Mathematics education in the United States

Attendance and completion rates

For many students, passing algebra is often a Herculean challenge, so much so that many students have dropped out of high school because of it. The greatest obstacle for excelling in algebra is fluency with fractions, something many Americans do not have. Without mastery of high-school algebra—Algebra I and II—students will not be able to pursue collegiate STEM courses. In fact, the lack of adequate preparation in mathematics is part of the reason why the rate of attrition in STEM is so high. From 1986 to 2012, though more students were completing Algebra II, their average performance has fallen. Indeed, students who had passed high-school courses, including those labeled "honors" courses, might still fail collegiate placement exams and had to take remedial courses. As for Algebra I, the number of 13-year-olds enrolled fell from 34% in 2012 to 24% in 2023.

Longitudinal analysis shows that the number of students completing high-school courses on calculus and statistics, including AP courses, have declined before 2019. Data taken from students' transcripts (${\displaystyle N=15,188}$) from the late 2000s to the mid-2010s reveals that majorities of students had completed Algebra I (96%), Geometry (76%), and Algebra II (62%). But not that many took Precalculus (34%), Trigonometry (16%), Calculus (19%), or Statistics (11%) and only an absolute minority took Integrated Mathematics (7%). Overall, female students were more likely to complete all mathematics courses, except Statistics and Calculus. Asian Americans were the most likely to take Precalculus (55%), Statistics (22%), and Calculus (47%) while African Americans were the least likely to complete Calculus (8%) but most likely to take Integrated Mathematics (10%) in high school. Among students identified as mathematically proficient by the PSAT, Asians are much more likely than blacks to attend an honors or Advanced Placement course in mathematics. Asians are also the most likely to have scored at least a 3 on the AP Calculus exams. Students of lower socioeconomic status were less likely to pass Precalculus, Calculus, and Statistics. While boys and girls are equally likely to take AP Statistics and AP Calculus AB, boys are the majority in AP Calculus BC (59%), as well as some other highly mathematical subjects, such as AP Computer Science A (80%), AP Physics C: Mechanics (74%) and AP Physics C: Electricity and Magnetism (77%). Although undergraduate men and women score the same grades in Calculus I (in college) on average, women are more likely than men to drop out because of mathematical anxiety. Perceptions and stereotypes of girls being less mathematically able than boys begin as early as second grade, and they affect how girls actually perform in class or in a competition, such as the International Mathematical Olympiad. Among university students who have taken calculus, engineering disciplines are the most popular among men and biology among women.

During the 1970s and 1980s, the number of students taking remedial courses in college rose substantially, partly due to the de-emphasis of calculus in high school, leading to less exposure to pre-calculus topics. In the twenty-first century, American community colleges require 60% of their students to pass at least one course in mathematics, depending on the program. But around 80% fail to meet this requirement, and 60% require remedial courses. Many students at these schools drop out after failing even in remedial courses, such as (the equivalent of) Algebra II. On the other hand, four-year institutions have seen increased student interest in STEM programs, including mathematics and statistics.

Controversies and issues

Mathematics education has been a topic of debate among academics, parents, as well as educators. Majorities agree that mathematics is crucial, but there has been many divergent opinions on what kind of mathematics should be taught and whether relevance to the "real world" or rigor should be emphasized. Another source of contention is the decentralized nature of American education, making it difficult to introduce standard curriculum implemented nationwide, despite the benefits of such a program as seen from the experience of other countries, such as Italy. In the early 2020s, the decision by some educators to include the topics of race and sexuality into the mathematical curriculum has also met with stiff resistance.

Progressive education

During the first half of the twentieth century, there was a movement aimed at systematically reforming American public education along more "progressive" grounds. William Heard Kilpatrick, one of the most vocal proponents of progressive education, advocated for the de-emphasis of intellectual "luxuries" such as algebra, geometry, and trigonometry, calling them "harmful rather than helpful to the kind of thinking necessary for ordinary living." He recommended that more advanced topics in mathematics should only be taught to the select few. Indeed, prior to the Second World War, it was common for educationists to argue against the teaching of academic subjects and in favor of more utilitarian concerns of "home, shop, store, citizenship, and health," presuming that a majority of high school students could not embark on a path towards higher education but were instead, destined to become unskilled laborers or their wives.

By the 1940s, however, the deficiency in mathematical skills among military recruits became a public scandal. Admiral Chester Nimitz himself complained about the lack of skills that should have been taught in public schools among officers in training and volunteers. In order to address this issue, the military had to open courses to teach basic skills such as arithmetic for bookkeeping or gunnery.

Indeed, many parents opposed the progressive reforms, criticizing the lack of contents. By mid-century, technological marvels, such as radar, nuclear energy, and the jet engine, made progressive education untenable.

New Math

Under the New Math initiative, created after the successful launch of the Soviet satellite Sputnik in 1957, conceptual abstraction rather than calculation gained a central role in mathematics education. The educational status quo was severely criticized as a source of national humiliation and reforms were demanded, prompting Congress to introduce the National Defense Education Act of 1958. The U.S. federal government under President Dwight D. Eisenhower realized it needed thousands of scientists and engineers to match the might of its ideological rival the Soviet Union and started pouring enormous sums of money into research and development as well as education. Conceived in response to the lack of emphasis on content of the progressive education and the technological advances of World War II, New Math was part of an international movement influenced by the Nicholas Bourbaki school in France, attempting to bring the mathematics taught in schools closer to what research mathematicians actually use. Students received lessons in set theory, which is what mathematicians actually use to construct the set of real numbers, normally taught to advanced undergraduates in real analysis (see Dedekind cuts and Cauchy sequences). Arithmetic with bases other than ten was also taught (see binary arithmetic and modular arithmetic). Other topics included number theory, probability theory, and analytic geometry.

However, this educational initiative soon faced strong opposition, not just from teachers, who struggled to understand the new material, let alone teach it, but also from parents, who had problems helping their children with homework. It was criticized by experts, too. In a 1965 essay, physicist Richard Feynman argued, "first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worthwhile teaching such material." In his 1973 book, Why Johnny Can't Add: the Failure of the New Math, mathematician and historian of mathematics Morris Kline observed that it was "practically impossible" to learn new mathematical creations without first understanding the old ones, and that "abstraction is not the first stage, but the last stage, in a mathematical development." Kline criticized the authors of the 'New Math' textbooks, not for their mathematical faculty, but rather their narrow approach to mathematics, and their limited understanding of pedagogy and educational psychology. Mathematician George F. Simmons wrote in the algebra section of his book Precalculus Mathematics in a Nutshell (1981) that the New Math produced students who had "heard of the commutative law, but did not know the multiplication table."

By the early 1970s, this movement was defeated. Nevertheless, some of the ideas it promoted still lived on. One of the key contributions of the New Math initiative was the teaching of calculus in high school.

Standards-based reforms and the NCTM

From the late twentieth century to the early twenty-first, there has been a fierce debate over how mathematics should be taught. On one hand, some campaign for a more traditional teacher-led curriculum, featuring algorithms and some memorization. On the other hand, some prefer a conceptual approach, with a focus on problem-solving and the sense of numbers. However, some instructors and researchers have cast doubt on the utility of "classroom struggles" in helping students understand the concepts. Moreover, as mathematician Hung-Hsi Wu explained, the apparent dichotomy between basic skills and understanding of mathematical concepts is a delusion.

In 1989 the National Council for Teachers of Mathematics (NCTM) produced the Curriculum and Evaluation Standards for School Mathematics. Widespread adoption of the new standards notwithstanding, the pedagogical practice changed little in the United States during the 1990s. In fact, mathematics education became a hotly debated subject in the 1990s and early 2000s. This debate pitted mathematicians (like UC Berkeley mathematician Hung-Hsi Wu) and parents, many of whom with substantial knowledge of mathematics (such as the Institute for Advanced Study physicist Chiara R. Nappi), who opposed the NCTM's reforms against educational professionals, who wanted to emphasized what they called "conceptual understanding." In many cases, however, educational professionals did not understand mathematics as well as their critics. This became apparent with the publication of the book Knowing and Teaching Elementary Mathematics (1999) by Liping Ma. The author gave evidence that even though most Chinese teachers had only 11 or 12 years of formal education, they understood basic mathematics better than did their U.S. counterparts, many of whom were working on their master's degrees.

In 1989, the more radical NCTM reforms were eliminated. Instead, greater emphasis was put on substantive mathematics. In some large school districts, this came to mean requiring some algebra of all students by ninth grade, compared to the tradition of tracking only the college-bound and the most advanced junior high school students to take algebra. A challenge with implementing the Curriculum and Evaluation Standards was that no curricular materials at the time were designed to meet the intent of the Standards. In the 1990s, the National Science Foundation funded the development of curricula such as the Core-Plus Mathematics Project. In the late 1990s and early 2000s, the so-called math wars erupted in communities that were opposed to some of the more radical changes to mathematics instruction. Some students complained that their new math courses placed them into remedial math in college. However, data provided by the University of Michigan registrar at this same time indicate that in collegiate mathematics courses at the University of Michigan, graduates of Core-Plus did as well as or better than graduates of a traditional mathematics curriculum, and students taking traditional courses were also placed in remedial mathematics courses. Mathematics instructor Jaime Escalante dismissed the NCTM standards as something written by a PE teacher.

In 2001 and 2009, NCTM released the Principles and Standards for School Mathematics (PSSM) and the Curriculum Focal Points which expanded on the work of the previous standards documents. Particularly, the PSSM reiterated the 1989 standards, but in a more balanced way, while the Focal Points suggested three areas of emphasis for each grade level. Refuting reports and editorials that it was repudiating the earlier standards, the NCTM claimed that the Focal Points were largely re-emphasizing the need for instruction that builds skills and deepens student mathematical understanding. These documents repeated the criticism that American mathematics curricula are a "mile wide and an inch deep" in comparison to the mathematics of most other nations, a finding from the Second and Third International Mathematics and Science Studies.

Integrated mathematics

As previously stated, American children usually follow a unique sequence of mathematics courses in secondary school (grades 6 to 12), learning one subject at a time. They take two years of Algebra punctuated by a year of Geometry. Geometry, hitherto a collegiate course, was introduced into high schools in the nineteenth century. In Europe, schools followed Felix Klein's call for Geometry to be integrated with other math subjects. In 1892 the American Committee of Ten recommended the same strategy for the United States, but American teachers had already been developing the habit of teaching Geometry as a separate course. The American high-school geometry curriculum was eventually codified in 1912 and developed a distinctive American style of geometric demonstration for such courses, known as "two-column" proofs. This remains largely true today, with Geometry as a proof-based high-school math class. On the other hand, many countries around the world from Israel to Italy teach mathematics according to what Americans call an integrated curriculum, familiarizing students with various aspects of calculus and prerequisites throughout secondary school. In fact, many topics in Algebra and Geometry that Americans typically learn in high school are taught in middle school in Europe, making it possible for European countries to require and to teach calculus in high school. In France and Germany, calculus was brought into the secondary-school curriculum thanks to the advocacy of famous mathematicians, such as Henri Poincaré and Felix Klein, respectively. However, as the Singaporean case demonstrates, early exposure to the concepts of calculus does not necessarily translates to actual understanding among high school students. In the U.S., this is reflected in the concerns voiced by many university professors, according to whom their students lack sufficient preparation in pre-calculus mathematics. Proponents of teaching the integrated curriculum believe that students would better understand the connections between the different branches of mathematics. On the other hand, critics—including parents and teachers—prefer the traditional American approach both because of their familiarity with it and because of their concern that certain key topics might be omitted, leaving the student ill-prepared for college. As mentioned above, only 7% of American high school students take Integrated Mathematics.

Preparation for college

Beginning in 2011, most states have adopted the Common Core Standards for mathematics, which were partially based on NCTM's previous work. Controversy still continues as critics point out that Common Core standards do not fully prepare students for college and as some parents continue to complain that they do not understand the mathematics their children are learning. Indeed, even though they may have expressed an interest in pursuing science, technology, engineering, and mathematics (STEM) in high school, many university students find themselves ill-equipped for rigorous STEM education in part because of their inadequate preparation in mathematics. Meanwhile, Chinese, Indian, and Singaporean students are exposed to high-level mathematics and science at a young age. About half of STEM students in the U.S. dropped out of their programs between 2003 and 2009. On top of that, many mathematics schoolteachers were not as well-versed in their subjects as they should be, and might well be uncomfortable with mathematics themselves. An emphasis on speed and rote memorization gives as many as one-third of students aged five and over mathematical anxiety.

Parents and high school counselors consider it crucial that students pass Calculus if they aim to be admitted to a competitive university. Private school counselors are especially likely to make this recommendation while admissions officers are generally less inclined to consider it a requirement. Moreover, there has been a movement to de-emphasize the traditional pathway with Calculus as the final mathematics class in high school in favor of Statistics and Data Science for those not planning to major in a STEM subject in college. Nevertheless, Calculus remains the most recommended course for ambitious students. But in the case of Utah, as of 2023, students may skip the final required course for high-school graduation—one that combines elements of Algebra II, Trigonometry, Precalculus, and Statistics—if they submit a letter signed by their parents acknowledging that this decision could jeopardize their chances of university matriculation.

By the mid-2010s, only a quarter of American high school seniors are able to do grade-level math, yet about half graduate from high school as A students, prompting concerns of grade inflation. Strong performance in Algebra I, Geometry, and Algebra II predict good grades in university-level Calculus even better than taking Calculus in high school.

Another issue with mathematics education has been integration with science education. This is difficult for public schools to do because science and math are taught independently. The value of the integration is that science can provide authentic contexts for the math concepts being taught and further, if mathematics is taught in synchrony with science, then the students benefit from this correlation.

Enrichment programs and accelerated tracks

Growing numbers of parents have opted to send their children to enrichment and accelerated learning after-school or summer programs in mathematics, leading to friction with school officials who are concerned that their primary beneficiaries are affluent white and Asian families, prompting parents to pick private institutions or math circles. Some public schools serving low-income neighborhoods even denied the existence of mathematically gifted students. In fact, American educators tend to focus on poorly performing students rather than those at the top, unlike their Asian counterparts. Parents' proposal for an accelerated track for their children are oftentimes met with hostility by school administrators. Conversely, initiatives aimed at de-emphasizing certain core subjects, such as Algebra I, triggered strong backlash from parents and university faculty members. Students identified by the Study of Mathematically Precocious Youth as top scorers on the mathematics (and later, verbal) sections of the SAT often became highly successful in their fields. By the mid-2010s, some public schools have begun offering enrichment programs to their students.

Similarly, while some school districts have proposed to stop separating students by mathematical ability in order to ensure they begin high school at the same level, parents of gifted children have pushed back against this initiative, fearing that it would jeopardize their children's future college admissions prospects, especially in the STEM fields. In San Francisco, for example, such a plan was dropped due to a combination of mixed results and public backlash.

Shortage of instructors

A shortage of qualified mathematics schoolteachers has been a serious problem in the United States for many years. In order to rectify this problem, the amount of instructional hours dedicated towards mathematical contents has been increased in undergraduate programs aimed at training elementary teachers. Teachers oftentimes unknowingly transmit their own negative attitude towards mathematics to their students, damaging the quality of instruction.

Standardized tests

In 2002, President George W. Bush signed into law the No Child Left Behind Act, holding schools accountable for how their students performed on exams. It was repealed in 2015 with the passage of the Every Student Succeeds Act, transferring power back to the states, resulting in more relaxed standards.

The Program for International Student Assessment (PISA) is held every three years for 15-year-old students worldwide. In 2012, the United States earned average scores in science and reading. It performed better than other progressive nations in mathematics, ranking 36 out of 65 other countries. The PISA assessment examined the students' understanding of mathematics as well as their approach to this subject and their responses. These indicated three approaches to learning. Some of the students depended mainly on memorization. Others were more reflective on newer concepts. Another group concentrated more on principles that they have not yet studied. The U.S. had a high proportion of memorizers compared to other developed countries. During the 2015 testing, the United States failed to make it to the top 10 in all categories including mathematics. More than 540,000 teens from 72 countries took the exam. American students' average score in mathematics declined by 11 points compared to the previous testing. The 2022 PISA test showed that U.S. national average in mathematics remained behind those of other industrialized nations and remained below the OECD average. Furthermore, one third of American students did not meet the requirements for basic proficiency in mathematics.

However, European- and especially Asian-American students perform above the OECD average. See chart below.

According to a 2021 report by the National Science Foundation (NSF), American students' mathematical literacy ranks 25th out of 37 nations of the Organization for Economic Cooperation and Development (OECD).

During the 2000s and 2010s, as more and more college-bound students take the SAT, scores have gone down. (See chart below.) This is in part because some states have required all high school students to take the SAT, regardless of whether or not they were going to college.

In 2015, educational psychologist Jonathan Wai of Duke University analyzed average test scores from the Army General Classification Test in 1946 (10,000 students), the Selective Service College Qualification Test in 1952 (38,420), Project Talent in the early 1970s (400,000), the Graduate Record Examination between 2002 and 2005 (over 1.2 million), and the SAT Math and Verbal in 2014 (1.6 million). Wai identified one consistent pattern: those with the highest test scores tended to pick mathematics and statistics, the natural and social sciences, and engineering as their majors while those with the lowest were more likely to choose healthcare, education, and agriculture. (See the two charts below.)

Results from the National Assessment of Educational Progress (NAEP) test show that scores in mathematics have been leveling off in the 2010s, but with a growing gap between the top and bottom students. The COVID-19 pandemic, which forced schools to shut down and lessons to be given online, further widened the divide, as the best students lost fewer points compared to the worst and therefore could catch up more quickly. While students' scores fell for all subjects, mathematics was the hardest hit, with a drop of eight points, the steepest decline in 50 years. Scores dropped for students of all races, sexes, socioeconomic classes, types of schools, and states with very few exceptions. This might be because mathematics education is more dependent upon the classroom experience than reading, as students who were allowed to return to in-person classes generally did better, more so in mathematics than in reading. However, on the topics of statistics and probability, student performance had already declined before the pandemic. As consequence, the entire cohort of college students in the 2022-23 academic year have lower average grades and mathematical standards. But there were noticeable improvements in mathematical proficiency among schoolchildren across the United States between 2022 and 2025.

A 2023 comparison between parents' views and standardized test scores revealed a significant gap; most parents overestimated their children's academic aptitude. In mathematics, only 26% were proficient, even though 90% of the parents asked thought their children met grade standards. Having a higher NAEP math score in eighth grade is correlated with high academic standing, higher income, lower rates of adolescent parenthood, and lower chances of criminality.

Advanced Placement Mathematics

There was considerable debate about whether or not calculus should be included when the Advanced Placement (AP) Mathematics course was first proposed in the early 1950s. AP Mathematics has eventually developed into AP Calculus thanks to physicists and engineers, who convinced mathematicians of the need to expose students in these subjects to calculus early on in their collegiate programs. By the mid-2020s, AP Calculus AB and AP Statistics are within the top ten most popular AP exams.

In the early 21st century, there has been a demand for the creation of AP Multivariable Calculus and indeed, a number of American high schools have begun to offer this class, giving colleges trouble in placing incoming students.

As of 2021, AP Precalculus was under development by the College Board, though there were concerns that universities and colleges would not grant credit for such a course, given that students had previously been expected to know this material prior to matriculation. AP Precalculus launched in Fall 2023. Attendance grew rapidly during the first few years. According to the College Board, majorities of students to took AP Precalculus went on to take AP Calculus AB or BC and at least on AP course in STEM or some other subject area.

Conferences

Mathematics education research and practitioner conferences include: NCTM's Regional Conference and Exposition and Annual Meeting and Exposition; The Psychology of Mathematics Education's North American Chapter annual conference; and numerous smaller regional conferences.