Mathematics curriculum, K–12

Summary: Curricular standards of mathematics have undergone a series of changes in the twentieth and twenty-first centuries in response to various national concerns.

The term “curriculum” has been variously defined as a coherent program of study in a specific subject area consisting of a set of courses and learning experiences provided by an educational institution. Since the beginning of the twentieth century, the school curriculum (grades K–12) of the United States has undergone numerous changes, particularly in mathematics. Indeed, the U.S. school curriculum has been more profoundly influenced by the demands of society than any other country in the world.

Twenty-first-century debates over the school curriculum are essentially debates on how best to prepare students to live in a just, democratic society; be competitive in a global economy; and thrive in a technologically literate workforce. Employers in both explicitly technical and nontechnical fields seek candidates who can use mathematics content, logical reasoning, and other problem-solving skills that are ideally acquired by the end of high school. Both government and private employers have publicized the various requirements for different career paths. Differences of opinion often stem from opposing views about the nature of learning, the needs of society, and the purposes of schooling. The evolution of the school curriculum in the United States and the changing nature of its mathematics component are most easily understood from an historical context.

Change in the Mathematics Curriculum

Since the inception of the United States, the predominant view has been that education is necessary for the common good of society and the survival of democracy. Although mathematics has always been recognized as an essential component of the K–12 curriculum, the role of mathematics in public schools and the nature of its content have fluctuated over the years. Benjamin Franklin was one of the nation’s first leaders to understand the need for mathematics instruction beyond basic arithmetic and measurement. For example, in 1751, he helped institute an academy in which geometry and algebra were among curricula designed to meet the practical needs of merchants, seamen, builders, and artisans.

Common School Movement

Throughout the 1800s, the curricular and educational trends of Europe influenced the mathematics curriculum of the United States. The focus of the mathematics curriculum was on basic arithmetic skills in the early grades and algebra and geometry in the upper grades. In 1837, the Common School movement was instituted by Horace Mann (1796–1859) from Massachusetts. He worked to develop a statewide common-school (public school) system. The philosophy was that education is a major “human equalizer” that balances the social structure of a country. For this reason, Horace Mann is often considered the “father of American public education.” The curriculum of the common schools of the 1800s served to reflect the values and needs of a democratic society and instituted free education for all U.S. citizens, making mathematics education much more broadly available

The Progressive Movement

For the majority of the twentieth century, U.S. educators consistently promoted a “Progressive Education” agenda, spearheaded by John Dewey (1859–1952). Progressive educators believed that the school curriculum should be determined primarily by the needs and interests of children. Dewey advocated a school curriculum that encouraged students to be thinkers and problem solvers. He encouraged instructional methods that were experiential and child-centered, covering content arising naturally within the child’s environment. This method is in contrast to traditional instruction which is usually classroom-based and teacher-centered, covering predetermined content. During the Progressive movement, mathematics instruction emerged primarily when needed within the real-life experiences of the child and was thus widely varying

By the 1940s, an alternate version of Progressive Education called “Life Adjustment” had gained popularity among some U.S. schools. The curriculum of Life Adjustment schools was designed to prepare many of the students for the working world and everyday living, though some opponents claim it was motivated by anti-intellectual philosophies. These students focused on practical concerns, such as home budgeting, consumerism, taxation, health, and citizenship, and mathematics courses, such as algebra, geometry, and trigonometry, were deemphasized.

By the end of World War II, rapid societal and technological changes abruptly came to the forefront. Public knowledge of the impact of atomic energy, radar, cryptography, and other scientific and technological advances underscored the need for a strong national curriculum in mathematics and science to maintain national security, to retain the nation’s lead in technology, and to prepare students for jobs in the sciences. As a consequence, Progressive Education came under severe attack following World War II.

New Math

A momentous event occurred in 1957 that impacted the nation’s mathematics and science curricula at all levels. U.S. society was stunned by the launching of the first space satellite, Sputnik, by the Soviet Union. Sputnik was considered a national embarrassment and a potential security threat. Its mere existence suggested that the Soviet Union was technologically superior and had a military capacity of launching offensive missiles at the United States. It also underscored an overall weakness in the U.S. educational system, particularly in mathematics, science, and technology.

The U.S. Congress responded to the nation’s panic and the implications of a security threat by passing the 1958 National Defense Education Act, intended to increase the quantity and quality of mathematics and science professionals. That same year, the American Mathematical Society (AMS) established the School Mathematics Study Group (SMSG), headed by Edward G. Begle of Yale University, to develop a new mathematics curriculum for the nation’s high schools. The aim was to produce the most highly capable mathematics students in the world, with a view toward regaining the nation’s technological superiority and bolstering its defense system against the Soviet Union. This marked the beginning of the New Math movement.

Funded by the National Science Foundation, the SMSG created a new, more-rigorous high school mathematics curriculum for college-bound students and wrote textbooks supporting the new curriculum. The SMSG curriculum was developed by mathematics professionals consisting of working mathematicians, university professors, high school teachers, and school supervisors. The SMSG soon expanded its curriculum to include mathematics for grades K–12. Similar mathematics curricula emerged in the early 1960s, modeled after the original work of SMSG. These curricula were products of other federally funded projects such as the Ball State Project, Greater Cleveland Mathematics Program, the University of Maryland Mathematics Project, and the Minnesota School Science and Mathematics Center. Each curriculum mirrored the rigorous mathematics content and educational philosophy of the New Math movement.

The New Math curriculum included advanced content that had never before been covered in public schools, such as set theory, Boolean algebra, base arithmetic, field axioms, algebraic structures, and formal math language and symbolism. The curriculum was designed to provide the theoretical foundations for studying calculus and abstract algebra in college, with the intent of producing as many mathematics, science, and engineering majors as possible.

Even though there were numerous successes in the New Math movement, after a decade of implementation, it was slowly removed from the nation’s public schools. Some believed its downfall was a result of excessive rigor and mathematical formalism at the expense of basic skills and problem solving. Many parents and school administrators were confused by the unfamiliar mathematics content and advanced symbolism. More importantly, a large number of the nation’s older teachers were unable to implement the New Math curriculum because they, themselves, were not academically prepared to deal with the content. Notwithstanding, 12 years following the launching of Sputnik, the United States succeeded in placing the first man on the moon with its 1969 Apollo 11 mission.

Back to Basics

By the early 1970s, the New Math movement was over. The National Science Foundation discontinued its funding of New Math programs, and the U.S. public called for a return of a “Back to Basics” curriculum, under a Progressive agenda. Mathematical rigor and advanced symbolism were discouraged; teachers experimented with child-centered instructional approaches, such as Individualized Instruction, Continuous Progress, and Open School environments. The failures of such practices were soon exposed as standardized mathematics test scores of U.S. students steadily declined throughout the decade.

Fueled by the failure of the Back to Basics movement, the lowering of college entrance requirements, and reduced enrollments in higher-level mathematics courses, by the end of the 1970s, the U.S. school curriculum was once again under severe public attack. Two publications by U.S. federal agencies had a significant impact on public perceptions of the U.S. educational system, An Agenda for Action and A Nation at Risk.

An Agenda for Action

Based on multiple national assessments, the National Council of Teachers of Mathematics (NCTM) produced the 1980 An Agenda for Action, which provided recommendations for reform in mathematics education. Its primary recommendation was that problem solving should be the primary focus of the mathematics curriculum, supported by the following instructional practices:

• Calculators and computers should be used in K–12 classrooms.
• Estimation and approximation should be an integral part of instruction.
• Team efforts in problem solving should be encouraged in the elementary classroom.
• Manipulatives should be used to develop new mathematical concepts and skills.
• Instructional strategies should provide for situations requiring student discovery and inquiry.
• Mathematics programs and student performance should be evaluated on a broader range of measures than conventional testing.

A Nation at Risk

Although An Agenda for Action provided innovative and lofty recommendations for reform in the school mathematics, it was overshadowed by the 1983 publication of A Nation at Risk, a report by the National Commission on Excellence in Education. In graphic terms, it warned Americans, “The educational foundations of our society are presently being eroded by a rising tide of mediocrity that threatens our very future as a nation and a people,” and, “If an unfriendly foreign power had attempted to impose on America the mediocre educational performance that exists today, we might well have viewed it as an act of war.”

A variety of educational issues and specific weaknesses in the mathematics curriculum were addressed. Specifically, the commission found that the textbooks used for instruction were void of rigorous content, the curriculum lacked continuity and depth, and high school teachers were typically underprepared in mathematics and academically weak. Despite out-dated information, A Nation at Risk is still often quoted in the twenty-first century and remains an influential publication.

National Standards-Based Curriculum

The publication of A Nation at Risk and similar reports of the dismal performance of U.S. students on international assessments have all served to provoke U.S. society and government to demand higher academic standards in public schools. International assessments provided strong evidence that mathematics teaching and its school curriculum must change if U.S. students are to be competitive in the global economy and able to deal with the complex decisions they will confront as responsible citizens and members of a technologically literate workforce.

In 1989, NCTM took a giant step in recommending a national agenda for curriculum reform, resulting in NCTM’s 1989 publication of Curriculum and Evaluation Standards for School Mathematics. This document initiated a national standards-based curriculum movement, influenced by its earlier work reported in An Agenda for Action.

Within five years, NCTM also produced two supporting documents: the 1991 Professional Standards for Teaching Mathematics and the 1995 Assessment Standards for School Mathematics. These documents recommended teaching standards, instructional methodologies, and an array of assessment strategies for accommodating the new standards-based curriculum.

An updated version of NCTM’s original 1989 Standards was published in 2000, having a new title: Principles and Standards for School Mathematics. These three standards documents continue to be profoundly influential in the twenty-first century in matters of curriculum and assessment decisions for U.S. school mathematics. Specifically, Principles and Standards for School Mathematics provides six principles for school mathematics, five process standards, and five content standards.

The six principles for school mathematics are as follows:

• Equity—high expectations and strong support for all
• Curriculum—courses and learning experiences focused on important mathematics, well articulated across the grades
• Teaching—instruction that is challenging, supportive, and focused on what students know and need to learn
• Learning—develops understanding by building new knowledge on students’ experiences and prior knowledge
• Assessment—provides useful information to teachers and students and supports the learning of important mathematics
• Technology—essential for teaching and learning mathematics, influences the mathematics that is taught, and enhances student learning

The five process standards considered essential for teaching all mathematics, are problem solving, reasoning and proof, communication, connections, and representation. These processes were expected to be integrated into the teaching of all of mathematics, regardless of the topic or the grade level.

The five content standards include each of the following: number and operations, algebra, geometry, measurement, and data analysis and probability. Each content area is expected to be covered to some degree of integrity at every grade level.

Overall, given the breadth of mathematics applications found in modern society, both in work and everyday life, schools were encouraged to widen their mathematics offerings. For example, one recommendation for the high school curriculum was that calculus should not always be the primary goal for the mathematics curriculum. Instead, discrete mathematics, probability, and statistics should also be considered valuable goals. The curriculum must prepare students for a variety of career paths that use mathematics; for example, actuarial science (probability), engineering and electronics (technical mathematics), economics and behavioral science (statistics and decision theory), theoretical or nuclear physicist (calculus), and numerous others.

In 2006, NCTM released another supporting document, Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, which articulated the specific topics that should be the focus for each grade level pre-K–8. The curriculum focal points acknowledged that NCTM’s five content standards are not equally weighted and should have greater emphasis at different grade levels. These topics are identified in this document.

In 2009, NCTM released Focus in High School Mathematics: Reasoning and Sense Making. This publication was designed to provide teachers with curriculum guidance and content focal points for high school mathematics, modeled after NCTM’s 2006 pre-K–8 document, Curriculum Focal Points. In this 2009 document, NCTM stresses that reasoning and sense making should be the focus of all high school mathematics, spanning all content areas, and evident in the teachers’ instructional strategies and assessment practices. The goal is for mathematics to be viewed as a logical, problem-solving tool, rather than a set of meaningless procedures, disconnected from everyday life and decision making. It is stressed that students should have experiences with reasoning and sense making within a broad curriculum that may deviate from the textbook. Such experiences should be designed to meet students’ future needs and prepare them for citizenship, the workplace, and future careers.

Twenty-First-Century Mathematics Curriculum

History has shown that as national needs and societal perceptions change, so does the school curriculum. Pervasive and radical changes have occurred throughout the world since the 1990s, such as genetic engineering, nanotechnologies, global economies, environmental disasters, global warming, depleting energy sources, and countless others. It is clear that U.S. citizens must be prepared to deal creatively and competently with a multitude of rapid changes and to tackle complex problem situations. The school curriculum must respond accordingly to provide students with the content knowledge, problem-solving skills, and learning experiences that are necessary for students to meet these immense challenges.

Unfortunately, the U.S. mathematics curriculum still has a long way to go in preparing students to meet these challenges. In fact, international assessments report serious deficiencies in the mathematical performance of U.S. students. In 2009, the Programme for International Student Assessment (PISA) reported that 15-year-olds from the United States ranked 18 among 33 developed nations in mathematical literacy and problem solving. In sum, compared to other developed nations, students from U.S. schools score in the lower 50th percentile in mathematics. Furthermore, the National Center for Education Statistics reported in 2003 significant racial achievement gaps in the United States. Societal concerns for economic stability, national security, and equity in instruction all demand immediate and substantial reforms in the U.S. mathematics curriculum and educational system as a whole.

No Child Left Behind

Research has shown that the school curriculum is closely tied to assessment. One governmental attempt to address the school curriculum and the lagging academic achievement of U.S. students is the No Child Left Behind (NCLB) Act of 2002. NCLB includes a number of mandates designed to promote significant gains in student achievement and to hold states and schools accountable for meeting curricular goals. NCLB supports a standards-based curriculum and was founded on the belief that setting high standards and measurable objectives would result in improved teaching and learning in the nation’s schools.

As a provision for federal funding, the NCLB Act requires that states develop assessments in basic skills for students at specific grade levels and that each state set its own curriculum, content standards, and achievement benchmarks. The Act further mandates that 100% of the students in each school be proficient in reading and mathematics by the year 2014. As a consequence, any school not showing significant progress toward meeting these goals will be subject to sanctions, culminating in the closing of the school and termination of the faculty and staff.

The underlying theory is that schools will show significant improvement if children in grades 3–8 are held accountable for their academic achievement, as measured by their test scores every year. As of 2011, more than 10,000 schools have been labeled as “failures”; thousands of teachers have been fired; and numerous schools, heretofore considered “very good,” are being forced to close. To meet NCLB goals, many schools have eliminated studies in art, history, science, foreign languages, physical education, and geography from their offerings. The majority of school time is now devoted to preparing students for high-stakes tests in the basic skills, the results of which will determine if the school remains open for the following year.

The consequences of students’ test performance are so punitive that some districts have experienced record amounts of cheating. Some states have even lowered the passing score on their annual mathematics exams to increase the pass rates for their schools. Reactions such as these to the mandates of the NCLB Act underscore the fact that testing alone will not increase student achievement nor improve instruction. Regardless of how well the curriculum is constructed, meaningful instruction will be abandoned for the sake of test preparation.

Common Core State Standards

Several of the nations to which the United States is often compared academically do have national curricula, such as Great Britain, Germany, France, and Japan. Even though NCTM has provided national guidelines for mathematics education, until 2010, nearly every state had its own unique set of mathematics standards and curriculum for each grade level. In some cases, decisions about curricula were made by county and local school districts and boards. Consequently, state mathematics standards have varied considerably from state to state, and valid comparisons are difficult to make with respect to student performance. Because of the absence of a common set of standards among states, 48 of the nation’s state governors and their chief school officers set forth to create the Common Core State Standards (CCSS), released in 2009.

The CCSS were developed in collaboration with content experts, college professors, public school teachers, school administrators, and parents. They are designed for a curriculum that includes rigorous content and applications; requires high-order thinking skills; and prepares students to succeed in a global economy. They are also aligned with the mathematics curricula of top-performing countries in the world. As of 2011, 41 of the 50 states have adopted CCSS.

Race to the Top

The rapid adoption of the CCSS by nearly every state in the nation was surely spurred by the Race to the Top (RTTT) program funded the Educational Recovery Act of 2009. RTTT is a $4.35 billion U.S. Department of Education program offering competitive grants designed to promote educational reforms in state education. The underlying federal agenda is to establish national standards, tests, and curricula. Even though the principle of states’ rights ensure that individual states have total control over their educational systems, the promise of RTTT’s discretionary funding of hundreds of millions of dollars is a huge incentive for states to adopt the CCSS, which is prerequisite to RTTT funding. When states receive RTTT discretionary federal funding, they must agree to implement the CCSS as well as comply with other stipulations.

Consistent with NCLB, state assessments for RTTT’s funding are highly reliant on students’ test scores as the sole measure of student achievement. Additionally, many states intend to use students’ test scores to evaluate their teachers’ performance and determine salaries and bonuses. As of 2011, there is also a rising movement among state governors to push for an end to teachers’ unions, tenure, and rights to due process, many of which have existed since at least the early twentieth century. It appears that if these movements continue in the twenty-first century, teachers will soon have no organized voice for addressing teaching conditions, budgetary concerns, or program and curricular issues.

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