Mathematics education in the United States
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From kindergarten through high school, mathematics education in public schools in the United States has historically varied widely from state to state, and often even varies considerably within individual states. There has been considerable disagreement on the style and contents of mathematics teaching, including the question of whether or not there should be any national standards at all.[1][2][3] Moreover, manyhttps://anshlivevk.blogspot.com/2023/03/mathematics-education-in-united-states.html students take alternatives to the traditional pathways, including accelerated tracks. As of 2023, twenty-seven states require students to pass three math courses before graduation from high school, but seventeen states and the District of Columbia require four.[4]
With the adoption of the Common Core Standards most states and the District of Columbia, mathematics content across the country is moving into closer agreement for each grade level. The SAT, a standardized university entrance exam, has been reformed to better reflect the contents of the Common Core.[5] However, Alabama,[6] Arizona,[7] Florida,[8] Indiana,[9] New Jersey,[10] Oklahoma,[11] South Carolina,[12] and Tennessee[13] have repealed the Common Core. Florida[14] and New Jersey[10] have introduced new standards while Oklahoma[11] has restored its own. Minnesota has chosen to adopt Common Core standards for English language arts but not mathematics.[15] Nebraska,[16] Texas,[17] and Virginia[18] never signed on.
Curricular content[edit]
Each state sets its own curricular standards and details are usually set by each local school district. Although there are no federal standards, since 2015 most states have based their curricula on the Common Core State Standards in mathematics. The stated goal of the mathematics standards is to achieve greater focus and coherence in the curriculum.[19] This is largely in response to the criticism that American mathematics curricula are "a mile wide and an inch deep."[20] The National Council of Teachers ofhttps://anshlivevk.blogspot.com/2023/03/mathematics-education-in-united-states.html Mathematics published educational recommendations in mathematics education in 1991 and 2000 which have been highly influential, describing mathematical knowledge, skills and pedagogical emphases from kindergarten through high school. The 2006 NCTM Curriculum Focal Points have also been influential for its recommendations of the most important mathematical topics for each grade level through grade 8.
In the United States, mathematics curriculum in elementary and middle school is integrated, while in high school it traditionally has been separated by topic, like Algebra I, Geometry, Algebra II, each topic usually lasting for the whole school year. However, from 2013-14 onward, some school districts and s
Curricular content[edit]
Each state sets its own curricular standards and details are usually set by each local school district. Although there are no federal standards, since 2015 most states have based their curricula on the Common Core State Standards in mathematics. The stated goal of the mathematics standards is to achieve greater focus and coherence in the curriculum.[19] This is largely in response to the criticism that American mathematics curricula are "a mile wide and an inch deep."[20] The National Council of Teachers of Mathematics published educational recommendations in mathematics education in 1991 and 2000 which have been highly influential, describing mathematical knowledge, skills and pedagogical emphases from kindergarten through high school. The 2006 NCTM Curriculum Focal Points have also been influential for its recommendations of the most important mathematical topics for each grade level through grade 8.
In the United States, mathematics curriculum in elementary and middle school is integrated, while in high school it traditionally has been separated by topic, like Algebra I, Geometry, Algebra II, each topic usually lasting for the whole school year. However, from 2013-14 onward, some school districts and states have switched to an integrated curriculum.[21][22]
Primary school
Secondary school
Pre-algebra can be taken in middle school. Students learn about real numbers and some more arithmetic (prime numbers, prime factorization, and the fundamental theorem of arithmetic), the rudiments of algebra and geometry (areas of plane figures, the Pythagorean theorem, and the distance formula), and introductory trigonometry (definitions of the trigonometric functions).
Algebra I is the first course students take in algebra. Historically, this class was offered in high school but could be taken as early as the seventh grade but more traditionally in eighth or ninth grades, after the student has 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.
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), thehttps://anshlivevk.blogspot.com/2023/03/mathematics-education-in-united-states.html 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, andhttps://anshlivevk.blogspot.com/2023/03/mathematics-education-in-united-states.html
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, matrix determinants, Cramer's rule, and the inverse of a 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.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, University of California requires three years of "college-preparatory mathematics that include the topics covered in elementary and advanced algebra and two- and three-dimensional geometry"[23] to be admitted. After California Department of Education adopted Common Core, the University of California clarified that "approved integrated math courses may be used to fulfill part or all"[23] of this admission requirement.
Pre-calculus 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 sectio 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 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. Pre-calculus ends with an introduction to limits of a function. Some instructors might give lectures on mathematical induction and combinatorics in this course.[24]https://anshlivevk.blogspot.com/2023/03/mathematics-education-in-united-states.htmlhttps://anshlivevk.blogspot.com/2023/03/mathematics-education-in-united-states.htmlDepending on the school district, several courses may be compacted and combined within one school year, either studied sequentially or simultaneously. Without such acceleration, it may be not possible to take more advanced classes like calculus in high school. Students may also receive lessons on set theory at various grade levels throughout secondary school.
College algebra is offered at many community colleges as remedial courses.[25] It should not be confused with abstract algebra and linear algebra, taken by students who major in mathematics and allieelds (such as computer science) in
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 take 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.[26][27]
In this class, students learn
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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), 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), 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 binomial series, Euler's complex identity, polar representation of complex numbers, parametric equations, and curves in polar coordinates.[28][29][30][31]
Depending on the course and instructor, special topics in introductory calculus might include the classical d
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