

Understanding the Relationship Between Mathematics and Science Coursework Patternsby Xin Ma  2009 Background/Context: There has been little research on the relationship between mathematics and science coursework in secondary school. Purpose of Study: The present analysis explored the patterns of science coursetaking in relation to the patterns of mathematics coursetaking among high school graduates. Research Design: Using data from the 2000 High School Transcript Study (N = 20,368), secondary analysis was performed in the form of multilevel models with students nested within schools to document a strong relationship between mathematics and science coursework patterns. Findings/Results: Results highlighted that (1) taking more courses in advanced mathematics was related to taking more courses in advanced science (this relationship remained strong even after adjustment for studentlevel and schoollevel variables); (2) the more courses that students took in advanced mathematics, the more likely it was that student and school characteristics would join in to select students into taking more courses in advanced science; (3) many high school graduates complied with graduation requirements by taking limited nonadvanced mathematics and science coursework during high school; and (4) mathematics coursework was necessary but insufficient to promote advanced science coursework. Conclusions/Recommendations: State governments are encouraged to prescribe not only the number but also the content of mathematics and science courses required for high school graduation. School personnel such as career counselors are encouraged to help promote better coursework of students in mathematics and science.
Both mathematics and science have been labeled as a critical filter in that mathematics and science effectively screen students for prestigious occupations (Beane, 1988; Sells, 1973). The United States was described as a nation of scientific illiterates, with one main concern being the inadequate mathematics and science coursework in secondary school (American Association for the Advancement of Science [AAAS], 1990; National Commission on Excellence in Education, 1983). For example, the College Entrance Examination Board (2003) reported a steady trend in student coursetaking patterns: Among collegebound high school seniorsthe most advanced group in each years graduation populationonly 15% have taken more than 4 years of mathematics, and only 9% have taken more than 4 years of science in secondary school (i.e., middle and high school). The present study is a direct response to the educational challenge created through the inadequate coursework in mathematics and science, based on the premise that engagement in advanced courses is an important educational experience because access to course content can be viewed as a fundamental key to learning (Finn, Gerber, & Wang, 2002, pp. 337338). It attempts to understand the interaction between mathematics and science coursetaking as it relates to both mathematics and science coursework. The goal was to provide empirical evidence leading to educational policy and practice that encourage students to pursue adequate advanced mathematics and science coursework. Mathematics and science have a long history of alliance, dating back many centuries (AAAS, 1990). Although each has a character and history of its own, each depends on and reinforces the other. It is the union of mathematics and science that forms the scientific endeavor and makes it so successful, according to the AAAS. This union (relationship) can be understood from three perspectives: practical, conceptual, and educational. MATHEMATICS AND SCIENCE ARE HIGHLY CONNECTED PRACTICALLY Mathematics provides the language in which science aspires to describe and analyze the universe; this natural relationship between mathematics and science is the reason that schools, universities, and governments usually group mathematics and science together as a single entity (Ainsley, 1990). The AAAS (1990) stated that science provides mathematics with interesting problems to investigate, and mathematics provides science with powerful tools to use in analyzing data. . . . Science and mathematics are both trying to discover general patterns and relationships, and in this sense they are part of the same endeavor (p. 17). This relationship between mathematics and science is established through both content and process (National Council of Teachers of Mathematics [NCTM], 2000). According to the AAAS (1990), Mathematics is essentially a process of thinking that involves building and applying abstract, logically connected networks of ideas. These ideas arise from the need to solve problems in science, technology, and everyday lifeproblems ranging from how to model certain aspects of a complex scientific problem to how to balance a checkbook. (p. 129) Underhill, Abdi, and Peters (1994) asserted that many real world problem solving situations require the application of both mathematical and scientific skills (p. 26). MATHEMATICS AND SCIENCE ARE HIGHLY CONNECTED CONCEPTUALLY McBride and Silverman (1991) have provided an adequate description of the conceptual relationship between mathematics and science. Mathematics and science share similar conceptual systems of thoughts and are naturally correlated conceptually in the physical world. Mathematics helps achieve a deeper understanding of scientific concepts, providing ways to quantify and explain scientific concepts and relationships, whereas science provides concrete examples for abstract mathematical concepts and thoughts, creating relevancy and motivation for studying mathematical concepts and relationships. Similar arguments abound. Lehrer and Schauble (2002) believe that there is a form of symbolic communication that is particularly shared between mathematics and science in terms of concepts and relationships. Conceptual symbols and processes in mathematics gain practical meanings in the context of science (NCTM, 1989). Heflich, Dixon, and Davis (2001) argued that the learning of scientific concepts can be greatly enhanced by exposure to realworld phenomena that can be expressed in mathematical concepts. Sheety (2002) even documented many similarities between mathematics and science teachers in their conceptual thinking about mathematics and science curriculum (or content). According to the AAAS (1990), the conceptual relationship between mathematics and science is enhanced by many features that mathematics and science share, including a belief in understandable order; an interplay of imagination and rigorous logic; ideals of honesty and openness; the critical importance of peer criticism; the value placed on being the first to make a key discovery; being international in scope; and even, with the development of powerful electronic computers, being able to use technology to open up new fields of investigation. (p. 18) MATHEMATICS AND SCIENCE ARE HIGHLY CONNECTED EDUCATIONALLY The interconnection between mathematics and science is a matter of standard for both mathematics and science instruction (NCTM, 1989; National Science Teachers Association [NSTA], 1992). The coordination between mathematics and science education is critical to the learning of both mathematics and science (Meier, Cobbs, & Nicol, 1998). The National Research Council (1990) believes that students ability to understand the relationship between mathematics and science strengthens their mastery of content knowledge in both mathematics and science. Crissey and Wilkinson (2005) stated that unlike other types of subjects, such as humanities courses, math and science courses build upon one another (p. 7). Because mathematics provides the grammar of sciencethe rules for analyzing scientific ideas and data rigorously (AAAS, 1990, p. 18), the science teacher is not able to cover most topics without calling on mathematical concepts and skills (Frykholm & Meyer, 2002, p. 504). The reliance of science on mathematics certainly varies across the different fields of science. For example, physics places a much heavier demand on mathematical skills than does biology (Basson, 2002; Hanna & Jahnke, 2002; Meltzer, 2002; Orton & Roper, 2000). Stevenson, Schiller, and Schneider (1994) asserted that students opportunities to learn mathematics and science can be organized into sequences that span grades and schools and that such opportunity sequences are a form of stratification that links students future opportunities for learning mathematics and science with their earlier opportunities. Lederman and Niess (1998) believe that the current reforms have resulted in renewed interest in curriculum integration, especially between mathematics and science (p. 281). Sound coordination between mathematics and science education is associated with desirable outcomes in both cognitive and affective domains of mathematics and science (e.g., Berlin & Hillen, 1994; Meier, 1998). Many organizations have been encouraging mathematics and science educators to seek greater integration of mathematics and science curricula and instruction (NCTM, 2000; NRC, 1996; NSTA, 1992). CURRICULAR PRACTICE IN MATHEMATICS AND SCIENCE In the United States, state education authorities set minimum coursework requirement in mathematics and science for graduation from high school. In general, states require their students to take at least 3 years of mathematics (whereas many colleges and universities prefer 4 years of mathematics) and at least 3 years of science. The College Board recommends algebra I, algebra II, geometry, trigonometry, precalculus, or calculus in high school mathematics, and Earth science, space science, biology, chemistry, or physics in high school science. Many high schools permit students to choose their electives in mathematics and science as a way to shape their curricular experiences corresponding to their academic interests and career goals. Many high schools have Advanced Placement (AP) courses in mathematics and science whose pace resembles collegelevel courses. Each AP course has a test at the end of the year, and students who pass the test receive college credit for the course. Students who do not want to take AP courses can receive college credit by taking courses at a local community college. These courses can often be transferred to the college or university that students attend after high school. Curricular practice in mathematics and science becomes complicated when high schools practice curriculum tracking or ability grouping. Usually, there are advanced tracks for collegebound students, vocational tracks for students who may enter the labor market after high school, and general tracks for students in between. In many high schools, not all mathematics and science courses are available to students in all tracks. For example, AP courses are more likely to be available to students in the advanced track (and the general track in some cases) than to students in the vocational track. Although highly connected, mathematics and science education are different, with discipline (subject) playing a major role in science, without a parallel in mathematics. High schools have credentialed area specialists in science (e.g., biology, chemistry, or physics). Science teachers are specialists, with their content rooted in specific science disciplines. In contrast, high schools have no credentialed area specialists in mathematics (e.g., trigonometry, calculus, or statistics). Mathematics teachers are generalists, with their content rooted in multiple mathematics disciplines. Understanding the relationship between mathematics and science coursework in general, and the results of the present study in particular, requires attention to these curricular practices. AIMS OF THE PRESENT STUDY The close relationship between mathematics and science education makes research that simultaneously investigates educational issues in both mathematics and science beneficial, if not necessary. The need for such research is certainly clear in the present case of mathematics and science coursework. Both school mathematics and science are structured in a highly sequential fashion; access to mathematics and science courses is determined by prior success in particular courses (often referred to as prerequisites) that systematically regulate student academic progress in the learning of mathematics and science (see Oakes, 1990). Many problems in science coursework cannot be adequately resolved without examining their correlation with problems in mathematics coursework. The present study was unique on two counts. One was that, for the first time in research, mathematics and science coursetaking were investigated together in a systematic and comprehensive manner to explore the relationship between mathematics and science coursework patterns. In the present study, coursework pattern refers to the number of advanced mathematics and science courses that a student took during secondary schooling partitioned into two specific periods: Grades 8 and 9, and Grades 1012). The other unique aspect of the present study is related to the two competing approaches in the research literature to addressing mathematics and science coursework. Some researchers emphasize the number of courses taken (e.g., Sebring, 1987), whereas others emphasize the content of courses taken (e.g., Hoffer, 1997). The present study took a compromised approach in order to consider both the number and content of courses when establishing coursework patterns in mathematics and science. An underlying assumption was that favorable coursework patterns correlate between mathematics and science, even after adjusting for student and school characteristics. The present study was guided by three research questions: (1) How many distinct coursework patterns are there in mathematics and science? (2) To what extent are students coursework patterns in science related to their coursework patterns in mathematics? (3) What studentlevel and schoollevel variables influence the relationship between mathematics and science coursework patterns? Over the past few years, nationally representative data describing students and schoolsin particular, transcript data with a focus on mathematics and science educationhave become available, and advanced statistical techniques that can be employed to address research questions concerning mathematics and science coursework have been developed. Therefore, the research conditions for addressing the research questions regarding mathematics and science coursework patterns have matured to allow a meaningful investigation both analytically and practically. METHOD DATA SOURCE Data for the present study came from the 2000 High School Transcript Study (HSTS). In 2000, the National Center for Education Statistics conducted a national survey of high school transcripts of 12thgrade students to obtain a detailed record of mathematics and science courses that each student took during his or her entire secondary school career. A multistage probabilitybased sampling procedure was employed to obtain a nationally representative sample of high school seniors. Counties were sampled in the first stage, schools within counties were sampled in the second stage, and students within schools were sampled in the third stage (see National Center for Education Statistics [NCES], 2005, Chapter 2). The sample of HSTS 2000 contained 20,931 high school seniors from 277 public and nonpublic schools. In addition to authentic high school senior transcripts, HSTS also collected student information (student background, curriculum tracking, graduation requirement status, overall academic achievement, and coursework credits) and school information (school context, program policy, and school climate to be discussed in detail later on; see also NCES, 2005, Chapter 4). This research excluded high school seniors who failed to graduate in 2000 based on the assumption that their coursework patterns might be too unusual to be combined with those of high school graduates. The result was 20,368 high school graduates from 234 public and nonpublic schools. Although this research realized the concern that public and nonpublic students might have different coursetaking behaviors, it did not create separate analyses according to school sectors. Instead, this research identified school sectors as schoollevel variables to examine whether these sectors were able to account for the variation in the relationship between coursework patterns in mathematics and science. This joint analysis approach was preferred because it recognizes the correlation in coursework between school sectors (the separate analysis approach in which each school sector is analyzed in isolation ignores such correlation). MEASURES OF MATHMATICS AND SCIENCE COURSEWORK Because high school mathematics and science courses differ in content and level (even with the same title) across the country, a common course coding system needs to be developed. The Classification of Secondary School Courses (CSSC) is able to uniquely identify various mathematics and science courses and translate them into a common coding scale (see NCES, 2005, Chapter 5). Thus, each course on a student transcript is assigned a unique code based on content and level. HSTS 2000 contained a very detailed list of CSSC courses in mathematics and science, providing data to construct coursework patterns in mathematics and science. These CSSC courses were used as the primary measures of secondary mathematics and science coursework in the present study. For the research purpose of examining coursework patterns, these CSSC courses need to be classified according to the level and function of each course. Taxonomies of mathematics and science courses were developed in the present study (see Appendixes A and B). For consistency and continuity with previous classifications of mathematics and science courses, these taxonomies were established in close consultation with the Condition of Education 2004 Supplemental Note 6 (NCES, 2004). Specifically, mathematics courses were classified into Functional Mathematics, Basic Mathematics, General Mathematics, Applied Mathematics, Gatekeeper Mathematics, Unified Mathematics, Standard Mathematics, Traditional Advanced Mathematics, and Modern Advanced Mathematics. Science courses were classified into Primary Science, Secondary Science, General Biology, Chemistry 1 and Physics 1, and Chemistry 2, Physics 2, and Advanced Biology. Categories in each subject were separable in term of function but might overlap with one another in terms of level. A definition of advanced mathematics and science courses is inevitable when examining coursework patterns. A common sense approach that emphasizes expert opinions was adopted for this task. In the taxonomy of mathematics coursework (Appendix A), courses in the categories of Traditional Advanced Mathematics and Modern Advanced Mathematics were defined as advanced mathematics coursework. In the taxonomy of science coursework (Appendix B), courses in the category of Chemistry 2, Physics 2, and Advanced Biology were defined as advanced science coursework. These definitions allowed the identification of coursework patterns in mathematics and science in terms of the number of advanced courses that students took during the two periods of secondary school (addressing the first research question). Identified coursework patterns in science became the dependent measure that was predicted by identified coursework patterns in mathematics (the chief independent measure) with control over student and school characteristics to address the second and third research questions. MEASURES OF STUDENT AND SCHOOL CHARACTERISTICS Student and school characteristics were used as the independent variables to adjust the relationship between mathematics and science coursework patterns. Studentlevel variables came from the HSTS 2000 student questionnaire, highlighting five sets of student characteristics. Student background included gender (male, with female as the baseline), age, socioeconomic status (SES, measured through family Title I status), and raceethnicity (White, Black, Hispanic, and Asian, with other racialethnic backgrounds as the baseline). Curriculum tracking was measured through academic track (academic, vocational, and combined, with no academic track as the baseline). Graduation requirement status was measured through earned Carnegie units for graduation (greater than requirement, greater than 75% of requirement, and equal to 75%, with less than 75% as the baseline). The Carnegie unit is defined as the number of credits a student receives for a course taken every day, one period per day, for a full school year. Overall academic achievement was measured through grade point average (GPA). Mathematics coursework credits included eighthgrade credits in mathematics, 10thgrade credits in mathematics, and total secondary mathematics credits, all measured in Carnegie units. Schoollevel variables came from the HSTS 2000 school questionnaire, highlighting school characteristics in terms of school context, program policy, and school climate. School context included number of students (school enrollment size), number of teachers (school workforce size), school type (public, religious, and Catholic, with Bureau of Indian Affairs and Department of Defense as the baseline), school location (city and suburban, with rural region as the baseline), school socioeconomic composition (measured through the percentage of students eligible for the National School Lunch Program), school linguistic composition (measured through the percentage of students with English as the second language), and 4year high school (yes, with no as the baseline). Program policy included ability grouping in mathematics, ability grouping in science, 4year mathematics requirement, 2year mathematics requirement, 4year science requirement, 2year science requirement (yes, with no as the baseline, for all variables), and Carnegie units required to graduate. School climate included disciplinary climate (measured through the extent to which the following concerns exist: student absenteeism, student tardiness, conflicts among students, teacher absenteeism, racial conflicts, health problems, lack of parent involvement, use of alcohol, use of tobacco, use of drugs, gang activities, student misbehavior in class, student cheating, conflicts with teachers, and vandalism), academic pressure (measured through student attitude toward academic achievement, parental support for academic achievement, and teacher expectation on academic achievement), parental involvement (measured through the extent to which parents participate in PTA, attend open houses, attend conferences, make curriculum decisions, and volunteer in school activities), teacher attribution (measured through the percentage of teachers left), teacher morale, mathematics extracurricular activities, and science extracurricular activities. STATISTICAL PROCEDURES According to the research questions, the first task was to identify coursework patterns in mathematics and science, based on coursework measures across secondary school constructed from HSTS 2000. Once derived, multilevel analysis was used as the primary statistical technique to examine the relationship between mathematics and science coursework patterns. One of the rationales for a multilevel framework is that HSTS 2000 data are hierarchical in structure, with students nested within schools. Statistical analysis of HSTS data therefore requires a multilevel framework (see Raudenbush & Bryk, 2002). Because coursework patterns, when derived to reflect both quantity and quality of course participation, were ordinal in measurement, a twolevel ordinal model was used with students at the first level and schools at the second level. The multilevel software package used in the present study, Hierarchical Linear Modeling (HLM; Raudenbush, Bryk, Cheong, & Congdon, 2000), is able to use coursework patterns as the dependent variable (see Chapter 5). Data analysis started with the absolute relationship between mathematics and science coursework patterns. The model structure had science coursework patterns as the dependent variable and mathematics coursework patterns as the independent variable (at the first level). The outcome was used to demonstrate to what extent students science coursework patterns were associated with their mathematics coursework patterns (addressing the second research question). Studentlevel and schoollevel variables were then introduced to examine how student and school characteristics mediated the absolute relationship between mathematics and science coursework patterns. To simultaneously examine (or to analytically separate) student and school effects, one again needs multilevel data analysis (another rationale for a multilevel framework). The model structure had science coursework patterns as the dependent variable and mathematics coursework patterns and student characteristics as the independent variables at the first level, and school characteristics as the independent variables at the second level. The outcome was used to demonstrate whether students science coursework patterns were still associated with their mathematics coursework patterns once student and school characteristics were adjusted, and which student and school characteristics mediated the relationship in coursework patterns between mathematics and science (i.e., identifying individual differences in, and school effects on, the relationship, thus addressing the third research question). RESULTS MATHEMATICS AND SCIENCE COURSEWORK PATTERNS The present study attempted to consider both the number and content of mathematics and science courses when establishing coursework patterns. The content of mathematics and science courses for Grades 1012 was considered based on the definition of advanced mathematics and science coursework. For students without any advanced mathematics and science courses during that period, the number of mathematics and science courses during the eighth and ninth grades was considered. This approach revealed eight coursework patterns in both mathematics and science. These are presented in descending order in Table 1, reflecting how proactively students under each pattern pursued both mathematics and science courses. Table 1. High school graduates mathematics and science coursework patterns, Grade 8 (G8) to Grade 12 (G12)
Note. Grade 7 is omitted because few students take neither mathematics nor science courses in that grade. Under the current patterns, coursework in Grades 8 and 9 is omitted for students with advanced courses in Grades 1012. Students with more than four advanced courses in Grades 1012 are combined with students with four advanced courses. Students with more than three courses in Grades 8 and 9 are combined with students with three courses. Carnegie credits are used as the number of measurement units, with one unit as 100 Carnegie credits. Among high school graduates in 2000, 46% took at least one advanced mathematics course during Grades 1012, and 19% took at least one advanced science course during the same period. As the most popular coursework pattern, 1 in 3 high school graduates took two mathematics courses in the eighth and ninth grades and no advanced mathematics course during Grades 1012, and 2 in 5 high school graduates took two science courses in Grades 8 and 9 and no advanced science course during Grades 1012. Table 1 also contains information on all coursework credits in Carnegie units earned by high school graduates under each pattern of both mathematics and science coursework. Within each subject area, students with more advanced courses during Grades 1012 earned more (total) Carnegie units or took a larger number of (total) courses during their secondary schooling. A similar trend was also true for students without any advanced coursework during the same period (students with more courses in Grades 8 and 9 earned more total Carnegie units during their secondary schooling), and these students also demonstrated similar coursework behaviors between mathematics and science with close measures of Carnegie units. As a common phenomenon across mathematics and science, students who took no course in Grades 8 and 9 earned the least coursework credits by taking limited nonadvanced coursework during Grades 1012. Students with at least three advanced science courses also need to be singled out for their unique coursework behaviors (see the top two patterns in science). Together, these students represented only 3% of high school graduates. However, this small number of students took a substantial amount of coursework in science, with Carnegie units averaging 5.26 for students with four advanced science courses, and 4.69 for students with three advanced science courses. They actually earned more Carnegie units in science than their counterparts in mathematics (with Carnegie units averaging 4.54 for students with four advanced science courses, and 4.41 for students with three advanced science courses). Standard deviation (SD) measured how consistent Carnegie units were on all coursework credits among students under each pattern of both mathematics and science coursework. Among students with advanced coursework during Grades 1012, SD was larger (in some cases substantially larger) in science than mathematics for each pair of corresponding patterns. This finding indicated that, for example, students with four advanced science courses were a lot more heterogeneous in their total science coursework credits than students with four advanced mathematics courses in their total mathematics coursework credits. Similar phenomena (but to a lesser degree) were also true for students with coursework in Grades 8 and 9. RELATIONSHIP BETWEEN MATHEMATICS AND SCIENCE COURSEWORK PATTERNS The relationship between mathematics and science coursework patterns was examined in a descriptive way first, with the construction of a percentage distribution of each coursework pattern in one subject area across coursework patterns in the other subject area (see Tables 2 and 3). For ease of understanding and interpretation, these tables are actually a breakdown of a crosstabulation between mathematics and science coursework patterns. A chisquare test of the independence between mathematics and science coursework patterns was statistically significant, indicating that mathematics and science coursework patterns were dependent on each other. Table 2. Percentage distribution of each science coursework pattern across mathematics coursework patterns
Note. M_{1} is the sum of previous four columns, representing a percentage measure of students with at least one advanced mathematics course from Grade 10 to Grade 12 for each pattern in science. M_{2} is the sum of previous four columns, representing a percentage measure of students without any advanced mathematics course from Grade 10 to Grade 12 for each pattern in science. Tables 2 and 3 also provide a descriptive approach to unfolding the dependency between mathematics and science coursework patterns. For example, Table 2 indicates that 63% of students with four advanced courses in science during Grades 1012 also took four advanced courses in mathematics during the same period, and Table 3 indicates that 12% of students with four advanced courses in mathematics during Grades 1012 also took four advanced courses in science during the same period. A chisquare test confirmed the dependency between taking four advanced mathematics and science courses during the same period of Grades 1012. The two percentages offered a way to understand this dependency. Students with four advanced science courses were more likely to take four advanced mathematics courses, whereas students with four advanced mathematics courses were less likely to take four advanced science courses. Table 3. Percentage distribution of each mathematics coursework pattern across science coursework patterns
Note. M_{1} is the sum of previous four columns, representing a percentage measure of students with at least one advanced science course from Grade 10 to Grade 12 for each pattern in mathematics. M_{2} is the sum of previous four columns, representing a percentage measure of students without any advanced science course from Grade 10 to Grade 12 for each pattern in mathematics. In fact, a chisquare test revealed a general dependency between taking at least one advanced mathematics and science course during Grades 1012. A comparison of the M_{1} columns between both tables indicates that although students with advanced coursework in science were also students with advanced coursework in mathematics, the opposite was not necessarily true. For example, among students with four advanced courses in science, 92% took at least one advanced mathematics course (Table 2), whereas among students with four advanced courses in mathematics, 44% took at least one advanced science course (Table 3). In other words, advanced science students did take advanced mathematics courses, but advanced mathematics students did not necessarily take advanced science courses. Although this analysis by no means implied any causal dependence of science coursework on mathematics coursework, it did suggest that advanced science students tended to have an advanced mathematics background, but advanced mathematics students failed to have advanced science background. Why didnt many advanced mathematics students become advanced science students? How did the same mathematics coursework create different pathways for students in science education? Such thinking or assumption led to placing science coursework patterns as the dependent variable in the subsequent inferential analyses. If the mentioned descriptive analyses served to explore the relationship between mathematics and science coursework patterns, inferential analyses that follow tend to both confirm and quantify the relationship. Table 4 presents the absolute relationship between mathematics and science coursework patterns. Science coursework was the ordinal dependent variable in the multilevel analysis, with eight coursework patterns. For mathematics coursework, students without any advanced course during Grades 1012 were combined for focus and simplicity as the baseline against which students with advanced mathematics coursework during the same period were compared. Table 4. Results of the unconditional multilevel ordinal model estimating the effects of mathematics coursework patterns on science coursework patterns (Grades 812)
Note. All effects are statistically significant at the alpha level of 0.05. SE = standard error. OR = odds ratio. Odds ratio, the key estimate in the table, can be interpreted as the average change in the proportional odds, which has been accounted for by each mathematics coursework pattern (i.e., as a result of the effects of each mathematics coursework pattern). For example, students with four advanced courses in mathematics during Grades 1012 were about 8 times more likely to take four advanced courses in science during the same period. A descending order among odds ratios was evident across mathematics coursework patterns in the table. This finding indicates that overall, the more advanced mathematics courses that students took during Grades 1012, the more likely they were to take four advanced science courses during the same period. STUDENT AND SCHOOL CHARACTERISTICS AS MEDIATORS Table 5 presents statistical results of the final multilevel ordinal model that examined student and school characteristics as they mediated the relationship between mathematics and science coursework patterns. The relationship was adjusted at the student level for the effects of student background (in favor of White and Asian students), curriculum tracking (in favor of students in the academic and combined tracks), graduation requirements (in favor of Carnegie units greater than 100% and 75% of requirements), academic achievement (in favor of higher GPA), and mathematics coursework credits (in favor of higher Carnegie units in mathematics in the eighth grade) on science coursework patterns. At the school level, the relationship was adjusted for the effects of school context (in favor of students in public schools and students in urban schools), program policy (in favor of students in schools that require 4 or more years of science courses), and school climate (no statistically significant variable) over and above adjustment for the effects of student characteristics on science coursework patterns. Table 5. Results of the multilevel ordinal model estimating the effects of mathematics coursework patterns on science coursework patterns (Grades 8 to 12), conditional on student and school characteristics
Note. All effects are statistically significant at the alpha level of 0.05. SE = standard error. OR = odds ratio. Carnegie units in mathematics in Grade 8 are used as the number of measurement units, with half Carnegie units as one measurement unit. Even after adjustment for student and school characteristics, the relationship between mathematics and science coursework patterns remained statistically significant and practically strong. For example, students with four advanced mathematics courses during Grades 1012 were still about 5 times more likely to take four advanced science courses during the same period. Overall, the results showed substantial advantages in taking four advanced science courses for students who took three or four advanced mathematics courses. Even students who took two advanced mathematics courses showed some advantage in taking four advanced science courses. On the other hand, with adjustment for student and school effects, students who took only one advanced mathematics course displayed practically marginal, though statistically significant, advantage in taking four advanced science courses. A comparison between Tables 4 and 5 shows that the odds ratio of taking four advanced science courses decreased among students in every mathematics coursework pattern once student and school characteristics were adjusted. The extent of this decrease depended on the number of advanced mathematics courses (i.e., the more coursework the larger the decrease). Therefore, the largest effects of student and school characteristics (on taking four advanced science courses) occurred among students who took the most courses in advanced mathematics. In other words, the more courses that students took in advanced mathematics, the more likely it was that their student and school characteristics would join in to select students into taking four advanced science courses. Appeared to work most closely with mathematics coursework patterns to select students into taking four advanced science courses were graduation requirements and curriculum tracking (in this order) at the student level, and attending public schools at the school level. DISCUSSION COURSEWORK BEHAVIORS IN MATHMATICS AND SCIENCE The first part of the present analysis dealt mainly with high school graduates behaviors in taking mathematics and science courses. The main finding from Tables 1, 2, and 3 highlighted a great concern about high school graduates mathematics and, in particular, science coursework. If students who took at least three advanced courses during Grades 1012 (a period of 3 years) can be labeled as adequately trained, the percentage in 2000 was 15% in mathematics and 3% in science. American high schools prepared at most 15% of their graduates in 2000 to enter what is often referred to as the STEM areas (science, technology, engineering, and mathematics) without an overwhelming need for content remediation at the college level. Statistics like this are alarming to American mathematics and science educators. During the 1990s, all states prescribed increased graduation requirements for mathematics and science coursework as a way to augment mathematics and science standards. Conceptually, more mathematics and science courses deepen students understanding of mathematics and science for increased achievement in these subjects and create learning opportunities for improved participation in advanced mathematics and science coursework (e.g., Ma, 2000; Smith, 1996). Unfortunately, this policy change neither increased mathematics achievement (Hoffer, 1997) nor improved participation in advanced mathematics and science coursework, as shown in the present analysis. Hoffer attributed the lack of improvement in mathematics achievement to the fact that many schools responded to this policy change by offering more lowlevel mathematics courses (see also Ma), and he argued that it is time to move beyond simple coursecount requirements to an emphasis on specific curriculum content and actual learning outcomes (p. 48). Such a policy change means that state governments need to prescribe not only the number but also the content of mathematics and, particularly, science courses required for high school graduation. The present analysis shows that a common coping strategy among high school graduates to survive the mathematics and science requirements for high school graduation was to take limited nonadvanced courses during Grades 1012 (54% in mathematics and 71% in science). These coursework choices might be appropriate for many students, depending on their career plans. However, statistics from the status quo indicate that 25% of freshmen at 4year colleges are unable to make it to their sophomore year even with 50% having to take remedial content courses, and the percentage of freshmen who drop out in their sophomore year jumps to 50% at 2year colleges (Adelman, 1999). One effective way to promote adequate preparation in mathematics and science for these students is to prescribe advanced content of mathematics and particularly science coursework for graduation. Remedial or developmental courses so popular at the college level aim to overcome deficiencies in high school mathematics and science preparation. Perhaps the shift toward content requirement would make such courses more effective. These remedial or developmental courses may even become unnecessary so that students can instead take more collegelevel mathematics and science courses, which qualifies or adequately prepares more college students for the STEM majors. RELATIONSHIP BETWEEN MATHMATICS AND SCIENCE COURSEWORK PATTERNS One of the major findings of the present analysis is that science coursework patterns were significantly related to mathematics coursework patterns, supported by results from a chisquare test (Table 3) and a multilevel ordinal model (Table 4). Such a finding creates the policy premise that mathematics and science educators should work together for better participation of students in both mathematics and science courses. Results of multilevel ordinal models clearly quantified the advantage of taking more advanced mathematics coursework that was substantially associated with the odds of taking more advanced science coursework. Results from Tables 3 and 4 together indicate that advanced mathematics coursework appears necessary to advanced science coursework, but insufficient. It is necessary in that taking advanced mathematics coursework is associated with favorable odds of taking advanced science coursework (students with four advanced mathematics courses were 8 times more likely to take four advanced science courses; see Table 4). It is insufficient in that few students with four advanced mathematics courses took advantage of their advanced mathematics backgrounds to get adequately trained in advanced science (the minority [12%] of students with four advanced mathematics courses took four advanced science courses; see Table 3). Stated differently, students with four advanced mathematics courses were 8 times more likely to complete four advanced science courses successfully, but only 12% of those students took advantage of their advanced mathematics backgrounds. This finding indicates the need to lead or channel students with advanced mathematics backgrounds to advanced science coursework. High schools may need to have some sort of mechanism to persuade students with advanced mathematics coursework to consider advanced science coursework. One may reasonably speculate that one potential way is to use school career counselors to set up this mechanism. COMPLEXITY OF THE RELATIONSHIP BETWEEN MATHMATICS AND SCIENCE COURSEWORK PATTERNS The present analysis addressed the complexity issue by means of comparing the absolute relationship (Table 4) with the one adjusted for student and school characteristics (Table 5). The goal is to examine whether student and school characteristics work with mathematics coursework patterns to account for the variation in science coursework patterns among high school graduates. Such a strategy appeared successful in that the odds ratio decreased substantially once studentlevel and schoollevel variables were added (e.g., from 7.52 to 4.65 in the case of the effects of taking four advanced mathematics courses). The main finding is that the more courses that students took in advanced mathematics, the more likely it was that student and school characteristics would join in to select students into taking more advanced science courses. Specifically, studentlevel and schoollevel variables mediated the behavior in science coursework of adequately trained mathematics students (defined earlier as those taking at least three advanced mathematics courses during Grades 1012) to a much larger degree than that of students with fewer than three advanced mathematics courses during the same period. In other words, the behavior in science coursework of adequately trained mathematics students was dependent on student and school characteristics (adjustment for studentlevel and schoollevel variables decreased their advantage in taking advanced science coursework), whereas the behavior in science coursework of students with fewer than three advanced mathematics courses was largely independent on student and school characteristics (adjustment for studentlevel and schoollevel variables did not change their odds ratio much). These findings provide suggestions for the American school system to work effectively with adequately trained students in mathematics to encourage them to venture into advanced science coursework. According to results in Table 5, student background, curriculum tracking, graduation requirements, academic achievement, and mathematics coursework credits at the student level, as well as school context and program policy at the school level, are all tools that mathematics and science educators can use to promote proactive participation in advanced science coursework. For example, working with school counselors, mathematics and science educators may want to encourage adequately trained Black and Hispanic students in mathematics to take more advanced science courses in order to pursue prestigious college major and career options later on, or they may want to ensure that students build up adequate mathematics coursework credits in the eighth grade to prepare for advanced science courses in high school. Finally, some great effects in Table 5 are important to extend the research literature on science coursework. Secondary in effect only to graduation requirements, results associated with curriculum tracking supported the claim that a constrained curriculum composed of mainly academic (advanced) courses promotes higher academic attainment (e.g., Lee, Croninger, & Smith, 1997). In fact, even programs that mixed academic courses with vocational courses appeared to promote more proactive participation in advanced science coursework. The greatest effects on advanced science coursework came from graduation requirements. More demanding coursework requirements for graduation did lead to more proactive participation in advanced science courses. At the school level, graduates from public high schools in 2000 did excel in participation in advanced science coursework. This is perhaps a positive result of the public schools overwhelming emphasis on academic attainment, particularly in an era after A Nation at Risk, which alarmed the public about the failure of public schools to adequately prepare students in mathematics and science education (National Commission on Excellence in Education, 1983). That attending public schools better prepared students in advanced science coursework is one positive conclusion in the research literature, given the overwhelming academic advantage that, say, Catholic schools usually demonstrate over public schools (see Bryk, Lee, & Holland, 1993; Hunt, Joseph, & Nuzzi, 2004). References Adelman, C. (1999). Answers in the tool box: Academic intensity, attendance patterns, and Bachelors degree attainment. Washington, DC: U.S. Department of Education. Ainsley, R. (1990). Bluff your way in math. Lincoln, NE: Centennial. American Association for the Advancement of Science. (1990). Science for all Americans. Washington, DC: Author. Basson, I. (2002). Physics and mathematics as interrelated fields of thought development using acceleration as an example. International Journal of Mathematical Education in Science and Technology, 33, 679690. Beane, D. B. (1988). Mathematics and science: Critical filters for the future of minority students. Washington, DC: American University. , & (1994). Making connections in math and science: Identifying student outcomes. , 94, 283290. Bryk, A., Lee, V., & Holland, P. (1993). Catholic schools and the common good. Cambridge, MA: Harvard University Press. College Entrance Examination Board. (2003). 2003 profile of collegebound seniors. New York: Author. Crissey, S. R., & Wilkinson, L. (2005, March). The role of high school math and science course taking in the transition to first birth. Paper presented at the annual meeting of the Population Association of America, Philadelphia, PA. Finn, J. D., Gerber, S. B., & Wang, M. C. (2002). Course offerings, course requirements, and course taking in mathematics. Journal of Curriculum and Supervision, 17, 336366. Frykholm, J. A., & Meyer, M. R. (2002, May). Integrated instruction: Is it science? Is it mathematics? Mathematics Teaching in the Middle School, 502508. Hanna, G., & Jahnke, H. (2002). Arguments from physics in mathematical proofs: An educational perspective. For the Learning of Mathematics, 22(3), 3845. Heflich, D. A., Dixon, J. K., & Davis, K. S. (2001). Taking it to the field: The authentic integration of mathematics and technology in inquirybased science instruction. Journal of Computers in Mathematics and Science Teaching, 20, 99112. Hoffer, T. B. (1997). High school graduation requirements: Effects on dropping out and student achievement. Teachers College Record, 98, 584607. Hunt, T. C., Joseph, E. A., & Nuzzi, R. J. (Eds.). (2004). Catholic schools still make a difference: Ten years of research, 19912000. Washington, DC: National Catholic Educational Association. Lee, V. E., Croninger, R. G., & Smith, J. B. (1997). Coursetaking, equity, and mathematics learning: Testing the constrained curriculum hypothesis in U.S. secondary schools. Educational Evaluation and Policy Analysis, 19, 99121. Lederman, N. G., & Niess, M. L. (1998). 5 apples + 4 oranges = ? School Science and Mathematics, 98, 281284. Lehrer, R., & Schauble, L. (2002). Symbolic communication in mathematics and science: Coconstituting inscription and thought. In E. Amsel & J. P. Byrnes (Eds.), Language, literacy, and cognitive development: The development and consequences of symbolic communication (pp. 167192). Mahwah, NJ: Erlbaum. Ma, X. (2000). A longitudinal assessment of antecedent coursework in mathematics on subsequent mathematical attainment. Journal of Educational Research, 94, 1628. , & (1991). Integrating elementary/middle school science and mathematics. , 91, 285292. Meier, S. L. (1998). TIMSS and IMaST students: How do they fare? CeMaST Update, 3(1). Meier, S. L., Cobbs, G., & Nicol, M. (1998). Potential benefits and barriers to integration. School Science and Mathematics, 98, 438447. Meltzer, D. E. (2002). The relationship between mathematics preparation and conceptual learning gains in physics: A possible hidden variable in diagnostic pretest scores. American Journal of Physics, 70, 1259 1268. National Center for Education Statistics. (2004). The condition of education 2004. Washington, DC: Author. National Center for Education Statistics. (2005). The 2000 High School Transcript Study users guide and technical report. Washington, DC: Author. National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform. Washington, DC: U.S. Government Printing Office. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Research Council. (1990). Reshaping school mathematics: A philosophy and framework for curriculum. Washington, DC: National Academy Press. National Research Council. (1996). National science education standards. Washington, DC: National Academy Press. National Science Teachers Association. (1992). NSTA standards for science teacher preparation: Final draft. Reston, VA: Author. Oakes, J. (1990). Multiplying inequalities: The effects of race, social class, and tracking on opportunities to learn mathematics and science. Santa Monica, CA: RAND. Orton, T., & Roper, T. (2000). Science and mathematics: A relationship in need of counseling? Studies in Science Education, 35, 123154. Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models (2nd ed.). Newbury Park, CA: Sage. Raudenbush, S. W., Bryk, A. S., Cheong, Y. F., & Congdon, R. (2000). HLM5: Hierarchical linear and nonlinear modelling. Lincolnwood, IL: Scientific Software International. Sebring, P. A. (1987). Consequences of differential amounts of high school coursework: Will the new graduation requirements help? Educational Evaluation and Policy Analysis, 9, 258273. Sells, L. W. (1973). High school mathematics as the critical filter in the job market. Proceedings of the Conference on Minority Graduate Education. Berkeley: University of California. Sheety, A. S. (2002). How do teachers who teach different subject matter think how their students learn? Unpublished doctoral dissertation, Arizona State University, Tempe. Smith, J. B. (1996). Does an extra year make any difference? The impact of early access to algebra on longterm gains in mathematics attainment. Educational Evaluation and Policy Analysis, 18, 141153. Stevenson, D. L., Schiller, K. S., & Schneider, B. (1994). Sequences of opportunities for learning. Sociology of Education, 67, 184198 Underhill, R. G., Abdi, S. W., & Peters, P. F. (1994). The Virginia State Systemic Initiative: A brief overview of the lead teacher component and a description of the evolving mathematics and science integration outcomes. School Science and Mathematics, 94, 2629. APPENDIX A Taxonomy of Mathematics Courses
APPENDIX B Taxonomy of Science Courses





