

High School Graduate Requirements: Effects on Dropping Out and Student Achievementby Thomas B. Hoffer  1997 This study examines effects of legislation requiring high school students to complete at least three years of mathematics on three types of outcomes: the kinds of mathematics courses students complete during high school, high school dropout rates, and mathematics achievement test score gains during high school. Several state and local authorities have recently enacted legislation requiring high school students to complete at least three years of mathematics in order to graduate. This study examines effects of these policies on three types of outcomes: the kinds of mathematics courses students complete during high school, high school dropout rates, and mathematics achievement test score gains during high school. Two additional questions related to achievement are also addressed: whether requiring more courses affects the association of student socioeconomic status (SES) with test scores and dropping out, and whether requiring more courses reduces the effects of completing additional math courses on achievement. Results from an analysis of the nationally representative National Education Longitudinal Study of 1988 (NELS:88) data give little support for the notion that requiring more mathematic courses is generally beneficial or harmful. No effects are found on either the probability of dropping out or achievement gains, and the effects of SES are not reduced in the schools requiring three math courses. The threecourse requirement does lead to higher rates of course completions in geometry and algebra 2, but the higher requirements apparently dilute the effectiveness of completing the additional courses. The results thus give little credence to efforts to improve achievement outcomes by simply raising the number of math courses students must complete in order to graduate from high school. Interest in improving high school students levels of academic performance has steadily grown since the early 1980s. Vigorous debates continue over how learning outcomes should be defined and measured, and how best to improve those outcomes. The curriculum area that has received perhaps the most attention is mathematics. While much has been done by the main professional association, the National Council of Teachers of Mathematics, the most common policy focus has been on the number of courses students must complete in order to graduate. Several states and local districts have raised the numbers of required mathematics courses, in most cases from two years to three years of mathematics during the four years of high school (Blank & Gruebel, 1993). The primary goal of the increased requirements is improvement of students proficiency in mathematics. Proponents of these measures typically argue that many students are finishing high school without adequate mastery of mathematics, and that completing more courses will raise test scores. The policy is simple and relatively inexpensive, has teeth, and even has a credible, though circumstantial, basis in research. Requiring more mathematics in high school, if effective at all, may well have an equalizing effect on students levels of mathematics proficiency. The students who are least proficient are those who take the fewest mathematics courses, and these tend to be disproportionally students from lower socioeconomic backgrounds. National studies have found that highersocioeconomicstatus (SES) students complete higher numbers and levels of math courses in high school (see Hoffer, Rasinski, & Moore, 1995 for recent results). By preventing these students from opting outor from being pushed out, as the case may beof mathematics after two years, both the overall achievement inequalities and the inequalities tied to socioeconomic background should diminish. This study addresses these issues by examining evidence from a recent national sample survey of high school students and their schools. These data allow for a broader set of tests of the effects of graduation policy differences than the landmark study of Wilson and Rossman (1993), which was confined to five high schools in Maryland but which addressed many of the same themes. TAKING A BITE OUT OF UNDERACHIEVEMENT? There is no shortage of research evidence supporting the notion that requiring more courses will raise levels of math proficiency. Several analyses of nationally representative surveys of high school students over the past decade have found that students who complete more mathematics courses score higher on standardized tests of mathematics achievement (Gamoran, 1987; Hoffer, Rasinski, & Moore, 1995; Rock & Pollack, 1995; Sebring, 1987). While not experimental studies, most of these analyses sought to isolate the effects of coursetaking on achievement test scores by making statistical adjustments for the effects of differences in students social and academic backgrounds on the kinds of courses they complete and their test scores. In the view of some observers, the effectiveness of taking additional mathematics courses is the most solid policy implication that can be drawn from the big national education studies (Witte, 1992). Additional support for higher requirements is found in the recent historical trend in mathematics coursetaking among high school graduates. The large national studies of high school students also collected high school transcripts, which allow accurate comparisons of changes. From 1982 to 1992, the percentage of high school graduates completing at least a firstyear algebra course rose from about 68 to 79 percent. Percentages completing geometry and a secondyear algebra course increased even more sharply over that span: Graduates completing geometry rose from 48 to 70 percent, while the percent completing secondyear algebra rose from 37 to 56. Numbers of students completing a course in trigonometry also increased, from 12 to 21 percent of the high school graduates (National Science Board, 1996, p. 20). Part of these large increases is probably due to the increased graduation requirements of the 1980s and early 1990s. But higher requirements are probably not the whole story, since the percentages of students completing calculus have also grown. Calculus is typically taken as a fourthyear course, and would thus not be directly affected by an increased requirement from two to three years. Nonetheless, the increases in geometry, algebra 2, and trigonometry are certainly consistent with the policy trend. Have the increases in requirements and actual math course work resulted in higher levels of student achievement in mathematics? The best source of information on trends in student achievement is the National Assessment of Educational Progress (NAEP). The record from 1982 to 1992 shows significant increases in average mathematics proficiency of seventeenyearold students (Mullis et al., 1994). Unfortunately, it is not possible to link the NAEP achievement data with information about either the graduation requirements to which the students were subject or the actual courses the students completed. One cannot determine from the NAEP, then, whether the wave of higher graduation requirements led to higher achievement, or even if they led to the greater proportions of students completing algebra, geometry, and trigonometry. WILL UNDERACHIEVERS BITE BACK? While circumstantial, the case in favor of higher requirements is compelling. The extension of this research to school policy thus seems straightforward: If one wants to raise student mathematics achievement levels, then make students take more mathematics courses. But if social science teaches anything, it is that the consequences of purposive action are very rarely if ever fully anticipated. Are there ways in which requiring students to complete more mathematics courses may not lead to greater proficiency in mathematics? One possibility is that students affected by the higher requirements may be able to fulfill them with undemanding courses. Some observers have expressed concern that the new requirements may result in such diluted material that students will learn very little, and that the greater requirements will thus fail to improve achievement levels (Clune & White, 1992; Porter, 1995; Wilson & Rossman, 1993). This scenario could result from the schools making existing courses such as geometry or algebra 2 easier in order to accommodate the new studentsstudents, it should be remembered, who would not take the courses if left to their own devices. Or it may reflect schools that allow students to fulfill the requirements by taking basic courses that are redundant with ones they have already completed. In either case, the school has allowed the students a way to complete the requirements without learning much if anything. Why would a school in effect sabotage a policy designed to improve student learning? One reason may be that the school lacks the instructional resources to implement the policy properly. Moving more students into a third year of mathematics means either hiring additional math teachers or moving nonmath teachers over to teach math. Because of the expense of hiring additional teachers, or, when funds are available, a shortage of qualified math teachers, the school may wind up having lesswelltrained faculty handling the influx of students in mathematics. One would expect this to be a shortterm problem, but many schools are still in the short term with respect to higher requirements. Another reason may be that the school believes, rightly or wrongly, that the new students are unwilling or unable to handle more difficult mathematics. While the power of students to shape practice is often ignored in research, testimonials to its strength are often heard from teachers, parents, and principals. Schools do not like to fail students or to see them drop out. According to some observers, U.S. high schools typically exchange low demands on most students in return for attendance and civil behavior (Cusick, 1983; Powell, Farrar, & Cohen, 1985). By implication, a school that raises the demands may run the risk of disorder or dropout. If higher standards would diminish order or increase either the failure or dropout rates, a school may well back off from demanding more from the students (McDill, Natriello, & Pallas, 1986). These negative scenarios underscore the point that graduation policies for public schools are typically set at state or local district levels, rather than autonomously by the school. Schools thus may or may not embrace the authors intent for the policy, and schools that do not are probably less likely to implement it as intended (Wilson & Rossman, 1993). RESEARCH HYPOTHESES The analysis presented here focuses on four hypotheses about the effects of higher graduation requirements. Two are predictions of positive consequences and two point to negative outcomes. The first hypothesis is that, among students with comparable social and academic backgrounds, those attending high schools with higher graduation requirements will on average show greater achievement gains across the high school years. The mechanisms for this hypothesized effect are that students in the schools with higher graduation requirements will take more math courses, and that the additional course work they complete will result in greater learning. The second hypothesis is that higher graduation requirements will reduce the impact of students socioeconomic backgrounds on achievement growth during high school. High school graduation requirements are targeted toward students who would otherwise take fewer math courses, and these students are more likely to be students from lowerSES backgrounds. By reducing the differences among students in the numbers of math courses completed, higher graduation requirements may thus reduce the differentiating effect of SES with respect to achievement growth. Higher graduation requirements may have quite different effects, though. The possible negative consequences discussed in the previous section point to two additional hypotheses. One is that higher graduation requirements will represent an insurmountable hurdle for some students, leading them to drop out of high school. Alternatively, schools may allow students to fulfill the higher requirements by taking watereddown courses where little learning takes place. This would be reflected in weaker marginal effects of additional course work on achievement gains for students in schools requiring more math courses for graduation. FINDINGS FROM THE NELS:88 SURVEYS To assess these hypotheses, I draw on data from the National Education Longitudinal Study (NELS) of 1988, 1990, and 1992. The NELS:88 project began in 1988 with a sample of over 26,000 eighthgrade students drawn from about 1,200 schools. Approximately 16,500 of the original sample were resurveyed in 1990 and 1992, when most were sophomores and then seniors. Of these, 11,725 completed a math test in 1992 and had their transcripts collected by the study, and were thus used to define the sample for this analysis.1 The principals were asked in 1990 to indicate the number of years of mathematics students were required to complete in order to graduate. The distribution of responses showed that virtually all students (94 percent) were in schools that required either two or three years, and that the distribution is thus best treated as a simple dichotomy.2 The data collected from high school principals indicate that about 53 percent of the high school sophomores that year were enrolled in schools requiring two credits or fewer in math for graduation, and 47 percent were in schools requiring three or more credits to graduate. Since it follows a single cohort of students, the NELS:88 study does not allow one to examine how student outcomes changed with the advent of higher requirements. Beforeandafter comparisons at the level of schools or states (as illustrated by Wilson & Rossman, 1993) are thus not possible with these data. And since the data were not collected as part of a controlled experiment, it is important to see whether policy differences among schools are associated with social and economic characteristics of students, as well as with other school organizational differences that might confound the effort to assess the real effects of the policy difference at issue. It turns out that whether a student is enrolled in a school requiring three credits versus two credits is not strongly associated with conventional social or economic categories. High, middle, and lowSES students are all about equally likely to be enrolled in a school requiring three math credits for graduation. AfricanAmerican and Hispanic are somewhat more likely than nonHispanic white or Asian students to be in schools requiring three credits, but the differences are not extreme (55 percent of African Americans and 61 percent of Hispanics in schools requiring three years, versus 45 percent of whites and 41 percent of Asians). Two characteristics of the students high schools do, however, show strong relationships with math graduation requirements: whether the school is public, and the region of the country in which the school is located. Students in Catholic and other private schools are much more likely than their public school counterparts to have a threecredit requirement (74 percent versus about 45 percent). Similarly, students in schools in the Northeast and the South are much more likely than students in the Northcentral and the West to have three math credits required (68 percent in the Northeast and South versus only 16 percent in the Northcentral and 30 percent in the West). The high requirements faced by private, and particularly Catholic, school students are not surprising, in light of the studies documenting their high academic standards (see Bryk, Lee, & Holland, 1993). The large regional differences may be somewhat less transparent, though. The story here seems to be one of statelevel policymakers acting in concert, often through regional associations. The Northeast has a tradition of setting higher requirements for high school students than the rest of the country, and the prevalence of threemathcredit requirements there is consistent with that reputation. In the South, however, the higher requirements reflect a recent push to improve high school students achievement levels. The governors of several southern states formed a coalition in the mid1980s to seek solutions to a common problem of lower levels of academic achievement in their states compared with the rest of the nation. State elected officials and education bureaucracies in the Northcentral and the West have been less directive, ceding decisions on requirements to local authorities. GRADUATION REQUIREMENTS AND STUDENT COURSE WORK While the mathematics curriculum is currently undergoing major changes in some states and localities, the NELS:88 student transcript data (collected in fall of 1992) indicate that the overwhelming majority of recorded math courses were still the familiar categories of basic math, algebra 1, geometry, algebra 2, trigonometry, precalculus or analytic geometry, and calculus. Moreover, other analyses of these data suggest that most students follow the traditional sequencethe order just listedof those courses (Stevenson, Schiller, & Schneider, 1994). Students differ greatly in how far they progress in this standard sequence by the end of high school, and the differences depend both on the level at which the students start high school and the number of credits they complete. The different entry points are essentially ability tracks in most schools. The highesttrack students take algebra 1 in the eighth grade and start high school with geometry. The next track is still a collegepreparatory track, and starts high school with algebra 1. Lowerlevel students spend one or even two years in prealgebra math courses. If higher graduation requirements work as intended, students required to complete three math credits should be more likely to have completed courses that typically represent a third year of mathematics. The numbers shown in Table 1 certainly indicate that this holds for all courses above algebra 1: Students who attended a high school requiring three math credits for graduation are more likely to have completed geometry, algebra 2, and trigonometry. The numbers for basic mathematics and algebra 1 are subdivided into students who completed just one year and students who completed one and a half or two years of algebra 1. These two groups are distinguished in order to see if students subject to threecredit requirements tend to fulfill the extra requirement by simply taking more credits of basic math, or algebra 1 for two years instead of the usual one year. Many schools now allow students to do the latter, as evidenced by the 20 percent of seniors who completed more than one credit of algebra 1. However, there is no evidence here that students required to complete three math credits are more likely to earn the additional credits by taking the nominally easier courses. At the other extreme of the coursework hierarchy, precalculus and calculus, one does not expect to find an effect of requiring three versus two years of math. This is because these courses are typically the fourth year of the high school math curriculum, and are thus taken by students who have already completed the problematic third credit. Students enrolled in the schools requiring three years of math turn out to be slightly more likely to complete a calculus course than students in schools requiring two years (11 percent versus 8 percent, in Table 1), but this may reflect the effect of other variables associated with both taking calculus and being enrolled in a school that requires three math credits. As the findings for calculus suggest, the apparent advantages of attending a school that requires three credits may be due at least in part to other factors than the schools requirements. If one were to design an experiment to assess the effects of different graduation requirements on student course work and learning, schools would be randomly assigned to different graduation requirements and students would be randomly assigned to schools. In reality, neither assignment is random and graduation requirements may thus be correlated with other factors that influence course work and academic achievement. Multivariate analysis techniques are thus needed to make statistical adjustments that approximate the experimental controls achieved through random assignment of subjects to treatment and control groups.3 To determine whether the coursework differences in Table 1 might be explained by uncontrolled differences on such factors, I used logistic regression to estimate the effects of high school graduation requirements on the odds of completing at least onehalf credit in each of the courses listed in Table 1, controlling for measures of prior achievement, social background, and high school characteristics.4 Individuals who were eighth graders in 1988 but who later dropped out of high school are included in this and the rest of the analyses. As might be expected, dropouts finish their high school careers having completed much less math course work, and scoring at substantially lower levels on the achievement tests. Despite these large effects, the regressions do not include measures of whether the students dropped out. This is because higher graduation requirements may lead to a higher dropout rate: If dropping out were controlled for, then schools with higher graduation requirements might look better simply because they have driven out some portion of the students they were supposed to benefit. The coursework regression results are shown in Table 2. The numbers in the cells of this table are predicted odds ratios, calculated from logistic regressions. They reflect the change in the odds of taking the course in question associated with a unit difference on the independent variable, holding constant the other independent variables. A value of 1.0 represents no effect of the independent variable. A positive effect is signified by an odds ratio greater than one, and negative effects are reflected in odds ratios less than one. The numbers show significant positive effects of higher requirements on course work for geometry, algebra 2, and trigonometry. These results thus corroborate the simple breakdowns shown in Table 1, and support the first premise of the argument in favor of higher graduation requirements, that students will complete more high school mathematics courses. The coursework regressions include SESbygraduation requirement interaction terms, in order to assess whether the stratifying effect of SES on math coursetaking is lower in schools with higher graduation requirements. The interaction term is not statistically significant in any of the coursework regressions. This means that the gaps between high and lowSES students in the numbers of math courses and highest math course completed are not attenuated by higher requirements. The higher requirements thus appear to be affecting the coursetaking decisions of higherand lowerSES students about equally. GRADUATION REQUIREMENTS AND DROPPING OUT The coursework regressions included students who had dropped out of high school, but did not give any indication of how graduation requirements may affect the tendency of students to drop out. To address that issue, the next analysis uses the technique of logistic regression to estimate the effects of graduation requirements on dropping out, controlling for the same set of background variables used in the course work regressions. Overall, the NELS:88 data show that about 10 percent of the 1988 eighthgraders had dropped out and not returned to high school or a GED program by 1992. About onehalf of these students dropped out between 1990 and 1992. Since one cannot determine which high schools the students who dropped out between 1988 and 1990 would have attended, the analysis includes only those who left between 1990 and 1992. Without adjusting for the effects of student background differences and school characteristics, the dropout rates among students from high schools requiring two versus three years of mathematics are about equal. Results from a logistic regression of the dichotomous indicator of whether the student had dropped out of high school as of spring 1992 on the graduation requirement indicator, 1988 test scores, student background controls, and schoollevel controls are listed in Table 3. These estimates show that higher graduation requirements are not related to the likelihood of dropping out. GRADUATION REQUIREMENTS AND STUDENT ACHIEVEMENT GAINS The main goal of requiring students to complete more mathematics courses is to improve the students levels of proficiency in mathematics. NELS:88 assessed the sampled students proficiencies in broadranged tests of mathematics, using multiplechoice items tapping knowledge of basic arithmetic operations, probability, algebra, and geometry. The tests were administered in 1988, 1990, and 1992. The 1990 and 1992 tests were adaptive in that they varied in difficulty, so that students at different gross levels of proficiency (low, medium, and high) were given tests that contained more items appropriate to their proficiency level. By asking lowerperforming students more lowdifficulty items, and higherperforming students more higherdifficulty items, the tests could obtain more accurate measurement of the students abilities than would be possible with nonadaptive tests with the same testing time available. Despite the different test forms used, all tests contained a number of common items, which allowed NELS:88 to place all students on the same scale for all three time points. This was accomplished by means of the techniques based on Item Response Theory (IRT). To estimate the effects of graduation requirements on the students test scores, linear regression statistical techniques are used. The results of these regressions give estimates of how many test score points, if any, are attributable to requiring students to complete three as opposed to two math courses. Since a students final level of high school achievement is highly correlated with his or her initial level of achievement, social background, and other school characteristics, the regressions include controls for these factors. Specifically, control variables include eighthgrade achievement levels in math, reading, and science; gender; minority status; socioeconomic status; schoolaverage SES; and whether the school is public, Catholic, or other private. The first regression equation regressed the 1992 composite math scores on the requirement indicator and control variables (Table 4). Since the 1988 test scores are included as control variables in the achievement regressions, the effects of the other independent variables signify the impact of each variable on how much the students scores changed from 1988 to 1992. This is appropriate, because the main variable of interest, the graduation requirements indicator, would have whatever effects it may generate during this time span. The results are somewhat surprising, in light of the strong effects that graduation requirements have on the coursework variables. In contrast to these effects, Table 4 shows that students in schools requiring three years of math do not gain significantly more on the math achievement tests from 1988 to 1992. The achievement regression equation also includes an interaction term between student SES and the graduation requirement indicator. Coupled with the main effect of SES, the coefficient on this variable indicates whether there is any significant reduction in the effect of SES on achievement growth. This coefficient also proves to be statistically insignificant. In sum, requiring students to complete three years of mathematics appears neither to raise average math achievement levels, nor to reduce the impact of SES background on high school learning outcomes. WATEREDDOWN MATH COURSES? The positive effects of requiring three years of math on coursework completions (Table 2) coupled with the absence of any effect of the requirement on math achievement over the high school years (Table 4) present something of a paradox. To confirm that the coursework variables are indeed related to achievement outcomes as past research has claimed, I regressed the 1992 scores on the requirement indicator, control variables, and coursework measures. The coursework measures used here are a series of mutually exclusive 01 dichotomous variables indicating the highest level of mathematics course work attained by the student during high school. In order to count a student as having attained a level of course work in this analysis, that student must have earned at least 0.5 credits of the course in question. Levels are the same as those shown in Table 1 and are ordered, from highest to lowest, as follows: calculus, precalculus, trigonometry, algebra 2, geometry, algebra 1, and basic math. A student who reached precalculus but not calculus would thus have a value of one on the precalculus 01 indicator, and values of zero on the other indicators. The omitted reference category is geometry. Results in the first column of Table 5 strongly confirm the expectation that students math coursework completions are related to their math achievement scores. The coursework variables prove to have very strong effects on the students math achievement gains from 1988 to 1992. The ordering of estimated effects follows the assumed hierarchy, but the differences between levels are not all equal. On average, students who completed just geometry gained about eight points more on the NELS:88 math test from 1988 to 1992 than their otherwise comparable counterparts who did not complete algebra 1, and about ten points less than their counterparts who completed calculus. The analysis thus shows a paradoxical pattern of (a) graduation requirements affecting course work, and (b) course work affecting achievement, but (c) graduation requirements not affecting achievement. One way this could happen is that the math courses that students in the highrequirement schools take are less effective for promoting achievement scores. To assess this possibility, the next regression adds a variable to estimate whether the effects of the coursework variables on achievement gains depend on the schools graduation requirements. If there is a dilution effect of higher requirements, the regression should show that the effects of completing additional math courses are weaker among students required to complete three math courses. This dilution should be particularly evident in the areas of course work where the graduation requirements are most effective: geometry, algebra 2, and trigonometry. The results for the second model listed in Table 5 are summarized in Figure 1. The chart shows that the returns to additional math courses in geometry and algebra 2 are significantly lower in schools requiring three courses. Since geometry is the omitted reference category, the negative coefficient associated with a threemathcourse requirement means that students who stopped taking math after completing geometry gained significantly less in schools with the higher requirements. The results show that students in the schools requiring three courses gained about 2.4 points less than students in the other schools from completing geometry instead of just algebra 1. The loss in the returns to algebra 2 from the higher requirements is less, about 1.2 points. No significant differences between students subject to high and lower requirements are evident on at any of the other levels of course work. In sum, in the two areas where higher requirements are most successful in raising participation (geometry and algebra 2), the cognitive gains from the additional course work are sufficiently low that no net gains are evident. Only in trigonometry do higher requirements appear to raise participation with no associated reduction in the average learning gains from the course. POLICY EFFECTS ON STUDENTS MOST LIKELY TO TAKE TWO MATH COURSES The requirement that students complete three credits of mathematics is of course intended to affect only a subset of students: those who would otherwise complete only two math courses. From the research perspective, the ideal case would be to have those target students clearly identified in both schools that require three credits and those that require two. The achievement gains and graduation rates of the two groups could then be compared with each other, as well as with nontargeted students, and a stronger set of inferences drawn about the effects of the policy difference. One method of identifying student coursework preferences would be to ask them directly of the students. The drawback of doing so is that it would probably be very difficult to obtain accurate data from the students in the schools requiring three credits, because the requirements are likely to affect how students view things. One would have to ask them to answer in the abstract about a choice situation they have not experienced. An alternative method is suggested by Rosenbaum and Rubins (1983) propensity score analysis. For the case at hand, this involves using the data on students in the schools that require just two math credits to identify a set of variables that predict taking two versus three or more math courses during high school. If one then assumes that the same relationships between those variables and the students unobserved propensity to take two math courses holds in the schools that require three credits, then one can compare the outcomes of students in the two policy environments who are equally likely to take two courses. In practice, Rosenbaum and Rubin (1983) recommend using logistic regression to model the selection processin this case the odds of completing two versus three or more math courses. The logistic regression is estimated only for students in schools that require just two math courses. The regression results are then used to calculate predicted probabilities of taking just two courses for all students, both those in schools requiring only two as well as those in schools requiring three. Rosenbaum and Rubin recommend defining comparison groups by dividing the distribution of predicted probabilities into quintiles. The outcomes of students in the two policy settings are then compared within propensity quintiles. A summary of the results of this analysis are presented in Table 6. The cells of this table contain the estimated effects of a threemathcredit requirement versus a twocredit requirement on the math achievement gains. The effects are estimated by regressions of the 1992 math achievement variable on the students 1988 math score, educational expectations in 1990, SES, minority status, gender, and type of school attended (public, Catholic, or other private). The results show that the threecredit requirement is not effective in raising achievement levels of students who are the least likely to take three or more math credits on their own (quintiles 4 and 5). Nor do higher requirements lead to higher achievement scores among students who were more likely to take three math courses anyway (quintiles 13). These findings thus confirm the generally negative results found for the sample as a whole. The propensity score analysis indicates that the effects of the policy do not depend on the background of the students subject to the policy. CONCLUSIONS The results presented here generally do not support the notion that requiring high school students to complete more mathematics courses in order to graduate is sufficient to raise achievement scores. Average achievement scores were not higher among students who attended the schools requiring three years than among those whose schools required only two years. This suggests that the additional courses that students in schools requiring three years of mathematics take are not sufficiently demanding to improve achievement. Perhaps reflecting this, the analysis also showed that students who attended the schools requiring three years were not more likely to drop out of high school. No support was found for the hypothesis that higher graduation requirements would attenuate the effects of student SES on achievement growth, for the effects of SES did not differ significantly for students in the two types of schools. The analysis also indicates that the effects of additional math course completions on achievement growth are lower in schools with higher graduation requirements. The results thus give little credence to recent efforts to require students to complete more mathematics courses in order to graduate from high school. While national survey data are at best relatively crude measures of instructional dynamics, there is some indication here that higher requirements are associated with less effective courses. The higher rates of mathematics coursework completions produced by the higher requirements thus do not translate into higher levels of math achievement. This suggests that more specific standards will be needed in order to improve learning outcomes among high school students. Current proposals in that direction include the National Council of Teachers of Mathematics (NCTM) standards for curriculum content, instruction, and teacher professional training and development in secondaryschool mathematics (NCTM, 1989, 1991); and efforts by some states to establish outcome standards (Elmore, Abelmann, & Fuhrman, 1996). The consequences of graduation policies may also vary significantly across schools, so that the overall, summary effects actually include a mix of positive and negative outcomes. Wilson and Rossmans (1993) comparative case studies show that the effects of state requirements can have quite different effects from one school to another, depending on how the policies are implemented. As the introductory discussion implied, certain conditions are likely to affect the degree to which the policy works as intended. One is that, at least in the short term, some schools may not have adequate staff resources to provide solid mathematics courses to the expanded number of students taking three years of mathematics. Another is that schools may be in a relatively weak power position and thus unable to make additional demands on some of their lowerachieving students. This may be the case for noncollegebound students, who may rightly believe that academic achievement will have little consequence for their vocational interests. It may also be true for collegebound students who do not have family backgrounds that are strongly supportive of academic achievement. All of these possible explanations can be pursued further with the data from the NELS:88 survey. New studies are also needed, however, for it is quite possible that high school and presecondary mathematics education have undergone important changes since the beginning of the decade. 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