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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. With the adoption of the Common Core Standards by 45 states, mathematics content across the country is moving into closer agreement for each grade level. Furthermore, the SAT, a standardized university entrance exam, has been reformed to better reflect the contents of the Common Core.[1]

Curricular content

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 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.[2][3]

Secondary school

James Garfield's proof of the Pythagorean theorem.

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 has been a high school level course that is often offered as early as the seventh grade but more traditionally in eighth or ninth grades, after the student has taken Pre-algebra. The course is also offered in community colleges as a basic skill or remedial course. 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 concept of rigor in mathematics by way of some basic concepts in mainly Euclidean geometry. Students learn about 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. Geometry 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.

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.

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

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"[4] 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"[4] 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 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 and 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.[5]

Depending 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.

College algebra is offered at many community colleges (as a remedial course). 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 Newton's method for numerical root extraction.

Calculus is usually taken by high-school seniors or university freshmen, but can occasionally be taken as early as tenth grade. 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. In this class, students learn about limits and continuity (the intermediate and mean value theorems), differentiation and its applications (implicit differentiation, logarithmic differentiation, related rates, optimization, 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.[6][7]

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. Some American high schools today also offer multivariable calculus[8] (partial differentiation, the multivariable chain rule and Clairault's theorem; constrained optimization and Lagrange multipliers; multidimensional integration, Fubini's theorem, change of variables, and Jacobian determinants, gradients, directional derivatives, divergence, curl, the fundamental theorem of gradients, Green's theorem, Stokes' theorem, and Gauss's theorem).[9]

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).

Controversies

New Math

Under the 'New Math' initiative, created after the successful launch of the Soviet satellite Sputnik in 1957, conceptual abstraction gained a central role in mathematics education.[10] It 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.[11] 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). However, this educational initiative 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.[10] 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."[12] 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."[13] 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.[14] 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."[15]

Standards-based reforms and the NCTM

Main articles: Math wars and NCTM

In the 20th century, reforms to mathematics education were proposed, based on ideas originating from the 1980s, when research began to support an emphasis on problem-solving, mathematical reasoning, conceptual understanding, and student-centered learning and a de-emphasis on rote memorization. About the same time as the development of a number of controversial standards across reading, science and history, 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.[16] In fact, mathematics education became a hotly debated subject, and after the initial adoption of standards-based curricula, some schools and districts supplemented or replaced standards-based curricula in the late 1990s and early 2000s.

In standards-based education reform all students, not only the college-bound, must take 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.[17] 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.[18]

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[19] 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

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.[2]

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.[20][21] Meanwhile, Chinese, Indian, and Singaporean students are exposed to high-level mathematics and science at a young age.[20] About half of STEM students dropped out of their programs between 2003 and 2009.[21] 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.[21][22] An emphasis on speed and rote memorization gives as many as one-third of students aged five and over mathematical anxiety.[23]

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.[24]

Enrichment programs

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. But by the mid-2010s, some public schools have begun offering enrichment programs to their students.[21]

Standardized tests

The Program for International Student Assessment (PISA) conducted the 2015 assessment test which is held every three years for 15-year-old students worldwide.[25] 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.[23] During the latest 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. Their average score in mathematics declined by 11 points.[26]

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.[27]

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.[8]

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.[8]

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.[8]

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.

See also

References

  1. ^ Lewin, Tamar (March 5, 2014). "A New SAT Aims to Realign With Schoolwork". The New York Times. Archived from the original on May 13, 2014. Retrieved May 14, 2014. He said he also wanted to make the test reflect more closely what students did in high school and, perhaps most important, rein in the intense coaching and tutoring on how to take the test that often gave affluent students an advantage.
  2. ^ a b Will, Madeline (November 10, 2014). "In Transition to Common Core, Some High Schools Turn to 'Integrated' Math". Education Week. Archived from the original on August 31, 2022. Retrieved August 31, 2022.
  3. ^ Fensterwald, John. "Districts confirm they're moving ahead with Common Core". EdSource. Retrieved 18 November 2013.
  4. ^ a b "University of California admission subject requirements". Retrieved 2018-08-24.
  5. ^ Demana, Franklin D.; Waits, Bert K.; Foley, Gregory D.; Kennedy, Daniel (2000). Precalculus: Graphical, Numerical, Algebraic (7th ed.). Addison-Wesley. ISBN 9780321356932.
  6. ^ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas's Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 978-0-321-58876-0.
  7. ^ Finney, Ross L.; Demana, Franklin D.; Waits, Bert K.; Kennedy, Daniel (2012). Calculus: Graphical, Numerical, Algebraic (4th ed.). Prentice Hall. ISBN 9780133178579.
  8. ^ a b c d Bressoud, David (July 1, 2022). "Thoughts on Advanced Placement Precalculus". MAA Blog. Retrieved September 13, 2022.
  9. ^ Stewart, James (2012). Calculus: Early Transcendentals (7th ed.). Brooks/Cole Cengage Learning. ISBN 978-0-538-49790-9.
  10. ^ a b Knudson, Kevin (2015). "The Common Core is today's New Math – which is actually a good thing". The Conversation. Retrieved September 9, 2015.
  11. ^ Gispert, Hélène. "L'enseignement des mathématiques au XXe siècle dans le contexte français". CultureMATH (in French). Archived from the original on July 15, 2017. Retrieved November 4, 2020.
  12. ^ Feynman, Richard P. (1965). "New Textbooks for the 'New' Mathematics" (PDF). Engineering and Science. XXVIII (6): 9–15. ISSN 0013-7812.
  13. ^ Kline, Morris (1973). Why Johnny Can't Add: The Failure of the New Math. New York: St. Martin's Press. pp. 17, 98. ISBN 0-394-71981-6.
  14. ^ Gillman, Leonard (May 1974). "Review of Why Johnny Can't Add". American Mathematical Monthly. 81 (5): 531–2. JSTOR 2318615.
  15. ^ Simmons, George F. (2003). "Algebra – Introduction". Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry: Geometry, Algebra, Trigonometry. Wipf and Stock Publishers. p. 33. ISBN 9781592441303.
  16. ^ Hiebert, James; Stigler, James W. (September 2000). "A proposal for improving classroom teaching: Lessons from the TIMSS video study". The Elementary School Journal. 101 (1): 3–20. doi:10.1086/499656.
  17. ^ Christian Science Monitor Archived 2008-05-09 at the Wayback Machine
  18. ^ "Frequently Asked Questions About the Core-Plus Mathematics Project". Archived from the original on 2010-08-21. Retrieved 2009-10-26.
  19. ^ Wall Street Journal, New York Times, Chicago Sun Times
  20. ^ a b Drew, Christopher (November 4, 2011). "Why Science Majors Change Their Minds (It's Just So Darn Hard)". Education Life. The New York Times. Archived from the original on 2011-11-04. Retrieved October 28, 2019.
  21. ^ a b c d Tyre, Peg (February 8, 2016). "The Math Revolution". The Atlantic. Archived from the original on June 28, 2020. Retrieved February 4, 2021.
  22. ^ Sparks, Sarah D. (January 7, 2020). "The Myth Fueling Math Anxiety". Education Week. Archived from the original on August 31, 2022. Retrieved August 31, 2022.
  23. ^ a b Boaler, Jo; Zoido, Pablo (2016-10-13). "Why Math Education in the U.S. Doesn't Add Up". Scientific American Mind. 27 (6): 18–19. doi:10.1038/scientificamericanmind1116-18. ISSN 1555-2284. Archived from the original on August 23, 2022.
  24. ^ Furner, Joseph M., and Kumar, David D. [1], "The Mathematics and Science Integration Argument: A Stand for Teacher Education," Eurasia Journal of Mathematics, Science & Technology Education, Vol. 3, Num. 3, August, 2007, accessed on 15 December 2013
  25. ^ "What the world can learn from the latest PISA test results". The Economist. Retrieved 2018-07-25.
  26. ^ "The latest ranking of top countries in math, reading, and science is out — and the US didn't crack the top 10". Business Insider. Retrieved 2018-07-25.
  27. ^ Mervosh, Sarah (September 1, 2022). "The Pandemic Erased Two Decades of Progress in Math and Reading". The New York Times. Archived from the original on September 1, 2022. Retrieved September 1, 2022.
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