Mathematics Skill Areas for Elementary Grade Students

ECLS measures student proficiency at nine specific mathematics skill levels. These skill levels were identified based on frameworks from other national assessments and advice from a panel of education experts and represent a progression of mathematics skills and knowledge. Levels 6, 7, and 8 were first assessed in third grade, and level 9 was first assessed in fifth grade. By the fifth grade, levels 1 through 4 were not assessed. Each level is labeled by the most sophisticated skill in the set.

Level 1 Number and shape: Recognize single-digit numbers and shapes.

Level 2 Relative size: Count beyond 10, recognize the sequence in basic patterns, and compare the relative size and dimensional relationship of objects.

Level 3 Ordinality and sequence: Recognize two-digit numbers, identify the next number in a sequence, identify the ordinal position of an object, and solve simple word problems.

Level 4 Add and subtract: Solve simple addition and subtraction items and identify relationships of numbers in sequence.

Level 5 Multiply and divide: Perform basic multiplication and division and recognize more complex number patterns.

Level 6 Place value: Demonstrate understanding of place value in integers to the hundreds place.

Level 7 Rate and measurement: Use knowledge of measurement and rate to solve word problems.

Level 8 Fractions: Solve problems using fractions.

Level 9 Area and volume: Solve problems using area and volume.

Achievement Negatively Correlated With Confidence in Learning Across Countries/Economies

TIMSS measured a concept less frequently reported with standardized test results: whether students are self-confident in learning. Correlating achievement with self-confidence reveals surprising results. When comparing mathematics score averages across countries/economies, those with higher percentages of students reporting higher confidence in learning mathematics scored lower than countries/economies with lower percentages of students reporting such confidence (Loveless 2006; Mullis et al. 2004).

On eighth grade mathematics assessments, 39% of U.S. students reported that they usually do well in mathematics, compared with 4% in Japan (table 1-4). However, the average national test score for the United States was 66 points lower than Japan's. Within a given country, however, students who were more self-confident in learning did score higher than other students in their country (Loveless 2006).

Links Between Coursetaking and Learning

Researchers have uncovered an association between courses completed and achievement scores, but not all have controlled for student ability. Students with strong academic skills are likely to take more challenging courses, but if they learn more than other students over time, researchers would like to know how much of the additional gain is attributable to skill and how much to coursework.

Two recent studies that applied controls for ability are described here. Using data from students who took its college entrance exams in 2004, an ACT study found that students who completed a recommended core curriculum scored higher on the ACT tests, regardless of sex, race/ethnicity, family income, or ability (ACT 2006). ACT defined that core curriculum as 3 years each of mathematics, science, and social studies and 4 years of English. Taking advanced courses beyond the core requirements, including additional courses in mathematics and science, was linked to larger score gains, even after controlling for students' prior achievement. Completing the core curriculum also led to higher rates of college enrollment and success in first-year courses like college algebra. Core curriculum graduates were also more likely to be prepared for further workforce training, according to tests of applied learning.

In another study, Bozick, Ingels, and Daniel (2007) used student 10th-grade mathematics proficiency scores as one control measure in examining associations between the mathematics courses taken in 11th and 12th grades and test score gains from 10th to 12th grades. The analysis found that mathematics achievement test scores in 12th grade and achievement gains from 10th to 12th grades were positively related to student mathematics course sequences during the last 2 years of high school. The largest overall gains, and the greatest gains in advanced skills such as derivations and making inferences from algebraic expressions, were made by students who took precalculus in 11th grade plus an additional mathematics course in 12th grade (in most cases, calculus). The largest gains in intermediate skills (such as simple operations and problem solving) were made by those who followed the geometry/algebra II sequence. The smallest gains were made by students who took one mathematics course or no mathematics courses during their last 2 years of secondary school. The analyses controlled for students' prior skill levels and demographic characteristics, including socioeconomic status, educational aspirations, family composition, and school sector.

Attitudes of Parents, Students, and School Staff Toward Standards

Prominent business and education organizations have continued to underscore the need for high schools to raise standards so that students will gain the skills and knowledge base required by employers and postsecondary institutions. Among these organizations are the Gates Foundation and the American Diploma Project (ADP), a consortium that includes Achieve, Inc., many state leaders, the Education Trust, and the Thomas B. Fordham Foundation. In addition, majorities of employers and professors surveyed in 1998–2002 reported that many or most high school graduates (depending on the specific question) lacked skills needed for successful job performance and course completion. For example, in 2001 nearly two-thirds of both groups thought that graduates' basic mathematics skills were fair or poor, and 73%–75% rated student writing ability fair or poor (Public Agenda 2002).

However, these views contrast with those of parents and students. A 2006 survey of parents and students in public school grades 6–12 showed that most do not believe that their local schools need much improvement or that more mathematics and science instruction is necessary. For example, 32% of parents thought their child's school should be teaching more mathematics and science, whereas 57% thought the current amounts were fine (Public Agenda 2006). At 70%, parents of high school students were the most likely (compared with parents of younger students) to think that no increases were needed. Concern about this issue has decreased since 1994, when 52% of parents identified not learning enough mathematics and science as a serious problem, compared with 32% in 2006. This change may partly reflect increases over time in student coursetaking in these subjects.

On academic standards, students in grades 6–12 also expressed some complacency. Only 35% thought it was a problem at their school that "academic standards are too low and kids are not expected to learn enough," and it was not a high priority among 13 problems rated by students. More were concerned about fellow students lacking respect and using bad language, cheating, skipping school/classes, and "too much pressure to make good grades." Even fewer parents (15%) identified "low academic standards and outdated curricula" as a source of the most pressing problems in schools (in a question with different wording).

Active support from school leaders and teachers is also necessary for reforms to be effective. However, many educators (particularly leaders) do not agree that schools need to raise standards or enact other fundamental reforms. Nearly 80% of both principals and superintendents called it "not a serious problem" that academic standards were too low and students were not expected to learn enough. On a related question, 93% of superintendents and 80% of principals evaluated current educational quality as better than the education they received.

Most parents rated their children's public schools highly in 2006. The majority believed that when their children graduate from high school they will have the skills needed for employment or success in college (61% and 69%, respectively). Nearly two-thirds (65%) of parents said that their children were learning more difficult material in school than they had in their school days, and 61% thought their children's schooling was better than their own at that age. Despite their satisfaction with schools overall, parents of different income levels tended to have divergent opinions. For example, over half of low-income parents in a 2002 survey (56%) worried a lot about the low quality of public schools, compared with just 38% of high-income parents (Public Agenda 2002).

The State of State Assessments

State-administered tests seek to demonstrate whether students are achieving at the level required by state standards; they are also used to track progress in meeting federal requirements for student proficiency. In the 2006 academic year, 47 states and the District of Columbia administered mathematics assessments aligned with state standards at the elementary, middle, and high school levels (Editorial Projects in Education 2007). The No Child Left Behind (NCLB) Act requires assessments in mathematics by academic year 2005 in each grade 3–8 and one in grades 10–12; and in science by 2007 in at least one grade in elementary, middle, and high school. State-approved science assessments were thus commonly given but somewhat less widespread in 2006; for example, 20 states lacked them at the high school level. In addition, to graduate from high school in many states, students must surpass a cutoff score on upper grades tests that include mathematics.

How closely tests are aligned with course standards and curriculums remains a contested issue (Barton 2006). The American Federation of Teachers (AFT) recently reviewed state assessments and concluded that in some states, some tests are not sufficiently aligned with the standards (AFT 2006). Students in these states may therefore be tested on some skills and material that their teachers either did not address or covered inadequately, and their test results would not accurately reflect learning differences among groups or gains over time. Even tests with closely aligned content may have other drawbacks, particularly in science. Although written tests can determine whether students understand elements like scientific concepts, methods of inquiry, and terminology, they cannot test hands-on laboratory skills.

Experts have also questioned the quality of state achievement tests, pointing to both the validity of test items and the scores set for reaching certain achievement levels. For example, critics charge that some states may set the minimum score for proficient too low (Petrilli and Finn 2006; Ravitch 2005). The percentage of students reaching proficient on many state tests is close to the percentage reaching the basic level on NAEP, whereas in other states, percentages for the two tests are similar (Center for Public Education 2006; NCES 2007c). (See chapter 8 for recent NAEP scores by state.) Moreover, in an effort to increase the percentage of students considered proficient (a measure specified in NCLB), and facing pressure to make continuing progress toward the goal of universal proficiency by 2014, some state agencies have lowered the proficient cutoff scores on their tests over time (Petrilli and Finn 2006), thus undermining progress toward higher student achievement.

Discrepancies existed between state and NAEP test results even before NCLB took effect (Fuller et al. 2006). Although setting and reviewing standards and developing aligned tests are widely viewed as effective mechanisms for increasing learning, the details of implementation may still need to be evaluated and improved over time.

Advanced Mathematics and Science Courses

Advanced courses referenced in this section are defined as courses that not all students complete and that are not, as a rule, required for graduation. However, whether all courses in certain categories should be categorized as advanced is debatable. For example, any chemistry course, even a standard college preparatory course, is included in the category "any chemistry." This point also applies to the categories any physics, any calculus, and any environmental science.

The "any advanced biology" category is slightly different from the other categories labeled "any" in that it includes second- and third-year biology courses and those designated honors, accelerated, or Advanced Placement/International Baccalaureate (AP/IB), plus a range of specialized courses like anatomy, physiology, and physical science of biotechnology, most of which are college-level courses. Advanced biology therefore does not include the standard first-year biology courses required of nearly all students. Similarly, earth science courses are not counted here because they are often (1) required and (2) not advanced, taking the form of basic survey courses that most students take in 9th or 10th grade. On the other hand, certain courses that are clearly advanced are not measured here because they are so rarely studied in high school (for example, space science/astronomy).

AP/IB courses are all advanced; they aim to teach college-level material and develop skills needed for college study. A school's AP/IB courses are included in the broader category for the relevant subject as well as in the separate AP/IB category, which isolates the subset of courses that meet either of these programs' guidelines.

Project Lead The Way

Some prominent STEM professionals have expressed concern that, as members of the current engineering and science workforce retire, they will not be replaced in adequate numbers (Business Roundtable 2005; Committee on Prospering in the Global Economy of the 21st Century 2006). In the former report, 15 leading business organizations called for the nation to double the number of STEM graduates by 2015.* These organizations argue that not only has the total number of engineering degrees awarded in the United States decreased in recent years (NSB 2006), but the proportion of doctoral degrees in engineering earned by U.S. citizens or permanent residents has also been dropping.†

Project Lead The Way (PLTW) is a pre-engineering program that aims to attract more students to engineering and train them for college study. It requires students to tackle challenging academic content in middle and high school to prepare for postsecondary study in engineering and related technologies. The program, started in 1997–98 in a few schools, has expanded to more than 1,300 schools in 45 states plus the District of Columbia.

PLTW seeks participation by students of both sexes and all racial/ethnic groups roughly in proportion to their share of the population. Evaluation data show that in 2004–05, Asian/Pacific Islander and white students were overrepresented, and black and Hispanic students underrepresented, when compared with their proportions in the sampled schools. However, compared with the distribution of students completing postsecondary degrees in engineering, each group (particularly Hispanics) had closer to proportional representation in PLTW. Females are seriously underrepresented among PLTW completers, constituting about 15% of the total. Program planners expect that female participation will increase as they introduce four new biomedical science courses in 2008–09. The biomedical courses will address topics in microbiology, physiology, public health, and legal issues.

The curriculums reinforce high-level mathematics and science content aligned with national standards using engineering applications in electronics, robotics, and manufacturing processes. PTLW participants are required to study college-preparatory mathematics every year in grades 9–12. Students work, often in teams and using computers, on challenging problemsolving and analysis tasks. Students can qualify for college credit through performance on course exams, final grades, and project portfolios. The project provides curriculums for five 9-week units for grades 6–8 and eight high school courses. Middle-grade units address topics such as modeling, electrons, automation, robotics, the science of technology, and flight. High school courses offered currently include foundation courses such as Principles of Engineering, Engineering Design, and Digital Electronics; and specialization courses including Civil Engineering and Architecture, Computer Integrated Manufacturing, Aerospace Engineering, and Biotechnical Engineering. A capstone course requires advanced students to develop a solution to a complex engineering problem with guidance from a mentor and to defend their project to external reviewers.

Demographic Characteristics of Mathematics and Science Teachers in U.S. Public Schools

In 2003, about 3.2 million teachers were employed in U.S. public elementary and secondary schools (table 1-7). About 231,000 were mathematics teachers and 208,000 were science teachers, based on main assignment field (the subject in which they taught the most classes).

The U.S. public school teaching force increased by 7% from 1999 to 2003; the numbers of mathematics and science teachers increased even more, by 11% and 14%, respectively. Most of these increases have occurred in middle schools or in schools with the highest concentrations of minority and poor students. In contrast, and to place these increased staffing levels in perspective, public school enrollment rose by 3%, from 46.9 million in 1999 to 48.5 million in 2003 (NCES 2006c).

In both 1999 and 2003, three of every four public school teachers were female (table 1-8). However, the predominance of female teachers was less pronounced at the high school level. In 2003, for example, 56% of public high school teachers were women. The sex distribution among public school mathematics and science teachers reflects the overall pattern.

Public school teachers were also predominantly white. In both 1999 and 2003, black and Hispanic teachers accounted for 8% and 6%, respectively, and other racial/ethnic groups accounted for less than 3%. The racial and ethnic distributions among middle and high school mathematics and science teachers resemble the overall pattern. Although the share of black and Hispanic teachers among middle and high school mathematics and science teachers appeared to increase between 1999 and 2003, these changes were not statistically significant.

The average age of the teacher workforce increased slightly over this period. In 1999, 29% of public school teachers were at least 50 years old; that percentage rose to 33% in 2003. Similar trends were also observed among middle and high school mathematics and science teachers. These trends suggest that more teachers are approaching retirement age and that recruitment needs may exceed recent levels.

In-Field and Out-of-Field Teaching

Different researchers (and previous editions of Indicators) have defined out-of-field teaching in different ways (Ingersoll 1999, 2003; McGrath, Holt, and Seastrom 2005; Seastrom et al. 2002). Estimates of how widespread out-of-field teaching is depend on how strictly the concept is defined. This section uses a four-level indicator of the linkage between preparation for teaching science and mathematics courses and the main teaching assignment reported by teachers in SASS.

In the following definitions full certification includes regular, advanced, or probationary certification status. Major refers to the field of study for an undergraduate or graduate degree. Unlike related concepts used in the research literature, this definition recognizes general preparation. State certification regulations vary about whether they treat middle-grade teachers more like elementary teachers (thus requiring a general education credential that covers some preparation in core academic subjects) or more like secondary teachers (requiring single-subject credentials). In some states, the most common type of certification for middle-grade teachers is a general elementary certificate.

The four levels of the indicator are as follows (in decreasing strength of linkage between teacher preparation and the teacher's main assignment field).

In-field. In-field teachers have either a major or full certification in their main teaching field, or both. For example, a mathematics teacher is in field if he or she majored in mathematics or is fully certified in mathematics.

Related-field. Related-field teachers have either a major or full certification in a field related to their main teaching field, or both. For example, a related-field mathematics teacher has a major or full certification in computer science, engineering, or physics.

General preparation. General preparation teachers have either a major or full certification in general elementary, middle, or secondary education. For example, a physics teacher has general preparation if he or she has a major or full certification in general elementary, middle, or secondary education.

Out-of-field. Out-of-field teachers have neither a major nor full certification in their main teaching field, a related field, or general elementary, middle, or secondary education. For example, a biology/life science teacher is teaching out-of-field if he or she has neither a major nor certification in biology, a related field (e.g., physics, chemistry, earth science), or general elementary, middle, or secondary education.

This indicator cannot be used as a gauge of teacher competence because indicators of quality teaching include many other characteristics that are difficult and costly to measure, such as commitment to the profession, sense of responsibility for student learning, and ability to motivate students and diagnose and remedy their learning difficulties. Nevertheless, research, policy, and legislation (e.g., NCLB) point to in-field teaching as a desirable national goal, and states, schools, and school systems administrators can look to this indicator as they engage in efforts to improve teaching.

The discussion in this section focuses on the polar categories of in-field and out-of-field teaching. Appendix table 1-15 also provides data on the nation's teachers of mathematics, biology/life science, and physical sciences who fall between these two extremes.

State Professional Development Policies for Teachers

For two decades, the U.S. government has made teacher professional development a component of its reform efforts (Little 1993; Porter et al. 2000), and many states have developed and implemented policies designed to promote participation in professional development (CCSSO 2005, 2007; Editorial Projects in Education 2006). A total of 48 states required professional development for teacher license renewal in both 2002 and 2006 (table 1-11). Between 2004 and 2006, the number of states that had standards in place for professional development increased from 35 to 40, as did those that financed professional development programs (37 to 39), provided professional development funds for all districts in the state (27 to 31), and required districts or schools to set aside teacher time for professional development (13 to 15). In 2006, 15 states also required and financed mentoring programs for all novice teachers.

Attrition From Teaching

Concerns about K–12 teacher shortages, teacher quality, and the cost of keeping high-quality instructors in the nation's schools have led policymakers to focus attention on teacher attrition and to identify it as one of the most serious problems occurring today in the teaching profession (NCTAF 2003). A recent national study revealed that 8% of all public K–12 school teachers in the 2003–04 academic year had left the teaching profession by the following year (Marvel et al. 2007). For public school mathematics and science teachers, about 6%–7% had left. Although the attrition rates of all teachers have continued to increase over time, the attrition rates for mathematics and science teachers appeared to level off in recent years (figure 1-18).

Another study (Henke, Cataldi, and Nevill forthcoming) focused on the attrition of a segment of new teachers (recent college graduates who taught any of grades K–12 immediately following receipt of a bachelor's degree) and compared their occupational stability with individuals in other occupations. The results of this study suggest that movement among different occupations is common and that teaching is actually one of the more stable occupations in terms of attrition. As shown in figure 1-19, among recent college graduates working in April 1994, 34% were working in the same occupational category in 2003, and 54% had made a change in occupation. In contrast, of those working as K–12 teachers in 1994, 61% were still doing so in 2003, and only 21% had left teaching for nonteaching jobs. Teachers were more likely to remain in the same occupation than most other professionals, including those with comparable education such as legal professionals and legal support personnel, engineers, scientists, laboratory and research assistants, and computer and technical workers. Although recent college graduates do not represent the teaching workforce as a whole, in this study they indicate the job stability of teachers relative to that of other professionals.

International Comparisons of High School Completion

How does the United States compare with other nations in terms of the rates at which young people graduate from high school? A 2006 report from the Organisation for Economic Co-Operation and Development (OECD) found that the United States is falling behind other industrialized nations on this indicator (OECD 2006). In 2004, the high school graduation rate was 75% in the United States, which was lower than the overall average rate of 81% for the 22 OECD countries with available data (figure 1-24). The United States ranked 17th in the overall high school graduation rate among OECD countries, behind such top-ranked countries as Norway, Germany, South Korea, Ireland, Japan, and Denmark.*