Chapter 3: Findings

As noted earlier, this report addresses three principal research questions. The first of these was:

How much does mathematics achievement change during the last 2 years of high school and are these changes related to student background and school characteristics?

Table 1 shows the IRT-estimated number-right scores in 10th and 12th grade and changes in those scores by student background and school characteristics, and the effect sizes associated with the changes. Table 2 shows the proficiency probability in the 10th and 12th grade, changes in those scores by student background and school characteristics, and the effect sizes associated with the changes. Standard deviations corresponding to the effect sizes are shown in table B-1b and in table B-2b.

On average, students improved their performance on the mathematics assessment by about five correctly answered questions—about a third of a standard deviation (e.g., an effect size of 0.33). Additionally, all subgroups yield gains (table 1). Students in the Catholic sector made the largest gain: the effect size associated with their learning gain is a little more than half a standard deviation (0.54). The proficiency scores augment this information by detailing the levels of skills that were learned. The averages at the top of table 2 indicate that the smallest gains were made at the lowest and highest levels while the largest gains took place at levels 3 and 4. In 10th grade, 53 percent of students were proficient at level 3 and 25 percent were proficient at level 4. By the end of 12th grade, 65 percent and 38 percent were proficient at level 3 and level 4, respectively. At the highest level, gains across the second half of high school were smaller than those made in levels 2, 3, and 4. By the end of senior year, only about 4 percent of seniors had mastered the skills to be considered proficient at level 5.¹²

Before entering the last 2 years of high school, there are no detectable differences between boys and girls in either their number-right scores or in their level of mastery at all five proficiency levels. As measured by both gains in the number-right scores and increases in proficiency levels, on average, there are no detectable differences between girls and boys in their rate of learning across the final 2 years of high school. Additionally, upon leaving high school, there are no detectable differences between boys and girls in their number-right score nor in the level of mastery at all five levels of proficiency. These findings corroborate other contemporary research that finds increasing parity in mathematics achievement (for information on trends, see Bae et al. 2000 and Cahalan et al. 2006; for a review of the scientific evidence of gender differences in mathematics learning, see Halpern et al. 2007.

With respect to race/ethnicity, differences noted in other national studies of achievement—for example, Hoffer, Rasinski, and Moore's (1995) analysis of the National Education Longitudinal Study of 1988 (NELS:88)—were detected here as well. Compared with their Black and Hispanic counterparts, Asian and White students had higher number-right scores at both time points. However, there were no differences detected among racial/ethnic groups in their number-right score gains across the second half of high school. In addition to overall aggregate differences, there are disparities in the content of their learning. Asian and White students entered the second of half of high school having mastered basic mathematics skills and concepts—for example, 83 percent of Asians and 82 percent of Whites were proficient in level 2 at the end of sophomore year (table 2). Black and Hispanic students, on the other hand, entered the second half of high school with a less solid foundation in mathematics: 46 percent of Black students and 54 percent of Hispanic students were proficient at level 2 at the end of their sophomore year. Given this disadvantage, Black and Hispanic students may be less likely than their Asian and White peers to acquire the most advanced mathematics skills before graduation. Indeed, the evidence from ELS:2002 suggests this is the case: when leaving high school, 52 percent of Asian students and 45 percent of White students are proficient at level 4, compared to 20 percent of Hispanic students and 12 percent of Black students.

Patterns across socioeconomic status (SES) follow suit. For the most part, students in the highest SES quartile make greater gains than students in the lowest SES quartile. Moreover, compared with students in the lowest SES quartile, high SES students are making gains in more advanced subject matter. For example, those in the highest SES quartile improved their proficiency at level 4 by 17 percentage points while those in the lowest SES quartile improved their proficiency at level 4 by 7 percentage points (table 2). High SES students are ahead in their learning of mathematics, with higher levels of achievement and growth in their understanding of advanced topics, while less affluent students are less prepared. The gains low SES students do make are concentrated in intermediate-level skills.

Much like the volume of literature comparing private and public schools (Coleman and Hoffer 1987; Morgan and Sorenson 1999), evidence from ELS:2002 shows that students in Catholic schools fare better in mathematics than their public school counterparts. On average, Catholic school students outgained public school students on the mathematics assessment (6 versus 4) and left high school with higher overall scores (58 versus 51) (table 1). There were no detected differences in the gains of students attending other private schools and public school students (5 versus 4). However, other private school students left high school with higher overall scores than their public school peers (59 versus 51). With the exception of level 4, there were no differences detected in the gains across different levels of proficiency between public school students and Catholic school students during the last 2 years of high school. At level 4, Catholic school students were about 10 percentage points ahead of their public school counterparts (table 2). There were no differences detected in the gains at the three lowest levels of proficiency between public school students and other private school students. However, other private school students outpaced their public school peers at level 4 and level 5.

Compared with public school students, other private school students left high school with greater proficiency at all five levels and Catholic school students left high school with greater proficiency at levels 1–4. There were no differences at the end of high school detected between Catholic school students and other private school students at levels 1–4. Other private school students, however, were more proficient than were Catholic school students at level 5 (11 percent versus 6 percent) (table 2).

Like school sector, family composition is also associated with learning in mathematics. While there were no differences detected in the gains of students who live with both parents in the family, single parent families, and stepparent families, students who live with both parents in the family left high school with higher number-right scores and greater proficiency at levels 2–4. For example, 43 percent of students who live with both parents in the family were proficient at level 4 compared with 31 percent of students living in stepfamilies (i.e., mother or father and guardian) and 28 percent of students living with single parents (table 2). Compared with students who live with both parents in the family, students in other family forms left high school with lower number-right scores and lower proficiencies at all five levels.

Lastly, the educational expectations of students were linked with mathematics learning. Those who expected a college degree performed better on the mathematics assessments than their peers who expected to complete high school or less—for example, those expecting a college degree answered an average of 53.6 questions correctly on the 12th-grade assessment compared with an average of 37.3 correct answers for those who expected high school or less (table 1). During the final 2 years of high school, those who expected to attend college outpaced their peers at the highest levels (level 4 and level 5), and finished high school with a greater understanding of mathematics concepts at all levels than their peers who expected a high school degree or less.

In the aggregate, these descriptive findings are neither new nor novel—educational researchers have long documented differences in mathematics achievement along key dimensions of student background. What this analysis does highlight, however, is that by mastering basic mathematics skills by their sophomore year, Asians, Whites, High-SES students, students who live with both parents in the family, students attending private schools, and students expecting a college degree are in a better position to acquire more intermediate and advanced skills than their peers. As the learning of mathematics skills and concepts—both basic and advanced—rests on the content and instruction received in different courses, this analysis now turns its focus to the curricular pathways that students follow.

What are the most common mathematics course sequences taken by students in the 11th and 12th grades and are these sequences related to student background and school characteristics?

As mentioned in the introduction, this study improves upon previous research on learning and coursetaking by identifying the actual course sequences students take between test administrations, rather then simply counting the number of credits earned or using broad curricular classification such as a track placement or a level (e.g., "highest mathematics"). The analysis in this report was accomplished using data from the ELS:2002 High School Transcript Study. First, all mathematics courses for which the student earned credit during 2002–03 and 2003–04,¹³ the 2 academic years between the 2 mathematics assessments, were classified in one of the following 16 hierarchical categories:

1. No Mathematics;
2. Basic Mathematics;
3. General Mathematics;
4. Applied Mathematics;
5. Prealgebra;
6. Algebra I;
7. Geometry;
8. Algebra II;
9. Trigonometry;
10. Other Advanced Mathematics;
11. Precalculus;
12. Statistics;
13. Advanced Placement/International Baccalaureate (AP/IB) Mathematics;
14. Calculus;
15. Advanced Placement/International Baccalaureate (AP/IB) Calculus; or
16. Other Mathematics.¹⁴

The course titles that comprise these categories are listed in appendix A. As the typical school curriculum permits one mathematics course each academic year, this study operationalizes course sequences in terms of a two-course sequence: mathematics course (if any) for which credit was earned in 2002–03 and mathematics course (if any) for which credit was earned in 2003–04.¹⁵ Of the 256 possible combinations of two course sequence courses based on the classifications above, 180 sequences were followed by ELS:2004 sample members. Only six sequences were followed by more than 5 percent of students.

This process of constructing mathematics sequences reveals that the secondary mathematics curriculum in the United States is more diverse than once thought. For instance, analyses of NELS:88 using broader categories to classify courses found that 75 percent of students fit into one of five predefined coursetaking patterns (Burkam and Lee 2003). Aggregating course titles into broad patterns obscures some of the heterogeneity of mathematics coursetaking. Despite the relatively sequential and hierarchical nature of the subject matter, students enroll in a range of mathematics courses in their final years of high school.

The large number of course sequences precludes a succinct analysis of the learning gains for the entire analytic sample. To facilitate interpretation and to produce efficient estimates of learning, course sequences followed by more than 200 students (approximately 3 percent of the unweighted sample) form the basis of this study.¹⁶ Nine course sequences meet this criterion and are listed in table 3. In preliminary analyses (not shown), students who had followed a geometry–no mathematics sequence had similar mathematics gains as those in geometry–geometry.¹⁷ Given that these students were only exposed to geometry during the interval and that geometry is sometimes taught over the course of 2 years, they were combined into one group. This course sequence is herein referred to as geometry–geometry/no mathematics. Similarly, as trigonometry is often embedded in the content of algebra II and preliminary analyses (not shown) find no differences in their learning gains, students who had followed an algebra II–trigonometry sequence were combined with students who followed an algebra II–algebra II sequence.¹⁸ This course sequence is herein referred to as algebra II–algebra II/trigonometry. Since these two course sequences were constructed by combining courses, they should be interpreted with caution.

These nine course sequences account for a little more than half of the sample (55 percent).¹⁹ The most common course sequence undertaken in the second half of high school is algebra II during junior year followed by no mathematics course during the senior year. This course pattern was undertaken by 13 percent of students. The most common course sequence involving two separate courses was algebra II during junior year followed by precalculus taken during senior year. This course pattern was undertaken by 7 percent of the students. Despite its centrality to preparation for postsecondary studies, 6 percent of students do not take any mathematics courses during the final 2 years of high school. Readers should note that with the exception of no mathematics, none of these sequences contain courses lower than geometry, and thus this study does not address the learning gains made by students who take general or basic mathematics courses. About 45 percent of students took course sequences taken by fewer than 200 students. The most common courses taken by students following these other course sequences were algebra II, no mathematics, and other advanced mathematics. The most common course sequences taken by students following these other course sequences were algebra I–no mathematics, other advanced mathematics–no mathematics, and algebra II–other advanced mathematics. Given the heterogeneity of courses in the "all other patterns" sequence, the learning gains of these students are not discussed in this report.²⁰

As discussed earlier, numerous studies have found that the types of courses students take are related to both their background characteristics and the types of schools they attend. To assess whether these patterns are present when using a new classification of curricular experiences and this cohort of high school students, table 4 shows the percentage of students taking mathematics course sequences by student and school characteristics.

Much like the analysis of mathematics assessment scores, sequences of mathematics courses taken during junior and senior year of high school differed across student characteristics. With respect to sex, there were no detectable differences between boys and girls in taking course sequences containing precalculus during the last 2 years of high school: 18 percent of boys and 19 percent of girls did so. With respect to race/ethnicity, 25 and 20 percent of Asian and White students, respectively, followed course sequences that contained precalculus, compared to 15 percent of their Hispanic peers and 12 percent of their Black peers (table 4). The differences are greater when examining socioeconomic status. Almost 30 percent of students in the highest SES quartile followed pathways that included precalculus, while 11 percent of those in the lowest SES did so. This descriptive evidence shows that Blacks, Hispanics, and less affluent students were reaching advanced mathematics courses less often than their Asian, White, and more affluent peers.

Turning next to school type, 28 percent of Catholic school students have taken a sequence that included precalculus, compared to 18 percent of their public school peers. Similar patterns were found for other private school students, who were more likely than public school students to have taken a sequence containing precalculus (30 percent versus 18 percent). At the most advanced course sequence in the present analysis, precalculus-AP/IB, there were no detectable differences among the three school sectors: 6 percent of public school students and 9 percent of Catholic school students and other private school students followed this sequence during the second half of high school.

In terms of family structure, there were no detectable differences between students living with both parents and students living in a stepfamily or with a single parent in taking course sequences containing precalculus during the last 2 years of high school: 21 percent of students living with both parents, 15 percent of students living in a stepfamily, and 14 percent of students living with a single parent. However, more students living with both parents (21 percent) have taken course sequences containing precalculus during the last 2 years of high school than their peers living in other family forms (10 percent).

Lastly, students' expectations for their future educational attainment were linked with differential curricular pathways: those who hold higher educational plans tend to take more advanced courses while those who set lower educational goals tend to take fewer advanced courses. For example, 7 percent of those who expected a college degree when enrolled in 10th grade later followed a precalculus–AP/IB calculus sequence while less than 1 percent of those who expected to attain some college and those who expected a high school degree or less did so. Moreover, 4 percent of students who expected a bachelor's degree took no mathematics courses during the latter part of high school, compared with 14 percent of students who expected a high school diploma or less.

What mathematics course sequences are associated most closely with mathematics achievement?

To answer this research question, both bivariate and multivariate techniques are used. First, learning gains in mathematics are examined descriptively for students taking different course sequences. Next, regression techniques are used to measure the association of mathematics courses with mathematics achievement at the end of high school, apart from key factors related with these curricular pathways. Together, these analyses identify course sequences most often associated with a student's mastery of advanced mathematics skills and concepts.

Table 5 shows average scores and gains for the number-right scores. Effect sizes for these gains are shown in parentheses; corresponding standard deviations are shown in table B-5b. In terms of number-right scores, students improved the most during their junior and senior years when they reached at least algebra II and took another advanced course, while students who took less than two mathematics courses or were still taking the geometry series improved the least. The gains for students in the most advanced course sequences—precalculus–calculus and precalculus–AP/IB calculus—are particularly large, with effect sizes nearing almost one standard deviation.

To highlight the relationships between coursework and learning, consider algebra II, the modal course taken by high school juniors.²¹ On average, students who took algebra II during their junior year followed by precalculus in their senior year improved by an average of 6.9 correct answers (table 5). On the other hand, those who took algebra II in their junior year but did not take a mathematics course during their senior year improved by an average of 4 correct answers. Their overall number-right scores at the end of high school were about 9 points apart (57.6 and 48.7, respectively).

By and large, these results complement past research that finds that more advanced coursetaking is associated with greater gains (Leow et al. 2004; Rock, Owings, and Lee 1994; Rock and Pollack 1995a; Scott et al. 1995; Wang and Goldschmidt 2003). However, are students who follow different mathematics course sequences learning different mathematics concepts and skills? The proficiency probability scores in ELS:2004 allow for such an assessment. Table 6 shows average scores, gains, and effect sizes for the gains for the proficiency probability scores for each of the nine major mathematics sequences. Standard deviations associated with the effect sizes are shown in table B-6a. The estimates in table 6 show that while advanced courses are associated with learning in the aggregate (as evidenced when using number-right scores), curricular pathways differentially predict the acquisition of mathematics skills and concepts.

At level 1, the basic level, improvements are highest for those students who took the geometry-geometry/no mathematics sequence. This is not surprising as this level measures the most basic mathematical concepts such as arithmetic and whole numbers. As most of the other major pathways shown are beyond basic mathematics and algebra I, which require a solid foundation in arithmetic, student mastery is near the ceiling for students in the other major course sequences—leaving little room for growth.

At the intermediate levels—level 2 and level 3—improvements are the highest for those who take algebra II paired with geometry or trigonometry. For example, the percentage of students who took geometry paired with algebra II who were proficient at level 2 improved by 14 percentage points. The percentage of these students who were proficient at level 3 improved by 23 percentage points (table 6). Learning gains at level 2 and level 3 are smaller for those in the most advanced course sequences as these students have already mastered these skills—for example, before starting their junior year, 99 percent of students taking the precalculus–calculus sequence were already proficient at level 2 and 93 percent were already proficient at level 3.

For the most part, the largest gains at the advanced levels are made by students who take precalculus paired with another course. For example, improvements at level 4 are highest for those who follow the algebra II-precalculus sequence. These students improved their proficiency at this level by 28 percentage points. At level 5, improvements are highest for those who follow the precalculus–AP/IB calculus sequence. Students following this sequence improved their proficiency by 22 percentage points. As these courses expose students to the most challenging skills and concepts in the high school mathematics curriculum, it is not surprising that the students taking them learn the most. Despite their rigor, however, there is still substantial room for learning: the majority of students (71 percent) taking precalculus–AP/IB calculus, arguably the most advanced sequence, are not proficient at level 5.²²

Table 7 summarizes the key relationships in table 6 by showing the two course sequences associated with the largest gain in learning at each proficiency level. Taken together, these descriptive findings suggest that gains in mathematics achievement and an understanding of more advanced topics are best achieved by juniors and seniors who take precalculus along with another course. However, it is possible that these observed improvements in learning are a reflection of the types of students who follow these course sequences. In other words, students who enroll in a precalculus–calculus sequence may be faring well in mathematics because they are affluent, ambitious, and attend private schools, not because of the courses they are taking. To separate out the influence of course sequences from background characteristics, a series of regression analyses are performed. The advantage of regression analysis is that it can show the relationship between a dependent variable and any individual independent variable, while holding the other independent variables constant. Despite this advantage, the regression analysis reported here cannot establish a causal relationship between coursetaking and mathematics achievement. Readers should keep this in mind when interpreting the results reported herein.

The availability of two mathematics assessments, one before students enter their junior year and one at the end of senior year, permits a more stringent assessment of learning gains. This study estimates a set of regression models where the test score from the 12th grade is used as the dependent variable and the test score from the 10th grade is used as a control variable. Using the 10th-grade test score as a control variable conditions the effects of the rest of the predictor variables on students' initial level of mathematics proficiency (i.e., the estimated relationship for a particular covariate is conditional upon the other variables in the model). This is often referred to as the covariate adjustment or regressor variable approach to analyzing change.²³

Two sets of models are estimated. First, the relationships between the course sequences and the 12th-grade number-right score are estimated. Second, the relationships between the course sequences and the five proficiency levels from the 12th grade are estimated. Both outcomes are continuous and thus, ordinary least squares (OLS) estimation is used. When a lagged version of the dependent variable (i.e., the 10th-grade score) is included in the model, as is the case in this study, the coefficients associated with the other predictor variables represent the amount of change in the dependent variable associated with a unit change in each independent variable in the model (Finkel 1995). With the level of achievement in 10th grade already controlled, any differences detected by the rest of the coefficients reflect achievement beyond what is already measured at the end of 10th grade—hence, the change interpretation. The first set of models is shown in table 8.²⁴

Model 1 includes only the 10th-grade number-right score and the student background characteristics, which are used as control variables in addition to the 10th-grade mathematics scores. All of the characteristics are categorical and are entered as a series of indicator (0, 1) variables. Reference categories include: White students (race/ethnicity), female students (sex), SES quartile 1 (low) (socioeconomic status), public school students (school sector), students who live with both parents in the family (family composition), and expecting a high school diploma or less (educational expectations).

As indicated by the adjusted R-squared, 81 percent of the variation in 12th-grade number-right scores is explained by sociodemographic characteristics and 10th-grade number-right scores—revealing that a substantial amount of the variation in learning at the end of high school is explained by factors that precede enrollment in mathematics courses during the last 2 years of high school. Most relationships shown in Model 1 are in accord with other multivariate analyses of learning gains using NCES data sets (e.g., Carbonaro 2005; Morgan and Sorenson 1999). When controlling for other background factors, most racial/ethnic differences disappear, with the exception of differences between Black and White students' test scores: all else equal, Black students' correct-answer gains were 0.8 less than their White counterparts (Model 1, table 8). Socioeconomic disparities persist: students in SES quartile 3 and quartile 4 had significantly higher gains in mathematics learning than their peers in the lowest quartile. Though there were no bivariate differences detected in the learning rates of boys and girls, once student and school background characteristics were controlled for, the correct-answer gains made by boys were significantly less than the gains made by girls. School sector effects are still evident, even with other student background characteristics controlled: Catholic school students gained 1.4 correct answers and other private school students gained 0.9 correct answers more than their public school peers. Lastly, students from stepfamilies and other family forms had smaller gains than did students who live with both parents in the family, and students who expected to earn a bachelor's degree had significantly higher gains than those who expected a high school degree or less.

Mathematics course sequences are added to student background characteristics in Model 2 on table 8. These are added as a set of indicator (0,1) variables indicating the different course sequences. The reference category is "algebra II–no mathematics" so that all comparisons could be made with respect to the modal pattern taken by contemporary high school students. Also included is a control for students who had taken more than three courses during the 2002–03 and 2003–04 school years.

Before looking at the effects of the individual coursetaking sequences, note that at a very general level, curricular pathways are important predictors of mathematics learning at the end of high school. F-tests are used to compare the fit of different models. An F-test comparing the fit of Model 1 (10th-grade scores + background characteristics) with the fit of Model 2 (10th-grade scores + background characteristics + course sequences) yields a significant test statistic (F = 64.61, df = 10, p < .01), indicating that course sequences significantly improve the fit of the model. Substantively, however, the explanatory power of course sequences is limited, explaining only 2 percent of the variation beyond what is accounted for by previous learning and background factors. Course sequences do matter, as evidenced by the significant F statistic, but achievement at the end of high school is largely predicted by factors that precede these years.

What course sequences are associated with the largest learning gains in mathematics at the end of high school? In accord with the descriptive analysis shown in table 5, with previous learning and background characteristics controlled, students who take more advanced course sequences gain more than do students taking lower-level course sequences. The relationship is mostly linear as the magnitude and direction of coefficients change from – to + when moving from sequences containing less advanced math courses to sequences containing more advanced math courses.²⁵

To provide a direct example, consider algebra II—the modal course taken by high school juniors. Some students will continue on to precalculus during their senior year while others will refrain from any mathematics course during their senior year. Though these different pathways are associated with students' achievement and sociodemographic characteristics, the relationship between an additional mathematics course and learning gains is better assessed using multiple regression as these possibly confounding factors are controlled. The estimates on table 8 show that all else equal, students who take precalculus following algebra II gain 2.1 correct answers on the mathematics assessment more than their peers who follow an algebra II–no mathematics sequence (Model 2, table 8). Students who are further along in the mathematics sequence have even higher gains. For example, those who take precalculus in their junior year and take either calculus or AP/IB calculus in their senior year gain about 4.2 correct answers more than their peers who follow an algebra II–no mathematics sequence.

Though the use of number-right scores provides a portrait of mathematics gains in the aggregate, the relationship between different course sequences and particular skills and concepts are unknown. The use of proficiency probability scores as dependent variables permits such an analysis. Table 9 shows five multiple regression models—one predicting each of the proficiency levels. As was the case for Models 1 and 2 on table 8, all models control for proficiency measured in the 10th grade as well as background characteristics.²⁶

Similar to the descriptive analysis, mathematics course sequences are generally unrelated to gains in proficiency at level 1. For example, with the exception of the coefficient for no mathematics–no mathematics, none of the coefficients associated with the major course sequences are significantly different from algebra II–no mathematics, the modal course sequence. This is likely because most students enter their junior year with a mastery of basic arithmetic and whole numbers; thus, there is little if any room for improvement. At the intermediate levels—level 2 and level 3—there is evidence that when compared to those who followed an algebra II–no mathematics sequence, the largest gains were made by those who followed an algebra II–precalculus sequence. They outgained their algebra II–no mathematics peers by 4 and 7 percentage points at levels 2 and 3, respectively (table 9).

In levels 1, 2, and 3, the most advanced course sequences—precalculus–calculus and precalculus–AP/IB calculus—have negative coefficients. While at first this may seem counterintuitive, recall that students in these sequences enter the second half of high school with almost complete mastery of these topics (see table 6). Therefore, they make fewer gains than their peers who are in the algebra II–no mathematics sequence.

At level 4, the largest gains relative to algebra II–no mathematics were made by those who followed the algebra II-algebra II/trigonometry sequence and those who followed one of the precalculus sequences. At the most advanced level—level 5—the largest gains were made by students who followed one of the precalculus sequences. For example, at level 5 students who followed the precalculus–calculus sequence outgained those who were in the algebra II–no mathematics sequence by 11 percentage points and students who followed the precalculus–AP/IB calculus sequence outgained those who were in the algebra II–no mathematics sequence by 20 percentage points. This complements the bivariate findings presented earlier: mastery of the most advanced skills is associated with credits earned in courses beyond algebra II, such as trigonometry, precalculus, and calculus.

Finally, in 4 of the 6 models in which course sequences were used to predict mathematics achievement outcomes in 12th grade (Model 2 in table 8 and models for levels 2, 3, and 4 in table 9), students who followed an algebra II–precalculus sequence had higher gains than their peers who followed an algebra II–no mathematics sequence. Moreover, none of the models showed any measurable differences between students who followed a geometry–algebra II sequence and those who followed an algebra II–no mathematics sequence. In essence, students who took algebra II in their junior year and did not take any mathematics courses in their senior year learned no more or less than their peers who did not reach algebra II until their senior year. On the other hand, students who reached algebra II in their junior year and then continued on to precalculus in their senior year made the most sizeable gains.

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¹² At the time of the F1 interview, 720 sample members who had participated in the BY interview were dropouts. These sample members are excluded from the analysis because they lack a F1 test score and thus cannot contribute to an analysis of learning gains. In accord with research in the dropout literature, these dropouts fared worse on the base-year mathematics assessment than their peers who were enrolled at the time of both the BY and F1 interviews. In the spring of 2002, these dropouts had a number-right score of 32.6. Their proficiency probability scores for the five levels were 80.7 (level 1), 37.9 (level 2), 17.1 (level 3), 4.4 (level 4), and 0.1 (level 5).
¹³ Earning course credit is defined by receiving a letter grade higher than an F or a "pass."
¹⁴ "Other mathematics" is a residual category and is not considered more or less rigorous than the other 15 course categories.
¹⁵ Given the structure of most school calendars and curricula, the average student enrolls in two courses during his or her junior and senior years. However, due to summer school, dual enrollment (e.g., business mathematics and geometry in the same term), or semester-long courses students could possibly take more than two courses in this time period. In the analytic sample, 12.7 percent (n = 1,200) had taken three or more mathematics courses during the 2002–03 and 2003–04 academic years. To classify these students in a way that was consistent with the majority of students who had a two-course sequence, these students were classified based sequentially on their two highest courses. For example, if the student had taken geometry, applied mathematics, and algebra II, his or her course sequence was classified as "geometry–algebra II." The effect of this classification rule on the estimates of learning should be negligible for two reasons. First, 62 percent of these students (n = 750) have sequences that fall under "all other patterns"—the residual group excluded from key comparisons; and second, all multiple regression models control for students who took more than two courses.
¹⁶ Selecting the most common course sequences yields the most pertinent information as these courses were experienced by a majority of students. These course sequences account for 55 percent of the sequences taken by students in ELS:2002. Including course sequences followed by fewer students would potentially threaten the efficiency of the estimates, particularly in a multiple regression analysis.
¹⁷ Differences in number-right score gains and differences in the proficiency probability score gains between students who followed a geometry–no mathematics sequence and students who followed a geometry–geometry sequence did not meet the .05 level for statistical significance required for this study.
¹⁸ Differences in number-right score gains and differences in the proficiency probability score gains between students who followed an algebra II–trigonometry sequence and students who followed an algebra II–algebra II sequence did not meet the .05 level for statistical significance required for this study.
¹⁹ There are no detectable differences in the sociodemographic composition of students who follow these nine course sequences (n = 5,300) and the sociodemographic composition of the analytic sample (n = 9,460). This is shown in section A.6 (Bias Analysis) of appendix A.
²⁰ As discussed earlier, dropouts are not included in the analysis due to a lack of an F1 mathematics test score and incomplete coursetaking records. Of the 720 dropouts with a BY test score, 130 have coursetaking information for the 2002–03 and 2003–04 school year. The most common coursetaking sequences for these dropouts include: no mathematics–no mathematics, no mathematics–basic mathematics, and algebra II–no mathematics.
²¹ Algebra II is the modal course taken in the 11th grade. In the analytic sample, 35 percent of students earned credit in algebra II in the 2003–04 school year.
²² Though the scaling of the test was designed to prevent a "ceiling effect" at this level, that no course sequence yields proficiency over 50 percent suggests that even the most highly prepared students leaving high school may need more training in mathematics to master advanced skills such as multistep problem solving and derivations.
²³ There are two general approaches to modeling change in a regression framework when the dependent variable is measured at two points in time: the change score approach and the covariate adjustment approach. In the change score approach, the dependent variable would be the difference between the F1 test score and the BY test score. This difference score would be regressed on the measures of course sequences and background characteristics. In the covariate adjustment approach, the F1 test score is used as the dependent variable and the BY test score is used as a predictor variable alongside the measures of course sequences and background variables. When the "treatment" occurs between the pretest and the posttest, and assignment to the treatment group is affected by the pretest, the covariate adjustment approach is preferable to the change score approach (Allison 1990; Maris 1998). In this study, course sequences are considered to be the "treatment" and they occur between the BY and F1 test administration. As evidenced in the descriptive statistics in tables 5 and 6, course sequences that students take at the end of high school are associated with their initial scores on the pretest. Additionally, the change score approach constrains the coefficient associated with the baseline score to 1, which is too restrictive. For these reasons, this study uses the covariate adjustment approach rather than the change score approach. In the absence of an experimental design, the covariate adjustment method greatly reduces the threat of endogeneity (i.e., that the omission of other unmeasured factors influencing both course sequences and 12th-grade test scores will bias the estimates) (Allison 1990; Maris 1998).
²⁴ None of the variables in any of the models presented in tables 8 and 9 yielded a variance inflation factor greater than 5.
²⁵ Establishing a hierarchy of less advanced to more advanced course sequences is not straightforward given that two separate courses comprise a sequence. Thus, some course sequences are not necessarily more or less advanced than one another (e.g., algebra II–precalculus and precalculus–no mathematics). However, it is largely the case that course sequences containing more (less) advanced courses correspond to larger (smaller) gains in contrast to algebra II–no mathematics.
²⁶ Since background characteristics are not the main focus of this report, the models predicting the proficiency levels including only background characteristics are not shown. Similar to the model comparisons shown in table 8, the addition of the coursetaking terms explain little of the variation in levels 1–4 beyond what is accounted for by previous learning and background factors. When the coursetaking terms are added, the adjusted R-squared remains the same (.49) for level 1, remains the same (.69) for level 2, improves from .62 to .64 for level 3, and improves from .69 to .72 for level 4. The largest growth in adjusted R-squared occurs for level 5, where the inclusion of the coursetaking terms improves 9 percent—from .38 to .47.