Mathematics education in the United States

Multiplication seen as scaling on the number line

Primary schoolchildren learn counting, arithmetic and properties of operations, geometry, measurement, statistics and probability. They typically begin studying fractions in third grade.

James Garfield's proof of the Pythagorean theorem.

Pre-algebra can be taken in middle school. Students typically begin by learning about real numbers and basic number theory (prime numbers, prime factorization, fundamental theorem of arithmetic, ratios, and percentages), topics needed for algebra (powers, roots, graphing, order of operations, variables, expressions, and scientific notation) and geometry (quadrilaterals, polygons, areas of plane figures, the Pythagorean theorem, distance formula, equations of a line, simple solids, their surface areas, and volumes), and sometimes introductory trigonometry (definitions of the trigonometric functions). Such courses usually then go into simple algebra with solutions of simple linear equations and inequalities.

Algebra I is the first course students take in algebra. Although some students take it as eighth graders, this class is most commonly taken in ninth or tenth grade,[30] after the students have taken Pre-algebra. Students learn about real numbers and the order of operations (PEMDAS), functions, linear equations, graphs, polynomials, the factor theorem, radicals, and quadratic equations (factoring, completing the square, and the quadratic formula), and power functions.

This course is considered a gatekeeper for those who want to pursue STEM[11] because taking Algebra I in eighth grade allows students to eventually take Calculus before graduating from high school.[31] As such, tracking students by their aptitude and deciding when they should take Algebra I has become topic of controversy in California[32] and Massachusetts.[33] Parents of high-performing students are among the most vocal critics of policies discouraging the taking of Algebra I in middle school.[32][33]

Geometry, usually taken in ninth or tenth grade, introduces students to the notion of rigor in mathematics by way of some basic concepts in mainly Euclidean geometry. Students learn the rudiments of propositional logic, methods of proof (direct and by contradiction), parallel lines, triangles (congruence and similarity), circles (secants, chords, central angles, and inscribed angles), the Pythagorean theorem, elementary trigonometry (angles of elevation and depression, the law of sines), basic analytic geometry (equations of lines, point-slope and slope-intercept forms, perpendicular lines, and vectors), and geometric probability. Students are traditionally taught to prove simple geometric theorems using two-column proofs, a method developed in the early 20th century in the U.S. specifically for this course, though other methods may also be used in some courses.[34] Depending on the curriculum and instructor, students may receive orientation towards calculus, for instance with the introduction of the method of exhaustion and Cavalieri's principle.

Visual proof of the double-angle identity for the sine

Algebra II has Algebra I as a prerequisite and is traditionally a high-school-level course. Course contents include inequalities, quadratic equations, power functions, exponential functions, logarithms, systems of linear equations, matrices (including matrix multiplication, ${\displaystyle 2\times 2}$ matrix determinants, Cramer's rule, and the inverse of a ${\displaystyle 2\times 2}$ matrix), the radian measure, graphs of trigonometric functions, trigonometric identities (Pythagorean identities, the sum-and-difference, double-angle, and half-angle formulas, the laws of sines and cosines), conic sections, among other topics.

Requiring Algebra II for high school graduation gained traction across the United States in the early 2010s.[35] The Common Core mathematical standards recognize both the sequential as well as the integrated approach to teaching high-school mathematics, which resulted in increased adoption of integrated math programs for high school. Accordingly, the organizations providing post-secondary education updated their enrollment requirements. For example, the University of California (UC) system requires three years of "college-preparatory mathematics that include the topics covered in elementary and advanced algebra and two- and three-dimensional geometry"[36] to be admitted. After the California Department of Education adopted the Common Core, the UC system clarified that "approved integrated math courses may be used to fulfill part or all"[36] of this admission requirement. On the other hand, in a controversial decision, the Texas Board of Education voted to remove Algebra II as a required course for high school graduation.[37]

In California, suggestions that Algebra II should be de-emphasized in favor of Data Science (a combination of algebra, statistics, and computer science) has faced severe criticism out of concerns that such a pathway would leave students ill-prepared for collegiate education. In 2023, the faculty of the University of California system voted to end an admissions policy that accepts Data Science in lieu of Algebra II.[32]

Pascal's arithmetic triangle appears in combinatorics as well as algebra via the binomial theorem.

Precalculus follows from the above, and is usually taken by college-bound students. Pre-calculus combines algebra, analytic geometry, trigonometry, and analytic trigonometry. Topics in algebra include the binomial theorem, complex numbers, the Fundamental Theorem of Algebra, root extraction, polynomial long division, partial fraction decomposition, and matrix operations. In the chapters on analytic geometry, students are introduced to polar coordinates and deepen their knowledge of conic sections. In the components of (analytic) trigonometry, students learn the graphs of trigonometric functions, trigonometric functions on the unit circle, the dot product, the projection of one vector onto another, and how to resolve vectors. If time and aptitude permit, students might learn Heron's formula, how to calculate the determinant of a ${\displaystyle 3\times 3}$ matrix via the rule of Sarrus, and the vector cross product. Students are introduced to the use of a graphing calculator to help them visualize the plots of equations and to supplement the traditional techniques for finding the roots of a polynomial, such as the rational root theorem and the Descartes rule of signs. Precalculus ends with an introduction to limits of a function. Some instructors might give lectures on mathematical induction and combinatorics in this course.[38][39][40]

AP Precalculus has only three required chapters. polynomial and rational functions, exponential and logarithmic functions, and trigonometric functions and polar curves. Optional materials include parametric equations, implicit functions, conic sections, vectors, and matrix algebra (${\displaystyle 2\times 2}$ matrix inversion, ${\displaystyle 2\times 2}$ determinants, and linear transformations).[41]

Depending on the school district, several courses may be compacted and combined within one school year, either studied sequentially or simultaneously. For example, in California, Algebra II and Precalculus may be taken as a single compressed course.[31] Without such acceleration, it may be not possible to take more advanced classes like calculus in high school.

In Oregon, high-school juniors and seniors may choose between three separate tracks, depending on their interests. Those aiming for a career in mathematics, the physical sciences, and engineering can pursue the traditional pathway, taking Algebra II and Precalculus. Those who want to pursue a career in the life sciences, the social sciences, or business can take Statistics and Mathematical Modelling. Students bound for technical training can take Applied Mathematics and Mathematical Modelling.[42] Students may also receive lessons on mathematical logic and set theory at various grade levels in high school, and this is the case in Florida, following the new 2020 reforms.[43] The new Floridian standards also promote financial literacy and emphasize how different mathematical topics from different grade levels are connected.[44]

College algebra is offered at many community colleges as remedial courses.[45] It should not be confused with abstract algebra and linear algebra, taken by students who major in mathematics and allied fields (such as computer science) in four-year colleges and universities.

Illustration of the epsilon-delta definition of the limit of a function

Calculus is usually taken by high-school seniors or university freshmen, but can occasionally be taken as early as tenth grade. Unlike many other countries from France to Israel to Singapore, which require high school students aiming for a career in STEM or placed in the track for advanced mathematics to study differential and integral calculus, the United States generally treats calculus as collegiate mathematics. A successfully completed college-level calculus course like one offered via Advanced Placement program (AP Calculus AB and AP Calculus BC) is a transfer-level course—that is, it can be accepted by a college as a credit towards graduation requirements. Prestigious colleges and universities are believed to require successful completion AP courses, including AP Calculus, for admissions.[46][47]

In this class, students learn about limits and continuity (the intermediate and mean value theorems), differentiation (the product, quotient, and chain rules) and its applications (implicit differentiation, logarithmic differentiation, related rates, optimization, concavity, Newton's method, L'Hôpital's rules), integration and the Fundamental Theorem of Calculus, techniques of integration (u-substitution, by parts, trigonometric and hyperbolic substitution, and by partial fractions decomposition), further applications of integration (calculating accumulated change, various problems in the sciences and engineering, separable ordinary differential equations, arc length of a curve, areas between curves, volumes and surface areas of solids of revolutions), improper integrals, numerical integration (the midpoint rule, the trapezoid rule, Simpson's rule), infinite sequences and series and their convergence (the nth-term, comparison, ratio, root, integral, p-series, and alternating series tests), Taylor's theorem (with the Lagrange remainder), Newton's generalized binomial theorem, Euler's complex identity, polar representation of complex numbers, parametric equations, and curves in polar coordinates.[48][49][50][51]

Depending on the course and instructor, special topics in introductory calculus might include the classical differential geometry of curves (arc-length parametrization, curvature, torsion, and the Frenet–Serret formulas), the epsilon-delta definition of the limit, first-order linear ordinary differential equations, Bernoulli differential equations.[48][50] Some American high schools today also offer multivariable calculus[52] (partial differentiation, the multivariable chain rule and Clairault's theorem; constrained optimization, Lagrange multipliers and the Hessian; multidimensional integration, Fubini's theorem, change of variables, and Jacobian determinants; gradients, directional derivatives, divergences, curls, the fundamental theorem of gradients, Green's theorem, Stokes' theorem, and Gauss' theorem).[48][50][51]

Other optional mathematics courses may be offered, such as statistics (including AP Statistics) or business math. Students learn to use graphical and numerical techniques to analyze distributions of data (including univariate, bivariate, and categorical data), the various methods of data collection and the sorts of conclusions one can draw therefrom, probability, and statistical inference (point estimation, confidence intervals, and significance tests).

High school students of exceptional ability may be selected to join a competition, such as the USA Mathematical Olympiad,[53][54] or the International Mathematical Olympiad.[55][56]

An illustration of Stokes' theorem in vector calculus

All students in STEM, especially mathematics, physics, chemistry, computer science, and engineering must take single-variable calculus unless they have Advanced Placement credits (or equivalents, such as IB Math HL). Students majoring in mathematics, the physical sciences,[57][58] and engineering[59] then take multivariable calculus,[48][50][51] linear algebra,[60][61][62] complex variables,[63][64] ordinary differential equations,[65][66][67] and partial differential equations.[68][69][70] In addition, mathematics students study abstract algebra,[71][72] number theory,[73][74] real analysis,[75][76][77][78] advanced calculus,[79][80][81] complex analysis,[82][83][84][85] probability theory,[86] statistics,[87][88] and advanced topics, such as set theory and mathematical logic,[89][90][91][92] stochastic processes,[93] integration and measure theory,[94][95][96] Fourier analysis,[97][98] functional analysis,[99] differential geometry,[100][101][102] and topology.[103] They may further choose courses in applied mathematics, such as mathematical modelling, game theory, or mathematical optimization. The calculus of variations,[104][105][106] the history of mathematics,[107][108][109][110] and topics in theoretical or mathematical physics (such as classical mechanics,[111][105][112][113] electrodynamics,[114][115] fluid mechanics,[116][117] quantum mechanics,[118][119][120] or general relativity[121][122][123][124]) may be taken as electives. Computer science majors must take courses on discrete mathematics[125][126] (such as combinatorics and graph theory), information theory,[127] theory of computation,[128][129] and cryptography. Those who study biomedical and social sciences have to study elementary probability[130] and statistics.[131] Students in computer science and economics might have the option of taking algorithmic game theory.[132] Students in the physical sciences and engineering need to understand error analysis for their laboratory sessions and courses.[133][134] Advanced undergraduates and beginning graduate students in physics may take a course on advanced mathematical methods for physics.[135][136][137][138][139][140] Exact requirements and available courses will depend on the institution in question.

At many colleges and universities, confident students may compete in the Integration Bee.[141][142][143][144] Exceptional undergraduates may participate in the annual William Lowell Putnam Mathematical Competition.[145][146] Many successful competitors have gone on to fruitful research careers in mathematics. Although doing well on the Putnam is not a requirement for becoming a mathematician, it encourages students to develop skills and hone intuitions that could help them become successful researchers.[145][147] Besides the monetary prize, the winners are virtually guaranteed acceptance to a prestigious graduate school.[148] Such competitions are one way for mathematical talents to stand out.[149]

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.[23]

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.[23]

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.[23]

Navajo students learning arithmetic (c. 1940). By the late 1950s, mathematics education had become more rigorous.

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.[24] 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.[23] 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.[164][165] Conceived in response to the lack of emphasis on content of the progressive education[23] and the technological advances of World War II,[166] 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).[167] Other topics included number theory, probability theory, and analytic geometry.[166]

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.[24] 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."[168] 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."[169] 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.[170] 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."[171]

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.[23]

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.[172] However, as mathematician Hung-Hsi Wu explained, the apparent dichotomy between basic skills and understanding of mathematical concepts is a delusion.[173]

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.[174] 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.[23]

In 1989, the more radical NCTM reforms were eliminated. Instead, greater emphasis was put on substantive mathematics.[23] 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.[175] 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.[176] Mathematics instructor Jaime Escalante dismissed the NCTM standards as something written by a PE teacher.[23]

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[177] 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.

Some geometric curves expressed algebraically in polar coordinates

Most countries teach mathematics according to what Americans call an integrated curriculum. Various aspects of calculus and prerequisites are taught throughout secondary school.[46][47] In fact, many topics in algebra and geometry that Americans typically learn in high school are taught in middle school in Europe,[162] 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.[46][47] However, as the Singaporean case demonstrates, early exposure to the concepts of calculus does not necessarily translates to actual understanding among high school students.[178][47] 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.[26] As mentioned above, only 7% of American high school students take Integrated mathematics.[154]

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.[12][179] Meanwhile, Chinese, Indian, and Singaporean students are exposed to high-level mathematics and science at a young age.[12] About half of STEM students in the U.S. dropped out of their programs between 2003 and 2009.[179] 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.[18][179][180] An emphasis on speed and rote memorization gives as many as one-third of students aged five and over mathematical anxiety.[6]

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.[31] 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.[181] Nevertheless, Calculus remains the most recommended course for ambitious students.[181]

By the mid-2010s, only a quarter of American high school seniors are able to do grade-level math,[182] yet about half graduate from high school as A students, prompting concerns of grade inflation.[183] Strong performance in Algebra I, Geometry, and Algebra II predict good grades in university-level Calculus even better than taking Calculus in high school.[30]

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.[184]

A math club at a preparatory school in Texas (2018)

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.[179] In fact, American educators tend to focus on poorly performing students rather than those at the top, unlike their Asian counterparts.[185] Parents' proposal for an accelerated track for their children are oftentimes met with hostility by school administrators.[186] Conversely, initiatives aimed at de-emphasizing certain core subjects, such as Algebra I, triggered strong backlash from parents and university faculty members.[32][33] 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.[187] By the mid-2010s, some public schools have begun offering enrichment programs to their students.[179]

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.[31][181] In San Francisco, for example, such a plan was dropped due to a combination of mixed results and public backlash.[31]

A shortage of qualified mathematics schoolteachers has been a serious problem in the United States for many years.[16][17] 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.[188] Teachers oftentimes unknowingly transmit their own negative attitude towards mathematics to their students, damaging the quality of instruction.[180]