Chapter 4: Conclusion

This study finds that at the end of their high school years, American students increase their knowledge in mathematics—mostly in intermediate-level skills and concepts. The majority of students enter their junior year with a solid proficiency in lower-level skills such as whole numbers, fractions, and decimals, and improve their understanding of intermediate skills and concepts such as algebraic relationships and logic. Despite these overall gains, most do not graduate with a solid foundation in the most advanced skills, such as derivations and multistep problems. At this level, only 4 percent are proficient, up 3 percentage points from the end of sophomore year.

This study, using data from the Education Longitudinal Study of 2002 (ELS:2002), yields mixed information on the relationships between coursetaking and learning. On the one hand, students who reach the most advanced courses such as precalculus and calculus before leaving high school are more likely to learn the most advanced skills and concepts. On the other hand, the lion's share of learning that takes place at the end of high school is predicted by background characteristics and curricular experiences that precede the start of junior year.²⁷ Each of these is discussed in turn.

Much like the bulk of previous research, this study finds that students who take advanced courses such as precalculus and algebra II learn more than their peers who take intermediate courses such as geometry and those who do not take mathematics courses. However, this study extends previous research by using scaled scores that permit an assessment of the specific skills and concepts being learned. While there is evidence of learning gains in mathematics during the last 2 years of high school across the board, students who follow a geometry–algebra II sequence or an algebra II–trigonometry sequence show the greatest improvement in intermediate skills, such as operations with whole numbers and basic algebraic expressions, while students who follow an algebra II–precalculus sequence or a precalculus–calculus sequence show the greatest improvements in advanced skills such as multistep analytical problems. These effects were robust when included in regression models that controlled for prior learning and key background characteristics.

While mathematics course sequences are associated with learning, most of 12th-grade achievement is explained by background factors and previous learning; mathematics courses at the end of high school explain little of the variation in achievement once background factors and previous achievement are included as controls. This is likely because these background factors and previous learning experiences are linked with their coursetaking patterns. Indeed, differences in coursetaking follow a pattern that is well documented by social scientists and replicated here in the ELS:2004 data. Students with more socioeconomic and educational resources—students from affluent families, students in two-parent homes, students who attend private schools, and students who have ambitious educational expectations—are more likely to reach the most advanced mathematics courses. Their experiences and instruction at earlier stages in schooling likely gives them the foundational skills to move through the hierarchy at a pace that ensures their enrollment in courses like precalculus and calculus.

Despite the strengths of this study—for example, the longitudinal design, course information from administrative records, and assessments scaled to different proficiency levels—ELS:2002, like many National Center for Education Statistics (NCES) data sets, is observational. As such, students were not randomly assigned to schools, classrooms, or course sequences—limiting the ability to establish a causal link between coursetaking and learning. Regression procedures were used to estimate the relationship between coursetaking and learning controlling for prior achievement and other observed characteristics known to shape students' placement in different tracks/courses. Though the effects of course sequences were robust when all control variables were included, there may have been other unmeasured characteristics (e.g., student motivation, teacher engagement, etc.) that may have caused both course sequence placement and learning, thus making the relationship between coursetaking and learning spurious.

Another limitation to the study is that the course sequences for 45 percent of the analytic sample could not be examined because there were too few cases to make valid generalizations. It could well be the case that some of these course sequences are more beneficial at improving mathematics proficiency than those identified in this report. The small sample sizes, however, preclude a thorough examination of every curricular pathway undertaken by American high school students. Additionally, by creating course sequences from two separate courses, it is not possible to assess the strength of the relationship of each individual course with mathematics achievement. Lastly, compared with the target population, the analytic sample used in this study has higher proportions of White students, students who expect to attain a bachelor's degree or higher, and students living with both their father and their mother. Readers should keep these caveats in mind when interpreting the results and conclusions of this report.

In closing, this study shows that in the last 2 years of high school, American students are improving in mathematics, but that there is room for further improvement. Proficiency scores show that by the end of high school some 38 percent had mastered intermediate mathematics skills, as compared to 25 percent of the sample 2 years before. However, only 4 percent had mastered advanced skills and concepts, compared to 1 percent 2 years earlier. This study shows that the mathematics achievement gains registered between sophomore and senior year are associated with specific coursetaking sequences. In turn, different sequences are related to the proficiency level at which the gains are taking place. The largest gains in intermediate skills were made by students who followed an algebra II–geometry sequence. The largest gains in advanced skills were made by students who followed a sequence that included precalculus paired with algebra II, calculus, or Advanced Placement/International Baccalaureate calculus. At the same time, this study shows that despite the significant relationship between mathematics coursetaking sequences and achievement gain, a greater amount of the variation in learning at the end of high school is explained by factors that precede enrollment in these coursetaking sequences.

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²⁷ Eighty-one percent of the variation in 12th-grade number-right scores is explained by sociodemographic characteristics and 10th-grade number-right scores.