & Greene, 2003) statistical analysis of remedial education at degree-granting postsecondary institutions in fall 2000, a higher proportion of degree-granting 2- and 4-year institutions offered remedial courses in mathematics (71%) as compared with the other two areas (68% for writing and 56% for reading). Similarly, the proportion of college freshmen who enrolled in remedial courses was the largest for mathematics (22%) as compared with writing (14%) and reading (11%).

Deficits in math readiness carry severe consequences not only for individual students but also for college institutions and the general public. Research (e.g., Wieschenberg, 1994) has found that over 40% of U.S. first-year college and university calculus students fail the course. Similarly, students in remedial courses also have high failure rates because of their weak mathematical foundation (Adelman, 1995). A large proportion of these students eventually leave college without earning a degree (Hagedorn, Siadat, Fogel, Pascarella, & Nora, 1999). Thus, the limited success of these remedial programs costs the society that funds them as well as the students themselves (Hoyt & Sorenson, 2001; Kozeracki, 2002).

These problems could likely be avoided if students were adequately prepared upon graduation from high school. Precollege education, as the National Science Board (2006) emphasizes, is the foundation for fostering leadership in science, technology, engineering, and mathematics (STEM). Naturally, one area that needs research is student mathematics achievement in the middle and high school years and how student achievement is affected by various individual and school compositional factors. The present study attempted to investigate how high school seniors get to where they are in terms of end-of-high-school mathematics attainment. In addition, the study explored what factors might predict students attainment and their growth trajectories in mathematics during secondary school years.

STUDYING MATHEMATICS ACHIEVEMENT: A LONGITUDINAL AND MULTILEVEL MODELING APPROACH

Many studies have examined students mathematics achievement in the middle and high school years. Methodologically, however, most of the studies on students mathematics achievement used cross-sectional data (Wilkins & Ma, 2002). Cross-sectional designs have both statistical and conceptual limitations. Statistically, cross-sectional designs can confound age and cohort differences with other differences because of their inability to separate different sources of variation (Becker & Forsyth, 1994). Conceptually, a cross-sectional design does not reflect the fact that the very notion of learning implies growth and change (Willet, 1988, p. 346) because the design assumes that student achievementthe outcome of learningcan be represented as a score measured at a single time point, as opposed to scores measured at multiple time points to show a growth pattern.

A few studies that used longitudinal data typically included two, and sometimes three, time points (Ma, 2000). A pure longitudinal design is problematic as well because it does not take into account the multilevel nature of data derived from educational settings. As Bryk and Raudenbush (1988) noted,

The central phenomenon of interest in educational research [is] . . . how personal characteristics of students, such as their ability and motivation, and aspects of their individual educational experiences, such as the amount of instruction, influence their academic growth. The learning chiefly takes place in the organizational settings of schools and classrooms. Features of these settings size, climate, resources, materials, and the likecan have substantial influence on the learning processes occurring within them. (pp. 6566)

Therefore, from a methodological perspective, research on student achievement should incorporate both the dynamic nature of learning (i.e., focus on growth) and the organizational context in which the learning takes place (i.e., pay attention to the compositional effect). Hierarchical linear models (HLMs) that combine longitudinal growth in outcomes and the multilevel organizational structures are the best suited to studying longitudinal and nested data that are common in educational settings (Wilkins & Ma, 2002). This methodological approach is employed in the present study to examine students growth in mathematics achievement during secondary school years and end-of-high-school mathematics attainment.

PREDICTORS OF MATHEMATICS ACHIEVEMENT: A SOCIAL-PSYCHOLOGICAL FRAMEWORK

In an attempt to account for differences in mathematics achievement, researchers have advanced different explanations, varying from affective/psychological factors (e.g., math attitude) to social factors (e.g., influences of parents, teachers, and peers). Affective factors play an important role in the learning and teaching of mathematics (McLeod, 1990). They can have an important impact on students decisions about how much mathematics they will need in the future and how they approach the mathematics content they study (Reyes, 1984). Sociologists argue, on the other hand, that parents, teachers, and peers constitute important social factors that might contribute to differences in mathematics achievement (Cesar, 1998). Though no one explanation has unequivocal support, each provides insights into the understanding of individual differences in mathematics achievement.

However, because of the division of psychology and sociology, subdivisions within these fields, and specialized individual research interests, a limitation of the research literature is that the variables are often studied in isolation rather than in concert (Walberg, 1982). A promising way to resolve this problem, as Walberg argued in his psychological theory of educational productivity, is to include the chief known correlates of educational achievement derived from experimental and nonexperimental research and simultaneously analyze panel data that are collected on many individuals over multiple time periods on variables such as age or developmental level; ability, including prior achievement; social environment for learning; and home environment. The present study attempts to integrate these perspectives in the context of examining mathematics achievement and attainment.

AFFECTIVE VARIABLES, GENDER, AND MATHEMATICS

Volumes of work have been devoted to the role of affective factors, particularly math attitude, in the gender¹ gap in mathematics achievement. One line of inquiry focuses on relationships between attitude and mathematics achievement (Brush, 1985; Fennema, 1974; Neale, 1969; Tocci & Engelhard, 1991) and gender differences in attitudes toward mathematics (Eccles, 1987; Fennema & Sherman, 1977, 1978). Results from these studies showed that attitude was a very important correlate of mathematics achievement. Furthermore, differences in both attitudes toward, and achievement in, mathematics are frequently found to favor boys over girls at the junior high level and beyond (Aiken, 1976). In addition to these studies, several theoretical models also were proposed in the 1980s to account for gender differences in mathematics achievement (e.g., Eccles; Fennema & Peterson, 1985). Many of these models highlight the contribution of affect and attitude variables to the gender gap in achievement (Frost, Hyde, & Fennema, 1994). The underlying assumption of these models is that gender differences in mathematics-related attitudes contribute to gender differences in achievement, course selection, and career choice (Frost et al.).

Another important affective variable in mathematics education, particularly in the context of understanding gender-related differences in mathematics, is confidence (Reyes, 1984). Confidence is a component of self-concept (Reyes), or in other words, self-esteema favorable or positive attitude toward oneself (Primavera, Simon, & Primavera, 1974). One reason that self-esteem is important is that it has a consistent, positive relationship with general academic achievement (Ross & Broh, 2000). More specific to mathematics achievement, several studies (e.g., Adrian, 1978; Pajares & Miller, 1994) have suggested a significant, positive relationship between self-esteem and mathematics achievement.

STUDENT MOTIVATION

In addition to affective factors such as attitude toward mathematics, motivation also plays a key role in students learning of mathematics. Studies (e.g., Chen & Stevenson, 1995; Schiefele & Csikszentmihalyi, 1995) on motivation and mathematics achievement suggest that high educational expectation is one of the factors associated with high achievement in mathematics. Another factor related to student motivation is behavior problems. Previous longitudinal research (e.g., Shane, Egeland, & Adrian, 1999) on achievement trajectories and factors associated with change found that behavior problems are one of the factors that predict deflections in achievement from first grade to age 16. The current study examined two key factors related to motivation, namely, student educational expectations and behavior problems.

OPPORTUNITIES TO LEARN MATHEMATICS

This study used three variables as proxy measures of students opportunities to learn mathematics, namely, types of mathematics courses taken at Grade 7, continual progress in taking higher level mathematics courses, and percent of class time devoted to teaching new materials. Long lines of research (notably by Gamoran 1986, 1987) have documented the relationship between secondary course-taking patterns and students opportunities to learn, and achievement in mathematics. For instance, previous studies (e.g., Gamoran, 1987; Porter, 1989; White, Gamoran, Smithson, & Porter, 1996) suggested that students who take lower level mathematics (e.g., general math) have fewer opportunities to learn than students who take more challenging mathematics (e.g., algebra). One key variable that the present study investigated, therefore, was the types of mathematics courses a student took in Grade 7, which served as a proxy measure of early tracking in secondary schools. In addition to this early tracking measure, the current study also created a dummy variable called continuous progress in taking higher level math courses. This course progress variable took on a value of 1 if, throughout middle and high school years, a student continued to take a higher level math course each year than the previous year as he or she moved on to a higher grade (0 otherwise). The secondary school mathematics curriculum is highly hierarchical in the sense that success in lower level mathematics is a prerequisite for moving on to higher level mathematics (Bozick & Ingels, 2007). Therefore, this course progress variable was a proxy measure of students progress and experience in mathematics during secondary school years. Finally, the variable percent class time covering new materials² was used to indicate students opportunities to learn new topics instead of repeating old materials. Research on mathematics curricula in the United States has documented the overly repetitive nature of course content, which prevents students from learning new materials and making progress (Porter, 1989).

STUDENT SOCIAL ECONOMIC AND DEMOGRAPHIC BACKGROUND

In addition to various social-psychological and opportunities to learn measures, key factors that also affect students participation and achievement in mathematics are their social, economic, and demographic backgrounds. The present study used mothers education and home resources as proxy measures that indexed the educational and economic aspects of students home environments. In terms of demographic background, the study focused on gender because extensive research has pointed to the relationship between gender, and participation and achievement in mathematics.

SCHOOL CONTEXTUAL FACTORS

Research on school effectiveness has pointed to the important role that school contextual factors play in students learning and achievement (Alexander & Eckland, 1975). One contextual factor that the present study focused on was percent minority students in a school. Previous research (e.g., Gamoran, 1987) has documented that students learning opportunities were stratified both between and within schools depending on students demographic composition. Students from more advantaged backgrounds (e.g., affluent, White) tended to have better access to challenging curriculum than students from more disadvantaged backgrounds (e.g., poor, minority). In addition to racial composition (i.e., percent minority students), the current study also included two other school contextual factors, namely, community commitment to math, and school safety. These two composite variables were proxy measures of general academic (i.e., parents and students commitment to mathematics) and physical (i.e., general safety and disciplines) environments in a school. Positive learning outcomes depend in part on positive academic and disciplinary environments (Fraser & Fisher, 1982).

Taken together, these studies provide a foundation for studying individual differences in secondary mathematics growth and end-of-high-school mathematics attainment and for exploring various psychological and social factors that might predict such differences from a longitudinal and multilevel perspective. The present study attempted to address these issues through longitudinal and multilevel analysis of national probability sampled longitudinal data, the Longitudinal Study of American Youth (LSAY) data, which followed a national probability sample of Grade 7 students until the end of high school. The following research questions guided the analysis in this study: (1) What were the growth patterns and end status (i.e., mathematics attainment at the end of high school) for students who were followed from Grade 7 to Grade 12? (2) What is the relationship between differences in math attainment and the growth patterns, and various individual and school compositional factors?

METHOD

DATA

Data for the current study came from the Longitudinal Study of American Youth (LSAY), a project funded by the National Science Foundation to examine the development of student attitudes toward, and achievement in, science and mathematics during middle school, high school, and the first 4 years after high school. The LSAY project consisted of two cohorts, namely the younger cohort and the older cohort. This study focuses on the younger cohort. The younger cohort of LSAY represents a national probability sample of approximately 3,116 seventh graders randomly selected from 52 public schools. This group of students was followed through the end of their high school years (i.e., from Grade 7 to Grade 12).

To date, the LSAY data are the only longitudinal data that annually tracked a national representative sample of middle school students as they progressed through secondary school years and therefore spanned almost the entire middle and high school grades. Though several other national longitudinal databases exist (e.g., HS&B, NELS:88, ELS:2002), these data do not track sampled students annually, and none started tracking students earlier than Grade 8. Therefore, the LSAY data provide a unique opportunity to assess students end-of-high-school mathematics achievement in relation to how they got where they were, even though the LSAY project was completed more than 10 years ago.

OUTCOME MEASURES

The outcome variables of the current study are math scores measured at each grade from Grade 7 to Grade 12, which were equated using the item response theory (IRT) technique (scale 0100). The math test, consisting mainly of 60 items from the National Assessment of Educational Progress (NAEP) tests, covered such areas as math skill and knowledge, routine application, problem-solving and understanding, and spatial visualization (reliabilities of these math scores ranged from 0.86 to 0.95).

PREDICTORS

Table 1 summarizes the background variables predicting students growth in mathematics achievement from Grade7 to Grade 12. The variables are grouped into three categories: student level, school level, and time-varying predictors. Time-varying predictors are variables whose values may change from year to year. For instance, a students attitude toward mathematics at Grade 7 may differ from that at Grade 8.

These three groups of predictors (i.e., student, school, and time-varying), thematically, were proxy measures of key affective (e.g., attitude toward mathematics), motivational (e.g., educational aspiration), opportunities to learn, student social economic and demographic, and school contextual factors (e.g., percent minority students). These various kinds of factors have been shown to be important determinants of students opportunities to learn and their achievement in mathematics based on prior research.

Table 1. Descriptions of the Variables

Predictors	Descriptions
Student Level Grade 7 mathematics course	Highest math course a student took at Grade 7. The types of math courses were: 0no math course; 1low/basic math; 2average math; 3high math; 4prealgebra; 5algebra; 6geometry; 7algebra2; 8trigonometry/precalculus; 9calculus.
Student progress in math courses	A students progress in math course from Grade 7 to Grade 121 if throughout 6 years, a student proceeded to take a higher type math course than the previous grade, 0 if not.
Student behavior problems	Student behavior problems at Grade 7 (scale 04). This composite variable, a developed by the LSAY project staff, measures student-reported problem behaviors at school. The student may indicate that he or she is having trouble keeping up with classes; is bored with school this semester; has cut classes now and then during the school year; has had disciplinary problems in school this year.
Student educational expectations	Student educational expectations at Grade 7. This variable reflects the highest level of education the students believe they will attain, ranging from 1 (high school only) to 6 (doctoral).
Student self-esteem	Student self-esteem at Grade 7 (scale 024). This composite variable measures student self-esteem based on responses to items like: I take a positive attitude toward myself; I feel I am a person of worth; I am able to do things as well as most other people; on the whole, I am satisfied with myself, and so on.
Student gender	Student gender, coded 0 for girls, 1 for boys.
Home resources	Home math and science resources (scale 06). This composite variable counts the number of science and math resources that a student reports having in their homes. The resources include a computer, an atlas or globe, a pocket calculator, more than 50 books, a microscope, and a telescope.
Mothers educational level	Mothers education. This variable measures the highest degree that a students mother has earned, ranging from 1 (some high school) to 5 (masters or doctoral degree).
School Level Composite variablesb from teacher survey
Community commitment to math	This composite variable measures parent and student commitment to math at a school. It is based on items that indicate parents and students interest in math and whether they think math is important, and students preparedness.
School safety	This composite variable captures a schools safety and disciplinary environment, such as the degree of absenteeism, theft or vandalism, drugs or alcohol, fighting or assault, verbal abuse of teachers, and racial relations.
Variables from the principal survey
Percent minority student	This variable indicates the percent of minority students in a school (0100%).
Time-varying covariates
Student attitude toward mathematics	Math attitude Grade 7Grade 12 (scale 016). This composite variable measures student attitude toward mathematics. The scale is based on the means of such components of mathematics attitude as interest and utility.
Percent class time covering new topics	Percent of class time devoted to teaching new math materials (scale 0100%).

Note. ^a All composite variables for the student-level data were developed by the LSAY project staff unless otherwise indicated. ^b These composite variables are based on teachers responses to 28 items on the teacher survey and are aggregated to the school level.

STATISTICAL MODELS: THREE-LEVEL LONGITUDINAL AND MULTILEVEL MODELS

This study adopted a three-level longitudinal and multilevel modeling approach (see the appendix for a general description of the statistical models). In this approach, the first level (Level 1) models individual growth trajectories and other time-varying variables; the second level (Level 2) captures the variation in growth parameters among students with a school; and the third level (Level 3) estimates the variation in growth parameters among schools.

Level 1 model: Individual growth model. Exploratory analysis of a sample of individual students growth patterns suggested a flattening trend between Grade 11 and Grade 12 (i.e., a slight decrease in math scores occurred during the senior year of high school). Therefore, a two-piece linear growth model was used to describe students growth, where one growth rate was parameterized as the base rate for the entire secondary grades (i.e., Grades 712) and the other as the decrement to the base rate between Grades 11 and 12. The intercept in the Level 1 model represents the end-of-high-school mathematics attainment.

Besides these three growth parameters (i.e., attainment at the end of Grade 12, annual learning rate between Grades 7 and 12, and decrement to the learning rate between Grades 11 and 12), the Level 1 model included two time-varying predictors measured each year. These two predictors were students attitude toward mathematics and percent class time devoted to teaching new materials. One point worth mentioning is that students were with different mathematics teachers each year. LSAY did not link individual students to teachers, making it impossible to explicitly model teacher or classroom level in the analysis. An alternative would be to include time-varying variables associated with teachers/classrooms at the Level 1 analysis, as was done in the current study.

Level 2 model: Between-student and within-school level. At Level 2, a persons end status (i.e., end-of-high-school mathematics attainment) and growth rates are modeled as varying from the average end status and growth rates as a function of background predictors. These predictors included Grade 7 mathematics course, progress in math courses, behavior problems, educational expectations, self-esteem, gender, home resources, and mothers educational level.

Level 3 model: School-level model. To study the variation in growth parameters among schools, an additional level (i.e., Level 3) is necessary. This Level 3 model is called a between-school model. In the Level 3 model, the mean end status and growth rates of a school are modeled as varying from the grand mean end status and growth rates of the whole sample as a function of school characteristic or compositional variables. These compositional variables included percent minority students, community commitment to mathematics, and school safety.  

DATA ANALYSIS

An earlier study using a smaller portion of the data (Ai, 2002) indicated that there were two different groups of students in the sample, namely low beginners and high beginners. The descriptive statistics of math scores (which were measured on a 0100 scale) at Grade 7 indicated that students mathematics achievement at this grade (the starting point of the study) ranged from 28.19 to 83.97. When the math scores at Grade 7 were recoded into 050, 5075, and 75100 categories, the frequency of these categories indicated that 45.9% of the students scored below 50; 53.7% scored between 50 and 75; and 0.5% scored above 75. Because the number of students who scored above 75 at Grade 7 was extremely low (15 out of 3,090), the sample was divided into two groups based on initial status in the following way, which was roughly a median split: the low group (those whose initial math scores were below 50) and the high group (those whose initial math scores were above 50).  

These two groups of students differed in terms of (a) the relationship between some background variables and the growth parameters, and (b) the observed gender differences. Therefore, to avoid the problem of combining groups when they differed with respect to the relationship between the variables being studied, the model analysis in the present study was conducted separately for low beginner girls (n = 713), low beginner boys (n = 768), high beginner girls (n = 879), and high beginner boys (n = 728) before combining girls and boys in the low and high group, respectively.   

The analysis proceeded in two steps as follows. In the first step, an unconditional model was estimated for each group. The results for this model gave an estimation of the mean growth trajectory. The second step in the analysis consisted of a set of models that focused on the relationship between math scores and various student and school compositional variables. Results from the analyses of this set of models helped us to see whether these background factors were related to the variation in the growth in mathematics achievement.

RESULTS

AVERAGE MATH ATTAINMENT AND GROWTH TRAJECTORIES: SENIOR SLUMP IN MATHEMATICS

Table 2 presents the average mathematics attainment at the end of high school and growth rates for the four groups.

Table 2. Average End Status and Growth Rates for Each Group

	Coefficient (SE) Low Girls           Low Boys           High Girls           High Boys
Average end status	54.06*              52.89*               71.91*                     73.41* (.61)                 (.70)                   (.55)                        (.75)
Average growth rate	3.22*                3.21*                  3.60*                       3.90* (.13)                 (.16)                   (.13)                        (.15)
Average decrement in growth	-5.58*             -5.37*                  -4.21*                      -4.84* (.32)                (.39)                   (.30)                        (.28)

* Statistically significant at 0.0001 level.

The results in Table 2 indicated a positive overall growth trajectory averaged across all students and schools for both boys and girls before the 12th grade. However, the positive growth trend declined between the 11th and 12th grades. Figure 1 graphically represents the average growth patterns for the four groups based on the results in Table 2.

Figure 1. Average growth patterns for each group

click to enlarge

As shown in Figure 1, for those who started low at Grade 7, boys average math achievement at the end of high school was slightly lower than that of the girls. However, both groups of students had about the same average growth rates, therefore, the average growth trajectories were approximately parallel; this means that on the average, boys math achievement in the seventh grade was also lower than the girls math achievement. The growth trajectories for those who started high in the seventh grade were almost the reverse of those who started low. For boys and girls who started high, boys had a very small advantage in the average math achievement at the end of high school andin contrast to the findings for the group that started low, in which no difference in growth rate was detectedalso had a slightly higher average growth rate. Figure 1 also indicates that there was a drop in the average math achievement between the 11th and 12th grades for all four groups of students.

RELATIONSHIP BETWEEN PREDICTORS, MATH ATTAINMENT, GROWTH RATE, AND SENIOR SLUMP

Table 3 summarizes the results from the final three-level conditional models for the low and high groups. The first section of Table 3 shows the predictors for the average math achievement at the end of high school.

Table 3. Growth Parameter Estimates and Various Student, Home, and School Predictors

	Low Initial Status	High Initial Status
Math Attainment: Intercept	54.95* (.57)a	66.09* (.44)
Student-level predictors:
Progress in math courses	11.71* (1.30)	3.60* (.45)
Grade 7 self-esteem	0.24* (.08)	0.12 (.06)
Grade 7 behavior problems	-0.77* (.28)	-0.55* (.21)
Gender	-0.20 (.57)	0.87* (.34)
Grade 7 math course	5.50* (.68)	4.00* (.19)
Grade 7 expectations	1.23* (.22)	1.02* (.15)
Mothers education	-- b	0.64* (.17)
Home resources	0.59* (.17)	0.49* (.12)
Grade7 math* gender	-1.85* (.85)	-- b
Grade 7 esteem* gender	-- b	-0.16 (.09)
School-level predictors:
Community commitment to math	-0.02 (.99)	-0.40 (.83)
School safety measures	1.62 (1.58)	1.07 (1.33)
Percent minority	-5.92* (2.31)	-5.12* (2.08)
Base Growth Rate	3.25* (.12)	3.52* (.12)
Student-level predictors:
Progress in math courses	2.34* (.32)	0.90* (.13)
Grade 7 self-esteem	0.01 (.02)	0.06* (.02)
Grade 7 behavior problems	-0.03 (.07)	-0.17* (.06)
Gender	0.16 (.14)	0.31* (.10)
Grade 7 math course	0.98* (.17)	0.21* (.06)
Grade 7 expectations	0.26* (.05)	0.15* (.04)
Mothers education	-- b	-0.03 (.05)
Home resources	0.08 (.04)	0.09* (.03)
Grade 7 math* gender	-0.36 (.21)	-- b
Grade 7 esteem* gender	-- b	-0.07* (.03)
School-level predictors:
Community commitment to math	0.01 (.19)	-0.26 (.23)
School safety measures	0.17 (.29)	0.01 (.37)
Percent minority	-1.07* (.43)	-0.55 (.58)
Drop in Math Growth During Senior Year	-4.01* (.25)	-3.38* (.23)
Student-level predictors:	-1.79* (.77)	-0.52 (.33)
Progress in math courses	0.02 (.05)	-0.09 (.05)
Grade 7 self-esteem	-.08 (.17)	0.01 (.16)
Grade 7 behavior problems	-0.12 (.34)	-0.35 (.27)
Gender	-1.23* (.40)	-0.11 (.14)
Grade 7 math course	-0.35* (.13)	-0.05 (.11)
Grade 7 expectations	-- b	0.06 (.13)
Mothers education	-0.14 (.10)	-0.15 (.09)
Home resources	0.20 (.51)	-- b
Grade7 math* gender	-- b	0.15* (.07)
Grade 7 esteem* gender
School-level predictors:	0.36 (.36)	0.28 (.37)
Community commitment to math	-0.43 (.54)	-0.31 (.63)
School safety measures	0.61 (.78)	0.27 (.96)
Percent minority
Effects of Time-Varying Predictors
Attitudes toward math:	0.19* (.04)	0.02 (.03)
Gender	-0.01 (.05)	0.17* (.05)
Cross-level interaction with
Community commitment to math	-.07 (.06)	-0.05 (.05)
School safety measures	-.07 (.09)	0.04 (.09)
Percent minority	-0.24* (.11)	-0.04 (.12)
% class time covering new topics	0.03* (.01)	0.01 (.01)
Gender	-0.02* (.009)	-0.01 (.01)
Cross-level interaction with
Community commitment to math	-0.02 (.02)	0.00 (.02)
School safety measures	0.05 (.03)	0.03 (.03)
Percent minority	-0.01 (.04)	0.03 (.04)

Note. ^a The numbers in the parentheses are the standard error estimates. ^b This predictor did not have any effect on any of the parameters for this group; therefore, it was excluded in the final three-level conditional model for this group of students.

Statistically significant at 0.05 level.

LOW GROUP: MATHEMATICS ATTAINMENT AND SIGNIFICANT PREDICTORS

Progress in math courses. Table 3 suggests that a students continual progress in math courses from Grades 712 has a substantial and significant effect on his or her math achievement at the end of high school. If throughout 6 years (i.e., from Grade 7 to Grade 12), a student continued to take a higher level math class when he or she moved on to a higher grade, his or her math achievement at the end of high school would be almost 12 points higher (11.71) than someone who did not make continual progress in math courses, holding constant all other predictors.

Early tracking. The type of math class a student took in the seventh grade also had a considerably positive association with his or her math achievement at the end of high school (i.e., 5.5). Imagine two students from the same school who are similar with respect to all the individual background variables, except that one student (A) is in a math class (e.g., high math) one level higher than the other student (B) (e.g., average math). On average, we would expect that Student As math achievement at the end of high school would be 5.5 points higher than Student Bs math achievement.

Affective factors. Other significant predictors of a students math achievement at the end of high school were self-esteem, student behavior problems, student educational expectations, and home resources. The effects of these predictors on a students math achievement at the end of high school were small to moderate. Specifically, for a 1-unit increase in self-esteem, holding constant all other predictors, there was a quarter point (i.e., 0.24) increase in math achievement at the end of high school. Student behavior problems had a negative effect on math achievement at the end of high school. A 1-unit increase in behavior problems would accompany about 0.77-point drop in math achievement at the end of high school, holding constant all other background variables. Student educational expectations had a moderately positive effect on math achievement at the end of high school. Holding all other predictors constant, for a 1-unit increase in student educational expectations, the expected math achievement at the end of high school would be increased by 1.23 points. Home math and science resources had a small significant effect on a students math achievement at the end of high school. Holding all other predictors constant, a 1-unit increase in home resources would see only about 0.59-point increase in math achievement at the end of high school. The coefficient for SEX (i.e., -0.20) in Table 3 was not statistically significant, which suggested that boys and girls in the low group had about the same average math achievement at the end of high school. However, there was a significant interaction between gender and the types of math courses a student took in Grade 7. Table 3 shows that this significant interaction effect was -1.85. Because boys were coded 1, the negative sign indicates that the effect of the types of math courses taken in the seventh grade on math achievement at the end of high school was approximately 1.85 points stronger for girls than for boys.

School composition: Percent minority students. Compared with the number of significant student-level predictors of math achievement at the end of high school, the only school-level predictor that was statistically significant was percent minority students in a school, which had a considerably negative effect. Specifically, a 1-unit increase in the percent of minority students would see 5.92-point drop in math achievement at the end of high school, holding constant all student background variables and other school-level variables. In other words, if we had two students (A and B) who were similar in all the background variables, but A was attending a school with a 1-unit-higher percentage of minority students than Bs school, then As math achievement at the end of high school would be about 6 points lower than Bs.

LOW GROUP: MATHEMATICS GROWTH AND SIGNIFICANT PREDICTORS

Progress in math courses. The second section of Table 3 shows the predictors of growth rates for the low group. Results indicated that a students continual progress in math courses from Grade 7 to Grade 12 had a significant effect on his or her math growth rate. If throughout 6 years (i.e., from Grade 7 to Grade 12), a student continued to take a higher level math class when he or she moved on to a higher grade, his or her math learning rate would be 2.34 points higher than someone who did not make continual progress in math courses, holding constant all other predictors.  

Early tracking. The type of math class a student took in seventh grade also had a considerably positive association with his or her math learning rate (i.e., 0.98). Again, imagine two students from the same school who are similar with respect to all the individual background variables except that one student (A) is in a math class (e.g., high math) one level higher than the other student (B) (e.g., low math). The results show that Student As math learning rate would, on average, be close to 1 point faster than Student Bs math learning rate per grade. Results also suggested that boys and girls in the low group had roughly the same math growth rates, and the effect of the types of math courses in seventh grade on math growth rates was the same for boys and girls.

Affective factors. Student educational expectations also had a positive but small effect on math growth rate. Holding constant other predictors, a 1-unit increase in educational expectations would see a 0.26-point-faster math growth rate.

School composition: Percent minority students. Percent minority students in a school had a negative effect on math learning rate. For a 1-unit increase in percent minority students, holding constant all other predictors, the expected math learning rate would decrease by 1.07 points.

LOW GROUP: SENIOR SLUMP AND SIGNIFICANT PREDICTORS

Progress in math courses. The third section of Table 3 shows the predictors of the drop in math growth between Grade 11 and Grade 12. Results indicated that a students continual progress in math courses from Grade 7 to Grade 12 had a significant and mitigating effect on his or her drop in math growth. If throughout 6 years (i.e., from Grade 7 to Grade 12), a student continued to take a higher level math class when he or she moved on to a higher grade, he or she would see about a 1.79-point-smaller drop in math growth during the senior year than someone who did not make continual progress in math courses, holding constant all other predictors.

Early tracking. The type of math class a student took in the seventh grade also had a negative effect on his or her math learning rate (i.e., 1.23). Again, imagine two students from the same school who were similar with respect to all the individual background variables, except that one student (A) was in a math class (e.g., high math) one level higher than the other student (B) (e.g., average math). The result showed that Student A would see a 1.23-point-smaller drop in math growth than Student B in the senior year of high school. There was no significant difference between boys and girls with regard to the drop in math growth associated with being in the low group and also with regard to the relationship between the course taken in seventh grade and rate of math growth.

Affective factors. Student educational expectations also had a small mitigating effect on drop in math growth. Holding constant other predictors, a 1-unit increase in educational expectations would see 0.35-point-smaller drop in math growth.

LOW GROUP: EFFECTS OF TIME-VARYING PREDICTORS

Math attitude. The last section in Table 3 shows the average effect of the two time-varying variables (i.e., math attitude and percent of class time devoted to teaching new math materials) on a students math scores and the relationship between the effect of these time-varying variables and other school-level variables. The first number in this section (i.e., 0.19) indicates that math attitude had a significant positive effect on a low girls math scores (when SEX = 0). Holding constant all other predictors, the expected math scores would increase by 0.19 point, with a 1-unit increase in math attitude. For instance, the expected math score for a girl (A) with average values on all the predictors at Grade 12 would be 54.95. Consider another girl (B) with the same values on all the predictors, except that her math attitude was 1 unit higher than that of Girl A. Girl Bs expected math score at Grade 12 would be 54.95, plus the effect of math attitude (0.19), which was about 55.14. This example showed that the effect of math attitude on math scores, albeit statistically significant, was practically very small. The coefficient for SEX (i.e., -0.01) was not statistically significant, which meant that the effect of math attitude on math scores was roughly the same for boys and girls. There were no other individual-level predictors moderating the math attitude effect because analyses showed that the effect of math attitude on math scores was essentially constant for all students within a school.

Effect of math attitude and school composition. The results suggested that for students who started low on math in the seventh grade, the effect of their math attitude on their math scores was significantly related to the percentage of minority students in their schools. In other words, the effect of math attitude, which was an individual-level effect, depended on the school-level predictors. This kind of relationship is called a cross-level interaction. The term cross-level here implies, for example, that the effect of a student-level variable interacts with (or depends on) a school-level variable. In this case, for a 1-unit increase in percent of minority students in a school, holding constant all other predictors, the effect of math attitude on a students math scores would be weakened by about 0.24 points. This means that on average, the larger the proportion of minority students in a school, the weaker the association between attitude and achievement.

 

Percent class time covering new materials. Table 3 indicates that the effect of the other time-varying variable, percent of class time devoted to teaching new math materials, on math scores was also significant but very small. Holding all other predictors constant, if a teacher spent 10% (i.e., 10 units increase) more class time teaching new math materials, students math scores would see only less than a third of a point increase (i.e., 10*0.03 = 0.3). The coefficient for SEX is -0.02, which was statistically significant. Because boys were coded 1, this suggested that the effect of percent of class time devoted to teaching new math materials on math scores was stronger for girls than for boys. No cross-level interactions were found for this time-varying variable. The variance of this time-varying variable (0.002) was highly significant (p = .0001), which indicated not only that percent of class time devoted to teaching new math materials had a significant effect on math scores on average, but also that this significant effect varied across schools. In other words, the positive effect of percent of class time devoted to teaching new math materials might be stronger in some school than in other schools.

RESULTS FOR THE HIGH GROUP IN COMPARISON WITH RESULTS FOR THE LOW GROUP

Similarities. The effects of various predictors on math achievement at the end of high school, math growth, and the drop in math growth for the high group (i.e., the second column of numbers in Table 3) could be interpreted in the same way as those for the low group. The comparison of the two columns of numbers in Table 3 indicates that the low and high groups were similar in terms of (a) the positive effect of progress in math courses on math attainment and growth rate, though the effect seemed stronger for the low group (i.e., 11.71 and 2.34, respectively) than for the high group (i.e., 3.60 and .90, respectively), and (b) the effect of Grade 7 math course on math attainment (5.50 for the low group and 4.00 for the high group) and growth rate (0.98 for the low group and 0.21 for high group). Again, the effect of early tracking into higher level math on attainment and growth rate was slightly stronger for the low group than for the high group.

Differences. Results in Table 3 also indicate that the low and high groups differed with respect to (a) the average growth parameter estimates (i.e., mathematics attainment, growth rate, and drop in growth during the senior year), which determined growth patterns in mathematics, (b) significant predictors of these growth parameters, and (c) gender gap in math. Table 4 summarizes these differences.

Table 4. Comparison of HLM Results for the Low Group Versus the High Group

	Low Initial Status	High Initial Status
Growth parameters
End of high school attainment	Lower	Higher
Average growth rate	Slower	Faster
Average drop	More drop	Less drop
Significant predictors of attainment
Grade 7 self-esteem	Significant (+)	No effect
Mothers education	No effect	Significant (+)
Significant predictors of growth
Grade 7 self-esteem	No effect	Significant (+)
Grade 7 behavior problems	No effect	Significant (-)
Home resources	No effect	Significant (+)
% minority in the school	Significant (-)	No effect
Significant predictors of drop
Progress in math courses	Significant (+)	No effect
Grade 7 math course	Significant (+)	No effect
Student educational expectation	Significant (+)	No effect
Effects of time-varying predictors
Attitude toward math	Significant (+)	No effect
% class time teaching new topics	Significant (+)	No effect
Gender differences
End of high school attainment	None	Favor boys
Average growth rate	None	Favor boys

Attainment, growth, and senior slump. As Table 4 suggests, those who started high on math achievement in the seventh grade also had higher math attainment at the end of high school and faster math growth rates than those who started low. Conversely, those who started high saw a smaller senior year decrement in math growth than those who started low.

Course progress, early tracking, and school composition. A students continual progress in math courses and the type of math course a student took in the seventh grade had significant effects on the drop in math growth for those who started low, but not for those who started high. Specifically, continual course progress and early tracking into higher level mathematics courses could help to alleviate the senior slump for the low group.

The school compositional factor, percent minority students in a school, was a significant predictor of the growth rate for those who started low, but not for those who started high. The higher the percent minority students in a school, the lower the average growth rate for students who started low on mathematics at Grade 7.

DISCUSSION

Policy makers have a strong interest in reforming math and science education because these subjects are seen as critical to economic competitiveness given their impact on innovation in scientific research and technology (National Mathematics Advisory Panel, 2008). Yet studies on transition between high school and college indicated that many high school seniors are underprepared for college-level math upon entering colleges and universities (e.g., Hoyt & Sorenson, 2001). Precollege education, as the National Science Board (2006) emphasized, is the foundation for fostering leadership in science, technology, engineering, and mathematics (STEM). The present study attempts to understand how high school graduates got to where they were by investigating their mathematics growth trajectories in the middle and high school years in relation to several key social-psychological and opportunities-to-learn factors.

One of the important findings of this study was that on average, there was a drop in mathematics achievement during the senior year of high school for students in the sample regardless of student mathematics achievement in Grade 7. In other words, although students mathematics achievement exhibited an average upward growth trend between Grades 7 and 11, the drop that occurred between Grades 11 and 12 pulled down mathematics achievement, and therefore mathematics attainment, at the end of high school. That is, on average, student achievement in Grade 12 was slightly lower than that at the end of Grade 11 (i.e., the average upward growth rate was smaller than the decrement in growth between Grades 11 and 12). This drop in achievement during the senior year conforms with popular wisdom referred to by Kirst (2001), who noted that the senior slump after midyear grades is an American tradition. This phenomenon of students slacking off in mathematics in their senior year deserves our attention because of its immediate impact on postsecondary education. The slump in mathematics not only leaves students ill prepared in terms of quantitative skills, thereby setting a limit on their access to quantitative and science-related college majors, but it also forces colleges to spend billions of dollars on remedial education to help these students overcome this hurdle.  

What can we do to improve students mathematics attainment by the end of high school? The empirical evidence presented in this study helps to partly address this issue in two ways. First, the study examined the factors that might be predictive of the drop in mathematics achievement during the senior year of high school. Results indicated that among the variables this study focused on, the following had significant effects on the drop in mathematics achievement in the senior year: (a) a students continual progress in math courses between Grades 7 and 12, (b) the types of math courses a student took at Grade 7, and (c) student educational expectations at Grade 7. Specifically, if a student continued to progress in the math course sequence throughout the middle and high school years, took higher level math courses than basic math, and had higher educational expectations, he or she would see a smaller drop in mathematics achievement in the senior year of high school. This finding suggests that efforts to improve students mathematics achievement by specifying a more rigorous mathematics curriculum and setting higher instructional standards may help reduce the severity of the senior slump.

In addition to identifying significant predictors of the drop in mathematics achievement during the senior year, the study examined the significant factors that directly predicted students end-of-high-school mathematics attainment. These factors included (a) a students continual progress in math courses, (b) the types of math courses a student took in Grade 7, (c) self-esteem, (d) student educational expectations, (e) student behavior problems, (f) attitudes toward mathematics, and (g) percent of class time devoted to teaching new math materials. These results have implications for program interventions that aim to promote students mathematics achievement. One point worth mentioning is that self-esteem, educational expectations, and behavior problems were all measured at Grade 7. This finding points to the importance of early intervention and education because of their far-reaching effects on later achievement.

Another important finding was related to the inequity in mathematics achievement. The results of this study showed that there were two different groups of students in the sample who differed substantially in terms of mathematics achievement at Grade 7 (i.e., initial differences). These two groups of students also differed subsequently in mathematics achievement patterns in the later years of middle and high schools. The results indicated that those who started low on mathematics in Grade 7 on average had relatively smaller growth rates and a bigger drop in mathematics achievement during the senior year than those who started high at Grade 7.  As a result, those who started low at Grade 7 ended up far behind those who started high with respect to the mathematics attainment at the end of high school. In other words, initial differences in these two groups of students were retained or increased at the end of high school. This less equitable distribution of achievement at the end of high school associated with initial differences deserves educators and researchers close attention. The challenge for researchers and policy makers is to figure out how achievement can be boosted among these initially disadvantaged students. This study produced several empirical findings that may suggest ways to think about how to address this challenge.

To begin with, the study found two proxy indicators of opportunities to learn mathematics that had stronger positive impact on students who initially started low on mathematics in Grade 7 as compared with students who initially started high. These two opportunities-to-learn indicator variables are: (a) the type of math courses a student took at Grade 7, and (b) a students continual progress to higher level math courses as he or she progressed to higher grade levels. This finding calls into question the wisdom of practices such as within-school segregation through tracking without ensuring that these disadvantaged students have the opportunity to take higher level mathematics as they progress through the school system.

To a certain extent, the finding that early tracking in mathematics has significant a effect on outcomes confirms earlier research on the effects of stratification in secondary schools by various scholars, particularly Adam Gamoran (e.g., Gamoran & Mare, 1989). The present study, however, extends previous scholarship in several ways. First, the study examined the effect of early tracking not only on end-of-high-school mathematics attainment, but also on growth trajectories and on drop in mathematics achievement during the senior year. In other words, the study examined the effect of earlier tracking on learning outcomes from a multidimensional, dynamic perspective, taking advantage of the longitudinal feature of the available data. Second, the study examined the effect of early tracking for students who had different starting points (i.e., initial status) and revealed a nuanced relationship between early tracking and outcomes, depending on students initial math ability (using Grade 7 math scores as a proxy measure). Results suggested that although early tracking affected end-of-high-school math attainment and growth trajectories during secondary school years for both low and high students, the effect of early tracking was stronger for students who started low than for students who started high on math at Grade 7. In addition, the effect of early tracking on senior slump was observed among those who started low but not among those who started high on math at Grade 7. Specifically, students with relatively low math achievement who were placed in higher level mathematics courses in Grade 7 would see a smaller drop in mathematics achievement during the senior year than students with a similar level of math achievement who were placed in lower level math courses.

Alongside the effect of early tracking, the present study also found a positive effect of progress in math courses on math attainment and growth rate, and again the effect seemed stronger for the low group than for the high group. Similar to the effect of early tracking, continual progress in mathematics courses between Grades 7 and 12 could help to alleviate senior slump for the low group. In other words, students with relatively low math achievement in Grade 7 who were able to move up to a higher level of math the following year would see a smaller drop in mathematics achievement during the senior year than students with a similar level of math achievement in Grade 7 who did not advance through math courses each year (i.e., taking a higher level math course upon moving to the next grade level).

In addition to the effect of early tracking and course progress, this study also observed a similar pattern of relationships between the school compositional effect of percent minority students in a school on students end-of-high-school math attainment, and overall growth in math. The effect of this school compositional variable was demonstrated in two ways. First, percent minority students in a school was significantly related to students end-of-high-school math attainment and to the overall growth rate in math, but the effect seemed stronger for those who started low on math than for those who started high on math at Grade 7. Second, this school compositional variable moderates the effect of students attitudes toward math on achievement and was statistically significant for those who started low on math. Specifically, the higher the concentration of minority students in a school, the weaker the effect of attitude on math achievement.

Taken as a whole, these results are indicative of the powerful effect of school composition and opportunities to learn, such as early tracking and course progress, on students mathematics achievement. Furthermore, the observed patterns of relationships suggested that students with relatively low initial math achievement were particularly vulnerable to the effects of these forces. When low starters took challenging courses and were able to make continual progress each year to higher level math courses as they progressed through the school system, their end-of-high-school attainment and math trajectories were better. In a similar manner, school composition (i.e., percent minority students) seemed to have a stronger impact on students who started relatively low than on students who started relatively high. Given that low-income students and students of minority background tend to be placed in lower tracks than their White and middle-class counterparts, these findings have important implications. The findings not only point to the detrimental effect of practices such as early tracking of children into less challenging curricular paths who are vulnerable to such practices, but also reinforce the notion that all children could potentially benefit from a challenging curricular pathway regardless of where they start in Grade 7 (see also Gamoran & Hannigan, 2000; Meyer, 1999).          

A few caveats are worth discussing. To begin with, this study was motivated by a desire to understand how high school graduates got to where they were in terms of mathematics attainment. Capitalizing on the six-panel LSAY data, the study adopted a three-level HLM framework (Raudenbush & Bryk, 2002) that combined the longitudinal and multilevel features of the data. Under this framework, the intercept in the Level 1 growth equation represented end-of-high-school math attainment, namely, end status instead of initial status, as is typically done in longitudinal growth modeling analysis. As such (i.e., modeling end status as the intercept), only graduates with end-of-high-school math scores were included in the model. Consequently, interpretations of the findings should not be generalized to all high school graduates in the United States even though the LSAY began with a national representative sample of seventh graders. On the other hand, the focus on attainment helps to avoid problems associated with missing data. Future research could compare high school graduates in LSAY with graduates sampled in other national longitudinal surveys (e.g., HS&B, NELS:88, ELS:2002) to determine the extent to which results may or may not generalize to the population of American high school graduates.

Second, the current study contained two measures of students opportunities to learn mathematics, namely, the type of math courses students took in Grade 7 (as a rough proxy for early tracking) and progress in math courses. The type of math courses students took in Grade 7 is self-explanatory, meaning it was simply a variable labeling what math course a student took at the beginning of the LSAY study. The other variable, progress in math courses, was created based on the math course a student took each year and took on a value of 1 if the student was progressively taking higher level math in subsequent years (0 otherwise). This progress variable attempted to capture, to some extent, the qualitative aspect of students experiences in math in terms of opportunities to learn progressively advanced math topics. Neither of these variables (i.e., early tracking proxy and course progress) may explain whether challenging courses were available at a particular school,³ nor do these variables account for self-selection into courses. Nonetheless, these two opportunities-to-learn measures provided a chance to examine student course-taking patterns beyond simply counting the number of math courses a student took in high school, which in many cases does not capture the qualitative aspect of math course-taking.

Third, because student race/ethnicity was not available at the individual student level, this study could not include individual student race/ethnicity in the analysis (but available at the school level as percent minority students in a school used in the analysis). To compensate for this lack of individual race/ethnicity variable, though not perfect, the study included mothers education and home resources because these social economic measures tend to correlate highly with race/ethnicity.

Finally, this study did not find many significant effects of school characteristics and academic climate variables on students mathematics achievement. As the results indicated, the only school-level variable that had a significant effect on students mathematics achievement was percent minority students in a school. One possible explanation is that the LSAY data contained only 52 schools, and these schools were all public schools. The small sample size of schools might make the effects of some school-level variables difficult to detect, even if they are present. Additionally, the relative homogeneity of schools (i.e., all public schools and no private or catholic schools), with respect to the school-level variables, does not maximize the variation in these variables and therefore masks their effects. Of course, percent minority students in a school can be a proxy measure of many aspects of a school. Schools with a higher percentage of minority students may have lower expectations of students and may have instructional programs of lower quality as compared with schools with a lower percentage of minority students. It is important, therefore, to conduct further in-depth studies to understand the schooling process of these schools and identify the kinds of things that can be done to improve students mathematics achievement and eventually reduce the achievement disparity in mathematics.

Despite these caveats, this study produced several important empirical findings on the relationship between student achievement in mathematics, and various individual and school compositional variables. These empirical findings point to the further directions we may take to promote student achievement in mathematics.

Acknowledgments

This research was supported by a grant from the American Educational Research Association, which receives funds for its AERA Grants Program from the National Science Foundation and the National Center for Education Statistics (U.S. Department of Education) under NSF Grant No. RED-9452861. Opinions reflect those of the author and do not necessarily reflect those of the granting agencies. The author thanks four anonymous reviewers who provided constructive feedback.

Notes

1. Another important variable would be student race or ethnicity. Unfortunately, LSAY does not have this information at the individual student level but has a school-level measure of percent minority students in a school.

2. Percent class time covering new materials was measured at each grade and therefore was a time-varying variable.  Including this time-varying variable has a methodological advantage, which is discussed in the Method section.

3. In schools where policies regarding course-taking are known, researchers could use statistical techniques such as propensity score matching (Leow, Marcus, Zanutto, & Boruch, 2004) to formally model the causal effect of course-taking policy on achievement, comparing students who follow the policy with those who do not.

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APPENDIX

General Description of Three-Level Longitudinal and Multilevel Statistical Models

Level 1 model: Piecewise linear model. The Level 1 equation describing the growth pattern for each student is as follows:

 (1) Y_tij = p_0ij + p_1ij (T1)_tij + p_2ij (T2) + e_tij              e_tij ~ N (0, s²)

Where

Y_tij

is the math score at grade t for student i in school j;

(T1)_tij

is coded 0 at Grade 12, -1 at Grade 11, -2 at Grade 10, -3 at Grade 9, -4 at Grade 8, and -5 at Grade 7; this coding scheme is used so that the intercept (p_0ij) represents a students mathematics attainment at the end of high school; in addition, the growth rate, p₁ , represents the base learning rate for the entire period;

(T2)_tij

is coded 0 at Grade 12, and -1 otherwise; this coding scheme is used so that the growth rate, p_2ij , represents the decrement to the base rate during the senior year of high school;

p_0ij

is the end of high school mathematics attainment for student i in school j (i.e., when T1 & T2 =0);

p_1ij

is the base math learning rate for student i in school j over 1 year;

p_2ij

is the decrement to base math learning rate for student i in school j between Grade 11 and Grade 12 (or during the senior year of high school);

e_tij

is the residual associated with an individuals score at a specific time point, which is assumed to be normally distributed with a mean of zero and variance of s².

One point worth mentioning is that within the hierarchical linear modeling (HLM) framework, time-varying covariates (e.g., math attitude) are modeled at the Level 1 equation.

Level 2 model: Between-student and within-school level. The Level 2 equations without any predicators (referred to as unconditional Level 2 model) are as follows:

(2) p_0ij = b_00j + r_0ij                                                                r_0ij ~ N (0, t_0ij)

(3) p_1ij  = b_10j + r_1ij                                          r_1ij ~ N (0, t_1ij)

(4)  p_2ij  = b_20j + r_2ij                                         r_2ij ~ N (0, t_2ij)

where

b_00j  represents the mean math end status for school j;

b_10j  represents the mean math growth rate for school j;

b_20j  represents the mean decrement to math growth for school j during senior year;

r_0ij is the variation in end status among students within school;

r_1ij is the variation in base growth rate among students within school; and

r_2ij is the variation in decrement to base growth rate among students within school.

In Equations 2, 3, and 4, r_0ij, r_1ij and r_2ij are assumed to be multivariate normal, each with a mean of zero, some variance (i.e., t_0ij, t_1ij and t_2ij respectively) and covariance among them (e.g., t_01ij).  Student-level predictors (e.g., course progress, early tracking, gender, and so on) are modeled in Equations 24.

Level 3 model: School-level model.  For simplicity, let us consider the Level 3 equations without school composite variables.  In this case, the average school end status (i.e., b_00j) and growth rates (i.e., b_10j, b_20j), as represented in Equations 2, 3, and 4, are modeled at Level 3 as varying from the grand mean end status and growth rates, respectively.  Because there are three coefficients (i.e., b_00j, b_10j, and b_20j) at the Level 2 model, we need three equations at Level 3.  The Level 3 equations are:

(5) b_00j = g₀₀₀ + u_00j                           u_00j ~ N (0, t_00j)

(6) b_10j  = g₁₀₀ + u_10j                           u_10j ~ N (0, t_10j)

(7) b_20j  = g₂₀₀ + u_20j                           u_20j ~ N (0, t_20j)

where

g₀₀₀ represents the grand mean math end status;

g₁₀₀ represents the grand mean math base learning rate;

g₂₀₀ represents the grand mean decrement to base math learning rate;

u_00j is the variation in end of high school mathematics attainment across schools;

u_10j is the variation in base growth rate across schools; and

u_20j is the variation in decrement to base growth rate across schools.

In Equations 5, 6, and 7, u_00j, u_10j and u_20j are assumed to be multivariate normal, each with a mean of zero, some variance (i.e., t_00j, t_10j and t_20j respectively) and covariance among them (e.g., t_01j).  School-level predictors (e.g., percent minority students in a school) are modeled in Equations 57.

To summarize, the three-level HLM framework described here combines longitudinal and multilevel features; Level 1 describes each individuals growth in mathematics achievement over time (i.e., the longitudinal feature), Level 2 captures the variation across individuals in growth parameters (i.e., end status and growth rates) within a school, and Level 3 captures variation in growth parameters among schools (i.e., the multilevel features).

Cite This Article as: Teachers College Record Volume 112 Number 4, 2010, p. 1064-1095 https://www.tcrecord.org ID Number: 15660, Date Accessed: 1/28/2022 11:12:51 PM Purchase Reprint Rights for this article or review