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id="toc-Attendance_and_completion_rates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Controversies_and_issues" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Controversies_and_issues"> <div class="vector-toc-text"> 3 Controversies and issues </div> </a> <button aria-controls="toc-Controversies_and_issues-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Controversies and issues subsection </button> <ul id="toc-Controversies_and_issues-sublist" class="vector-toc-list"> <li id="toc-Progressive_education" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Progressive_education"> <div class="vector-toc-text"> 3.1 Progressive education </div> </a> <ul id="toc-Progressive_education-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-New_Math" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#New_Math"> <div class="vector-toc-text"> 3.2 New Math </div> </a> <ul id="toc-New_Math-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Standards-based_reforms_and_the_NCTM" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standards-based_reforms_and_the_NCTM"> <div class="vector-toc-text"> 3.3 Standards-based reforms and the NCTM </div> </a> <ul id="toc-Standards-based_reforms_and_the_NCTM-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integrated_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integrated_mathematics"> <div class="vector-toc-text"> 3.4 Integrated mathematics </div> </a> <ul id="toc-Integrated_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Preparation_for_college" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Preparation_for_college"> <div class="vector-toc-text"> 3.5 Preparation for college </div> </a> <ul id="toc-Preparation_for_college-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enrichment_programs_and_accelerated_tracks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Enrichment_programs_and_accelerated_tracks"> <div class="vector-toc-text"> 3.6 Enrichment programs and accelerated tracks </div> </a> <ul id="toc-Enrichment_programs_and_accelerated_tracks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shortage_of_instructors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Shortage_of_instructors"> <div class="vector-toc-text"> 3.7 Shortage of instructors </div> </a> <ul id="toc-Shortage_of_instructors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Standardized_tests" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Standardized_tests"> <div class="vector-toc-text"> 4 Standardized tests </div> </a> <ul id="toc-Standardized_tests-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Advanced_Placement_Mathematics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Advanced_Placement_Mathematics"> <div class="vector-toc-text"> 5 Advanced Placement Mathematics </div> </a> <ul id="toc-Advanced_Placement_Mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conferences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conferences"> <div class="vector-toc-text"> 6 Conferences </div> </a> <ul id="toc-Conferences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" 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plainlist" style="width:22.0em; border: 4px double #d69d36;"><tbody><tr><td class="sidebar-pretitle">This article is part of <a href="/wiki/Category:Education_in_the_United_States" title="Category:Education in the United States">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle" style="background:#002868; border: 1px double #8C959A;"><a href="/wiki/Education_in_the_United_States" title="Education in the United States">Education in the United States</a></th></tr><tr><th class="sidebar-heading" style="background:#bf0a30; color:#FFF; border: 1px double #8C959A;"> Summary</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Category:Education_in_the_United_States_by_state_or_territory" title="Category:Education in the United States by state or territory">By state</a> and <a href="/wiki/Category:Education_in_insular_areas_of_the_United_States" title="Category:Education in insular areas of the United States">in insular areas</a></li> <li><a href="/wiki/Category:Education_in_the_United_States_by_subject" title="Category:Education in the United States by subject">By subject area</a></li> <li><a href="/wiki/History_of_education_in_the_United_States" title="History of education in the United States">History of education in the United States</a></li> <li><a href="/wiki/History_of_education_in_Chicago" title="History of education in Chicago">History of education in Chicago</a></li> <li><a href="/wiki/History_of_education_in_Kentucky" title="History of education in Kentucky">History of education in Kentucky</a></li> <li><a href="/wiki/History_of_education_in_Massachusetts" title="History of education in Massachusetts">History of education in Massachusetts</a></li> <li><a href="/wiki/History_of_education_in_Missouri" title="History of education in Missouri">History of education in Missouri</a></li> <li><a href="/wiki/History_of_education_in_New_York_City" title="History of education in New York City">History of education in New York City</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#bf0a30; color:#FFF; border: 1px double #8C959A;"> Curriculum topics</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Literacy_in_the_United_States" title="Literacy in the United States">Literacy</a></li> <li><a href="/wiki/Normal_schools_in_the_United_States" title="Normal schools in the United States">Normal schools</a></li> <li><a href="/wiki/Apprenticeship_degree" title="Apprenticeship degree">Apprenticeship degrees</a></li> <li><a href="/wiki/Art_education_in_the_United_States" title="Art education in the United States">Art education</a></li> <li><a href="/wiki/Civic_education_in_the_United_States" title="Civic education in the United States">Civic education</a></li> <li><a href="/wiki/Music_education_and_programs_within_the_United_States" title="Music education and programs within the United States">Music education</a></li> <li><a href="/wiki/Legal_education_in_the_United_States" title="Legal education in the United States">Legal education</a> <ul><li><a href="/wiki/Law_school_in_the_United_States" title="Law school in the United States">Law school</a></li></ul></li> <li><a href="/wiki/Medical_education_in_the_United_States" title="Medical education in the United States">Medical education</a> <ul><li><a href="/wiki/Medical_school_in_the_United_States" title="Medical school in the United States">Medical school</a></li> <li><a href="/wiki/Nursing_degrees_in_the_United_States" title="Nursing degrees in the United States">Nursing degrees</a></li></ul></li> <li><a href="/wiki/Environmental_education_in_the_United_States" title="Environmental education in the United States">Environmental education</a></li> <li><a href="/wiki/Language_education_in_the_United_States" title="Language education in the United States">Language education</a></li> <li><a class="mw-selflink selflink">Mathematics education</a></li> <li><a href="/wiki/Sex_education_in_the_United_States" title="Sex education in the United States">Sex education</a></li> <li><a href="/wiki/Vocational_education_in_the_United_States" title="Vocational education in the United States">Vocational education</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#bf0a30; color:#FFF; border: 1px double #8C959A;"> <a href="/wiki/Education_policy_of_the_United_States" title="Education policy of the United States">Education policy issues</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Educational_accreditation" title="Educational accreditation">Accreditation</a> <ul><li><a href="/wiki/Pre-tertiary-education_accreditation" title="Pre-tertiary-education accreditation">Primary and secondary</a></li> <li><a href="/wiki/Higher_education_accreditation_in_the_United_States" title="Higher education accreditation in the United States">Post-secondary</a></li></ul></li> <li><a href="/wiki/Category:Education_finance_in_the_United_States" title="Category:Education finance in the United States">Financing</a></li> <li><a href="/wiki/Educational_attainment_in_the_United_States" title="Educational attainment in the United States">Educational attainment</a></li> <li><a href="/wiki/Issues_in_higher_education_in_the_United_States" title="Issues in higher education in the United States">Post-secondary issues</a> <ul><li><a href="/wiki/Higher_education_bubble_in_the_United_States" title="Higher education bubble in the United States">Bubble</a></li> <li><a href="/wiki/Cost_and_financing_issues_facing_higher_education_in_the_United_States" class="mw-redirect" title="Cost and financing issues facing higher education in the United States">Cost and financing</a></li> <li><a href="/wiki/Educational_inflation" title="Educational inflation">Credentialism</a></li> <li><a href="/wiki/Elite_overproduction" title="Elite overproduction">Elite overproduction</a></li> <li><a href="/wiki/Graduate_unemployment#United_States" title="Graduate unemployment">Graduate unemployment</a></li> <li><a href="/wiki/Student_financial_aid_in_the_United_States" title="Student financial aid in the United States">Student financial aid</a></li> <li><a href="/wiki/Student_loans_in_the_United_States" title="Student loans in the United States">Student loans</a></li></ul></li> <li><a href="/wiki/Education_reform##Contemporary_issues_(United_States)" title="Education reform">Reform</a> <ul><li><a href="/wiki/Charter_schools_in_the_United_States" title="Charter schools in the United States">Charter schools</a></li> <li><a href="/wiki/Educational_inequality_in_the_United_States" title="Educational inequality in the United States">Inequality</a></li> <li><a href="/wiki/Achievement_gaps_in_the_United_States" title="Achievement gaps in the United States">Achievement gaps</a></li> <li><a href="/wiki/Racial_achievement_gap_in_the_United_States" title="Racial achievement gap in the United States">Racial achievement gap</a></li> <li><a href="/wiki/Desegregation_busing" title="Desegregation busing">Desegregation busing</a></li> <li><a href="/wiki/Sex_differences_in_education_in_the_United_States" title="Sex differences in education in the United States">Gender achievement gap</a></li> <li><a href="/wiki/Head_Start_(program)" title="Head Start (program)">Head Start</a></li> <li><a href="/wiki/School_choice" title="School choice">School choice</a></li> <li><a href="/wiki/Racial_diversity_in_United_States_schools" title="Racial diversity in United States schools">Racial diversity</a></li> <li><a href="/wiki/School_segregation_in_the_United_States" title="School segregation in the United States">School segregation</a></li> <li><a href="/wiki/Standards-based_education_reform_in_the_United_States" title="Standards-based education reform in the United States">Standards-based reform</a></li></ul></li> <li><a href="/wiki/School_corporal_punishment_in_the_United_States" title="School corporal punishment in the United States">School corporal punishment</a></li> <li><a href="/wiki/School_meal_programs_in_the_United_States" title="School meal programs in the United States">School meals</a></li> <li><a href="/wiki/School_violence_in_the_United_States" title="School violence in the United States">School violence</a></li> <li><a href="/wiki/Sexual_harassment_in_education_in_the_United_States" title="Sexual harassment in education in the United States">Sexual harassment</a></li> <li><a href="/wiki/Qatari_involvement_in_higher_education_in_the_United_States" title="Qatari involvement in higher education in the United States">Foreign involvement</a></li> <li><a href="/wiki/Special_education_in_the_United_States" title="Special education in the United States">Special education</a></li> <li><a href="/wiki/Apprenticeship_in_the_United_States" title="Apprenticeship in the United States">Apprenticeship</a> <ul><li><a href="/wiki/School-to-work_transition" title="School-to-work transition">School-to-work transition</a></li></ul></li> <li><a href="/wiki/Community_colleges_in_the_United_States" title="Community colleges in the United States">Community colleges</a></li> <li><a href="/wiki/For-profit_higher_education_in_the_United_States" title="For-profit higher education in the United States">For-profit higher education</a> <ul><li><a href="/wiki/For-profit_colleges_in_the_United_States" title="For-profit colleges in the United States">For-profit colleges</a></li></ul></li> <li><a href="/wiki/List_of_research_universities_in_the_United_States" title="List of research universities in the United States">Research universities</a></li> <li><a href="/wiki/Community_school_(United_States)" title="Community school (United States)">Community school</a> <ul><li><a href="/wiki/Full-service_community_schools_in_the_United_States" title="Full-service community schools in the United States">Full-service community schools</a></li></ul></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#bf0a30; color:#FFF; border: 1px double #8C959A;"> Levels of education</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Early_childhood_education_in_the_United_States" title="Early childhood education in the United States">Early childhood</a></li> <li><a href="/wiki/K%E2%80%9312_education_in_the_United_States" title="K–12 education in the United States">K–12</a> <ul><li><a href="/wiki/Primary_education_in_the_United_States" title="Primary education in the United States">Primary</a></li> <li><a href="/wiki/Secondary_education_in_the_United_States" title="Secondary education in the United States">Secondary</a></li></ul></li> <li><a href="/wiki/Higher_education_in_the_United_States" title="Higher education in the United States">Post-secondary</a></li></ul> </div> <ul><li><a href="/wiki/Category:Educational_organizations_based_in_the_United_States" title="Category:Educational organizations based in the United States">Organizations</a></li></ul></td> </tr><tr><td class="sidebar-below" style="border-top:#aaa 1px solid; border-bottom:#aaa 1px solid;"> <a href="/wiki/File:Diploma_icon.png" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Diploma_icon.png/16px-Diploma_icon.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Diploma_icon.png/24px-Diploma_icon.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Diploma_icon.png/32px-Diploma_icon.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portal:Education" class="mw-redirect" title="Portal:Education">Education portal</a> <img alt="flag" src="//upload.wikimedia.org/wikipedia/en/thumb/a/a4/Flag_of_the_United_States.svg/16px-Flag_of_the_United_States.svg.png" decoding="async" width="16" height="8" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/a/a4/Flag_of_the_United_States.svg/24px-Flag_of_the_United_States.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/a/a4/Flag_of_the_United_States.svg/32px-Flag_of_the_United_States.svg.png 2x" data-file-width="1235" data-file-height="650" /> <a href="/wiki/Portal:United_States" title="Portal:United States">United States portal</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Education_in_the_U.S." title="Template:Education in the U.S."><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Education_in_the_U.S." title="Template talk:Education in the U.S."><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Education_in_the_U.S." title="Special:EditPage/Template:Education in the U.S."><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Mathematics education in the United States varies considerably from one state to the next, and even within a single state. However, with the adoption of the <a href="/wiki/Common_Core" title="Common Core">Common Core</a> Standards in most states and the <a href="/wiki/District_of_Columbia" class="mw-redirect" title="District of Columbia">District of Columbia</a> beginning in 2010, mathematics content across the country has moved into closer agreement for each grade level. The <a href="/wiki/SAT" title="SAT">SAT</a>, a standardized university entrance exam, has been reformed to better reflect the contents of the Common Core.<a href="#cite_note-1">[1]</a> However, many 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 (grades 9 to 12, for students typically aged 14 to 18), while seventeen states and the District of Columbia require four.<a href="#cite_note-2">[2]</a> A typical sequence of secondary-school (grades 6 to 12) courses in mathematics reads: Pre-Algebra (7th or 8th grade), Algebra I, Geometry, Algebra II, Pre-calculus, and Calculus or Statistics. However, some students enroll in integrated programs<a href="#cite_note-:1-3">[3]</a> while many complete high school without passing Calculus or Statistics.<a href="#cite_note-:3-4">[4]</a><a href="#cite_note-:47-5">[5]</a> At the other end, counselors at competitive public or private high schools usually encourage talented and ambitious students to take Calculus regardless of future plans in order to increase their chances of getting admitted to a prestigious university<a href="#cite_note-:16-6">[6]</a> and their parents enroll them in enrichment programs in mathematics.<a href="#cite_note-:67-7">[7]</a> Secondary-school algebra proves to be the turning point of difficulty many students struggle to surmount,<a href="#cite_note-:27-8">[8]</a><a href="#cite_note-:28-9">[9]</a><a href="#cite_note-:19-10">[10]</a><a href="#cite_note-:20-11">[11]</a> and as such, many students are ill-prepared for collegiate programs in the sciences, technology, engineering, and mathematics (<a href="/wiki/Science,_technology,_engineering,_and_mathematics" title="Science, technology, engineering, and mathematics">STEM</a>),<a href="#cite_note-:19-10">[10]</a><a href="#cite_note-:20-11">[11]</a><a href="#cite_note-:56-12">[12]</a> or future high-skilled careers.<a href="#cite_note-:26-13">[13]</a><a href="#cite_note-:31-14">[14]</a> According to a 1997 report by the <a href="/wiki/United_States_Department_of_Education" title="United States Department of Education">U.S. Department of Education</a>, passing rigorous high-school mathematics courses predicts successful completion of university programs regardless of major or family income.<a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a> Meanwhile, the number of eighth-graders enrolled in Algebra I has fallen between the early 2010s and early 2020s.<a href="#cite_note-:45-17">[17]</a> Across the United States, there is a shortage of qualified mathematics instructors.<a href="#cite_note-:34-18">[18]</a><a href="#cite_note-:37-19">[19]</a> Despite their best intentions, parents may transmit their <a href="/wiki/Mathematical_anxiety" title="Mathematical anxiety">mathematical anxiety</a> to their children, who may also have school teachers who fear mathematics,<a href="#cite_note-:35-20">[20]</a><a href="#cite_note-21">[21]</a> and they overestimate their children's mathematical proficiency.<a href="#cite_note-:24-22">[22]</a> About one in five American adults are functionally innumerate.<a href="#cite_note-:34-18">[18]</a> While an overwhelming majority agree that mathematics is important, many, especially the young, are not confident of their own mathematical ability.<a href="#cite_note-:34-18">[18]</a><a href="#cite_note-23">[23]</a> On the other hand, high-performing schools may offer their students accelerated tracks (including the possibility of taking collegiate courses after calculus)<a href="#cite_note-:2-24">[24]</a> and nourish them for mathematics competitions.<a href="#cite_note-:48-25">[25]</a><a href="#cite_note-:49-26">[26]</a> At the tertiary level, student interest in STEM has grown considerably.<a href="#cite_note-:81-27">[27]</a> However, many students find themselves having to take remedial courses for high-school mathematics<a href="#cite_note-:50-28">[28]</a><a href="#cite_note-:17-29">[29]</a> and many drop out of STEM programs due to deficient mathematical skills.<a href="#cite_note-:56-12">[12]</a><a href="#cite_note-:67-7">[7]</a> Compared to other developed countries in the Organization for Economic Co-operation and Development (<a href="/wiki/OECD" title="OECD">OECD</a>), the average level of <a href="/wiki/Numeracy" title="Numeracy">mathematical literacy</a> of American students is mediocre.<a href="#cite_note-:27-8">[8]</a><a href="#cite_note-:68-30">[30]</a><a href="#cite_note-:23-31">[31]</a><a href="#cite_note-:0-32">[32]</a><a href="#cite_note-:8-33">[33]</a> As in many other countries, math scores dropped during the COVID-19 pandemic.<a href="#cite_note-:25-34">[34]</a> However, Asian- and European-American students are above the OECD average.<a href="#cite_note-:03-35">[35]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Curricular_content_and_standards">Curricular content and standards</h2>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=1" title="Edit section: Curricular content and standards">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Status_of_Common_Core_Standards_vector.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Status_of_Common_Core_Standards_vector.svg/350px-Status_of_Common_Core_Standards_vector.svg.png" decoding="async" width="350" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Status_of_Common_Core_Standards_vector.svg/525px-Status_of_Common_Core_Standards_vector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Status_of_Common_Core_Standards_vector.svg/700px-Status_of_Common_Core_Standards_vector.svg.png 2x" data-file-width="959" data-file-height="593" /></a><figcaption>A map showing states in the U.S. which have either adopted, not adopted, partially adopted, or repealed the Common Core State Standards as of 2016:<style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><div class="legend">  States that have adopted the Standards</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend">  States that have partially adopted the Standards</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend">  States that adopted but later repealed the Standards</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend">  States that never adopted the Standards</div></figcaption></figure> Each U.S. 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 <a href="/wiki/Common_Core#Mathematics_standards" title="Common Core">Common Core State Standards</a> in mathematics. The stated goal of the Common Core mathematics standards is to achieve greater focus and coherence in the curriculum.<a href="#cite_note-36">[36]</a> This is largely in response to the criticism that American mathematics curricula are "a mile wide and an inch deep."<a href="#cite_note-37">[37]</a><a href="#cite_note-:5-38">[38]</a> The <a href="/wiki/National_Council_of_Teachers_of_Mathematics" title="National Council of Teachers of Mathematics">National Council of Teachers of Mathematics</a> published educational recommendations in mathematics education in 1989 and 2000 which have been highly influential, describing mathematical knowledge, skills and pedagogical emphases from kindergarten through high school. The 2006 NCTM <a rel="nofollow" class="external text" href="http://www.nctm.org/focalpoints/">Curriculum Focal Points</a> have also been influential for its recommendations of the most important mathematical topics for each grade level through grade 8. However, some states have either abandoned, or never adopted, the Common Core standards, but instead instituted their own. (See <a href="/wiki/Common_Core_implementation_by_state" title="Common Core implementation by state">Common Core implementation by state</a>.) In fact, there has been considerable disagreement on the style and content of mathematics teaching, including the question of whether or not there should be any national standards at all.<a href="#cite_note-:3-4">[4]</a><a href="#cite_note-:18-39">[39]</a><a href="#cite_note-:5-38">[38]</a> In the United States, mathematics <a href="/wiki/Curriculum" title="Curriculum">curriculum</a> in elementary and middle school is <a href="/wiki/Integrated_mathematics" title="Integrated mathematics">integrated</a>, while in high school it traditionally has been separated by topic, with each topic usually lasting for the whole school year. However, some districts have integrated curricula, or decided to try integrated curricula after Common Core was adopted.<a href="#cite_note-:1-3">[3]</a><a href="#cite_note-40">[40]</a> Since the days of the <a href="/wiki/Sputnik_crisis" title="Sputnik crisis">Sputnik</a> in the 1950s, the sequence of mathematics courses in secondary school has not changed: Pre-algebra, Algebra I, Geometry, Algebra II, Pre-calculus (or Trigonometry), and Calculus. Trigonometry is usually integrated into the other courses. Calculus is only taken by a select few.<a href="#cite_note-:3-4">[4]</a><a href="#cite_note-:10-41">[41]</a> Some schools teach Algebra II before Geometry.<a href="#cite_note-:10-41">[41]</a> Success in middle-school mathematics courses is correlated with having an understanding of numbers by the start of first grade.<a href="#cite_note-42">[42]</a> This traditional sequence assumes that students will pursue STEM programs in college, though, in practice, only a minority are willing and able to take this option.<a href="#cite_note-:3-4">[4]</a> Often a course in Statistics is also offered.<a href="#cite_note-:34-18">[18]</a> While a majority of schoolteachers base their lessons on a core curriculum, they do not necessarily follow them to the letter. Many also take advantage of additional resources not provided to them by their school districts.<a href="#cite_note-43">[43]</a> <div class="mw-heading mw-heading3"><h3 id="Primary_school">Primary school</h3>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=2" title="Edit section: Primary school">edit</a>]</div> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Multiplication_as_scaling_integers.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/02/Multiplication_as_scaling_integers.gif" decoding="async" width="400" height="200" class="mw-file-element" data-file-width="400" data-file-height="200" /></a><figcaption>Multiplication seen as scaling on the number line</figcaption></figure> Primary schoolchildren learn counting, arithmetic and properties of operations, geometry, measurement, statistics and probability. They typically begin studying fractions in third grade. <div class="mw-heading mw-heading3"><h3 id="Secondary_school">Secondary school</h3>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=3" title="Edit section: Secondary school">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Aa_garfield_pythag.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Aa_garfield_pythag.svg/220px-Aa_garfield_pythag.svg.png" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Aa_garfield_pythag.svg/330px-Aa_garfield_pythag.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Aa_garfield_pythag.svg/440px-Aa_garfield_pythag.svg.png 2x" data-file-width="473" data-file-height="328" /></a><figcaption><a href="/wiki/James_A._Garfield" title="James A. Garfield">James Garfield</a>'s proof of the Pythagorean theorem.</figcaption></figure> <a href="/wiki/Pre-algebra" title="Pre-algebra">Pre-algebra</a> can be taken in middle school by seventh or eighth graders. Students typically begin by learning about real numbers and basic number theory (<a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, prime factorization, <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>, ratios, and percentages), topics needed for algebra (powers, roots, graphing, order of operations, variables, expressions, and <a href="/wiki/Scientific_notation" title="Scientific notation">scientific notation</a>) and geometry (quadrilaterals, polygons, areas of plane figures, the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>, 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,<a href="#cite_note-:32-44">[44]</a> after the students have taken Pre-algebra. Students learn about real numbers and the <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a> (PEMDAS), functions, linear equations, graphs, polynomials, the <a href="/wiki/Factor_theorem" title="Factor theorem">factor theorem</a>, <a href="/wiki/Nth_root" title="Nth root">radicals</a>, and <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equations</a> (factoring, <a href="/wiki/Completing_the_square" title="Completing the square">completing the square</a>, and the <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a>), and power functions. This course is considered a gatekeeper for those who want to pursue STEM<a href="#cite_note-:20-11">[11]</a> because taking Algebra I in eighth grade allows students to eventually take Calculus before graduating from high school.<a href="#cite_note-:15-45">[45]</a> As such, tracking students by their aptitude and deciding when they should take Algebra I has become topic of controversy in California<a href="#cite_note-:43-46">[46]</a> and Massachusetts.<a href="#cite_note-:44-47">[47]</a> Parents of high-performing students are among the most vocal critics of policies discouraging the taking of Algebra I in middle school.<a href="#cite_note-:43-46">[46]</a><a href="#cite_note-:44-47">[47]</a> Geometry, usually taken in ninth or tenth grade, introduces students to the notion of <a href="/wiki/Rigour#Mathematics" title="Rigour">rigor in mathematics</a> by way of some basic concepts in mainly <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. Students learn the rudiments of <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional logic</a>, methods of proof (<a href="/wiki/Direct_proof" title="Direct proof">direct</a> and by <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">contradiction</a>), <a href="/wiki/Parallel_lines" class="mw-redirect" title="Parallel lines">parallel lines</a>, triangles (<a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruence</a> and <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity</a>), circles (<a href="/wiki/Secant_line" title="Secant line">secants</a>, tangents, <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">chords</a>, <a href="/wiki/Central_angle" title="Central angle">central angles</a>, and <a href="/wiki/Inscribed_angle" title="Inscribed angle">inscribed angles</a>), the Pythagorean theorem, elementary trigonometry (angles of elevation and depression, the <a href="/wiki/Law_of_sines" title="Law of sines">law of sines</a>), basic analytic geometry (<a href="/wiki/Linear_equation" title="Linear equation">equations of lines</a>, point-slope and slope-intercept forms, perpendicular lines, and <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a>), and geometric probability.<a href="#cite_note-:11-48">[48]</a> Students are traditionally taught to demonstrate simple geometric theorems using <a href="/wiki/Mathematical_proof#Two-column_proof" title="Mathematical proof">two-column proofs</a>, a method developed in the early 20th century in the U.S. specifically for this course, though other methods may also be used.<a href="#cite_note-:51-49">[49]</a><a href="#cite_note-50">[50]</a> Depending on the curriculum and instructor, students may receive orientation towards calculus, for instance with the introduction of the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> and <a href="/wiki/Cavalieri%27s_principle" title="Cavalieri's principle">Cavalieri's principle</a>.<a href="#cite_note-:11-48">[48]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg/220px-Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg.png" decoding="async" width="220" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg/330px-Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg/440px-Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg.png 2x" data-file-width="750" data-file-height="370" /></a><figcaption>Visual proof of the double-angle identity for the sine</figcaption></figure> Algebra II has Algebra I as a prerequisite and is traditionally a high-school-level course. Course contents include <a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">inequalities</a>, function notation, quadratic equations, power functions, <a href="/wiki/Exponential_function" title="Exponential function">exponential functions</a>, <a href="/wiki/Logarithm" title="Logarithm">logarithms</a>, systems of linear equations, matrices (including matrix multiplication, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"> matrix <a href="/wiki/Determinant" title="Determinant">determinants</a>, <a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a>, and the inverse of a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"> matrix), the <a href="/wiki/Radian" title="Radian">radian</a> measure, graphs of trigonometric functions, <a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">trigonometric identities</a> (Pythagorean identities, the sum-and-difference, double-angle, and half-angle formulas, the laws of sines and <a href="/wiki/Law_of_cosines" title="Law of cosines">cosines</a>), <a href="/wiki/Conic_section" title="Conic section">conic sections</a>, among other topics.<a href="#cite_note-51">[51]</a> Requiring Algebra II for high school graduation gained traction across the United States in the early 2010s.<a href="#cite_note-52">[52]</a> 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 <a href="/wiki/University_of_California" title="University of California">University of California</a> (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"<a href="#cite_note-uc-req-53">[53]</a> to be admitted. After the <a href="/wiki/California_Department_of_Education" title="California Department of Education">California Department of Education</a> adopted the Common Core, the UC system clarified that "approved integrated math courses may be used to fulfill part or all"<a href="#cite_note-uc-req-53">[53]</a> of this admission requirement. On the other hand, in a controversial decision, the <a href="/wiki/Texas_Board_of_Education" class="mw-redirect" title="Texas Board of Education">Texas Board of Education</a> voted to remove Algebra II as a required course for high school graduation.<a href="#cite_note-54">[54]</a> 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.<a href="#cite_note-:43-46">[46]</a> Students interested in taking <a href="/wiki/AP_Computer_Science_A" title="AP Computer Science A">AP Computer Science A</a><a href="#cite_note-55">[55]</a> or <a href="/wiki/AP_Computer_Science_Principles" title="AP Computer Science Principles">AP Computer Science Principles</a><a href="#cite_note-:04-56">[56]</a> must have taken at least one course on algebra in high school. <a href="/wiki/AP_Chemistry" title="AP Chemistry">AP Chemistry</a> specifically requires Algebra II.<a href="#cite_note-57">[57]</a><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PascalTriangleAnimated2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/PascalTriangleAnimated2.gif/220px-PascalTriangleAnimated2.gif" decoding="async" width="220" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/0d/PascalTriangleAnimated2.gif 1.5x" data-file-width="260" data-file-height="240" /></a><figcaption>Pascal's <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">arithmetic triangle</a> appears in combinatorics as well as algebra via the binomial theorem.</figcaption></figure><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a> follows from the above, and is usually taken by college-bound students. Pre-calculus combines algebra, analytic geometry, and trigonometry. Topics in algebra include the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a>, <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">Fundamental Theorem of Algebra</a>, <a href="/wiki/Nth_root" title="Nth root">root extraction</a>, <a href="/wiki/Polynomial_long_division" title="Polynomial long division">polynomial long division</a>, <a href="/wiki/Partial_fraction_decomposition" title="Partial fraction decomposition">partial fraction decomposition</a>, and <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> operations. In chapters on trigonometry, students learn about <a href="/wiki/Radian" title="Radian">radian</a> angle measure, are shown the <a href="/wiki/Sine_and_cosine" title="Sine and cosine">sine and cosine</a> functions as coordinates on the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>, relate the six common <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> and <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">their inverses</a> and plot their graphs, solve equations involving trigonometric functions, and practice manipulating <a href="/wiki/Trigonometric_identities" class="mw-redirect" title="Trigonometric identities">trigonometric identities</a>. In the chapters on analytic geometry, students are introduced to <a href="/wiki/Polar_coordinate_system" title="Polar coordinate system">polar coordinates</a> and deepen their knowledge of conic sections. Some courses include the basics of <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> geometry, including the <a href="/wiki/Dot_product" title="Dot product">dot product</a> and the <a href="/wiki/Vector_projection" title="Vector projection">projection</a> of one vector onto another. If time and aptitude permit, students might learn <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a> or the <a href="/wiki/Cross_product" title="Cross product">vector cross product</a>. 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 <a href="/wiki/Rational_root_theorem" title="Rational root theorem">rational root theorem</a> and the <a href="/wiki/Descartes%27_rule_of_signs" title="Descartes' rule of signs">Descartes rule of signs</a>. Precalculus ends with an introduction to <a href="/wiki/Limit_of_a_function" title="Limit of a function">limits</a> of a function. Some instructors might give lectures on <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a> and <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> in this course.<a href="#cite_note-58">[58]</a><a href="#cite_note-59">[59]</a><a href="#cite_note-60">[60]</a> Precalculus is a prerequisite for <a href="/wiki/AP_Physics_1" title="AP Physics 1">AP Physics 1</a> and <a href="/wiki/AP_Physics_2" title="AP Physics 2">AP Physics 2</a> (formerly <a href="/wiki/AP_Physics_B" title="AP Physics B">AP Physics B</a>).<a href="#cite_note-61">[61]</a><a href="#cite_note-62">[62]</a> 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 (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"> matrix inversion, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"> determinants, and <a href="/wiki/Linear_map" title="Linear map">linear transformations</a>).<a href="#cite_note-63">[63]</a> According to the College Board, "AP Precalculus may be the last mathematics course of a student's secondary education, the course is structured to provide a coherent capstone experience and is not exclusively focused on preparation for future courses."<a href="#cite_note-pdf-64">[64]</a> 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.<a href="#cite_note-:15-45">[45]</a> 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.<a href="#cite_note-65">[65]</a> In Florida, students may also receive lessons on mathematical logic and <a href="/wiki/Set_theory" title="Set theory">set theory</a> at various grade levels in high school following the new 2020 reforms.<a href="#cite_note-66">[66]</a> The new Floridian standards also promote <a href="/wiki/Financial_literacy" title="Financial literacy">financial literacy</a> and emphasize how different mathematical topics from different grade levels are connected.<a href="#cite_note-:41-67">[67]</a> In Utah, the final required mathematics course in high school incorporates elements of Algebra II, Trigonometry, Precalculus, and Data Science. However, as of 2023, students may opt out of this class with a signed letter from their parents, and about half do.<a href="#cite_note-:46-68">[68]</a> College algebra is offered at many community colleges as <a href="/wiki/Remedial_education" title="Remedial education">remedial courses</a> for students who did not pass courses before Calculus.<a href="#cite_note-Bailey-69">[69]</a> It should not be confused with <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> and <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, taken by students who major in mathematics and allied fields (such as computer science) in four-year colleges and universities. <figure typeof="mw:File/Thumb"><a href="/wiki/File:Limes_Definition_Vektorgrafik.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Limes_Definition_Vektorgrafik.svg/330px-Limes_Definition_Vektorgrafik.svg.png" decoding="async" width="330" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Limes_Definition_Vektorgrafik.svg/495px-Limes_Definition_Vektorgrafik.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Limes_Definition_Vektorgrafik.svg/660px-Limes_Definition_Vektorgrafik.svg.png 2x" data-file-width="533" data-file-height="485" /></a><figcaption>Illustration of the epsilon-delta definition of the limit of a function</figcaption></figure> 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 calculus, the United States generally treats calculus as collegiate mathematics. A successfully completed college-level calculus course like one offered via <a href="/wiki/Advanced_Placement" title="Advanced Placement">Advanced Placement</a> program (<a href="/wiki/AP_Calculus" title="AP Calculus">AP Calculus</a> 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.<a href="#cite_note-:12-70">[70]</a><a href="#cite_note-:13-71">[71]</a> Calculus is a prerequisite or a corequisite for <a href="/wiki/AP_Physics_C:_Mechanics" title="AP Physics C: Mechanics">AP Physics C: Mechanics</a> and <a href="/wiki/AP_Physics_C:_Electricity_and_Magnetism" title="AP Physics C: Electricity and Magnetism">AP Physics C: Electricity and Magnetism</a>.<a href="#cite_note-72">[72]</a> Since the 1990s, the role of calculus in the high school curriculum has been a topic of controversy.<a href="#cite_note-:3-4">[4]</a> In this class, students learn about limits and continuity (the <a href="/wiki/Intermediate_value_theorem" title="Intermediate value theorem">intermediate</a> and <a href="/wiki/Mean_value_theorem" title="Mean value theorem">mean value theorems</a>), differentiation (the <a href="/wiki/Product_rule" title="Product rule">product</a>, <a href="/wiki/Quotient_rule" title="Quotient rule">quotient</a>, and <a href="/wiki/Chain_rule" title="Chain rule">chain rules</a>) and its applications (<a href="/wiki/Implicit_function" title="Implicit function">implicit differentiation</a>, <a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">logarithmic differentiation</a>, <a href="/wiki/Related_rates" title="Related rates">related rates</a>, <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">optimization</a>, <a href="/wiki/Second_derivative" title="Second derivative">concavity</a>, <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>, <a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's rules</a>), integration and the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental Theorem of Calculus</a>, techniques of integration (<a href="/wiki/Integration_by_substitution" title="Integration by substitution">u-substitution</a>, <a href="/wiki/Integration_by_parts" title="Integration by parts">by parts</a>, <a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a> and hyperbolic substitution, and by <a href="/wiki/Partial_fraction_decomposition" title="Partial fraction decomposition">partial fractions decomposition</a>), further applications of integration (calculating accumulated change, various problems in the sciences and engineering, <a href="/wiki/Separation_of_variables#Ordinary_differential_equations_(ODE)" title="Separation of variables">separable ordinary differential equations</a>, <a href="/wiki/Arc_length" title="Arc length">arc length</a> of a curve, areas between curves, volumes and surface areas of <a href="/wiki/Solid_of_revolution" title="Solid of revolution">solids of revolutions</a>), <a href="/wiki/Improper_integrals" class="mw-redirect" title="Improper integrals">improper integrals</a>, numerical integration (the midpoint rule, the <a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">trapezoid rule</a>, <a href="/wiki/Simpson%27s_rule" title="Simpson's rule">Simpson's rule</a>), infinite sequences and series and their convergence (the <a href="/wiki/Term_test" class="mw-redirect" title="Term test">nth-term</a>, <a href="/wiki/Direct_comparison_test" title="Direct comparison test">comparison</a>, <a href="/wiki/Ratio_test" title="Ratio test">ratio</a>, <a href="/wiki/Root_test" title="Root test">root</a>, <a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">integral</a>, p-series, and <a href="/wiki/Alternating_series_test" title="Alternating series test">alternating series tests</a>), <a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a> (with the Lagrange remainder), <a href="/wiki/Binomial_series" title="Binomial series">Newton's generalized binomial theorem</a>, <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's complex identity</a>, polar representation of complex numbers, <a href="/wiki/Parametric_equation" title="Parametric equation">parametric equations</a>, and curves in polar coordinates.<a href="#cite_note-:7-73">[73]</a><a href="#cite_note-74">[74]</a><a href="#cite_note-:6-75">[75]</a><a href="#cite_note-:9-76">[76]</a> Depending on the course and instructor, special topics in introductory calculus might include the classical differential geometry of curves (<a href="/wiki/Differentiable_curve" title="Differentiable curve">arc-length parametrization</a>, <a href="/wiki/Curvature" title="Curvature">curvature</a>, <a href="/wiki/Torsion_of_a_curve" title="Torsion of a curve">torsion</a>, and the <a href="/wiki/Frenet%E2%80%93Serret_formulas" title="Frenet–Serret formulas">Frenet–Serret formulas</a>), the <a href="/wiki/Limit_of_a_function#(ε,_δ)-definition_of_limit" title="Limit of a function">epsilon-delta definition</a> of the limit, <a href="/wiki/Linear_differential_equation#First-order_equation_with_variable_coefficients" title="Linear differential equation">first-order linear ordinary differential equations</a>, <a href="/wiki/Bernoulli_differential_equation" title="Bernoulli differential equation">Bernoulli differential equations</a>.<a href="#cite_note-:7-73">[73]</a><a href="#cite_note-:6-75">[75]</a> Some American high schools today also offer multivariable calculus<a href="#cite_note-:2-24">[24]</a> (partial differentiation, the multivariable chain rule and <a href="/wiki/Symmetry_of_second_derivatives" title="Symmetry of second derivatives">Clairault's theorem</a>; constrained optimization, <a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multipliers</a> and the <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a>; multidimensional integration, <a href="/wiki/Fubini%27s_theorem" title="Fubini's theorem">Fubini's theorem</a>, change of variables, and <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian determinants</a>; <a href="/wiki/Gradient" title="Gradient">gradients</a>, <a href="/wiki/Directional_derivative" title="Directional derivative">directional derivatives</a>, <a href="/wiki/Divergence" title="Divergence">divergences</a>, <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curls</a>, the fundamental theorem of gradients, <a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's theorem</a>, <a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a>, and <a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss' theorem</a>).<a href="#cite_note-:7-73">[73]</a><a href="#cite_note-:6-75">[75]</a><a href="#cite_note-:9-76">[76]</a> Other optional mathematics courses may be offered, such as statistics (including <a href="/wiki/AP_Statistics" title="AP Statistics">AP Statistics</a>) or business math. Students learn to use graphical and numerical techniques to analyze distributions of data (including <a href="/wiki/Univariate" title="Univariate">univariate</a>, <a href="/wiki/Bivariate_data" title="Bivariate data">bivariate</a>, and <a href="/wiki/Categorical_data" class="mw-redirect" title="Categorical data">categorical</a> data), the various methods of <a href="/wiki/Data_collection" title="Data collection">data collection</a> and the sorts of conclusions one can draw therefrom, probability, and <a href="/wiki/Statistical_inference" title="Statistical inference">statistical inference</a> (<a href="/wiki/Point_estimation" title="Point estimation">point estimation</a>, <a href="/wiki/Confidence_intervals" class="mw-redirect" title="Confidence intervals">confidence intervals</a>, and <a href="/wiki/Significance_test" class="mw-redirect" title="Significance test">significance tests</a>). High school students of exceptional ability may be selected to join a competition, such as the <a href="/wiki/United_States_of_America_Mathematical_Olympiad" title="United States of America Mathematical Olympiad">USA Mathematical Olympiad</a>,<a href="#cite_note-eligibilityUSAMO-77">[77]</a><a href="#cite_note-:48-25">[25]</a> or the <a href="/wiki/International_Mathematical_Olympiad" title="International Mathematical Olympiad">International Mathematical Olympiad</a>.<a href="#cite_note-:49-26">[26]</a><a href="#cite_note-78">[78]</a> <div class="mw-heading mw-heading3"><h3 id="Tertiary_school">Tertiary school</h3>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=4" title="Edit section: Tertiary school">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Excessive_citations plainlinks metadata ambox ambox-style ambox-citation_style" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This section may contain an <a href="/wiki/Wikipedia:Citation_overkill" title="Wikipedia:Citation overkill">excessive number of citations</a>. The details given are: These citations do not verify the article text. They should be replaced with citations to college and university curricula, or secondary sources. Please help <a href="/wiki/Wikipedia:Citation_overkill#How_to_trim_excessive_citations" title="Wikipedia:Citation overkill">remove low-quality or irrelevant citations</a>. (May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Curl_theorem_in_EM.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Curl_theorem_in_EM.svg/220px-Curl_theorem_in_EM.svg.png" decoding="async" width="220" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Curl_theorem_in_EM.svg/330px-Curl_theorem_in_EM.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Curl_theorem_in_EM.svg/440px-Curl_theorem_in_EM.svg.png 2x" data-file-width="184" data-file-height="178" /></a><figcaption>An illustration of Stokes' theorem in vector calculus</figcaption></figure> 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 <a href="/wiki/Ib_math_hl" class="mw-redirect" title="Ib math hl">IB Math HL</a>). Students majoring in mathematics, the physical sciences,<a href="#cite_note-79">[79]</a><a href="#cite_note-80">[80]</a> and engineering<a href="#cite_note-81">[81]</a> then take multivariable calculus,<a href="#cite_note-:7-73">[73]</a><a href="#cite_note-:6-75">[75]</a><a href="#cite_note-:9-76">[76]</a> linear algebra,<a href="#cite_note-82">[82]</a><a href="#cite_note-83">[83]</a><a href="#cite_note-84">[84]</a> complex variables,<a href="#cite_note-85">[85]</a><a href="#cite_note-86">[86]</a><a href="#cite_note-87">[87]</a> <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a>,<a href="#cite_note-88">[88]</a><a href="#cite_note-89">[89]</a><a href="#cite_note-90">[90]</a> and <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>.<a href="#cite_note-91">[91]</a><a href="#cite_note-92">[92]</a><a href="#cite_note-93">[93]</a> Mathematics majors may take a course offering a rigorous introduction to the concepts of modern mathematics<a href="#cite_note-94">[94]</a><a href="#cite_note-95">[95]</a><a href="#cite_note-96">[96]</a> before they tackle abstract algebra,<a href="#cite_note-97">[97]</a><a href="#cite_note-98">[98]</a><a href="#cite_note-99">[99]</a> number theory,<a href="#cite_note-100">[100]</a><a href="#cite_note-101">[101]</a><a href="#cite_note-102">[102]</a> <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>,<a href="#cite_note-103">[103]</a><a href="#cite_note-104">[104]</a><a href="#cite_note-105">[105]</a><a href="#cite_note-106">[106]</a> <a href="/wiki/Calculus_on_Euclidean_space" title="Calculus on Euclidean space">advanced calculus</a>,<a href="#cite_note-107">[107]</a><a href="#cite_note-108">[108]</a><a href="#cite_note-109">[109]</a> <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>,<a href="#cite_note-110">[110]</a><a href="#cite_note-111">[111]</a><a href="#cite_note-112">[112]</a><a href="#cite_note-113">[113]</a> probability theory,<a href="#cite_note-114">[114]</a><a href="#cite_note-115">[115]</a> statistics,<a href="#cite_note-116">[116]</a><a href="#cite_note-117">[117]</a> and advanced topics, such as set theory and <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>,<a href="#cite_note-118">[118]</a><a href="#cite_note-119">[119]</a><a href="#cite_note-120">[120]</a><a href="#cite_note-121">[121]</a> <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic processes</a>,<a href="#cite_note-122">[122]</a> integration and <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure theory</a>,<a href="#cite_note-123">[123]</a><a href="#cite_note-124">[124]</a><a href="#cite_note-125">[125]</a><a href="#cite_note-126">[126]</a> <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>,<a href="#cite_note-127">[127]</a><a href="#cite_note-128">[128]</a> <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>,<a href="#cite_note-129">[129]</a> <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>,<a href="#cite_note-130">[130]</a><a href="#cite_note-131">[131]</a><a href="#cite_note-132">[132]</a> and <a href="/wiki/Topology" title="Topology">topology</a>.<a href="#cite_note-133">[133]</a><a href="#cite_note-134">[134]</a> They may further choose courses in applied mathematics, such as mathematical modelling, <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>,<a href="#cite_note-135">[135]</a> <a href="/wiki/Game_theory" title="Game theory">game theory</a>,<a href="#cite_note-136">[136]</a><a href="#cite_note-137">[137]</a><a href="#cite_note-138">[138]</a> or <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a>. The <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>,<a href="#cite_note-139">[139]</a><a href="#cite_note-:21-140">[140]</a><a href="#cite_note-141">[141]</a> the <a href="/wiki/History_of_mathematics" title="History of mathematics">history of mathematics</a>,<a href="#cite_note-142">[142]</a><a href="#cite_note-143">[143]</a><a href="#cite_note-144">[144]</a><a href="#cite_note-145">[145]</a> and topics in theoretical or mathematical physics (such as classical mechanics,<a href="#cite_note-146">[146]</a><a href="#cite_note-:21-140">[140]</a><a href="#cite_note-147">[147]</a><a href="#cite_note-148">[148]</a> electrodynamics,<a href="#cite_note-149">[149]</a><a href="#cite_note-150">[150]</a> nonlinear dynamics,<a href="#cite_note-151">[151]</a> fluid mechanics,<a href="#cite_note-152">[152]</a><a href="#cite_note-153">[153]</a> quantum mechanics,<a href="#cite_note-154">[154]</a><a href="#cite_note-155">[155]</a><a href="#cite_note-156">[156]</a> or general relativity<a href="#cite_note-157">[157]</a><a href="#cite_note-158">[158]</a><a href="#cite_note-159">[159]</a><a href="#cite_note-160">[160]</a>) may be taken as electives. Computer science majors must study <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a><a href="#cite_note-161">[161]</a><a href="#cite_note-162">[162]</a> (such as combinatorics and <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>), <a href="/wiki/Information_theory" title="Information theory">information theory</a>,<a href="#cite_note-163">[163]</a> the <a href="/wiki/Theory_of_computation" title="Theory of computation">theory of computation</a>,<a href="#cite_note-164">[164]</a><a href="#cite_note-165">[165]</a> and <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>. Students in computer science and economics might have the option of taking <a href="/wiki/Algorithmic_game_theory" title="Algorithmic game theory">algorithmic game theory</a>.<a href="#cite_note-166">[166]</a> Those who study biomedical and social sciences have to study elementary probability<a href="#cite_note-167">[167]</a> and statistics.<a href="#cite_note-168">[168]</a> Students in the physical sciences and engineering need to understand <a href="/wiki/Error_analysis_(mathematics)" title="Error analysis (mathematics)">error analysis</a> for their laboratory sessions and courses.<a href="#cite_note-169">[169]</a><a href="#cite_note-170">[170]</a> Advanced undergraduates and beginning graduate students in physics may take a course on advanced mathematical methods for physics, which may cover <a href="/wiki/Contour_integration" title="Contour integration">contour integration</a>, the theory of distributions (<a href="/wiki/Generalized_function" title="Generalized function">generalized functions</a>), Fourier analysis, <a href="/wiki/Green%27s_function" title="Green's function">Green's functions</a>, <a href="/wiki/Special_functions" title="Special functions">special functions</a> (especially Euler's <a href="/wiki/Gamma_function" title="Gamma function">gamma</a> and <a href="/wiki/Beta_function" title="Beta function">beta functions</a>; <a href="/wiki/Bessel_function" title="Bessel function">Bessel functions</a>; <a href="/wiki/Legendre_polynomials" title="Legendre polynomials">Legendre polynomials</a>; <a href="/wiki/Hermite_polynomials" title="Hermite polynomials">Hermite polynomials</a>; <a href="/wiki/Laguerre_polynomials" title="Laguerre polynomials">Laguerre polynomials</a>; and the <a href="/wiki/Hypergeometric_function" title="Hypergeometric function">hypergeometric series</a>), <a href="/wiki/Asymptotic_expansion" title="Asymptotic expansion">asymptotic series expansions</a>, the calculus of variations, <a href="/wiki/Tensor" title="Tensor">tensors</a>, and <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> theory.<a href="#cite_note-171">[171]</a><a href="#cite_note-172">[172]</a><a href="#cite_note-173">[173]</a><a href="#cite_note-174">[174]</a><a href="#cite_note-175">[175]</a><a href="#cite_note-176">[176]</a> Exact requirements and available courses will depend on the institution in question. At many colleges and universities, confident students may compete in the <a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a>.<a href="#cite_note-:02-177">[177]</a><a href="#cite_note-:210-178">[178]</a><a href="#cite_note-:73-179">[179]</a><a href="#cite_note-:36-180">[180]</a> Exceptional undergraduates may participate in the annual <a href="/wiki/William_Lowell_Putnam_Mathematical_Competition" title="William Lowell Putnam Mathematical Competition">William Lowell Putnam Mathematical Competition</a>.<a href="#cite_note-:33-181">[181]</a><a href="#cite_note-182">[182]</a> 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 <a href="/wiki/Logical_intuition" title="Logical intuition">intuitions</a> that could help them become successful researchers.<a href="#cite_note-:33-181">[181]</a><a href="#cite_note-183">[183]</a> Besides the monetary prize, the winners are virtually guaranteed acceptance to a prestigious graduate school.<a href="#cite_note-184">[184]</a> Such competitions are one way for mathematical talents to stand out.<a href="#cite_note-185">[185]</a> <div class="mw-heading mw-heading2"><h2 id="Attendance_and_completion_rates">Attendance and completion rates</h2>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=5" title="Edit section: Attendance and completion rates">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Algebra_I_by_Sex_or_Race_(2015-16).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Algebra_I_by_Sex_or_Race_%282015-16%29.png/500px-Algebra_I_by_Sex_or_Race_%282015-16%29.png" decoding="async" width="500" height="252" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Algebra_I_by_Sex_or_Race_%282015-16%29.png/750px-Algebra_I_by_Sex_or_Race_%282015-16%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/28/Algebra_I_by_Sex_or_Race_%282015-16%29.png 2x" data-file-width="762" data-file-height="384" /></a><figcaption>Significant race or sex differences exist in the completion of Algebra I.<a href="#cite_note-:39-186">[186]</a></figcaption></figure> For many students, passing algebra is often a Herculean challenge,<a href="#cite_note-:27-8">[8]</a><a href="#cite_note-:19-10">[10]</a><a href="#cite_note-:20-11">[11]</a> so much so that many students have dropped out of high school because of it.<a href="#cite_note-:27-8">[8]</a> The greatest obstacle for excelling in algebra is fluency with fractions, something many Americans do not have.<a href="#cite_note-:28-9">[9]</a> Without mastery of high-school algebra—Algebra I and II—students will not be able to pursue collegiate STEM courses.<a href="#cite_note-:20-11">[11]</a><a href="#cite_note-:19-10">[10]</a><a href="#cite_note-:39-186">[186]</a> In fact, the lack of adequate preparation in mathematics is part of the reason why the rate of attrition in STEM is so high.<a href="#cite_note-:56-12">[12]</a> 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.<a href="#cite_note-:50-28">[28]</a> As for Algebra I, the number of 13-year-olds enrolled fell from 34% in 2012 to 24% in 2023.<a href="#cite_note-:45-17">[17]</a> Longitudinal analysis shows that the number of students completing high-school courses on calculus and statistics, including AP courses, have declined before 2019.<a href="#cite_note-:47-5">[5]</a><a href="#cite_note-187">[187]</a> Data taken from students' transcripts (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=15,188}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>15</mn> <mo>,</mo> <mn>188</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=15,188}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5b6ff7fea51b82018ceb78d5a8f4a20a3e8b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.008ex; height:2.509ex;" alt="{\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.<a href="#cite_note-:14-188">[188]</a> Among students identified as mathematically proficient by the <a href="/wiki/PSAT/NMSQT" title="PSAT/NMSQT">PSAT</a>, Asians are much more likely than blacks to attend an honors or Advanced Placement course in mathematics.<a href="#cite_note-189">[189]</a> Asians are also the most likely to have scored at least a 3 on the AP Calculus exams.<a href="#cite_note-:12-70">[70]</a> Students of lower socioeconomic status were less likely to pass Precalculus, Calculus, and Statistics.<a href="#cite_note-:14-188">[188]</a> 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 <a href="/wiki/AP_Computer_Science_A" title="AP Computer Science A">AP Computer Science A</a> (80%), <a href="/wiki/AP_Physics_C:_Mechanics" title="AP Physics C: Mechanics">AP Physics C: Mechanics</a> (74%) and <a href="/wiki/AP_Physics_C:_Electricity_and_Magnetism" title="AP Physics C: Electricity and Magnetism">AP Physics C: Electricity and Magnetism</a> (77%).<a href="#cite_note-190">[190]</a> 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.<a href="#cite_note-191">[191]</a> Perceptions and stereotypes of girls being less mathematically able than boys begin as early as second grade, and they affect <a href="/wiki/Self-fulfilling_prophecy" title="Self-fulfilling prophecy">how girls actually perform</a> in class or in a competition, such as the International Mathematical Olympiad.<a href="#cite_note-192">[192]</a> Among university students who have taken calculus, engineering disciplines are the most popular among men and biology among women.<a href="#cite_note-:12-70">[70]</a> 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.<a href="#cite_note-:3-4">[4]</a> 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.<a href="#cite_note-:17-29">[29]</a> But around 80% fail to meet this requirement,<a href="#cite_note-:17-29">[29]</a> and 60% require remedial courses.<a href="#cite_note-:19-10">[10]</a> Many students at these schools drop out after failing even in remedial courses, such as (the equivalent of) Algebra II.<a href="#cite_note-193">[193]</a> On the other hand, four-year institutions have seen increased student interest in STEM programs, including mathematics and statistics.<a href="#cite_note-:81-27">[27]</a> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:U.S._college_major_change_between_2011_and_2018.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/U.S._college_major_change_between_2011_and_2018.png/700px-U.S._college_major_change_between_2011_and_2018.png" decoding="async" width="700" height="394" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/U.S._college_major_change_between_2011_and_2018.png/1050px-U.S._college_major_change_between_2011_and_2018.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/39/U.S._college_major_change_between_2011_and_2018.png 2x" data-file-width="1214" data-file-height="683" /></a><figcaption>Healthcare and <a href="/wiki/Science,_technology,_engineering,_and_mathematics" title="Science, technology, engineering, and mathematics">STEM</a> subjects, including mathematics and statistics, have grown in popularity while the liberal arts and social studies, especially history, have declined due to market forces.<a href="#cite_note-:81-27">[27]</a><a href="#cite_note-:132-194">[194]</a></figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Controversies_and_issues">Controversies and issues</h2>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=6" title="Edit section: Controversies and issues">edit</a>]</div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Boy_in_school,_1961_(52338002867).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Boy_in_school%2C_1961_%2852338002867%29.jpg/220px-Boy_in_school%2C_1961_%2852338002867%29.jpg" decoding="async" width="220" height="278" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Boy_in_school%2C_1961_%2852338002867%29.jpg/330px-Boy_in_school%2C_1961_%2852338002867%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Boy_in_school%2C_1961_%2852338002867%29.jpg/440px-Boy_in_school%2C_1961_%2852338002867%29.jpg 2x" data-file-width="1456" data-file-height="1843" /></a><figcaption>A schoolboy in Seattle (1961). The contents of mathematical education has been a topic of debate for decades.</figcaption></figure> Mathematics education has been a topic of debate among academics, parents, as well as educators.<a href="#cite_note-:3-4">[4]</a><a href="#cite_note-:28-9">[9]</a><a href="#cite_note-Banchero-195">[195]</a><a href="#cite_note-:5-38">[38]</a> 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.<a href="#cite_note-:15-45">[45]</a><a href="#cite_note-:26-13">[13]</a> 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.<a href="#cite_note-:40-196">[196]</a> 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.<a href="#cite_note-197">[197]</a> <div class="mw-heading mw-heading3"><h3 id="Progressive_education">Progressive education</h3>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=7" title="Edit section: Progressive education">edit</a>]</div> During the first half of the twentieth century, there was a movement aimed at systematically reforming American public education along more "<a href="/wiki/Progressive_education" title="Progressive education">progressive</a>" grounds. <a href="/wiki/William_Heard_Kilpatrick" title="William Heard Kilpatrick">William Heard Kilpatrick</a>, 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.<a href="#cite_note-:3-4">[4]</a> By the 1940s, however, the deficiency in mathematical skills among military recruits became a public scandal. Admiral <a href="/wiki/Chester_W._Nimitz" title="Chester W. Nimitz">Chester Nimitz</a> 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.<a href="#cite_note-:3-4">[4]</a> Indeed, many parents opposed the progressive reforms, criticizing the lack of contents. By mid-century, technological marvels, such as <a href="/wiki/Radar" title="Radar">radar</a>, <a href="/wiki/Nuclear_power" title="Nuclear power">nuclear energy</a>, and the <a href="/wiki/Jet_engine" title="Jet engine">jet engine</a>, made progressive education untenable.<a href="#cite_note-:3-4">[4]</a> <div class="mw-heading mw-heading3"><h3 id="New_Math">New Math</h3>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=8" title="Edit section: New Math">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Navajo_students_studying_mathematics_at_day_school_-_NARA_-_295155.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Navajo_students_studying_mathematics_at_day_school_-_NARA_-_295155.jpg/220px-Navajo_students_studying_mathematics_at_day_school_-_NARA_-_295155.jpg" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Navajo_students_studying_mathematics_at_day_school_-_NARA_-_295155.jpg/330px-Navajo_students_studying_mathematics_at_day_school_-_NARA_-_295155.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Navajo_students_studying_mathematics_at_day_school_-_NARA_-_295155.jpg/440px-Navajo_students_studying_mathematics_at_day_school_-_NARA_-_295155.jpg 2x" data-file-width="3000" data-file-height="2304" /></a><figcaption>Navajo students learning arithmetic (c. 1940). By the late 1950s, mathematics education had become more rigorous.</figcaption></figure> Under the '<a href="/wiki/New_Math" title="New Math">New Math</a>' initiative, created after the successful launch of the Soviet satellite <a href="/wiki/Sputnik_1" title="Sputnik 1">Sputnik</a> in 1957, conceptual <a href="/wiki/Abstraction_(mathematics)" title="Abstraction (mathematics)">abstraction</a> rather than calculation gained a central role in mathematics education.<a href="#cite_note-:18-39">[39]</a> The educational status quo was severely criticized as a source of national humiliation and reforms were demanded, prompting Congress to introduce the <a href="/wiki/National_Defense_Education_Act" title="National Defense Education Act">National Defense Education Act</a> of 1958.<a href="#cite_note-:3-4">[4]</a> The U.S. federal government under President <a href="/wiki/Dwight_D._Eisenhower" title="Dwight D. Eisenhower">Dwight D. Eisenhower</a> 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.<a href="#cite_note-Garraty-1991-198">[198]</a><a href="#cite_note-199">[199]</a> Conceived in response to the lack of emphasis on content of the progressive education<a href="#cite_note-:3-4">[4]</a> and the technological advances of World War II,<a href="#cite_note-:4-200">[200]</a> New Math was part of an international movement influenced by the <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicholas Bourbaki</a> school in France, attempting to bring the mathematics taught in schools closer to what research mathematicians actually use. Students received lessons in <a href="/wiki/Set_theory" title="Set theory">set theory</a>, which is what mathematicians actually use to construct the set of real numbers, normally taught to advanced undergraduates in <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a> (see <a href="/wiki/Dedekind_cut" title="Dedekind cut">Dedekind cuts</a> and <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a>). Arithmetic with bases other than ten was also taught (see <a href="/wiki/Binary_number" title="Binary number">binary arithmetic</a> and <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>).<a href="#cite_note-:22-201">[201]</a> Other topics included <a href="/wiki/Number_theory" title="Number theory">number theory</a>, <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, and analytic geometry.<a href="#cite_note-:4-200">[200]</a> 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.<a href="#cite_note-:18-39">[39]</a> It was criticized by experts, too. In a 1965 essay, physicist <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a> 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."<a href="#cite_note-202">[202]</a> In his 1973 book, <a href="/wiki/Why_Johnny_Can%27t_Add:_the_Failure_of_the_New_Math" class="mw-redirect" title="Why Johnny Can't Add: the Failure of the New Math">Why Johnny Can't Add: the Failure of the New Math</a>, mathematician and historian of mathematics <a href="/wiki/Morris_Kline" title="Morris Kline">Morris Kline</a> 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."<a href="#cite_note-Kline2-203">[203]</a> 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.<a href="#cite_note-204">[204]</a> Mathematician <a href="/wiki/George_F._Simmons" title="George F. Simmons">George F. Simmons</a> 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 <a href="/wiki/Commutative_law" class="mw-redirect" title="Commutative law">commutative law</a>, but did not know the <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication table</a>."<a href="#cite_note-205">[205]</a> 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.<a href="#cite_note-:3-4">[4]</a> <div class="mw-heading mw-heading3"><h3 id="Standards-based_reforms_and_the_NCTM">Standards-based reforms and the NCTM</h3>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=9" title="Edit section: Standards-based reforms and the NCTM">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Math_wars" title="Math wars">Math wars</a> and <a href="/wiki/NCTM" class="mw-redirect" title="NCTM">NCTM</a></div> 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.<a href="#cite_note-206">[206]</a> However, as mathematician Hung-Hsi Wu explained, the apparent dichotomy between basic skills and understanding of mathematical concepts is a delusion.<a href="#cite_note-207">[207]</a> In 1989 the National Council for Teachers of Mathematics (NCTM) produced the <a href="/wiki/National_Council_of_Teachers_of_Mathematics#1989_Curriculum_and_Evaluation_Standards_for_School_Mathematics" title="National Council of Teachers of Mathematics">Curriculum and Evaluation Standards for School Mathematics</a>. Widespread adoption of the new standards notwithstanding, the pedagogical practice changed little in the United States during the 1990s.<a href="#cite_note-208">[208]</a> In fact, mathematics education became a hotly debated subject in the 1990s and early 2000s. This debate pitted mathematicians (like <a href="/wiki/University_of_California,_Berkeley" title="University of California, Berkeley">UC Berkeley</a> mathematician Hung-Hsi Wu) and parents, many of whom with substantial knowledge of mathematics (such as the <a href="/wiki/Institute_for_Advanced_Study" title="Institute for Advanced Study">Institute for Advanced Study</a> physicist <a href="/wiki/Chiara_Nappi" title="Chiara Nappi">Chiara R. Nappi</a>), 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.<a href="#cite_note-:3-4">[4]</a> In 1989, the more radical NCTM reforms were eliminated. Instead, greater emphasis was put on substantive mathematics.<a href="#cite_note-:3-4">[4]</a> 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 <a href="/wiki/Core-Plus_Mathematics_Project" title="Core-Plus Mathematics Project">Core-Plus Mathematics Project</a>. In the late 1990s and early 2000s, the so-called <a href="/wiki/Math_wars" title="Math wars">math wars</a> 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.<a href="#cite_note-Christian_Science_Monitor-209">[209]</a> However, data provided by the <a href="/wiki/University_of_Michigan" title="University of Michigan">University of Michigan</a> 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.<a href="#cite_note-210">[210]</a> Mathematics instructor <a href="/wiki/Jaime_Escalante" title="Jaime Escalante">Jaime Escalante</a> dismissed the NCTM standards as something written by a PE teacher.<a href="#cite_note-:3-4">[4]</a> In 2001 and 2009, NCTM released the <a href="/wiki/Principles_and_Standards_for_School_Mathematics" title="Principles and Standards for School Mathematics">Principles and Standards for School Mathematics</a> (PSSM) and the <a href="/wiki/Principles_and_Standards_for_School_Mathematics#Curriculum_Focal_Points" title="Principles and Standards for School Mathematics">Curriculum Focal Points</a> 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<a href="#cite_note-211">[211]</a> 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. <div class="mw-heading mw-heading3"><h3 id="Integrated_mathematics">Integrated mathematics</h3>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=10" title="Edit section: Integrated mathematics">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Inverse_Curves_Parabola_Cardioid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Inverse_Curves_Parabola_Cardioid.svg/220px-Inverse_Curves_Parabola_Cardioid.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Inverse_Curves_Parabola_Cardioid.svg/330px-Inverse_Curves_Parabola_Cardioid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Inverse_Curves_Parabola_Cardioid.svg/440px-Inverse_Curves_Parabola_Cardioid.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption>Some geometric curves expressed algebraically in polar coordinates</figcaption></figure> 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 <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>'s call for Geometry to be integrated with other math subjects. In 1892 the American <a href="/wiki/Committee_of_Ten" title="Committee of Ten">Committee of Ten</a> recommended the same strategy for the United States, but American teachers had already been developing the habit of teaching Geometry has 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.<a href="#cite_note-:51-49">[49]</a> 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 <a href="/wiki/Integrated_mathematics" title="Integrated mathematics">integrated curriculum</a>, familiarizing students with various aspects of calculus and prerequisites throughout secondary school.<a href="#cite_note-:12-70">[70]</a><a href="#cite_note-:13-71">[71]</a> In fact, many topics in Algebra and Geometry that Americans typically learn in high school are taught in middle school in Europe,<a href="#cite_note-:40-196">[196]</a> 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 <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> and Felix Klein, respectively.<a href="#cite_note-:12-70">[70]</a><a href="#cite_note-:13-71">[71]</a> However, as the Singaporean case demonstrates, early exposure to the concepts of calculus does not necessarily translates to actual understanding among high school students.<a href="#cite_note-212">[212]</a><a href="#cite_note-:13-71">[71]</a> 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.<a href="#cite_note-:12-70">[70]</a> 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.<a href="#cite_note-:1-3">[3]</a> As mentioned above, only 7% of American high school students take Integrated Mathematics.<a href="#cite_note-:14-188">[188]</a> <div class="mw-heading mw-heading3"><h3 id="Preparation_for_college">Preparation for college</h3>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=11" title="Edit section: Preparation for college">edit</a>]</div> 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 (<a href="/wiki/Science,_technology,_engineering,_and_mathematics" title="Science, technology, engineering, and mathematics">STEM</a>) in high school, many university students find themselves ill-equipped for rigorous STEM education in part because of their inadequate preparation in mathematics.<a href="#cite_note-:56-12">[12]</a><a href="#cite_note-:67-7">[7]</a> Meanwhile, Chinese, Indian, and Singaporean students are exposed to high-level mathematics and science at a young age.<a href="#cite_note-:56-12">[12]</a> About half of STEM students in the U.S. dropped out of their programs between 2003 and 2009.<a href="#cite_note-:67-7">[7]</a> 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.<a href="#cite_note-:35-20">[20]</a><a href="#cite_note-:67-7">[7]</a><a href="#cite_note-:38-213">[213]</a> An emphasis on speed and rote memorization gives as many as one-third of students aged five and over <a href="/wiki/Mathematical_anxiety" title="Mathematical anxiety">mathematical anxiety</a>.<a href="#cite_note-:0-32">[32]</a> 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.<a href="#cite_note-:15-45">[45]</a> 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.<a href="#cite_note-:16-6">[6]</a> Nevertheless, Calculus remains the most recommended course for ambitious students.<a href="#cite_note-:16-6">[6]</a> 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.<a href="#cite_note-:46-68">[68]</a> By the mid-2010s, only a quarter of American high school seniors are able to do grade-level math,<a href="#cite_note-214">[214]</a> yet about half graduate from high school as A students, prompting concerns of <a href="/wiki/Grade_inflation" title="Grade inflation">grade inflation</a>.<a href="#cite_note-215">[215]</a> Strong performance in Algebra I, Geometry, and Algebra II predict good grades in university-level Calculus even better than taking Calculus in high school.<a href="#cite_note-:32-44">[44]</a> 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.<a href="#cite_note-216">[216]</a> <div class="mw-heading mw-heading3"><h3 id="Enrichment_programs_and_accelerated_tracks">Enrichment programs and accelerated tracks</h3>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=12" title="Edit section: Enrichment programs and accelerated tracks">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Math_Club_at_Lakehill_Preparatory.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Math_Club_at_Lakehill_Preparatory.jpg/220px-Math_Club_at_Lakehill_Preparatory.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Math_Club_at_Lakehill_Preparatory.jpg/330px-Math_Club_at_Lakehill_Preparatory.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Math_Club_at_Lakehill_Preparatory.jpg/440px-Math_Club_at_Lakehill_Preparatory.jpg 2x" data-file-width="3264" data-file-height="2448" /></a><figcaption>A math club at a preparatory school in Texas (2018)</figcaption></figure> 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 <a href="/wiki/Math_circle" title="Math circle">math circles</a>. Some public schools serving low-income neighborhoods even denied the existence of mathematically <a href="/wiki/Intellectual_giftedness" title="Intellectual giftedness">gifted</a> students.<a href="#cite_note-:67-7">[7]</a> In fact, American educators tend to focus on poorly performing students rather than those at the top, unlike their Asian counterparts.<a href="#cite_note-:412-217">[217]</a> Parents' proposal for an accelerated track for their children are oftentimes met with hostility by school administrators.<a href="#cite_note-218">[218]</a> Conversely, initiatives aimed at de-emphasizing certain core subjects, such as Algebra I, triggered strong backlash from parents and university faculty members.<a href="#cite_note-:43-46">[46]</a><a href="#cite_note-:44-47">[47]</a> Students identified by the <a href="/wiki/Study_of_Mathematically_Precocious_Youth" title="Study of Mathematically Precocious Youth">Study of Mathematically Precocious Youth</a> as top scorers on the mathematics (and later, verbal) sections of the SAT often became highly successful in their fields.<a href="#cite_note-219">[219]</a> By the mid-2010s, some public schools have begun offering enrichment programs to their students.<a href="#cite_note-:67-7">[7]</a> 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.<a href="#cite_note-:15-45">[45]</a><a href="#cite_note-:16-6">[6]</a> In San Francisco, for example, such a plan was dropped due to a combination of mixed results and public backlash.<a href="#cite_note-:15-45">[45]</a> <div class="mw-heading mw-heading3"><h3 id="Shortage_of_instructors">Shortage of instructors</h3>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=13" title="Edit section: Shortage of instructors">edit</a>]</div> A shortage of qualified mathematics schoolteachers has been a serious problem in the United States for many years.<a href="#cite_note-:34-18">[18]</a><a href="#cite_note-:37-19">[19]</a> 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.<a href="#cite_note-220">[220]</a> Teachers oftentimes unknowingly transmit their own negative attitude towards mathematics to their students, damaging the quality of instruction.<a href="#cite_note-:38-213">[213]</a> <div class="mw-heading mw-heading2"><h2 id="Standardized_tests">Standardized tests</h2>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=14" title="Edit section: Standardized tests">edit</a>]</div> <ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 544px"> <div class="thumb" style="width: 542px;"><a href="/wiki/File:PISA_average_Mathematics_scores_2018.png" class="mw-file-description" title="PISA average mathematics scores (2018)"><img alt="PISA average mathematics scores (2018)" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/PISA_average_Mathematics_scores_2018.png/813px-PISA_average_Mathematics_scores_2018.png" decoding="async" width="542" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/PISA_average_Mathematics_scores_2018.png/1221px-PISA_average_Mathematics_scores_2018.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/PISA_average_Mathematics_scores_2018.png/1626px-PISA_average_Mathematics_scores_2018.png 2x" data-file-width="7192" data-file-height="3318" /></a></div> <div class="gallerytext"><a href="/wiki/Programme_for_International_Student_Assessment" title="Programme for International Student Assessment">PISA</a> average mathematics scores (2018)</div> </li> <li class="gallerybox" style="width: 544px"> <div class="thumb" style="width: 542px;"><a href="/wiki/File:TIMSS_4th_grade_average_Mathematics_scores_2019.png" class="mw-file-description" title="TIMSS 4th grade average mathematics scores (2019)"><img alt="TIMSS 4th grade average mathematics scores (2019)" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/TIMSS_4th_grade_average_Mathematics_scores_2019.png/813px-TIMSS_4th_grade_average_Mathematics_scores_2019.png" decoding="async" width="542" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/TIMSS_4th_grade_average_Mathematics_scores_2019.png/1221px-TIMSS_4th_grade_average_Mathematics_scores_2019.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/TIMSS_4th_grade_average_Mathematics_scores_2019.png/1626px-TIMSS_4th_grade_average_Mathematics_scores_2019.png 2x" data-file-width="7192" data-file-height="3318" /></a></div> <div class="gallerytext"><a href="/wiki/Trends_in_International_Mathematics_and_Science_Study" title="Trends in International Mathematics and Science Study">TIMSS</a> 4th grade average mathematics scores (2019)</div> </li> </ul>The <a href="/wiki/Programme_for_International_Student_Assessment" title="Programme for International Student Assessment">Program for International Student Assessment</a> (PISA) is held every three years for 15-year-old students worldwide.<a href="#cite_note-221">[221]</a> 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.<a href="#cite_note-:0-32">[32]</a> 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.<a href="#cite_note-:23-31">[31]</a> 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.<a href="#cite_note-222">[222]</a> Furthermore, one third of American students did not meet the requirements for basic proficiency in mathematics.<a href="#cite_note-223">[223]</a> However, European- and especially Asian-American students perform above the OECD average. See chart below.<a href="#cite_note-:03-35">[35]</a> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:PISA_Math_Scores_(2018).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/ed/PISA_Math_Scores_%282018%29.png" decoding="async" width="845" height="573" class="mw-file-element" data-file-width="845" data-file-height="573" /></a><figcaption></figcaption></figure> According to a 2021 report by the <a href="/wiki/National_Science_Foundation" title="National Science Foundation">National Science Foundation</a> (NSF), American students' <a href="/wiki/Numeracy" title="Numeracy">mathematical literacy</a> ranks 25th out of 37 nations of the Organization for Economic Cooperation and Development (<a href="/wiki/OECD" title="OECD">OECD</a>).<a href="#cite_note-224">[224]</a> During the 2000s and 2010s, as more and more college-bound students take the SAT, scores have gone down.<a href="#cite_note-:29-225">[225]</a><a href="#cite_note-:8-33">[33]</a><a href="#cite_note-:68-30">[30]</a> (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.<a href="#cite_note-:29-225">[225]</a> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Historical_Average_SAT_Scores_(Vector).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Historical_Average_SAT_Scores_%28Vector%29.svg/680px-Historical_Average_SAT_Scores_%28Vector%29.svg.png" decoding="async" width="680" height="453" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Historical_Average_SAT_Scores_%28Vector%29.svg/1020px-Historical_Average_SAT_Scores_%28Vector%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Historical_Average_SAT_Scores_%28Vector%29.svg/1360px-Historical_Average_SAT_Scores_%28Vector%29.svg.png 2x" data-file-width="512" data-file-height="341" /></a><figcaption>Historical average SAT math scores reached a nadir in 1980, declined between 2005 and 2016 and after the 2016 re-scaling.</figcaption></figure> In 2015, educational psychologist Jonathan Wai of Duke University analyzed average test scores from the <a href="/wiki/Army_General_Classification_Test" title="Army General Classification Test">Army General Classification Test</a> in 1946 (10,000 students), the Selective Service College Qualification Test in 1952 (38,420), <a href="/wiki/Project_Talent" title="Project Talent">Project Talent</a> in the early 1970s (400,000), the <a href="/wiki/Graduate_Record_Examinations" title="Graduate Record Examinations">Graduate Record Examination</a> 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.)<a href="#cite_note-Wai-2015-226">[226]</a><a href="#cite_note-227">[227]</a> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Uni_Major_and_SAT_Averages.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/72/Uni_Major_and_SAT_Averages.png" decoding="async" width="853" height="480" class="mw-file-element" data-file-width="853" data-file-height="480" /></a><figcaption></figcaption></figure> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Graduate_Majors_and_GRE_Averages.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/1f/Graduate_Majors_and_GRE_Averages.png" decoding="async" width="842" height="451" class="mw-file-element" data-file-width="842" data-file-height="451" /></a><figcaption></figcaption></figure> Results from the <a href="/wiki/National_Assessment_of_Educational_Progress" title="National Assessment of Educational Progress">National Assessment of Educational Progress</a> (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.<a href="#cite_note-:25-34">[34]</a> While students' scores fell for all subjects, mathematics was the hardest hit, with a drop of eight points,<a href="#cite_note-228">[228]</a> the steepest decline in 50 years.<a href="#cite_note-:45-17">[17]</a> Scores dropped for students of all races, sexes, socioeconomic classes, types of schools, and states with very few exceptions.<a href="#cite_note-:30-229">[229]</a><a href="#cite_note-:42-230">[230]</a> This might be because mathematics education is more dependent upon the classroom experience than reading,<a href="#cite_note-:42-230">[230]</a> as students who were allowed to return to in-person classes generally did better, more so in mathematics than in reading.<a href="#cite_note-231">[231]</a> However, on the topics of statistics and probability, student performance had already declined before the pandemic.<a href="#cite_note-:31-14">[14]</a> As consequence, the entire cohort of college students in the 2022-23 academic year have lower average grades and mathematical standards.<a href="#cite_note-232">[232]</a> 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.<a href="#cite_note-:24-22">[22]</a> Having a higher NAEP math score in eighth grade is correlated with high academic standing, higher income, lower rates of <a href="/wiki/Teenage_pregnancy_in_the_United_States" title="Teenage pregnancy in the United States">adolescent parenthood</a>, and lower chances of criminality.<a href="#cite_note-233">[233]</a> <div class="mw-heading mw-heading2"><h2 id="Advanced_Placement_Mathematics">Advanced Placement Mathematics</h2>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=15" title="Edit section: Advanced Placement Mathematics">edit</a>]</div> There was considerable debate about whether or not calculus should be included when the <a href="/wiki/Advanced_Placement" title="Advanced Placement">Advanced Placement</a> (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.<a href="#cite_note-:2-24">[24]</a> 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.<a href="#cite_note-:2-24">[24]</a> As of 2021, <a href="/wiki/AP_Precalculus" title="AP Precalculus">AP Precalculus</a> 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.<a href="#cite_note-:2-24">[24]</a> AP Precalculus launched in Fall 2023.<a href="#cite_note-234">[234]</a> <div class="mw-heading mw-heading2"><h2 id="Conferences">Conferences</h2>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=16" title="Edit section: Conferences">edit</a>]</div> Mathematics education research and practitioner conferences include: <a href="/wiki/National_Council_of_Teachers_of_Mathematics" title="National Council of Teachers of Mathematics">NCTM</a>'s <a rel="nofollow" class="external text" href="http://www.nctm.org/conferences">Regional Conference and Exposition and Annual Meeting and Exposition</a>; The Psychology of Mathematics Education's North American Chapter <a rel="nofollow" class="external text" href="http://www.pmena.org">annual conference</a>; and numerous smaller regional conferences. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=17" title="Edit section: See also">edit</a>]</div> <style 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Spring. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-09503-5" title="Special:BookSources/978-0-387-09503-5"><bdi>978-0-387-09503-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods%3A+For+Students+of+Physics+and+Related+Fields&rft.edition=2nd&rft.pub=Spring&rft.date=2008&rft.isbn=978-0-387-09503-5&rft.aulast=Hassani&rft.aufirst=Sadri&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRileyHobsonBence2006" class="citation book cs1">Riley, K.F.; Hobson, Michael P.; Bence, S.J. (2006). Mathematical Methods for Physics and Engineering. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-67971-8" title="Special:BookSources/978-0-521-67971-8"><bdi>978-0-521-67971-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+for+Physics+and+Engineering&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-67971-8&rft.aulast=Riley&rft.aufirst=K.F.&rft.au=Hobson%2C+Michael+P.&rft.au=Bence%2C+S.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrang2016" class="citation book cs1"><a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Strang, Gilbert</a> (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-980-23277-6" title="Special:BookSources/978-0-980-23277-6"><bdi>978-0-980-23277-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Linear+Algebra&rft.edition=5th&rft.pub=Wellesley-Cambridge+Press&rft.date=2016&rft.isbn=978-0-980-23277-6&rft.aulast=Strang&rft.aufirst=Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxler2014" class="citation book cs1"><a href="/wiki/Sheldon_Axler" title="Sheldon Axler">Axler, Sheldon</a> (2014). Linear Algebra Done Right. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-11079-0" title="Special:BookSources/978-3-319-11079-0"><bdi>978-3-319-11079-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.pub=Springer&rft.date=2014&rft.isbn=978-3-319-11079-0&rft.aulast=Axler&rft.aufirst=Sheldon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos2017" class="citation book cs1"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul</a> (2017). Finite-Dimensional Vector Spaces (2nd ed.). Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-81486-5" title="Special:BookSources/978-0-486-81486-5"><bdi>978-0-486-81486-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite-Dimensional+Vector+Spaces&rft.edition=2nd&rft.pub=Dover+Publications&rft.date=2017&rft.isbn=978-0-486-81486-5&rft.aulast=Halmos&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpiegelLipschutzSchillerSpellman2009" class="citation book cs1">Spiegel, Murray R.; Lipschutz, Seymour; Schiller, John J.; Spellman, Dennis (2009). Schaum's Outline of Complex Variables (2nd ed.). McGraw-Hill Companies. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-071-61569-3" title="Special:BookSources/978-0-071-61569-3"><bdi>978-0-071-61569-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schaum%27s+Outline+of+Complex+Variables&rft.edition=2nd&rft.pub=McGraw-Hill+Companies&rft.date=2009&rft.isbn=978-0-071-61569-3&rft.aulast=Spiegel&rft.aufirst=Murray+R.&rft.au=Lipschutz%2C+Seymour&rft.au=Schiller%2C+John+J.&rft.au=Spellman%2C+Dennis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-86"><a href="#cite_ref-86">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKwok2010" class="citation book cs1">Kwok, Yue Kuen (2010). Applied Complex Variables for Scientists and Engineers (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-70138-9" title="Special:BookSources/978-0-521-70138-9"><bdi>978-0-521-70138-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Complex+Variables+for+Scientists+and+Engineers&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-70138-9&rft.aulast=Kwok&rft.aufirst=Yue+Kuen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrantz2008" class="citation book cs1">Krantz, Steven G. (2008). A Guide to Complex Variables. <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-883-85338-2" title="Special:BookSources/978-0-883-85338-2"><bdi>978-0-883-85338-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Guide+to+Complex+Variables&rft.pub=Mathematical+Association+of+America&rft.date=2008&rft.isbn=978-0-883-85338-2&rft.aulast=Krantz&rft.aufirst=Steven+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-88"><a href="#cite_ref-88">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZillWright2013" class="citation book cs1">Zill, Dennis G.; Wright, Warren S. (2013). Differential Equations with Boundary-Value Problems (8th ed.). Brooks/Cole Cengage Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-111-82706-9" title="Special:BookSources/978-1-111-82706-9"><bdi>978-1-111-82706-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Equations+with+Boundary-Value+Problems&rft.edition=8th&rft.pub=Brooks%2FCole+Cengage+Learning&rft.date=2013&rft.isbn=978-1-111-82706-9&rft.aulast=Zill&rft.aufirst=Dennis+G.&rft.au=Wright%2C+Warren+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyceDiPrima2012" class="citation book cs1">Boyce, William E.; DiPrima, Richard C. (2012). Elementary Differential Equations and Boundary Value Problems (10th ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-45831-0" title="Special:BookSources/978-0-470-45831-0"><bdi>978-0-470-45831-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Differential+Equations+and+Boundary+Value+Problems&rft.edition=10th&rft.pub=Wiley&rft.date=2012&rft.isbn=978-0-470-45831-0&rft.aulast=Boyce&rft.aufirst=William+E.&rft.au=DiPrima%2C+Richard+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-90"><a href="#cite_ref-90">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArnold1978" class="citation book cs1"><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Arnold, Vladimir</a> (1978). Ordinary Differential Equations. Translated by Silverman, Richard. The MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-51018-9" title="Special:BookSources/978-0-262-51018-9"><bdi>978-0-262-51018-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ordinary+Differential+Equations&rft.pub=The+MIT+Press&rft.date=1978&rft.isbn=978-0-262-51018-9&rft.aulast=Arnold&rft.aufirst=Vladimir&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBleeckerCsordas1997" class="citation book cs1">Bleecker, David D.; Csordas, George (1997). Basic Partial Differential Equations. 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Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-134-68960-9" title="Special:BookSources/978-0-134-68960-9"><bdi>978-0-134-68960-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.edition=2nd&rft.pub=Pearson&rft.date=2017&rft.isbn=978-0-134-68960-9&rft.aulast=Artin&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDummitFoote2003" class="citation book cs1">Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-43334-7" title="Special:BookSources/978-0-471-43334-7"><bdi>978-0-471-43334-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+Algebra&rft.edition=3rd&rft.pub=Wiley&rft.date=2003&rft.isbn=978-0-471-43334-7&rft.aulast=Dummit&rft.aufirst=David+S.&rft.au=Foote%2C+Richard+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPinter2010" class="citation book cs1">Pinter, Charles C. (2010). A Book of Abstract Algebra (2nd ed.). Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47417-5" title="Special:BookSources/978-0-486-47417-5"><bdi>978-0-486-47417-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Book+of+Abstract+Algebra&rft.edition=2nd&rft.pub=Dover+Publications&rft.date=2010&rft.isbn=978-0-486-47417-5&rft.aulast=Pinter&rft.aufirst=Charles+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-100"><a href="#cite_ref-100">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScharlauOpolka2010" class="citation book cs1">Scharlau, Winfried; Opolka, Hans (2010). From Fermat to Minkowski: Lectures on the Theory of Numbers and Its Historical Development. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-441-92821-4" title="Special:BookSources/978-1-441-92821-4"><bdi>978-1-441-92821-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Fermat+to+Minkowski%3A+Lectures+on+the+Theory+of+Numbers+and+Its+Historical+Development&rft.pub=Springer-Verlag&rft.date=2010&rft.isbn=978-1-441-92821-4&rft.aulast=Scharlau&rft.aufirst=Winfried&rft.au=Opolka%2C+Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGranville2019" class="citation book cs1">Granville, Andrew (2019). Number Theory Revealed: A Masterclass. Rhode Island: American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-6370-0" title="Special:BookSources/978-1-4704-6370-0"><bdi>978-1-4704-6370-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory+Revealed%3A+A+Masterclass&rft.place=Rhode+Island&rft.pub=American+Mathematical+Society&rft.date=2019&rft.isbn=978-1-4704-6370-0&rft.aulast=Granville&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDudley2008" class="citation book cs1">Dudley, Underwood (2008). Elementary Number Theory (2nd ed.). Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-46931-7" title="Special:BookSources/978-0-486-46931-7"><bdi>978-0-486-46931-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Number+Theory&rft.edition=2nd&rft.pub=Dover+Publications&rft.date=2008&rft.isbn=978-0-486-46931-7&rft.aulast=Dudley&rft.aufirst=Underwood&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMattuck2013" class="citation book cs1"><a href="/wiki/Arthur_Mattuck" title="Arthur Mattuck">Mattuck, Arthur</a> (2013). Introduction to Analysis. 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John Wiley & Sons, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-43331-6" title="Special:BookSources/978-0-471-43331-6"><bdi>978-0-471-43331-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Real+Analysis&rft.edition=4th&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=2011&rft.isbn=978-0-471-43331-6&rft.aulast=Bartle&rft.aufirst=Robert+G.&rft.au=Sherbert%2C+Donald+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-105"><a href="#cite_ref-105">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbbott2016" class="citation book cs1">Abbott, Stephen (2016). Understanding Analysis (2nd ed.). 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Advanced Calculus (revised ed.). World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9-814-58393-0" title="Special:BookSources/978-9-814-58393-0"><bdi>978-9-814-58393-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Calculus&rft.edition=revised&rft.pub=World+Scientific&rft.date=2014&rft.isbn=978-9-814-58393-0&rft.aulast=Loomis&rft.aufirst=Lynn+Harold&rft.au=Sternberg%2C+Shlomo+Zvi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-109"><a href="#cite_ref-109">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarsdenTromba2011" class="citation book cs1"><a href="/wiki/Jerrold_E._Marsden" title="Jerrold E. Marsden">Marsden, Jerrold E.</a>; Tromba, Anthony J. (2011). Vector Calculus (6th ed.). W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-429-21508-4" title="Special:BookSources/978-1-429-21508-4"><bdi>978-1-429-21508-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+Calculus&rft.edition=6th&rft.pub=W.+H.+Freeman&rft.date=2011&rft.isbn=978-1-429-21508-4&rft.aulast=Marsden&rft.aufirst=Jerrold+E.&rft.au=Tromba%2C+Anthony+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhlfors1978" class="citation book cs1"><a href="/wiki/Lars_Valerian_Ahlfors" class="mw-redirect" title="Lars Valerian Ahlfors">Ahlfors, Lars Valerian</a> (1978). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. 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Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95069-3" title="Special:BookSources/978-0-387-95069-3"><bdi>978-0-387-95069-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Analysis&rft.pub=Springer&rft.date=2001&rft.isbn=978-0-387-95069-3&rft.aulast=Gamelin&rft.aufirst=Theodore+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinShakarchi2003" class="citation book cs1"><a href="/wiki/Elias_M._Stein" title="Elias M. Stein">Stein, Elias M.</a>; Shakarchi, Rami (2003). Complex Analysis. 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All of Statistics: A Concise Course in Statistical Inference. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-40272-7" title="Special:BookSources/978-0-387-40272-7"><bdi>978-0-387-40272-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=All+of+Statistics%3A+A+Concise+Course+in+Statistical+Inference&rft.pub=Springer&rft.date=2003&rft.isbn=978-0-387-40272-7&rft.aulast=Wasserman&rft.aufirst=Larry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-118"><a href="#cite_ref-118">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLipschutz1998" class="citation book cs1">Lipschutz, Seymour (1998). Schaum's Outline of Set Theory and Related Topics. McGraw-Hill Companies. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-070-38159-9" title="Special:BookSources/978-0-070-38159-9"><bdi>978-0-070-38159-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schaum%27s+Outline+of+Set+Theory+and+Related+Topics&rft.pub=McGraw-Hill+Companies&rft.date=1998&rft.isbn=978-0-070-38159-9&rft.aulast=Lipschutz&rft.aufirst=Seymour&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-119"><a href="#cite_ref-119">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStoll1979" class="citation book cs1">Stoll, Robert Roth (1979). Set Theory and Logic. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-63829-4" title="Special:BookSources/978-0-486-63829-4"><bdi>978-0-486-63829-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+Logic&rft.pub=Dover+Publications&rft.date=1979&rft.isbn=978-0-486-63829-4&rft.aulast=Stoll&rft.aufirst=Robert+Roth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-120"><a href="#cite_ref-120">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1968" class="citation book cs1">Halmos, Paul R. (1968). Naive Set Theory. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90092-6" title="Special:BookSources/978-0-387-90092-6"><bdi>978-0-387-90092-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Naive+Set+Theory&rft.pub=Springer&rft.date=1968&rft.isbn=978-0-387-90092-6&rft.aulast=Halmos&rft.aufirst=Paul+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-121"><a href="#cite_ref-121">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRautenberg2006" class="citation book cs1">Rautenberg, Wolfgang (2006). A Concise Introduction to Mathematical Logic. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-30294-2" title="Special:BookSources/978-0-387-30294-2"><bdi>978-0-387-30294-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Concise+Introduction+to+Mathematical+Logic&rft.pub=Springer&rft.date=2006&rft.isbn=978-0-387-30294-2&rft.aulast=Rautenberg&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-122"><a href="#cite_ref-122">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDobrow2016" class="citation book cs1">Dobrow, Robert P. (2016). Introduction to Stochastic Processes with R. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-74065-1" title="Special:BookSources/978-1-118-74065-1"><bdi>978-1-118-74065-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Stochastic+Processes+with+R&rft.pub=Wiley&rft.date=2016&rft.isbn=978-1-118-74065-1&rft.aulast=Dobrow&rft.aufirst=Robert+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-123"><a href="#cite_ref-123">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBartle2001" class="citation book cs1"><a href="/wiki/Robert_G._Bartle" title="Robert G. Bartle">Bartle, Robert G.</a> (2001). A Modern Theory of Integration. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-821-80845-0" title="Special:BookSources/978-0-821-80845-0"><bdi>978-0-821-80845-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Modern+Theory+of+Integration&rft.pub=American+Mathematical+Society&rft.date=2001&rft.isbn=978-0-821-80845-0&rft.aulast=Bartle&rft.aufirst=Robert+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-124"><a href="#cite_ref-124">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinShakarchi2005" class="citation book cs1"><a href="/wiki/Elias_M._Stein" title="Elias M. Stein">Stein, Elias M.</a>; Shakarchi, Rami (2005). Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11386-9" title="Special:BookSources/978-0-691-11386-9"><bdi>978-0-691-11386-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+Analysis%3A+Measure+Theory%2C+Integration%2C+and+Hilbert+Spaces&rft.pub=Princeton+University+Press&rft.date=2005&rft.isbn=978-0-691-11386-9&rft.aulast=Stein&rft.aufirst=Elias+M.&rft.au=Shakarchi%2C+Rami&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-125"><a href="#cite_ref-125">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFolland2007" class="citation book cs1"><a href="/wiki/Gerald_B._Folland" class="mw-redirect" title="Gerald B. Folland">Folland, Gerald B.</a> (2007). Real Analysis: Modern Techniques and Their Applications (2nd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-31716-6" title="Special:BookSources/978-0-471-31716-6"><bdi>978-0-471-31716-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+Analysis%3A+Modern+Techniques+and+Their+Applications&rft.edition=2nd&rft.pub=Wiley&rft.date=2007&rft.isbn=978-0-471-31716-6&rft.aulast=Folland&rft.aufirst=Gerald+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-126"><a href="#cite_ref-126">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohn2015" class="citation book cs1">Cohn, Donald L. (2015). Measure Theory (2nd ed.). Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-489-99762-3" title="Special:BookSources/978-1-489-99762-3"><bdi>978-1-489-99762-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Measure+Theory&rft.edition=2nd&rft.pub=Birkh%C3%A4user&rft.date=2015&rft.isbn=978-1-489-99762-3&rft.aulast=Cohn&rft.aufirst=Donald+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-127"><a href="#cite_ref-127">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinShakarchi2003" class="citation book cs1"><a href="/wiki/Elias_M._Stein" title="Elias M. Stein">Stein, Elias M.</a>; Shakarchi, Rami (2003). Fourier Analysis: An Introduction. 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An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-09128-2" title="Special:BookSources/978-0-521-09128-2"><bdi>978-0-521-09128-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Fourier+Analysis+and+Generalised+Functions&rft.pub=Cambridge+University+Press&rft.date=1958&rft.isbn=978-0-521-09128-2&rft.aulast=Lighthill&rft.aufirst=M.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-129"><a href="#cite_ref-129">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinShakarchi2009" class="citation book cs1"><a href="/wiki/Elias_M._Stein" title="Elias M. 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Introduction to Differential Geometry of Space Curves and Surfaces. CreateSpace. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-546-68183-0" title="Special:BookSources/978-1-546-68183-0"><bdi>978-1-546-68183-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Differential+Geometry+of+Space+Curves+and+Surfaces&rft.pub=CreateSpace&rft.date=2017&rft.isbn=978-1-546-68183-0&rft.aulast=Sochi&rft.aufirst=Taha&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-131"><a href="#cite_ref-131">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDo_Carmo2016" class="citation book cs1">Do Carmo, Manfredo P. (2016). Differential Geometry of Curves and Surfaces. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-80699-0" title="Special:BookSources/978-0-486-80699-0"><bdi>978-0-486-80699-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Geometry+of+Curves+and+Surfaces&rft.pub=Dover+Publications&rft.date=2016&rft.isbn=978-0-486-80699-0&rft.aulast=Do+Carmo&rft.aufirst=Manfredo+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-132"><a href="#cite_ref-132">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPressley2010" class="citation book cs1">Pressley, Andrew (2010). Elementary Differential Geometry (2nd ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-848-82890-2" title="Special:BookSources/978-1-848-82890-2"><bdi>978-1-848-82890-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Differential+Geometry&rft.edition=2nd&rft.pub=Springer&rft.date=2010&rft.isbn=978-1-848-82890-2&rft.aulast=Pressley&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-133"><a href="#cite_ref-133">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunkres2000" class="citation book cs1"><a href="/wiki/James_Munkres" title="James Munkres">Munkres, James R.</a> (2000). Topology (2nd ed.). Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-131-81629-9" title="Special:BookSources/978-0-131-81629-9"><bdi>978-0-131-81629-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology&rft.edition=2nd&rft.pub=Pearson&rft.date=2000&rft.isbn=978-0-131-81629-9&rft.aulast=Munkres&rft.aufirst=James+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-134"><a href="#cite_ref-134">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendelson1990" class="citation book cs1">Mendelson, Bert (1990). Introduction to Topology (3rd ed.). Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-66352-4" title="Special:BookSources/978-0-486-66352-4"><bdi>978-0-486-66352-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Topology&rft.edition=3rd&rft.pub=Dover+Publications&rft.date=1990&rft.isbn=978-0-486-66352-4&rft.aulast=Mendelson&rft.aufirst=Bert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-135"><a href="#cite_ref-135">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSüliMayers2003" class="citation book cs1">Süli, Endre; Mayers, David (2003). An Introduction to Numerical Analysis. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-00794-8" title="Special:BookSources/978-0-521-00794-8"><bdi>978-0-521-00794-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Numerical+Analysis&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-00794-8&rft.aulast=S%C3%BCli&rft.aufirst=Endre&rft.au=Mayers%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-136"><a href="#cite_ref-136">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOsborneRubinstein1994" class="citation book cs1">Osborne, Michael J.; Rubinstein, Ariel (1994). A Course in Game Theory. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-65040-3" title="Special:BookSources/978-0-262-65040-3"><bdi>978-0-262-65040-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Game+Theory&rft.pub=MIT+Press&rft.date=1994&rft.isbn=978-0-262-65040-3&rft.aulast=Osborne&rft.aufirst=Michael+J.&rft.au=Rubinstein%2C+Ariel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-137"><a href="#cite_ref-137">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTadelis2013" class="citation book cs1">Tadelis, Steven (2013). Game Theory: An Introduction. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12908-2" title="Special:BookSources/978-0-691-12908-2"><bdi>978-0-691-12908-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Game+Theory%3A+An+Introduction&rft.pub=Princeton+University+Press&rft.date=2013&rft.isbn=978-0-691-12908-2&rft.aulast=Tadelis&rft.aufirst=Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-138"><a href="#cite_ref-138">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGibbons1992" class="citation book cs1">Gibbons, Robert (1992). Game Theory for Applied Economists. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-00395-5" title="Special:BookSources/978-0-691-00395-5"><bdi>978-0-691-00395-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Game+Theory+for+Applied+Economists&rft.pub=Princeton+University+Press&rft.date=1992&rft.isbn=978-0-691-00395-5&rft.aulast=Gibbons&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-139"><a href="#cite_ref-139">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKot2014" class="citation book cs1">Kot, Mark (2014). A First Course on the Calculus of Variations. American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-1495-5" title="Special:BookSources/978-1-4704-1495-5"><bdi>978-1-4704-1495-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+on+the+Calculus+of+Variations&rft.pub=American+Mathematical+Society&rft.date=2014&rft.isbn=978-1-4704-1495-5&rft.aulast=Kot&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-:21-140">^ <a href="#cite_ref-:21_140-0">a</a> <a href="#cite_ref-:21_140-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLanczos1986" class="citation book cs1"><a href="/wiki/Cornelius_Lanczos" title="Cornelius Lanczos">Lanczos, Cornelius</a> (1986). The Variational Principles of Mechanics (4th ed.). Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65067-8" title="Special:BookSources/978-0-486-65067-8"><bdi>978-0-486-65067-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Variational+Principles+of+Mechanics&rft.edition=4th&rft.pub=Dover+Publications&rft.date=1986&rft.isbn=978-0-486-65067-8&rft.aulast=Lanczos&rft.aufirst=Cornelius&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-141"><a href="#cite_ref-141">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelfandFomin2000" class="citation book cs1">Gelfand, Israel M.; Fomin, S.V. (2000). Calculus of Variations. Translated by Silverman, Richard. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-41448-5" title="Special:BookSources/978-0-486-41448-5"><bdi>978-0-486-41448-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+of+Variations&rft.pub=Dover+Publications&rft.date=2000&rft.isbn=978-0-486-41448-5&rft.aulast=Gelfand&rft.aufirst=Israel+M.&rft.au=Fomin%2C+S.V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-142"><a href="#cite_ref-142">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillwell2010" class="citation book cs1"><a href="/wiki/John_Stillwell" title="John Stillwell">Stillwell, John</a> (2010). Mathematics and Its History (3rd ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-441-96052-8" title="Special:BookSources/978-1-441-96052-8"><bdi>978-1-441-96052-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Its+History&rft.edition=3rd&rft.pub=Springer&rft.date=2010&rft.isbn=978-1-441-96052-8&rft.aulast=Stillwell&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-143"><a href="#cite_ref-143">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz2008" class="citation book cs1"><a href="/wiki/Victor_J._Katz" title="Victor J. Katz">Katz, Victor</a> (2008). A History of Mathematics: An Introduction (3rd ed.). Addison Wesley Longman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-38700-4" title="Special:BookSources/978-0-321-38700-4"><bdi>978-0-321-38700-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics%3A+An+Introduction&rft.edition=3rd&rft.pub=Addison+Wesley+Longman&rft.date=2008&rft.isbn=978-0-321-38700-4&rft.aulast=Katz&rft.aufirst=Victor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-144"><a href="#cite_ref-144">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyerMerzbach1991" class="citation book cs1"><a href="/wiki/Carl_Boyer" class="mw-redirect" title="Carl Boyer">Boyer, Carl B.</a>; Merzbach, Uta C. (1991). A History of Mathematics. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-54397-8" title="Special:BookSources/978-0-471-54397-8"><bdi>978-0-471-54397-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.pub=Wiley&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.aulast=Boyer&rft.aufirst=Carl+B.&rft.au=Merzbach%2C+Uta+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-145"><a href="#cite_ref-145">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline1972" class="citation book cs1"><a href="/wiki/Morris_Kline" title="Morris Kline">Kline, Morris</a> (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-195-01496-9" title="Special:BookSources/978-0-195-01496-9"><bdi>978-0-195-01496-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Thought+from+Ancient+to+Modern+Times&rft.place=New+York&rft.pub=Oxford+University+Press&rft.date=1972&rft.isbn=978-0-195-01496-9&rft.aulast=Kline&rft.aufirst=Morris&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-146"><a href="#cite_ref-146">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor2005" class="citation book cs1"><a href="/wiki/John_R._Taylor" title="John R. Taylor">Taylor, John R.</a> (2005). Classical Mechanics. University Science Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-891-38922-1" title="Special:BookSources/978-1-891-38922-1"><bdi>978-1-891-38922-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.pub=University+Science+Books&rft.date=2005&rft.isbn=978-1-891-38922-1&rft.aulast=Taylor&rft.aufirst=John+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-147"><a href="#cite_ref-147">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldsteinPooleSafko2001" class="citation book cs1"><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Goldstein, Herbert</a>; Poole, Charles; Safko, John (2001). <a href="/wiki/Classical_Mechanics_(Goldstein)" title="Classical Mechanics (Goldstein)">Classical Mechanics</a> (3rd ed.). Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-65702-9" title="Special:BookSources/978-0-201-65702-9"><bdi>978-0-201-65702-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.edition=3rd&rft.pub=Pearson&rft.date=2001&rft.isbn=978-0-201-65702-9&rft.aulast=Goldstein&rft.aufirst=Herbert&rft.au=Poole%2C+Charles&rft.au=Safko%2C+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-148"><a href="#cite_ref-148">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArnold1978" class="citation book cs1"><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Arnold, Vladimir</a> (1978). Mathematical Methods of Classical Mechanics. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90314-9" title="Special:BookSources/978-0-387-90314-9"><bdi>978-0-387-90314-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+of+Classical+Mechanics&rft.pub=Springer-Verlag&rft.date=1978&rft.isbn=978-0-387-90314-9&rft.aulast=Arnold&rft.aufirst=Vladimir&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-149"><a href="#cite_ref-149">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPurcellMorin2013" class="citation book cs1"><a href="/wiki/Edward_M._Purcell" class="mw-redirect" title="Edward M. Purcell">Purcell, Edward M.</a>; Morin, David J. (2013). <a href="/wiki/Electricity_and_Magnetism_(book)" title="Electricity and Magnetism (book)">Electricity and Magnetism</a> (3rd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-01402-2" title="Special:BookSources/978-1-107-01402-2"><bdi>978-1-107-01402-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electricity+and+Magnetism&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-1-107-01402-2&rft.aulast=Purcell&rft.aufirst=Edward+M.&rft.au=Morin%2C+David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-150"><a href="#cite_ref-150">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths2017" class="citation book cs1"><a href="/wiki/David_J._Griffiths" title="David J. Griffiths">Griffiths, David J.</a> (2017). <a href="/wiki/Introduction_to_Electrodynamics" title="Introduction to Electrodynamics">Introduction to Electrodynamics</a> (4th ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-42041-9" title="Special:BookSources/978-1-108-42041-9"><bdi>978-1-108-42041-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Electrodynamics&rft.edition=4th&rft.pub=Cambridge+University+Press&rft.date=2017&rft.isbn=978-1-108-42041-9&rft.aulast=Griffiths&rft.aufirst=David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-151"><a href="#cite_ref-151">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrogatz1994" class="citation book cs1"><a href="/wiki/Steven_Strogatz" title="Steven Strogatz">Strogatz, Steven H.</a> (1994). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-367-09206-1" title="Special:BookSources/978-0-367-09206-1"><bdi>978-0-367-09206-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+Dynamics+and+Chaos%3A+With+Applications+to+Physics%2C+Biology%2C+Chemistry%2C+and+Engineering&rft.pub=CRC+Press&rft.date=1994&rft.isbn=978-0-367-09206-1&rft.aulast=Strogatz&rft.aufirst=Steven+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-152"><a href="#cite_ref-152">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBatchelor2000" class="citation book cs1"><a href="/wiki/G._K._Batchelor" class="mw-redirect" title="G. K. Batchelor">Batchelor, G. K.</a> (2000). An Introduction to Fluid Dynamics. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-66396-0" title="Special:BookSources/978-0-521-66396-0"><bdi>978-0-521-66396-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Fluid+Dynamics&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-66396-0&rft.aulast=Batchelor&rft.aufirst=G.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-153"><a href="#cite_ref-153">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1987" class="citation book cs1"><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, Lev D.</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, Evgeny</a> (1987). Fluid Mechanics (2nd ed.). Butterworth-Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-750-62767-2" title="Special:BookSources/978-0-750-62767-2"><bdi>978-0-750-62767-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics&rft.edition=2nd&rft.pub=Butterworth-Heinemann&rft.date=1987&rft.isbn=978-0-750-62767-2&rft.aulast=Landau&rft.aufirst=Lev+D.&rft.au=Lifshitz%2C+Evgeny&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-154"><a href="#cite_ref-154">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTownsend2012" class="citation book cs1">Townsend, John S. (2012). A Modern Approach to Quantum Mechanics (2nd ed.). University Science Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-891-38978-8" title="Special:BookSources/978-1-891-38978-8"><bdi>978-1-891-38978-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Modern+Approach+to+Quantum+Mechanics&rft.edition=2nd&rft.pub=University+Science+Books&rft.date=2012&rft.isbn=978-1-891-38978-8&rft.aulast=Townsend&rft.aufirst=John+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-155"><a href="#cite_ref-155">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShankar2012" class="citation book cs1"><a href="/wiki/Ramamurti_Shankar" title="Ramamurti Shankar">Shankar, Ramamurti</a> (2012). <a href="/wiki/Principles_of_Quantum_Mechanics" title="Principles of Quantum Mechanics">Principles of Quantum Mechanics</a> (2nd ed.). Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-475-70578-2" title="Special:BookSources/978-1-475-70578-2"><bdi>978-1-475-70578-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Quantum+Mechanics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=2012&rft.isbn=978-1-475-70578-2&rft.aulast=Shankar&rft.aufirst=Ramamurti&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-156"><a href="#cite_ref-156">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSakuraiNapolitano2020" class="citation book cs1"><a href="/wiki/J._J._Sakurai" title="J. J. Sakurai">Sakurai, J. J.</a>; Napolitano, Jim (2020). <a href="/wiki/Modern_Quantum_Mechanics" title="Modern Quantum Mechanics">Modern Quantum Mechanics</a> (3rd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-47322-4" title="Special:BookSources/978-1-108-47322-4"><bdi>978-1-108-47322-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Quantum+Mechanics&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2020&rft.isbn=978-1-108-47322-4&rft.aulast=Sakurai&rft.aufirst=J.+J.&rft.au=Napolitano%2C+Jim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-157"><a href="#cite_ref-157">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartle2002" class="citation book cs1"><a href="/wiki/James_Hartle" title="James Hartle">Hartle, James B.</a> (2002). Gravity: An Introduction to Einstein's General Relativity. Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-805-38662-2" title="Special:BookSources/978-0-805-38662-2"><bdi>978-0-805-38662-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravity%3A+An+Introduction+to+Einstein%27s+General+Relativity&rft.pub=Pearson&rft.date=2002&rft.isbn=978-0-805-38662-2&rft.aulast=Hartle&rft.aufirst=James+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-158"><a href="#cite_ref-158">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarroll2019" class="citation book cs1"><a href="/wiki/Sean_Carroll_(physicist)" class="mw-redirect" title="Sean Carroll (physicist)">Carroll, Sean</a> (2019). Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-48839-6" title="Special:BookSources/978-1-108-48839-6"><bdi>978-1-108-48839-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spacetime+and+Geometry%3A+An+Introduction+to+General+Relativity&rft.pub=Cambridge+University+Press&rft.date=2019&rft.isbn=978-1-108-48839-6&rft.aulast=Carroll&rft.aufirst=Sean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-159"><a href="#cite_ref-159">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisnerThorneWheeler2017" class="citation book cs1"><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner, Charles</a>; <a href="/wiki/Kip_S._Thorne" class="mw-redirect" title="Kip S. 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Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-17779-3" title="Special:BookSources/978-0-691-17779-3"><bdi>978-0-691-17779-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pub=Princeton+University+Press&rft.date=2017&rft.isbn=978-0-691-17779-3&rft.aulast=Misner&rft.aufirst=Charles&rft.au=Thorne%2C+Kip&rft.au=Wheeler%2C+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-160"><a href="#cite_ref-160">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWald1984" class="citation book cs1"><a href="/wiki/Robert_Wald" title="Robert Wald">Wald, Robert</a> (1984). <a href="/wiki/General_Relativity_(book)" title="General Relativity (book)">General Relativity</a>. The University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-87033-5" title="Special:BookSources/978-0-226-87033-5"><bdi>978-0-226-87033-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Relativity&rft.pub=The+University+of+Chicago+Press&rft.date=1984&rft.isbn=978-0-226-87033-5&rft.aulast=Wald&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-161"><a href="#cite_ref-161">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrahamKnuthPatashnik1994" class="citation book cs1">Graham, Ronald L.; <a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a>; Patashnik, Oren (1994). <a href="/wiki/Concrete_Mathematics" title="Concrete Mathematics">Concrete Mathematics: A Foundation for Computer Science</a> (2nd ed.). Addison-Wesley Professional. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-55802-9" title="Special:BookSources/978-0-201-55802-9"><bdi>978-0-201-55802-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concrete+Mathematics%3A+A+Foundation+for+Computer+Science&rft.edition=2nd&rft.pub=Addison-Wesley+Professional&rft.date=1994&rft.isbn=978-0-201-55802-9&rft.aulast=Graham&rft.aufirst=Ronald+L.&rft.au=Knuth%2C+Donald&rft.au=Patashnik%2C+Oren&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-162"><a href="#cite_ref-162">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosen2018" class="citation book cs1">Rosen, Kenneth H. (2018). Discrete Mathematics and Its Applications (8th ed.). McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-259-67651-2" title="Special:BookSources/978-1-259-67651-2"><bdi>978-1-259-67651-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+and+Its+Applications&rft.edition=8th&rft.pub=McGraw-Hill&rft.date=2018&rft.isbn=978-1-259-67651-2&rft.aulast=Rosen&rft.aufirst=Kenneth+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-163"><a href="#cite_ref-163">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoverThomas2006" class="citation book cs1">Cover, Thomas M.; Thomas, Joy A. (2006). Elements of Information Theory (2nd ed.). Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-24195-9" title="Special:BookSources/978-0-471-24195-9"><bdi>978-0-471-24195-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Information+Theory&rft.edition=2nd&rft.pub=Wiley-Interscience&rft.date=2006&rft.isbn=978-0-471-24195-9&rft.aulast=Cover&rft.aufirst=Thomas+M.&rft.au=Thomas%2C+Joy+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-164"><a href="#cite_ref-164">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSipser1996" class="citation book cs1">Sipser, Michael (1996). Introduction to the Theory of Computation (3rd ed.). Cengage Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-133-18779-0" title="Special:BookSources/978-1-133-18779-0"><bdi>978-1-133-18779-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+of+Computation&rft.edition=3rd&rft.pub=Cengage+Learning&rft.date=1996&rft.isbn=978-1-133-18779-0&rft.aulast=Sipser&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-165"><a href="#cite_ref-165">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormenLeisersonRivestStein2009" class="citation book cs1">Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2009). Introduction to Algorithms (3rd ed.). The MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-03384-8" title="Special:BookSources/978-0-262-03384-8"><bdi>978-0-262-03384-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Algorithms&rft.edition=3rd&rft.pub=The+MIT+Press&rft.date=2009&rft.isbn=978-0-262-03384-8&rft.aulast=Cormen&rft.aufirst=Thomas+H.&rft.au=Leiserson%2C+Charles+E.&rft.au=Rivest%2C+Ronald+L.&rft.au=Stein%2C+Clifford&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-166"><a href="#cite_ref-166">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoughgarden2016" class="citation book cs1">Roughgarden, Tim (2016). Twenty Lectures on Algorithmic Game Theory. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-17266-1" title="Special:BookSources/978-1-107-17266-1"><bdi>978-1-107-17266-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Twenty+Lectures+on+Algorithmic+Game+Theory&rft.pub=Cambridge+University+Press&rft.date=2016&rft.isbn=978-1-107-17266-1&rft.aulast=Roughgarden&rft.aufirst=Tim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-167"><a href="#cite_ref-167">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrossHarrisRiehl2019" class="citation book cs1"><a href="/wiki/Benedict_Gross" title="Benedict Gross">Gross, Benedict</a>; <a href="/wiki/Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, Joseph</a>; <a href="/wiki/Emily_Riehl" title="Emily Riehl">Riehl, Emily</a> (2019). Fat Chance: Probability from 0 to 1. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-72818-8" title="Special:BookSources/978-1-108-72818-8"><bdi>978-1-108-72818-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fat+Chance%3A+Probability+from+0+to+1&rft.pub=Cambridge+University+Press&rft.date=2019&rft.isbn=978-1-108-72818-8&rft.aulast=Gross&rft.aufirst=Benedict&rft.au=Harris%2C+Joseph&rft.au=Riehl%2C+Emily&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-168"><a href="#cite_ref-168">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnsonKuby2003" class="citation book cs1">Johnson, Robert; Kuby, Patricia (2003). Just the Essentials of Elementary Statistics (3rd ed.). Thomson Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-38472-2" title="Special:BookSources/0-534-38472-2"><bdi>0-534-38472-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Just+the+Essentials+of+Elementary+Statistics&rft.edition=3rd&rft.pub=Thomson+Brooks%2FCole&rft.date=2003&rft.isbn=0-534-38472-2&rft.aulast=Johnson&rft.aufirst=Robert&rft.au=Kuby%2C+Patricia&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-169"><a href="#cite_ref-169">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor1996" class="citation book cs1"><a href="/wiki/John_R._Taylor" title="John R. Taylor">Taylor, John R.</a> (1996). Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements (2nd ed.). University Science Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-93570-275-0" title="Special:BookSources/978-0-93570-275-0"><bdi>978-0-93570-275-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Error+Analysis%3A+The+Study+of+Uncertainties+in+Physical+Measurements&rft.edition=2nd&rft.pub=University+Science+Books&rft.date=1996&rft.isbn=978-0-93570-275-0&rft.aulast=Taylor&rft.aufirst=John+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-170"><a href="#cite_ref-170">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHughesHase2013" class="citation book cs1">Hughes, Ifan G.; Hase, Thomas P. A. (2013). Measurements and their Uncertainties: A Practical Guide to Modern Error Analysis. Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-956633-4" title="Special:BookSources/978-0-19-956633-4"><bdi>978-0-19-956633-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Measurements+and+their+Uncertainties%3A+A+Practical+Guide+to+Modern+Error+Analysis&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=2013&rft.isbn=978-0-19-956633-4&rft.aulast=Hughes&rft.aufirst=Ifan+G.&rft.au=Hase%2C+Thomas+P.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> <li id="cite_note-171"><a href="#cite_ref-171">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeberHarrisArfken2012" class="citation book cs1">Weber, Hans J.; Harris, Frank E.; <a href="/wiki/George_B._Arfken" title="George B. 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Education Week. Archived from <a rel="nofollow" class="external text" href="https://www.edweek.org/teaching-learning/a-new-ap-precalculus-course-aims-to-diversify-the-math-pipeline/2022/05">the original</a> on March 24, 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Education+Week&rft.atitle=A+New+AP+Precalculus+Course+Aims+to+Diversify+the+Math+Pipeline&rft.date=2022-05-19&rft.aulast=Najarro&rft.aufirst=Ileana&rft_id=https%3A%2F%2Fwww.edweek.org%2Fteaching-learning%2Fa-new-ap-precalculus-course-aims-to-diversify-the-math-pipeline%2F2022%2F05&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=19" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarelickWilson2022" class="citation book cs1">Garelick, Barry; Wilson, JR (2022). Traditional Math: An effective strategy that teachers feel guilty using. John Catt Educational. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-91526-154-0" title="Special:BookSources/978-1-91526-154-0"><bdi>978-1-91526-154-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Traditional+Math%3A+An+effective+strategy+that+teachers+feel+guilty+using&rft.pub=John+Catt+Educational&rft.date=2022&rft.isbn=978-1-91526-154-0&rft.aulast=Garelick&rft.aufirst=Barry&rft.au=Wilson%2C+JR&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics+education+in+the+United+States" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Mathematics_education_in_the_United_States&action=edit&section=20" title="Edit section: External links">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="https://mathisyourfuture.com/articles/1">Math courses with “Math Is Your Future”</a>; an article about studying math with the use of Internet technologies</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=l9byN0skNKY">Math is amazing and we have to start treating it that way</a>, <a href="/wiki/Eugenia_Cheng" title="Eugenia Cheng">Eugenia Cheng</a> for the PBS Newshour.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output 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href="https://commons.wikimedia.org/wiki/Category:Mathematics_education" class="extiw" title="commons:Category:Mathematics education">Commons</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="American_mathematics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:American_mathematics" title="Template:American 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href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a></li> <li><a href="/wiki/American_Mathematical_Association_of_Two-Year_Colleges" title="American Mathematical Association of Two-Year Colleges">AMATYC</a></li> <li><a href="/wiki/Association_for_Women_in_Mathematics" title="Association for Women in Mathematics">AWM</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Institutions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/American_Institute_of_Mathematics" title="American Institute of Mathematics">AIM</a></li> <li><a href="/wiki/Institute_for_Advanced_Study" title="Institute for Advanced Study">IAS</a></li> <li><a href="/wiki/Institute_for_Computational_and_Experimental_Research_in_Mathematics" title="Institute for Computational and Experimental Research in Mathematics">ICERM</a></li> <li><a href="/wiki/Institute_for_Mathematics_and_its_Applications" title="Institute for Mathematics and its Applications">IMA</a></li> <li><a href="/wiki/Institute_for_Pure_and_Applied_Mathematics" title="Institute for Pure and Applied Mathematics">IPAM</a></li> <li><a href="/wiki/Mathematical_Biosciences_Institute" title="Mathematical Biosciences Institute">MBI</a></li> <li><a href="/wiki/Simons_Laufer_Mathematical_Sciences_Institute" title="Simons Laufer Mathematical Sciences Institute">SLMath</a></li> <li><a href="/wiki/Statistical_and_Applied_Mathematical_Sciences_Institute" title="Statistical and Applied Mathematical Sciences Institute">SAMSI</a></li> <li><a href="/wiki/Geometry_Center" title="Geometry Center">Geometry Center</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Competitions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> 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