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American Mathematical Society

Journals High quality journals covering a broad range of mathematical disciplines.

Notices of the American Mathematical Society

Current issue · All issues

Notices of the American Mathematical Society ISSN 1088-9477 (online) ISSN 0002-9920 (print) MCQ: 0.45

Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society ISSN 1088-9485 (online) ISSN 0273-0979 (print) MCQ: 0.47

Abstracts of Papers Presented to the American Mathematical Society

All issues : 2009 - Present

Abstracts of Papers Presented to the American Mathematical Society ISSN 2689-4831 (online) ISSN 0192-5857 (print) MCQ: 0.00

MCQ Info The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period.

Communications of the American Mathematical Society

Current volume · All volumes

Communications of the American Mathematical Society ISSN 2692-3688 MCQ: 0.47

Journal of the American Mathematical Society

Journal of the American Mathematical Society ISSN 1088-6834 (online) ISSN 0894-0347 (print) MCQ: 4.79

Representation Theory

Representation Theory ISSN 1088-4165 MCQ: 0.7

Proceedings of the American Mathematical Society

Proceedings of the American Mathematical Society ISSN 1088-6826 (online) ISSN 0002-9939 (print) MCQ: 0.85

Proceedings of the American Mathematical Society Series B

Proceedings of the American Mathematical Society Series B ISSN 2330-1511 MCQ: 0.84

Mathematics of Computation

Mathematics of Computation ISSN 1088-6842 (online) ISSN 0025-5718 (print) MCQ: 1.98

Conformal Geometry and Dynamics

Conformal Geometry and Dynamics ISSN 1088-4173 MCQ: 0.5

Memoirs of the American Mathematical Society

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Memoirs of the American Mathematical Society ISSN 1947-6221 (online) ISSN 0065-9266 (print) MCQ: 0.51

Transactions of the American Mathematical Society

Transactions of the American Mathematical Society ISSN 1088-6850 (online) ISSN 0002-9947 (print) MCQ: 1.43

Transactions of the American Mathematical Society Series B

Transactions of the American Mathematical Society Series B ISSN 2330-0000 MCQ: 1.79

Electronic Research Announcements

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Electronic Research Announcements ISSN 1079-6762 MCQ: 0.00

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St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal ISSN 1547-7371 (online) ISSN 1061-0022 (print) MCQ: 0.54

Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society ISSN 1547-738X (online) ISSN 0077-1554 (print) MCQ: 0.51

Sugaku Expositions

Sugaku Expositions ISSN 2473-585X (online) ISSN 0898-9583 (print) MCQ: 0.10

Annales Scientifiques de l'Ecole Normale Superieure

Annales Scientifiques de l'École Normale Supérieure ISSN: 1088-4173 MCQ: 2.09

Asterisque

Astérisque ISSN: 0303-1179 MCQ: 0.45

Bulletin de la Societe Mathematique de France

Bulletin de la Société Mathématique de France ISSN 0037-9484 MCQ: 0.70

Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics ISSN 1547-7363 (online) ISSN 0094-9000 (print) MCQ: 0.12

Journal of Algebraic Geometry

Journal of Algebraic Geometry ISSN 1534-7486 (online) ISSN 1056-3911 (Print) MCQ: 1.37

JOT

ISSN 0379-4024 (print) MCQ: 0.60

Quarterly of Applied Mathematics

Quarterly of Applied Mathematics ISSN 1552-4485 (online) ISSN 0033-569X (print) MCQ: 0.60

Moscow Mathematical Journal

Moscow Mathematical Journal 1609-3321 (print) MCQ: 0.61

Mmoires de la Socit Mathmatique de France

Mémoires de la Société Mathématique de France ISSN 0249-633X MCQ: 1.83

Journal of the Ramanujan Mathematical Society

Journal of the Ramanujan Mathematical Society ISSN 0970-1249 MCQ: 0.24

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How to Effectively Write a Mathematics Research Paper

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Mathematics research papers are different from standard academic research papers in important ways, but not so different that they require an entirely separate set of guidelines. Mathematical papers rely heavily on logic and a specific type of language, including symbols and regimented notation. There are two basic structures of mathematical research papers: formal and informal exposition .

Structure and Style

Formal Exposition

The author must start with an outline that develops the logical structure of the paper. Each hypothesis and deduction should flow in an orderly and linear fashion using formal definitions and notation. The author should not repeat a proof or substitute words or phrases that differ from the definitions already established within the paper. The theorem-proof format, definitions, and logic fall under this style.

Informal Exposition

Informal exposition complements the formal exposition by providing the reasoning behind the theorems and proofs. Figures, proofs, equations, and mathematical sentences do not necessarily speak for themselves within a mathematics research paper . Authors will need to demonstrate why their hypotheses and deductions are valid and how they came to prove this. Analogies and examples fall under this style.

Conventions of Mathematics

Clarity is essential for writing an effective mathematics research paper. This means adhering to strong rules of logic, clear definitions, theorems and equations that are physically set apart from the surrounding text, and using math symbols and notation following the conventions of mathematical language. Each area incorporates detailed guidelines to assist the authors.

Related: Do you have questions on language, grammar, or manuscript drafting? Get personalized answers on the FREE Q&A Forum!

Logic is the framework upon which every good mathematics research paper is built. Each theorem or equation must flow logically.

Definitions

In order for the reader to understand the author’s work, definitions for terms and notations used throughout the paper must be set at the beginning of the paper. It is more effective to include this within the Introduction section of the paper rather than having a stand-alone section of definitions.

Theorems and Equations

Theorems and equations should be physically separated from the surrounding text. They will be used as reference points throughout, so they should have a well-defined beginning and end.

Math Symbols and Notations

Math symbols and notations are standardized within the mathematics literature. Deviation from these standards will cause confusion amongst readers. Therefore, the author should adhere to the guidelines for equations, units, and mathematical notation, available from various resources .

Protocols for mathematics writing get very specific – fonts, punctuation, examples, footnotes, sentences, paragraphs, and the title, all have detailed constraints and conventions applied to their usage. The American Mathematical Society is a good resource for additional guidelines.

LaTeX and Wolfram

Mathematical sentences contain equations, figures, and notations that are difficult to typeset using a typical word-processing program. Both LaTeX and Wolfram have expert typesetting capabilities to assist authors in writing.

LaTeX is highly recommended for researchers whose papers constitute mathematical figures and notation. It produces professional-looking documents and authentically represents mathematical language.

Wolfram Language & System Documentation Center’s Mathematica has sophisticated and convenient mathematical typesetting technology that produces professional-looking documents.

The main differences between the two systems are due to cost and accessibility. LaTeX is freely available, whereas Wolfram is not. In addition, any updates in Mathematica will come with an additional charge. LaTeX is an open-source system, but Mathematica is closed-source.

Good Writing and Logical Constructions

Regardless of the document preparation system selected, publication of a mathematics paper is similar to the publication of any academic research in that it requires good writing. Authors must apply a strict, logical construct when writing a mathematics research paper.

There are resources that provide very specific guidelines related to following sections to write and publish a mathematics research paper.

  • Concept of a math paper
  • Title, acknowledgment, and list of authors
  • Introduction
  • Body of the work
  • Conclusion, appendix, and references
  • Publication of a math paper
  • Preprint archive
  • Choice of the journal, submission
  • Publication

The critical elements of a mathematics research paper are good writing and a logical construct that allows the reader to follow a clear path to the author’s conclusions.

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A guide to mathematics resources.

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Most Frequently Used Databases for Math & Statistics

MathSciNet may be first choice although items are added slowly!

arXiv and Google Scholar are also highly recommended.

Free resource

  • MathSciNet (1940 - ) via American Mathematical Society (AMS) This link opens in a new window Index to Mathematical Reviews and Current Mathematical Publications; plus some cites 1826-1939 added from the World Digital Mathematics Library (WDML). See also Help and Tutorials .
  • MathSciNet (1940 -) via Ebscohost This link opens in a new window Reviews, abstracts and bibliographic information for mathematical sciences literature from the American Mathematical Society (AMS). Professional mathematicians write these reviews of the current published literature. Over 100,000 new items are added to the database each year.
  • Scopus This link opens in a new window The world’s largest abstract and citation database of peer-reviewed literature. Contains over 46 million records, 70% with abstracts, and also includes over 4.6 million conference papers. NOTE: Click "Institutions" icon in right corner. Change organization to Washington University in St Louis John M Olin Library, Danforth Campus.
  • Web of Science (1900 - ) This link opens in a new window If you receive an error message and cannot access the database, try clearing your browser cache and cookies. Science, social science, arts, and humanities citations for scholarly literature. Access the world’s leading scholarly literature in the sciences, social sciences, arts, and humanities and examine proceedings of international conferences, symposia, seminars, colloquia, workshops, and conventions. The Libraries subscribe to Science Citation Index Expanded (1900-present), Social Sciences Citation Index (1970-present), and Arts & Humanities Citation Index (1975-present).

For topics in applied math, consider tools for those disciplines, such as Engineering, Economics, Business, etc. See Find Articles sections of those Research Guides for tips.

_________________________________________________________________

  • AMS Open Math Notes Repository of freely downloadable mathematical works in progress hosted by the American Mathematical Society. It contains draft works include course notes, textbooks, and research expositions in progress.
  • Applied Science Full Text
  • EuDML European Digital Mathematics Library Much fulltext content is freely available.
  • Gröbner Bases Bibliography
  • Jahrbuch Project Electronic Research Archive for Mathematics Covers 1868-1942 primarily
  • JSTOR Complete back runs of almost 500 journal titles, from whenever each title began publication to a point 3 or 5 years ago. WU does not have access to all content.
  • Primo Web of Science, Scopus, catalog and other sources searched together. More info
  • NUMDAM: Search and download archives of mathematical journals Fulltext access to many French journals; most embargo most recent 5 years
  • Project Euclid: mathematics and statistics resources online WU has access to most content.
  • Real Algebraic and Analytic Geometry - Preprint Server from School of Mathematics, University of Manchester, UK
  • ScienceDirect Mostly Elsevier content; WU does NOT have access to all.
  • Zentralblatt MATH Free, citation-check only access; we only get 3 results on any search because we are not subscribers but this can be very useful if you have some info. 1868+ coverage.
  • << Previous: Start Here ⭐
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  • Last Updated: Feb 14, 2024 11:07 AM
  • URL: https://libguides.wustl.edu/math

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Writing math research papers: a guide for students and instructors.

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Robert Gerver

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Writing Math Research Papers  is primarily a guide for high school students that describes how to write aand present mathematics research papers. But it’s really much more than that: it’s a systematic presentation of a philosophy that writing about math helps many students to understand it, and a practical method to move students from the relatively passive role of someone doing what is assigned to them, to creative thinkers and published writers who contribute to the mathematical literature.

As experienced writers know, the actual writing is not the half of it. William Zinsser once taught a writing class at the New School for Social Research which involved no writing at all: students talked through their ideas in class and through that process discovered the real story which could be written from their tangle of experiences, hopes and dreams. The actual writing was secondary, once they understood how to find the story and organize it.

Gerver, an experienced high school mathematics teacher, takes a similar approach. The primary audience is high school students who want to prepare formal papers or presentations, for contests or for a “math day” at their high school. But the discovery, research and organizational processes involved in writing an original paper, as opposed to rehashing information from a reference book, can help any student learn and understand math, and the experience will be useful even if the paper is never written.

Gerver leads students through a discovery process beginning with examining their own knowledge of mathematics and reviewing the basics of problem solving. The “math annotation” project follows next, in which students organize their class notes for one topic for presentation to their peers, resulting in a product similar to a section of a textbook or handbook, complete with illustrations and the necessary background and review material. Practical advice about finding a topic, developing it by keeping a research journal, and creating a final product, either a research paper or oral presentation, follows.

Writing Math Research Papers  is directed primarily to students, and could be assigned as a supplementary textbook for high school mathematics classes. It will also be useful to teachers who incorporate writing into their classes or who serve as mentors to the math club, and for student teachers in similar situations. An appendix for teachers includes practical advice about helping students through the research and writing process, organizing consultations, and grading the student papers and presentations. Excerpts from student research papers are included as well, and more materials are available from the web site www.keypress.com/wmrp .

Robert Gerver, PhD, is a mathematics instructor at North Shore High School in New York. He received the Presidential Award for Excellence in Mathematical Teaching in 1988 and the Tandy Prize and Chevron Best Practices Award in Education in 1997. He has been publishing mathematics. Dr. Gerver has written eleven mathematics textbooks and numerous articles, and holds two U.S. patents for educational devices.

Sarah Boslaugh, ( [email protected] ) is a Performance Review Analyst for BJC HealthCare and an Adjunct Instructor in the Washington University School of Medicine, both in St. Louis, MO. Her books include An Intermediate Guide to SPSS Programming: Using Syntax for Data Management  (Sage, 2004), Secondary Data Sources for Public Health: A Practical Guide (Cambridge, 2007), and Statistics in a Nutshell (O'Reilly, forthcoming), and she is Editor-in-Chief of The Encyclopedia of Epidemiology (Sage, forthcoming).

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How to do Research on Mathematics

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Selected Subject Headings

Listed below is a sample of a few broad Library of Congress subject headings—made up of one word or more representing concepts under which all library holdings are divided and subdivided by subject—which you can search under and use as subject terms as well when searching online library catalogs for preliminary and/or additional research, such as books, audio and video recordings, and other references, related to your research paper topic. When researching materials on your topic, subject heading searching may be more productive than searching using simple keywords. However, keyword searching when using the right search method (Boolean, etc.) and combination of words can be equally effective in finding materials more closely relevant to the topic of your research paper.

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  • Business Mathematics
  • Game Theory
  • Mathematics—Philosophy
  • Women in Mathematics

Selected Keyword Search Strategies and Guides

Most online library indexes and abstracts and full-text article databases offer basic and advanced “keyword” searching of virtually every subject. In this case, combine keyword terms that best define your thesis question or topic using the Boolean search method (employing “and” or “or”) to find research most suitable to your research paper topic.

If your topic is “the importance of mathematics in the world,” for example, enter “importance” and “mathematics” with “and” on the same line to locate sources directly compatible with the primary focus of your paper. To find research on more specific aspects of your topic, alternate with one new keyword at a time with “and” in between (for example, “advancements and mathematics,” “contributions and mathematics,” “influence and mathematics,” etc.).

For additional help with keyword searching, navigation or user guides for online indexes and databases by many leading providers—including Cambridge Scientific Abstracts, EBSCO, H.W. Wilson, OCLC, Ovid Technologies, ProQuest, and Thomson Gale—are posted with direct links on library Web sites to guides providing specific instruction to using whichever database you want to search. They provide additional guidance on how to customize and maximize your search, including advanced searching techniques and grouping of words and phrases using the Boolean search method—of your topic, of bibliographic records, and of full-text articles, and other documents related to the subject of your research paper.

Selected Source and Subject Guides

Mathematics Research Guide 2

Guide to Information Sources in Mathematics and Statistics , by Martha A. Tucker and Nancy D. Anderson, 348 pages (Westport, Conn.: Libraries Unlimited, 2004)

Mathematics Education Research: A Guide for the Research Mathematician , by Curtis McKnight et al., 106 pages (Providence, R.I.: American Mathematical Society, 2000)

In addition to these sources of research, most college and university libraries offer online subject guides arranged by subject on the library’s Web page; others also list searchable course-related “LibGuides” by subject. Each guide lists more recommended published and Web sources—including books and references, journal, newspaper and magazines indexes, full-text article databases, Web sites, and even research tutorials—that you can access to expand your research on more specific issues and relevant to your subject.

Selected Books and References

Dictionaries.

The Concise Oxford Dictionary of Mathematics , by Christopher Clapham and James Nicholson, 4th ed., 528 pages (Oxford and New York: Oxford University Press, 2009)

This revised fourth edition of applied mathematics and statistics features more than 3,000 entries, arranged in alphabetical order and illustrated with charts, diagrams, and graphs, covering technical mathematical terms, from Achilles paradox to zero matrix. Includes free access to regularly updated online version of the book.

Encyclopedic Dictionary of Mathematics , 2nd ed., edited by Mathematical Society of Japan and Kiyosi Ito, 4 vols. (Cambridge, Mass.: MIT Press, 1987)

This unique four-volume encyclopedia of applied mathematics features 450 articles, including 70 new articles since its first edition, published in 1977, covering such categories as algebra; group theory; number theory; Euclidean and projective geometry; differential geometry; algebraic geometry; topology; analysis; complex analysis; functional analysis; differential, integral, and functional equations; special functions; numerical analysis; computer science and combinatorics; probability theory; statistics; mathematical programming and operations research; mechanics and theoretical physics; and the history of mathematics.

The Facts on File Dictionary of Mathematics , 4th ed., by John Daintith and Richard Rennie, 262 pages (New York: Checkmark Books, 2005)

This dictionary covers mathematical terms and concepts—some 320 entries in all—fully illustrated, including lists of Web sites and bibliographies of sources.

The Penguin Dictionary of Mathematics , 4th ed., edited by David Nelson, 496 pages (London and New York: Penguin Books, 2008)

Everything from algebra to number theory and statistics to mechanics is thoroughly covered in this updated reference encompassing more than 3,200 cross-referenced entries from all branches of pure and applied mathematics. Also includes biographies of more than 200 major figures in mathematics.

Encyclopedias

CRC Concise Encyclopedia of Mathematics , 2nd ed., by Eric W. Weisstein, 3,252 pages (Boca Raton, Fla.; London: Chapman & Hall/CRC, 2003)

This revised and expanded second edition—adding 1,000 pages of new illustrated material since its first edition—broadly covers mathematical definitions, formulas, figures, tabulations, and references on the subject.

Encyclopaedia of Mathematics , 11 vols., 5,400 pages (Dordrecht, Netherlands, and Boston: Reidel; Norwell, Mass.: Kluwer Academic Publishers, 1989–94; New York: Springer, 2005– )

This major unabridged 11-volume reference with index, hailed as “the most up-to-date, authoritative and comprehensive English-language work of reference in mathematics which exists today,” contains more than 7,000 cross-referenced entries covering all aspects of mathematics, including mathematical definitions, concepts, explanations, surveys, examples, terminology and methods, and more. In 2007, two new supplements—the first since the series was first published—were issued containing nearly 600 new entries in each written by experts in the field.

Encyclopedia of Mathematics Education , by Louise Grinstein and Sally I. Lipsey, 700 pages (New York: Routledge Falmer, 2001)

Designed for elementary, secondary, and post-secondary educators, this single-volume lists more than 400 alphabetically arranged entries covering all areas of mathematics education, including assessment, curriculum, enrichment, learning and instruction, and more.

Encyclopedia of Statistical Sciences , 2nd ed., edited by Samuel Kotz, et al., 16 vols., 9,686 pages (New York: Wiley, 2005)

Revised reference set expanded to 16 volumes and written by 600 experts detailing every area of statistical sciences, including its origin, new trends, and changes, and such areas as statistical theory and methods and application in biomedicine, computer science, economics, engineering, genetics, medicine, the environment, sociology, and more.

Guides and Handbooks

Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences , 2nd ed., by Ivor Grattan-Guiness, 2 vols., 976 pages (Baltimore, Md.: Johns Hopkins University Press, 2003)

This illustrated two-volume set features 176 concise articles divided into 12 sections covering the development, history, cultural importance, problems, and theories and techniques of math and its execution in related sciences, including astronomy, computer science, engineering, philosophy, and social sciences, from its early beginnings through the 20th century. Features annotated bibliographies of sources with each article.

Figures of Thought: Mathematics and Mathematical Texts , by David Reed, 208 pages (London and New York: Routledge, 1994)

This single reference traces the history and evolution of mathematics and the work of famous mathematicians throughout history, including Dedekind, Descartes, Grothendieck, Hilbert, Kronecker, and Weil and an understanding of their approaches to mathematical science.

Guide to Information Sources in Mathematics and Statistics , by Martha A. Tucker and Nancy D. Anderson, 368 pages (Englewood, Colo.: Libraries Unlimited, 2004)

Praised as a “useful resource for college librarians and those just getting started in mathematics and statistics research,” this revised edition encompasses how to locate and access hundreds of print and electronic sources on mathematical sciences from 1800s to date.

A History of Mathematics: An Introduction , 3rd ed., by Victor J. Katz (Boston: Addison-Wesley, 2009)

This updated third edition provides a historical and world perspective of mathematics—its early and modern history, its evolving techniques, and contributions to the art of mathematics from throughout the Western and non-Western world.

INSTAT, International Statistics Sources: Subject Guide to Sources of International Comparative Statistics , by M. C. Fleming and J. G. Nellis, 1,080 pages (London: Routledge, 1994)

Unprecedented in its coverage, this A-to-Z guide details statistical data sources on business, economics, and social sciences topics, including agriculture, employment, energy, environment, finance, health, manufacturing, population, and wages. A subject index to all topics is included.

Selected Full-Text Article Databases

ArticleFirst  (Dublin, Ohio: OCLC FirstSearch, 1990– )

Full-text articles and citations to more than 16,000 journals in all subjects, including science, technology, and others; also known as OCLC ArticleFirst Database.

ESBCOHost Academic Search Elite  (Ipswich, Mass.: ESBSCO Publishing, EBSCOHost, abstracting/indexing: 1984– , full text: 1990– )

A Web index of full-text articles from more than 1,250 journals, plus abstracts and citations from 3,200 journals covering general science, the social sciences, and more.

JSTOR  (Ann Arbor, Mich.: Journal Storage Project, 1800s—latest 3 to 5 years)

A Web archive of important scholarly journals, some in full text, including more than 37,000 articles from The American Mathematical Monthly (1894–2004), since the 1800s in economics, finance, and mathematical sciences.

ScienceDirect  (St. Louis, Mo.: Elsevier Science, 1995– )

Leading science database on the Web with full-text access to more than 9.5 million articles from more than 2,500 scientific, mathematical, technical, and social science journals.

Web of Science  (Philadelphia: Thomson Scientific, 1840– )

Contains detailed bibliographic records to more than 8,700 worldwide scientific journals and publications, with full-text articles from more than 250 scientific journals from 1840 to date; allows searching of material from other related databases, including Science Citation Index (1900– ), Social Sciences Citation Index (1956– ), Arts & Humanities Citation Index (1975– ), Index Chemicus (1993– ), and Current Chemical Reactions (1986– ).

Wilson Select Plus  (Bronx, N.Y.: H.W. Wilson Co., WilsonDisc/OCLC FirstSearch, 1994– )

On the Web, indexes and abstracts full-text articles from 2,621 journals, magazines, and newspapers covering such subjects as science, humanities, education, and business.

Selected Periodicals

Acta Mathematica Sinica  (Tokyo, Japan: Springer-Verlag Tokyo/Chinese Mathematical Society, 1936– )

This English-translated version of the popular quarterly journal published by the Chinese Mathematical Society since 1936 (originally titled, Journal of Chinese Mathematical Society, until it was renamed in 1952) publishes authoritative reviews of current citations with abstracts to articles from current and past issues searchable in such online databases as Academic OneFile, Current Abstracts, Current Contents/Physical, Chemical and Earth Sciences, International Abstracts in Operations Research, Journal Citation Reports/Science Edition, Mathematical Reviews, Science and Technology Collection, Science Citation Index Expanded, SCOPUS, TOC Premier, and Zentralblatt MATH.

American Journal of Mathematics   (Baltimore, Md.: Johns Hopkins University Press, 1878– )

As “the oldest mathematics journal in the Western Hemisphere,” published since 1878, this academic journal, one of the most respected and celebrated in its industry, publishes pioneering mathematical papers and articles about all areas of contemporary mathematics. Articles are indexed and abstracted in the following electronic databases: CompuMath Citation Index, Current Contents/Physical, Chemical and Earth Sciences, Current Mathematical Publications, General Science Index, Index to Scientific Reviews, Math-SciNet, Mathematical Reviews, Science Citation Index, Social Science Citation Index, and Zentralblatt MATH. To browse journals by subject or title, or search past issues, visit  http://www.press.jhu.edu/journals/american_journal_of_mathematics/ .

Annals of Applied Probability  (Beachwood, Ohio: Institute of Mathematical Statistics, February 1991– )

First published in February 1991, this scholarly journal publishes important and original research covering all facets of contemporary applications of probability. Electronic access to all issues of the journal is available through JSTOR (issues older than 3 years from the current year).

Annals of Combinatorics  (Singapore and New York: Springer-Verlag, 1997– )

This journal covers new developments, mathematical breakthroughs and mathematical theories in combinatorial mathematics, particularly its applications to computer science, biology, statistics, probability, physics, and chemistry, as well as representation theory, number theory topology, algebraic geometry, and more. Articles are indexed and fully searchable in Academic OneFile, Current Abstracts, Current Contents/Physical, Chemical and Earth Sciences, Journal Citation Reports/Science Edition, Mathematical Reviews, Science and Technology Collection, Science Citation Index Expanded, and others.

Foundations of Computational Mathematics  (New York: Springer-Verlag New York, 2001– )

Introduced in January 2001, this quarterly academic journal, published in association with the Foundations of Computational Mathematics, features articles discussing the connections between mathematics and computation, including the interfaces between pure and applied mathematics, numerical analysis, and computer science. Full-text articles from issues since 2001 can be viewed in PDF form at  http://link.springer.com/journal/10208 .

Historia Mathematica  (Amsterdam, Netherlands: Elsevier Science B.V., 1974– )

Launched in February 1974, this quarterly periodical of the International Commission on the History of Mathematics of the Division of the History of Science of the International Union of the History and Philosophy of Science covers all aspects of mathematical sciences, including mathematicians and their work, organizations and institutions, pure and applied mathematics, and the sociology of mathematics, as well as all cultures and historical periods of mathematics and its development. The primary aim and focus of each issue is topics in the history of math, including research articles, book reviews, and more. Full-text articles are accessible through ScienceDirect.

Journal of Mathematics and Statistics  (New York: Science Publications, 2005– )

Published since January/March 2005, this peer-reviewed, open-access international scientific journal presents original and valuable research in all areas of applied and theoretical mathematics and statistics. To view or search back issues, visit  http://thescipub.com/jmss.toc .

Journal of Pure and Applied Algebra  (Evanston, Ill.: Elsevier Science, 1971– )

This monthly journal published in association with Northwestern University’s math department focuses on the development and theories of pure and applied algebra. Also available in microform since its first issue in January 1971, full-text articles from all issues can be searched in Elsevier Science’s ScienceDirect online database.

The Journal of Symbolic Logic  (Poughkeepsie, N.Y.: Association for Symbolic Logic, 1936– )

Leading scientific journal founded in 1936 by the Association for Symbolic Logic (ASL) containing scholarly work and research on symbolic logic. The journal is distributed with two others published by the ASL: The Bulletin of Symbolic Logic and Review of Symbolic Logic. Full-text access to The Journal of Symbolic Logic (1936–2003) is available via JSTOR.

Journal of the American Mathematical Society (JAMS)  (Providence, R.I.: American Mathematical Society, January 1, 1988– )

Mathematics journal published quarterly by the American Mathematical Society reporting research in all areas of pure and applied mathematics. Journal articles are indexed in such subscription Web databases as Citation Index—Expanded, CompuMath Citation Index, and Current Contents, Physical, Chemical & Earth Sciences. Since January 1996, JAMS is also accessible online at  http://www.ams.org/publications/journals/journalsframework/jams .

Mathematical Physics, Analysis, and Geometry  (Norwell, Mass.: Kluwer Academic/Plenum Publishing Corp., 1998– )

Scientific journal covering concrete problems of mathematics and theoretical analysis and application of analysis on all math, from geometry to physics, including problems of statistical physics and fl uids; complex function theory; operators in function space, especially operator algebras; ordinary and partial differential equations; and differential and algebraic geometry. Journal is indexed and abstracted in many subject-specific online databases, including Academic OneFile, Current Abstracts, Google Scholar, Journal Citation Reports/Science Edition, Mathematical Reviews, and others.

SIAM Journal on Applied Mathematics  (Newark, Dela.: Society for Industrial & Applied Mathematics, 1953– )

Published by the Society for Industrial & Applied Mathematics since 1953, this quarterly journal reviews applied mathematics of physical, engineering, biological, medical, and social sciences, including research articles discussing problems and methods pertinent to physical, engineering, financial, and life sciences. Full bibliographic records with abstracts of articles from 1997 to the present can be searched on SIAMS Journals Online at  http://www.siam.org/journals/siap.php .

Selected Web Sites

American Mathematical Society  ( http://www.ams.org/home/page )

Association of professional mathematicians, headquartered in Providence, Rhode Island, reporting on mathematical research and education, conferences, surveys, publications, scholarship programs, and more.

American Statistical Association  ( http://www.amstat.org/ )

This Web site for the nation’s leading professional organization for statisticians and professors provides resources for visitors and members, including association news, membership information, educational opportunities, publications, meetings and events, and outreach programs.

Euler Archive—Dartmouth College  ( http://eulerarchive.maa.org/ )

Provides online access to Dartmouth College’s archive of 866 original works of pioneering Swiss mathematician Leonard Euler as well as other original publications and current research.

MacTutor History of Mathematics  ( http://www-history.mcs.st-andrews.ac.uk/history/index.html )

Features searchable historical mathematical topics, biographies of notable mathematicians from AD 500 to present, and an index of famous curves.

Math Archives  ( http://archives.math.utk.edu/ )

Online resource covering a wide range of mathematical topics and Internet resources arranged by subject.

Mathematical Atlas  ( http://www.math-atlas.org/ )

Gateway collection of articles discussing various mathematical concepts, with links to additional resources in all areas of mathematics.

Mathematics on the Web  ( http://www.mathontheweb.org/mathweb/ )

Online mathematical sources maintained by the American Mathematical Society and organized by subject, including article abstracts and databases, as well as bibliographies; books, journals, columns, and handbooks; math history and math topics; information about mathematics departments, institutes, centers, associations, societies, and organizations; and related software and tools.

Mathematics, Statistics, and Computational Science at NIST  ( http://math.nist.gov/ )

Site provided by the National Institute of Standards and Technology, offering information on NIST projects, events, and organizations; math software; statistical guides and handbooks; statistical data sets; and more.

Math Forum  ( http://mathforum.org/ )

This site, maintained by Drexel University’s School of Education, offers both resources and information on math and math education, along with access to the Internet Mathematics Library, discussion groups, and more.

Math on the Web  ( http://www.mathontheweb.org/mathweb/index.html )

Web portal produced by the American Mathematical Society providing access to mathematical news and information, journals, and reference materials.

Math World  ( http://mathworld.wolfram.com/ )

Deemed by its creators as “the Web’s most extensive mathematics resource,” this site includes a searchable math dictionary and encyclopedia, interactive tools, and information on the computational software program Mathematica.

+plus magazine  ( http://plus.maths.org/content/ )

Free access to this weekly online magazine featuring the latest mathematical news, articles by leading mathematicians and science writers, a browsable archive, and information on a variety of mathematical applications.

Probability Tutorials  ( http://www.probability.net/ )

Online math tutorials explaining probability, definitions, theorems, solutions, and more.

SIAM: Society for Industrial and Applied Mathematics  ( http://www.siam.org/ )

Official Web site for the Society for Industrial and Applied Mathematics offering information on books, careers and jobs, conferences, journals, proceedings, and the latest news.

Zentralblatt MATH  ( http://www.zentralblatt-math.org/zmath/en/ )

Searchable database of more than 2 million citations with abstracts to books, journals, conference reports, and more, from 1868 to present.

Careers Related to Mathematics

Science, Technology, Engineering, and Mathematics Career Cluster ( http://career.iresearchnet.com/career-clusters/science-technology-engineering-and-mathematics-career-cluster/ )

Science careers include jobs in biology, chemistry, geology, meteorology, or any other natural, physical, or earth science. Mathematics is the science and study of numbers and how they relate to each other. Engineering and technology encompasses many areas of study, such as aviation, environmental science, and robotics, just to name a few. All of these engineering fields employ unique and sometimes similar methods of research, development, and production to reach practical solutions to problems and questions.

Mathematics and Physics Career Field ( http://career.iresearchnet.com/career-fields/mathematics-and-physics-career-field/ )

Mathematics and physics are closely related natural sciences. Mathematics is the science and study of numbers and how they relate with each other. Physics is the study of the basic elements and laws of the universe.

Engineering Career Field  ( http://career.iresearchnet.com/career-fields/engineering-career-field/ )

A lot of brainpower goes into engineering—a lot of knowledge, creativity, thoughtfulness, and pure hard work. Humankind has been “engineering,” so to speak, since we realized we had opposable thumbs that we could use to handle tools. And from that point on we began our ceaseless quest to make, to build, to create tools and systems that helped us live our lives better. There were a lot of mistakes, but engineers and scientists learned from them and built a foundation of engineering laws and principles.

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best math research paper

Mathematics at MIT is administratively divided into two categories: Pure Mathematics and Applied Mathematics. They comprise the following research areas:

Pure Mathematics

  • Algebra & Algebraic Geometry
  • Algebraic Topology
  • Analysis & PDEs
  • Mathematical Logic & Foundations
  • Number Theory
  • Probability & Statistics
  • Representation Theory

Applied Mathematics

In applied mathematics, we look for important connections with other disciplines that may inspire interesting and useful mathematics, and where innovative mathematical reasoning may lead to new insights and applications.

  • Combinatorics
  • Computational Biology
  • Physical Applied Mathematics
  • Computational Science & Numerical Analysis
  • Theoretical Computer Science
  • Mathematics of Data

8. Appendices

In the appendices you should include any data or material that supported your research but that was too long to include in the body of your paper. Materials in an appendix should be referenced at some point in the body of the report.

Some examples:

• If you wrote a computer program to generate more data than you could produce by hand, you should include the code and some sample output.

• If you collected statistical data using a survey, include a copy of the survey.

• If you have lengthy tables of numbers that you do not want to include in the body of your report, you can put them in an appendix.

Sample Write-Up

Seating unfriendly customers, a combinatorics problem.

By Lisa Honeyman February 12, 2002

The Problem

In a certain coffee shop, the customers are grouchy in the early morning and none of them wishes to sit next to another at the counter.

1. Suppose there are ten seats at the counter. How many different ways can three early morning customers sit at the counter so that no one sits next to anyone else?

2. What if there are n seats at the counter?

3. What if we change the number of customers?

4. What if, instead of a counter, there was a round table and people refused to sit next to each other?

Assumptions

I am assuming that the order in which the people sit matters. So, if three people occupy the first, third and fifth seats, there are actually 6 (3!) different ways they can do this. I will explain more thoroughly in the body of my report.

Body of the Report

At first there are 10 seats available for the 3 people to sit in. But once the first person sits down, that limits where the second person can sit. Not only can’t he sit in the now-occupied seat, he can’t sit next to it either. What confused me at first was that if the first person sat at one of the ends, then there were 8 seats left for the second person to chose from. But if the 1 st person sat somewhere else, there were only 7 remaining seats available for the second person. I decided to look for patterns. By starting with a smaller number of seats, I was able to count the possibilities more easily. I was hoping to find a pattern so I could predict how many ways the 10 people could sit without actually trying to count them all. I realized that the smallest number of seats I could have would be 5. Anything less wouldn’t work because people would have to sit next to each other. So, I started with 5 seats. I called the customers A, B, and C.

With 5 seats there is only one configuration that works.

As I said in my assumptions section, I thought that the order in which the people sit is important. Maybe one person prefers to sit near the coffee maker or by the door. These would be different, so I decided to take into account the different possible ways these 3 people could occupy the 3 seats shown above. I know that ABC can be arranged in 3! = 6 ways. (ABC, ACB, BAC, BCA, CAB, CBA). So there are 6 ways to arrange 3 people in 5 seats with spaces between them. But, there is only one configuration of seats that can be used. (The 1 st , 3 rd , and 5 th ).

Next, I tried 6 seats. I used a systematic approach to show that there are 4 possible arrangements of seats. This is how my systematic approach works:

Assign person A to the 1 st seat. Put person B in the 3 rd seat, because he can’t sit next to person A. Now, person C can sit in either the 5 th or 6 th positions. (see the top two rows in the chart, below.) Next suppose that person B sits in the 4 th seat (the next possible one to the right.) That leaves only the 6 th seat free for person C. (see row 3, below.) These are all the possible ways for the people to sit if the 1 st seat is used. Now put person A in the 2 nd seat and person B in the 4 th . There is only one place where person C can sit, and that’s in the 6 th position. (see row 4, below.) There are no other ways to seat the three people if person A sits in the 2 nd seat. So, now we try putting person A in the 3 rd seat. If we do that, there are only 4 seats that can be used, but we know that we need at least 5, so there are no more possibilities.

Possible seats 3 people could occupy if there are 6 seats

Once again, the order the people sit in could be ABC, BAC, etc. so there are 4 * 6 = 24 ways for the 3 customers to sit in 6 seats with spaces between them.

I continued doing this, counting how many different groups of seats could be occupied by the three people using the systematic method I explained. Then I multiplied that number by 6 to account for the possible permutations of people in those seats. I created the following table of what I found.

Next I tried to come up with a formula. I decided to look for a formula using combinations or permutations. Since we are looking at 3 people, I decided to start by seeing what numbers I would get if I used n C 3 and n P 3 .

3 C 3 = 1   4 C 3 = 4   5 C 3 = 10   6 C 3 = 20

3 P 3 = 6   4 P 3 = 24   5 P 3 = 60   6 P 3 = 120

Surprisingly enough, these numbers matched the numbers I found in my table. However, the n in n P r and n C r seemed to be two less than the total # of seats I was investigating. 

Conjecture 1:

Given n seats at a lunch counter, there are n -2 C 3 ways to select the three seats in which the customers will sit such that no customer sits next to another one. There are n -2 P 3 ways to seat the 3 customers in such a way than none sits next to another.

After I found a pattern, I tried to figure out why n -2 C 3 works. (If the formula worked when order didn’t matter it could be easily extended to when the order did, but the numbers are smaller and easier to work with when looking at combinations rather than permutations.)

In order to prove Conjecture 1 convincingly, I need to show two things:

(1) Each n – 2 seat choice leads to a legal n seat configuration.

(2) Each n seat choice resulted from a unique n – 2 seat configuration.

To prove these two things I will show

And then conclude that these two procedures are both functions and therefore 1—1.

Claim (1): Each ( n – 2) -seat choice leads to a legal n seat configuration.

Suppose there were only n – 2 seats to begin with. First we pick three of them in which to put people, without regard to whether or not they sit next to each other. But, in order to guarantee that they don’t end up next to another person, we introduce an empty chair to the right of each of the first two people. It would look like this:

We don’t need a third “new” seat because once the person who is farthest to the right sits down, there are no more customers to seat. So, we started with n – 2 chairs but added two for a total of n chairs. Anyone entering the restaurant after this procedure had been completed wouldn’t know that there had been fewer chairs before these people arrived and would just see three customers sitting at a counter with n chairs. This procedure guarantees that two people will not end up next to each other. Thus, each ( n – 2)-seat choice leads to a unique, legal n seat configuration.

Therefore, positions s 1 ' s 2 ', and s 3 ' are all separated by at least one vacant seat.

This is a function that maps each combination of 3 seats selected from n – 2 seats onto a unique arrangement of n seats with 3 separated customers. Therefore, it is invertible.

Claim (2): Each 10-seat choice has a unique 8-seat configuration.

Given a legal 10-seat configuration, each of the two left-most diners must have an open seat to his/her right. Remove it and you get a unique 8-seat arrangement. If, in the 10-seat setting, we have q 1 > q 2 , q 3 ; q 3 – 1 > q 2 , and q 2 – 1 > q 1 , then the 8 seat positions are q 1 ' = q 2 , q 2 ' = q 2 – 1, and q 3 ' = q 3 – 2. Combining these equations with the conditions we have

q 2 ' = q 2 – 1 which implies q 2 ' > q 1 = q 1 '

q 3 ' = q 3 – 2 which implies q 3 ' > q 2 – 1 = q 2 '

Since q 3 ' > q 2 ' > q 1 ', these seats are distinct. If the diners are seated in locations q 1 , q 2 , and q 3 (where q 3 – 1 > q 2 and q 2 – 1 > q 1 ) and we remove the two seats to the right of q 1 and q 2 , then we can see that the diners came from q 1 , q 2 – 1, and q 3 – 2. This is a function that maps a legal 10-seat configuration to a unique 8-seat configuration.

The size of a set can be abbreviated s ( ). I will use the abbreviation S to stand for n separated seats and N to stand for the n – 2 non-separated seats.

therefore s ( N ) = s ( S ).

Because the sets are the same size, these functions are 1—1.

Using the technique of taking away and adding empty chairs, I can extend the problem to include any number of customers. For example, if there were 4 customers and 10 seats there would be 7 C 4 = 35 different combinations of chairs to use and 7 P 4 = 840 ways for the customers to sit (including the fact that order matters). You can imagine that three of the ten seats would be introduced by three of the customers. So, there would only be 7 to start with.

In general, given n seats and c customers, we remove c- 1 chairs and select the seats for the c customers. This leads to the formula n -( c -1) C c = n - c +1 C c for the number of arrangements.

Once the number of combinations of seats is found, it is necessary to multiply by c ! to find the number of permutations. Looking at the situation of 3 customers and using a little algebraic manipulation, we get the n P 3 formula shown below.

This same algebraic manipulation works if you have c people rather than 3, resulting in n - c +1 P c

Answers to Questions

  • With 10 seats there are 8 P 3 = 336 ways to seat the 3 people.
  • My formula for n seats and 3 customers is: n -2 P 3 .
  • My general formula for n seats and c customers, is: n -( c -1) P c = n - c +1 P c

_________________________________________________________________ _

After I finished looking at this question as it applied to people sitting in a row of chairs at a counter, I considered the last question, which asked would happen if there were a round table with people sitting, as before, always with at least one chair between them.

I went back to my original idea about each person dragging in an extra chair that she places to her right, barring anyone else from sitting there. There is no end seat, so even the last person needs to bring an extra chair because he might sit to the left of someone who has already been seated. So, if there were 3 people there would be 7 seats for them to choose from and 3 extra chairs that no one would be allowed to sit in. By this reasoning, there would be 7 C 3 = 35 possible configurations of chairs to choose and 7 P 3 = 840 ways for 3 unfriendly people to sit at a round table.

Conjecture 2: Given 3 customers and n seats there are n -3 C 3 possible groups of 3 chairs which can be used to seat these customers around a circular table in such a way that no one sits next to anyone else.

My first attempt at a proof: To test this conjecture I started by listing the first few numbers generated by my formula:

When n = 6    6-3 C 3 = 3 C 3 = 1

When n = 7    7-3 C 3 = 4 C 3 = 4

When n = 8    8-3 C 3 = 5 C 3 = 10

When n = 9    9-3 C 3 = 6 C 3 = 20

Then I started to systematically count the first few numbers of groups of possible seats. I got the numbers shown in the following table. The numbers do not agree, so something is wrong — probably my conjecture!

I looked at a circular table with 8 people and tried to figure out the reason this formula doesn’t work. If we remove 3 seats (leaving 5) there are 10 ways to select 3 of the 5 remaining chairs. ( 5 C 3 ).

The circular table at the left in the figure below shows the n – 3 (in this case 5) possible chairs from which 3 will be randomly chosen. The arrows point to where the person who selects that chair could end up. For example, if chair A is selected, that person will definitely end up in seat #1 at the table with 8 seats. If chair B is selected but chair A is not, then seat 2 will end up occupied. However, if chair A and B are selected, then the person who chose chair B will end up in seat 3 . The arrows show all the possible seats in which a person who chose a particular chair could end. Notice that it is impossible for seat #8 to be occupied. This is why the formula 5 C 3 doesn’t work. It does not allow all seats at the table of 8 to be chosen.

The difference is that in the row-of-chairs-at-a-counter problem there is a definite “starting point” and “ending point.” The first chair can be identified as the one farthest to the left, and the last one as the one farthest to the right. These seats are unique because the “starting point” has no seat to the left of it and the “ending point” has no seat to its right. In a circle, it is not so easy.

Using finite differences I was able to find a formula that generates the correct numbers:

Proof: We need to establish a “starting point.” This could be any of the n seats. So, we select one and seat person A in that seat. Person B cannot sit on this person’s left (as he faces the table), so we must eliminate that as a possibility. Also, remove any 2 other chairs, leaving ( n – 4) possible seats where the second person can sit. Select another seat and put person B in it. Now, select any other seat from the ( n – 5) remaining seats and put person C in that. Finally, take the two seats that were previously removed and put one to the left of B and one to the left of C.

The following diagram should help make this procedure clear.

In a manner similar to the method I used in the row-of-chairs-at-a-counter problem, this could be proven more rigorously.

An Idea for Further Research:

Consider a grid of chairs in a classroom and a group of 3 very smelly people. No one wants to sit adjacent to anyone else. (There would be 9 empty seats around each person.) Suppose there are 16 chairs in a room with 4 rows and 4 columns. How many different ways could 3 people sit? What if there was a room with n rows and n columns? What if it had n rows and m columns?

References:

Abrams, Joshua. Education Development Center, Newton, MA. December 2001 - February 2002. Conversations with my mathematics mentor.

Brown, Richard G. 1994. Advanced Mathematics . Evanston, Illinois. McDougal Littell Inc. pp. 578-591

The Oral Presentation

Giving an oral presentation about your mathematics research can be very exciting! You have the opportunity to share what you have learned, answer questions about your project, and engage others in the topic you have been studying. After you finish doing your mathematics research, you may have the opportunity to present your work to a group of people such as your classmates, judges at a science fair or other type of contest, or educators at a conference. With some advance preparation, you can give a thoughtful, engaging talk that will leave your audience informed and excited about what you have done.

Planning for Your Oral Presentation

In most situations, you will have a time limit of between 10 and 30 minutes in which to give your presentation. Based upon that limit, you must decide what to include in your talk. Come up with some good examples that will keep your audience engaged. Think about what vocabulary, explanations, and proofs are really necessary in order for people to understand your work. It is important to keep the information as simple as possible while accurately representing what you’ve done. It can be difficult for people to understand a lot of technical language or to follow a long proof during a talk. As you begin to plan, you may find it helpful to create an outline of the points you want to include. Then you can decide how best to make those points clear to your audience.

You must also consider who your audience is and where the presentation will take place. If you are going to give your presentation to a single judge while standing next to your project display, your presentation will be considerably different than if you are going to speak from the stage in an auditorium full of people! Consider the background of your audience as well. Is this a group of people that knows something about your topic area? Or, do you need to start with some very basic information in order for people to understand your work? If you can tailor your presentation to your audience, it will be much more satisfying for them and for you.

No matter where you are presenting your speech and for whom, the structure of your presentation is very important. There is an old bit of advice about public speaking that goes something like this: “Tell em what you’re gonna tell ’em. Tell ’em. Then tell ’em what you told ’em.” If you use this advice, your audience will find it very easy to follow your presentation. Get the attention of the audience and tell them what you are going to talk about, explain your research, and then following it up with a re-cap in the conclusion.

Writing Your Introduction

Your introduction sets the stage for your entire presentation. The first 30 seconds of your speech will either capture the attention of your audience or let them know that a short nap is in order. You want to capture their attention. There are many different ways to start your speech. Some people like to tell a joke, some quote famous people, and others tell stories.

Here are a few examples of different types of openers.

You can use a quote from a famous person that is engaging and relevant to your topic. For example:

• Benjamin Disraeli once said, “There are three kinds of lies: lies, damn lies, and statistics.” Even though I am going to show you some statistics this morning, I promise I am not going to lie to you! Instead, . . .

• The famous mathematician, Paul Erdös, said, “A Mathematician is a machine for turning coffee into theorems.” Today I’m here to show you a great theorem that I discovered and proved during my mathematics research experience. And yes, I did drink a lot of coffee during the project!

• According to Stephen Hawking, “Equations are just the boring part of mathematics.” With all due respect to Dr. Hawking, I am here to convince you that he is wrong. Today I’m going to show you one equation that is not boring at all!

Some people like to tell a short story that leads into their discussion.

“Last summer I worked at a diner during the breakfast shift. There were 3 regular customers who came in between 6:00 and 6:15 every morning. If I tell you that you didn’t want to talk to these folks before they’ve had their first cup of coffee, you’ll get the idea of what they were like. In fact, these people never sat next to each other. That’s how grouchy they were! Well, their anti-social behavior led me to wonder, how many different ways could these three grouchy customers sit at the breakfast counter without sitting next to each other? Amazingly enough, my summer job serving coffee and eggs to grouchy folks in Boston led me to an interesting combinatorics problem that I am going to talk to you about today.”

A short joke related to your topic can be an engaging way to start your speech.

It has been said that there are three kinds of mathematicians: those who can count and those who can’t.

All joking aside, my mathematics research project involves counting. I have spent the past 8 weeks working on a combinatorics problem.. . .

To find quotes to use in introductions and conclusions try: http://www.quotationspage.com/

To find some mathematical quotes, consult the Mathematical Quotation Server: http://math.furman.edu/~mwoodard/mquot.html

To find some mathematical jokes, you can look at the “Profession Jokes” web site: http://www.geocities.com/CapeCanaveral/4661/projoke22.htm

There is a collection of math jokes compiled by the Canadian Mathematical Society at http://camel.math.ca/Recreation/

After you have the attention of your audience, you must introduce your research more formally. You might start with a statement of the problem that you investigated and what lead you to choose that topic. Then you might say something like this,

“Today I will demonstrate how I came to the conclusion that there are n ( n  – 4)( n  – 5) ways to seat 3 people at a circular table with n seats in such a way that no two people sit next to each other. In order to do this I will first explain how I came up with this formula and then I will show you how I proved it works. Finally, I will extend this result to tables with more than 3 people sitting at them.”

By providing a brief outline of your talk at the beginning and reminding people where you are in the speech while you are talking, you will be more effective in keeping the attention of your audience. It will also make it much easier for you to remember where you are in your speech as you are giving it.

The Middle of Your Presentation

Because you only have a limited amount of time to present your work, you need to plan carefully. Decide what is most important about your project and what you want people to know when you are finished. Outline the steps that people need to follow in order to understand your research and then think carefully about how you will lead them through those steps. It may help to write your entire speech out in advance. Even if you choose not to memorize it and present it word for word, the act of writing will help you clarify your ideas. Some speakers like to display an outline of their talk throughout their entire presentation. That way, the audience always knows where they are in the presentation and the speaker can glance at it to remind him or herself what comes next.

An oral presentation must be structured differently than a written one because people can’t go back and “re-read” a complicated section when they are at a talk. You have to be extremely clear so that they can understand what you are saying the first time you say it. There is an acronym that some presenters like to remember as they prepare a talk: “KISS.” It means, “Keep It Simple, Student.” It may sound silly, but it is good advice. Keep your sentences short and try not to use too many complicated words. If you need to use technical language, be sure to define it carefully. If you feel that it is important to present a proof, remember that you need to keep things easy to understand. Rather than going through every step, discuss the main points and the conclusion. If you like, you can write out the entire proof and include it in a handout so that folks who are interested in the details can look at them later. Give lots of examples! Not only will examples make your talk more interesting, but they will also make it much easier for people to follow what you are saying.

It is useful to remember that when people have something to look at, it helps to hold their attention and makes it easier for them to understand what you are saying. Therefore, use lots of graphs and other visual materials to support your work. You can do this using posters, overhead transparencies, models, or anything else that helps make your explanations clear.

Using Materials

As you plan for your presentation, consider what equipment or other materials you might want use. Find out what is available in advance so you don’t spend valuable time creating materials that you will not be able to use. Common equipment used in talks include an over-head projector, VCR, computer, or graphing calculator. Be sure you know how to operate any equipment that you plan to use. On the day of your talk, make sure everything is ready to go (software loaded, tape at the right starting point etc.) so that you don’t have “technical difficulties.”

Visual aides can be very useful in a presentation. (See Displaying Your Results for details about poster design.) If you are going to introduce new vocabulary, consider making a poster with the words and their meanings to display throughout your talk. If people forget what a term means while you are speaking, they can refer to the poster you have provided. (You could also write the words and meanings on a black/white board in advance.) If there are important equations that you would like to show, you can present them on an overhead transparency that you prepare prior to the talk. Minimize the amount you write on the board or on an overhead transparency during your presentation. It is not very engaging for the audience to sit watching while you write things down. Prepare all equations and materials in advance. If you don’t want to reveal all of what you have written on your transparency at once, you can cover up sections of your overhead with a piece of paper and slide it down the page as you move along in your talk. If you decide to use overhead transparencies, be sure to make the lettering large enough for your audience to read. It also helps to limit how much you put on your transparencies so they are not cluttered. Lastly, note that you can only project approximately half of a standard 8.5" by 11" page at any one time, so limit your information to displays of that size.

Presenters often create handouts to give to members of the audience. Handouts may include more information about the topic than the presenter has time to discuss, allowing listeners to learn more if they are interested. Handouts may also include exercises that you would like audience members to try, copies of complicated diagrams that you will display, and a list of resources where folks might find more information about your topic. Give your audience the handout before you begin to speak so you don’t have to stop in the middle of the talk to distribute it. In a handout you might include:

• A proof you would like to share, but you don’t have time to present entirely.

• Copies of important overhead transparencies that you use in your talk.

• Diagrams that you will display, but which may be too complicated for someone to copy down accurately.

• Resources that you think your audience members might find useful if they are interested in learning more about your topic.

The Conclusion

Ending your speech is also very important. Your conclusion should leave the audience feeling satisfied that the presentation was complete. One effective way to conclude a speech is to review what you presented and then to tie back to your introduction. If you used the Disraeli quote in your introduction, you might end by saying something like,

I hope that my presentation today has convinced you that . . . Statistical analysis backs up the claims that I have made, but more importantly, . . . . And that’s no lie!

Getting Ready

After you have written your speech and prepared your visuals, there is still work to be done.

  • Prepare your notes on cards rather than full-size sheets of paper. Note cards will be less likely to block your face when you read from them. (They don’t flop around either.) Use a large font that is easy for you to read. Write notes to yourself on your notes. Remind yourself to smile or to look up. Mark when to show a particular slide, etc.
  • Practice! Be sure you know your speech well enough that you can look up from your notes and make eye contact with your audience. Practice for other people and listen to their feedback.
  • Time your speech in advance so that you are sure it is the right length. If necessary, cut or add some material and time yourself again until your speech meets the time requirements. Do not go over time!
  • Anticipate questions and be sure you are prepared to answer them.
  • Make a list of all materials that you will need so that you are sure you won’t forget anything.
  • If you are planning to provide a handout, make a few extras.
  • If you are going to write on a whiteboard or a blackboard, do it before starting your talk.

The Delivery

How you deliver your speech is almost as important as what you say. If you are enthusiastic about your presentation, it is far more likely that your audience will be engaged. Never apologize for yourself. If you start out by saying that your presentation isn’t very good, why would anyone want to listen to it? Everything about how you present yourself will contribute to how well your presentation is received. Dress professionally. And don’t forget to smile!

Here are a few tips about delivery that you might find helpful.

  • Make direct eye contact with members of your audience. Pick a person and speak an entire phrase before shifting your gaze to another person. Don’t just “scan” the audience. Try not to look over their heads or at the floor. Be sure to look at all parts of the room at some point during the speech so everyone feels included.
  • Speak loudly enough for people to hear and slowly enough for them to follow what you are saying.
  • Do not read your speech directly from your note cards or your paper. Be sure you know your speech well enough to make eye contact with your audience. Similarly, don’t read your talk directly off of transparencies.
  • Avoid using distracting or repetitive hand gestures. Be careful not to wave your manuscript around as you speak.
  • Move around the front of the room if possible. On the other hand, don’t pace around so much that it becomes distracting. (If you are speaking at a podium, you may not be able to move.)
  • Keep technical language to a minimum. Explain any new vocabulary carefully and provide a visual aide for people to use as a reference if necessary.
  • Be careful to avoid repetitive space-fillers and slang such as “umm”, “er”, “you know”, etc. If you need to pause to collect your thoughts, it is okay just to be silent for a moment. (You should ask your practice audiences to monitor this habit and let you know how you did).
  • Leave time at the end of your speech so that the audience can ask questions.

Displaying Your Results

When you create a visual display of your work, it is important to capture and retain the attention of your audience. Entice people to come over and look at your work. Once they are there, make them want to stay to learn about what you have to tell them. There are a number of different formats you may use in creating your visual display, but the underlying principle is always the same: your work should be neat, well-organized, informative, and easy to read.

It is unlikely that you will be able to present your entire project on a single poster or display board. So, you will need to decide which are the most important parts to include. Don’t try to cram too much onto the poster. If you do, it may look crowded and be hard to read! The display should summarize your most important points and conclusions and allow the reader to come away with a good understanding of what you have done.

A good display board will have a catchy title that is easy to read from a distance. Each section of your display should be easily identifiable. You can create posters such as this by using headings and also by separating parts visually. Titles and headings can be carefully hand-lettered or created using a computer. It is very important to include lots of examples on your display. It speeds up people’s understanding and makes your presentation much more effective. The use of diagrams, charts, and graphs also makes your presentation much more interesting to view. Every diagram or chart should be clearly labeled. If you include photographs or drawings, be sure to write captions that explain what the reader is looking at.

In order to make your presentation look more appealing, you will probably want to use some color. However, you must be careful that the color does not become distracting. Avoid florescent colors, and avoid using so many different colors that your display looks like a patch-work quilt. You want your presentation to be eye-catching, but you also want it to look professional.

People should be able to read your work easily, so use a reasonably large font for your text. (14 point is a recommended minimum.) Avoid writing in all-capitals because that is much harder to read than regular text. It is also a good idea to limit the number of different fonts you use on your display. Too many different fonts can make your poster look disorganized.

Notice how each section on the sample poster is defined by the use of a heading and how the various parts of the presentation are displayed on white rectangles. (Some of the rectangles are blank, but they would also have text or graphics on them in a real presentation.) Section titles were made with pale green paper mounted on red paper to create a boarder. Color was used in the diagrams to make them more eye-catching. This poster would be suitable for hanging on a bulletin board.

If you are planning to use a poster, such as this, as a visual aid during an oral presentation, you might consider backing your poster with foam-core board or corrugated cardboard. A strong board will not flop around while you are trying to show it to your audience. You can also stand a stiff board on an easel or the tray of a classroom blackboard or whiteboard so that your hands will be free during your talk. If you use a poster as a display during an oral presentation, you will need to make the text visible for your audience. You can create a hand-out or you can make overhead transparencies of the important parts. If you use overhead transparencies, be sure to use lettering that is large enough to be read at a distance when the text is projected.

If you are preparing your display for a science fair, you will probably want to use a presentation board that can be set up on a table. You can buy a pre-made presentation board at an office supply or art store or you can create one yourself using foam-core board. With a presentation board, you can often use the space created by the sides of the board by placing a copy of your report or other objects that you would like people to be able to look at there. In the illustration, a black trapezoid was cut out of foam-core board and placed on the table to make the entire display look more unified. Although the text is not shown in the various rectangles in this example, you will present your information in spaces such as these.

Don’t forget to put your name on your poster or display board. And, don’t forget to carefully proof-read your work. There should be no spelling, grammatical or typing mistakes on your project. If your display is not put together well, it may make people wonder about the quality of the work you did on the rest of your project.

For more information about creating posters for science fair competitions, see

http://school.discovery.com/sciencefaircentral/scifairstudio/handbook/display.html ,

http://www.siemens-foundation.org/science/poster_guidelines.htm ,

Robert Gerver’s book, Writing Math Research Papers , (published by Key Curriculum Press) has an excellent section about doing oral presentations and making posters, complete with many examples.

References Used

American Psychological Association . Electronic reference formats recommended by the American Psychological Association . (2000, August 22). Washington, DC: American Psychological Association. Retrieved October 6, 2000, from the World Wide Web: http://www.apastyle.org/elecsource.html

Bridgewater State College. (1998, August 5 ). APA Style: Sample Bibliographic Entries (4th ed) . Bridgewater, MA: Clement C. Maxwell Library. Retrieved December 20, 2001, from the World Wide Web: http://www.bridgew.edu/dept/maxwell/apa.htm

Crannell, Annalisa. (1994). A Guide to Writing in Mathematics Classes . Franklin & Marshall College. Retrieved January 2, 2002, from the World Wide Web: http://www.fandm.edu/Departments/Mathematics/writing_in_math/guide.html

Gerver, Robert. 1997. Writing Math Research Papers . Berkeley, CA: Key Curriculum Press.

Moncur, Michael. (1994-2002 ). The Quotations Page . Retrieved April 9, 2002, from the World Wide Web: http://www.quotationspage.com/

Public Speaking -- Be the Best You Can Be . (2002). Landover, Hills, MD: Advanced Public Speaking Institute. Retrieved April 9, 2002, from the World Wide Web: http://www.public-speaking.org/

Recreational Mathematics. (1988) Ottawa, Ontario, Canada: Canadian Mathematical Society. Retrieved April 9, 2002, from the World Wide Web: http://camel.math.ca/Recreation/

Shay, David. (1996). Profession Jokes — Mathematicians. Retrieved April 5, 2001, from the World Wide Web: http://www.geocities.com/CapeCanaveral/4661/projoke22.htm

Sieman’s Foundation. (2001). Judging Guidelines — Poster . Retrieved April 9, 2002, from the World Wide Web: http://www.siemens-foundation.org/science/poster_guidelines.htm ,

VanCleave, Janice. (1997). Science Fair Handbook. Discovery.com. Retrieved April 9, 2002, from the World Wide Web: http://school.discovery.com/sciencefaircentral/scifairstudio/handbook/display.html ,

Woodward, Mark. (2000) . The Mathematical Quotations Server . Furman University. Greenville, SC. Retrieved April 9, 2002, from the World Wide Web: http://math.furman.edu/~mwoodard/mquot.html

Making Mathematics Home | Mathematics Projects | Students | Teachers | Mentors | Parents | Hard Math Café |

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February 15, 2024

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Studies recommend increased research into achievement, engagement to raise student math scores

by La Trobe University

math homework

A new study into classroom practices, led by Dr. Steve Murphy, has found extensive research fails to uncover how teachers can remedy poor student engagement and perform well in math.

More than 3,000 research papers were reviewed over the course of the study, but only 26 contained detailed steps for teachers to improve both student engagement and results in math. The review is published in the journal Teaching and Teacher Education .

Dr. Murphy said the scarcity of research involving young children was concerning.

"Children's engagement in math begins to decline from the beginning of primary school while their mathematical identity begins to solidify," Dr. Murphy said.

"We need more research that investigates achievement and engagement together to give teachers good advice on how to engage students in mathematics and perform well.

"La Trobe has developed a model for research that can achieve this."

While teachers play an important role in making decisions that impact the learning environment , Dr. Murphy said parents are also highly influential in children's math education journeys.

"We often hear parents say, 'It's OK, I was never good at math,' but they'd never say that to their child about reading or writing," Dr. Murphy said.

La Trobe's School of Education is determined to improve mathematical outcomes for students, arguing it's an important school subject that is highly applicable in today's technologically rich society.

Previous research led by Dr. Murphy published in Educational Studies in Mathematics found many parents were unfamiliar with the modern ways of teaching math and lacked self-confidence to independently assist their children learning math during the COVID-19 pandemic.

"The implication for parents is that you don't need to be a great mathematician to support your children in math, you just need to be willing to learn a little about how schools teach math today," Dr. Murphy said.

"It's not all bad news for educators and parents. Parents don't need to teach math; they just need to support what their children's teacher is doing.

"Keeping positive, being encouraging and interested in their children's math learning goes a long way."

Steve Murphy et al, Parents' experiences of mathematics learning at home during the COVID-19 pandemic: a typology of parental engagement in mathematics education, Educational Studies in Mathematics (2023). DOI: 10.1007/s10649-023-10224-1

Provided by La Trobe University

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181 Mathematics Research Topics From PhD Experts

math research topics

If you are reading this blog post, it means you are looking for some exceptional math research topics. You want them to be original, unique even. If you manage to find topics like this, you can be sure your professor will give you a top grade (if you write a decent paper, that is). The good news is that you have arrived at just the right place – at the right time. We have just finished updating our list of topics, so you will find plenty of original ideas right on this page. All our topics are 100 percent free to use as you see fit. You can reword them and you don’t need to give us any credit.

And remember: if you need assistance from a professional, don’t hesitate to reach out to us. We are not just the best place for math research topics for high school students; we are also the number one choice for students looking for top-notch research paper writing services.

Our Newest Research Topics in Math

We know you probably want the best and most recent research topics in math. You want your paper to stand out from all the rest. After all, this is the best way to get some bonus points from your professor. On top of this, finding some great topics for your next paper makes it easier for you to write the essay. As long as you know at least something about the topic, you’ll find that writing a great paper or buy phd thesis isn’t as difficult as you previously thought.

So, without further ado, here are the 181 brand new topics for your next math research paper:

Cool Math Topics to Research

Are you looking for some cool math topics to research? We have a list of original topics for your right here. Pick the one you like and start writing now:

  • Roll two dice and calculate a probability
  • Discuss ancient Greek mathematics
  • Is math really important in school?
  • Discuss the binomial theorem
  • The math behind encryption
  • Game theory and its real-life applications
  • Analyze the Bernoulli scheme
  • What are holomorphic functions and how do they work?
  • Describe big numbers
  • Solving the Tower of Hanoi problem

Undergraduate Math Research Topics

If you are an undergraduate looking for some research topics for your next math paper, you will surely appreciate our list of interesting undergraduate math research topics:

  • Methods to count discrete objects
  • The origins of Greek symbols in mathematics
  • Methods to solve simultaneous equations
  • Real-world applications of the theorem of Pythagoras
  • Discuss the limits of diffusion
  • Use math to analyze the abortion data in the UK over the last 100 years
  • Discuss the Knot theory
  • Analyze predictive models (take meteorology as an example)
  • In-depth analysis of the Monte Carlo methods for inverse problems
  • Squares vs. rectangles (compare and contrast)

Number Theory Topics to Research

Interested in writing about number theory? It is not an easy subject to discuss, we know. However, we are sure you will appreciate these number theory topics:

  • Discuss the greatest common divisor
  • Explain the extended Euclidean algorithm
  • What are RSA numbers?
  • Discuss Bézout’s lemma
  • In-depth analysis of the square-free polynomial
  • Discuss the Stern-Brocot tree
  • Analyze Fermat’s little theorem
  • What is a discrete logarithm?
  • Gauss’s lemma in number theory
  • Analyze the Pentagonal number theorem

Math Research Topics for High School

High school students shouldn’t be too worried about their math papers because we have some unique, and quite interesting, math research topics for high school right here:

  • Discuss Brun’s constant
  • An in-depth look at the Brahmagupta–Fibonacci identity
  • What is derivative algebra?
  • Describe the Symmetric Boolean function
  • Discuss orders of approximation in limits
  • Solving Regiomontanus’ angle maximization problem
  • What is a Quadratic integral?
  • Define and describe complementary angles
  • Analyze the incircle and excircles of a triangle
  • Analyze the Bolyai–Gerwien theorem in geometry
  • Math in our everyday life

Complex Math Topics

If you want to give some complex math topics a try, we have the best examples below. Remember, these topics should only be attempted by students who are proficient in mathematics:

  • Mathematics and its appliance in Artificial Intelligence
  • Try to solve an unsolved problem in math
  • Discuss Kolmogorov’s zero-one law
  • What is a discrete random variable?
  • Analyze the Hewitt–Savage zero-one law
  • What is a transferable belief model?
  • Discuss 3 major mathematical theorems
  • Describe and analyze the Dempster-Shafer theory
  • An in-depth analysis of a continuous stochastic process
  • Identify and analyze Gauss-Markov processes

Easy Math Research Paper Topics

Perhaps you don’t want to spend too much time working on your next research paper. Who can blame you? Check out these easy math research paper topics:

  • Define the hyperbola
  • Do we need to use a calculator during math class?
  • The binomial theorem and its real-world applications
  • What is a parabola in geometry?
  • How do you calculate the slope of a curve?
  • Define the Jacobian matrix
  • Solving matrix problems effectively
  • Why do we need differential equations?
  • Should math be mandatory in all schools?
  • What is a Hessian matrix?

Logic Topics to Research

We have some interesting logical topics for research papers. These are perfect for students interested in writing about math logic. Pick one right now:

  • Discuss the reductio ad absurdum approach
  • Discuss Boolean algebra
  • What is consistency proof?
  • Analyze Trakhtenbrot’s theorem (the finite model theory)
  • Discuss the Gödel completeness theorem
  • An in-depth analysis of Morley’s categoricity theorem
  • How does the Back-and-forth method work?
  • Discuss the Ehrenfeucht–Fraïssé game technique
  • Discuss Aleph numbers (Aleph-null and Aleph-one)
  • Solving the Suslin problem

Algebra Topics for a Research Paper

Would you like to write about an algebra topic? No problem, our seasoned writers have compiled a list of the best algebra topics for a research paper:

  • Discuss the differential equation
  • Analyze the Jacobson density theorem
  • The 4 properties of a binary operation in algebra
  • Analyze the unary operator in depth
  • Analyze the Abel–Ruffini theorem
  • Epimorphisms vs. monomorphisms: compare and contrast
  • Discuss the Morita duality in algebraic structures
  • Idempotent vs. nilpotent in Ring theory
  • Discuss the Artin-Wedderburn theorem
  • What is a commutative ring in algebra?
  • Analyze and describe the Noetherian ring

Math Education Research Topics

There is nothing wrong with writing about math education, especially if your professor did not give you writing prompts. Here are some very nice math education research topics:

  • What are the goals a mathematics professor should have?
  • What is math anxiety in the classroom?
  • Teaching math in UK schools: the difficulties
  • Computer programming or math in high school?
  • Is math education in Europe at a high enough level?
  • Common Core Standards and their effects on math education
  • Culture and math education in Africa
  • What is dyscalculia and how does it manifest itself?
  • When was algebra first thought in schools?
  • Math education in the United States versus the United Kingdom

Computability Theory Topics to Research

Writing about computability theory can be a very interesting adventure. Give it a try! Here are some of our most interesting computability theory topics to research:

  • What is a multiplication table?
  • Analyze the Scholz conjecture
  • Explain exponentiating by squaring
  • Analyze the Myhill-Nerode theorem
  • What is a tree automaton?
  • Compare and contrast the Pushdown automaton and the Büchi automaton
  • Discuss the Markov algorithm
  • What is a Turing machine?
  • Analyze the post correspondence problem
  • Discuss the linear speedup theorem
  • Discuss the Boolean satisfiability problem

Interesting Math Research Topics

We know you want topics that are interesting and relatively easy to write about. This is why we have a separate list of our most interesting math research topics:

  • What is two-element Boolean algebra?
  • The life of Gauss
  • The life of Isaac Newton
  • What is an orthodiagonal quadrilateral?
  • Tessellation in Euclidean plane geometry
  • Describe a hyperboloid in 3D geometry
  • What is a sphericon?
  • Discuss the peculiarities of Borel’s paradox
  • Analyze the De Finetti theorem in statistics
  • What are Martingales?
  • The basics of stochastic calculus

Applied Math Research Topics

Interested in writing about applied mathematics? Our team managed to create a list of awesome applied math research topics from scratch for you:

  • Discuss Newton’s laws of motion
  • Analyze the perpendicular axes rule
  • How is a Galilean transformation done?
  • The conservation of energy and its applications
  • Discuss Liouville’s theorem in Hamiltonian mechanics
  • Analyze the quantum field theory
  • Discuss the main components of the Lorentz symmetry
  • An in-depth look at the uncertainty principle

Geometry Topics for a Research Paper

Geometry can be a very captivating subject, especially when you know plenty about it. Check out our list of geometry topics for a research paper and pick the best one today:

  • Most useful trigonometry functions in math
  • The life of Archimedes and his achievements
  • Trigonometry in computer graphics
  • Using Vincenty’s formulae in geodesy
  • Define and describe the Heronian tetrahedron
  • The math behind the parabolic microphone
  • Discuss the Japanese theorem for concyclic polygons
  • Analyze Euler’s theorem in geometry

Math Research Topics for Middle School

Yes, even middle school children can write about mathematics. We have some original math research topics for middle school right here:

  • Finding critical points in a graph
  • The basics of calculus
  • What makes a graph ultrahomogeneous?
  • How do you calculate the area of different shapes?
  • What contributions did Euclid have to the field of mathematics?
  • What is Diophantine geometry?
  • What makes a graph regular?
  • Analyze a full binary tree

Math Research Topics for College Students

As you’ve probably already figured out, college students should pick topics that are a bit more complex. We have some of the best math research topics for college students right here:

  • What are extremal problems and how do you solve them?
  • Discuss an unsolvable math problem
  • How can supercomputers solve complex mathematical problems?
  • An in-depth analysis of fractals
  • Discuss the Boruvka’s algorithm (related to the minimum spanning tree)
  • Discuss the Lorentz–FitzGerald contraction hypothesis in relativity
  • An in-depth look at Einstein’s field equation
  • The math behind computer vision and object recognition

Calculus Topics for a Research Paper

Let’s face it: calculus is not a very difficult field. So, why don’t you pick one of our excellent calculus topics for a research paper and start writing your essay right away:

  • When do we need to apply the L’Hôpital rule?
  • Discuss the Leibniz integral rule
  • Calculus in ancient Egypt
  • Discuss and analyze linear approximations
  • The applications of calculus in real life
  • The many uses of Stokes’ theorem
  • Discuss the Borel regular measure
  • An in-depth analysis of Lebesgue’s monotone convergence theorem

Simple Math Research Paper Topics for High School

This is the place where you can find some pretty simple topics if you are a high school student. Check out our simple math research paper topics for high school:

  • The life and work of the famous Pierre de Fermat
  • What are limits and why are they useful in calculus?
  • Explain the concept of congruency
  • The life and work of the famous Jakob Bernoulli
  • Analyze the rhombicosidodecahedron and its applications
  • Calculus and the Egyptian pyramids
  • The life and work of the famous Jean d’Alembert
  • Discuss the hyperplane arrangement in combinatorial computational geometry
  • The smallest enclosing sphere method in combinatorics

Business Math Topics

If you want to surprise your professor, why don’t you write about business math? We have some exceptional topics that nobody has thought about right here:

  • Is paying a loan with another loan a good approach?
  • Discuss the major causes of a stock market crash
  • Best debt amortization methods in the US
  • How do bank loans work in the UK?
  • Calculating interest rates the easy way
  • Discuss the pros and cons of annuities
  • Basic business math skills everyone should possess
  • Business math in United States schools
  • Analyze the discount factor

Probability and Statistics Topics for Research

Probability and statistics are not easy fields. However, you can impress your professor with one of our unique probability and statistics topics for research:

  • What is the autoregressive conditional duration?
  • Applying the ANOVA method to ranks
  • Discuss the practical applications of the Bates distribution
  • Explain the principle of maximum entropy
  • Discuss Skorokhod’s representation theorem in random variables
  • What is the Factorial moment in the Theory of Probability?
  • Compare and contrast Cochran’s C test and his Q test
  • Analyze the De Moivre-Laplace theorem
  • What is a negative probability?

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260 Interesting Math Topics for Essays & Research Papers

Mathematics is the science of numbers and shapes. Writing about it can give you a fresh perspective and help to clarify difficult concepts. You can even use mathematical writing as a tool in problem-solving.

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In this article, you will find plenty of interesting math topics. Besides, you will learn about branches of mathematics that you can choose from. And if the thought of letters and numbers makes your head swim, try our custom writing service . Our professionals will craft a paper for you in no time!

And now, let’s proceed to math essay topics and tips.

🔝 Top 10 Interesting Math Topics

✅ branches of mathematics, ✨ fun math topics.

  • 🏫 Math Topics for High School
  • 🎓 College Math Topics
  • 🤔 Advanced Math
  • 📚 Math Research
  • ✏️ Math Education
  • 💵 Business Math

🔍 References

  • Number theory in everyday life.
  • Logicist definitions of mathematics.
  • Multivariable vs. vector calculus.
  • 4 conditions of functional analysis.
  • Random variable in probability theory.
  • How is math used in cryptography?
  • The purpose of homological algebra.
  • Concave vs. convex in geometry.
  • The philosophical problem of foundations.
  • Is numerical analysis useful for machine learning?

What exactly is mathematics ? First and foremost, it is very old. Ancient Greeks and Persians were already utilizing mathematical tools. Nowadays, we consider it an interdisciplinary language.

Biologists, linguists, and sociologists alike use math in their work. And not only that, we all deal with it in our daily lives. For instance, it manifests in the measurement of time. We often need it to calculate how much our groceries cost and how much paint we need to buy to cover a wall.

Albert Einstein quote.

Simply put, mathematics is a universal instrument for problem-solving. We can divide pure math into three branches: geometry, arithmetic, and algebra. Let’s take a closer look:

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  • Geometry By studying geometry, we try to comprehend our physical surroundings. Geometric shapes can be simple, like a triangle. Or, they can form complicated figures, like a rhombicosidodecahedron.
  • Arithmetic Arithmetic deals with numbers and simple operations: subtraction, addition, division, and multiplication.
  • Algebra Algebra is used when the exact numbers are unclear. Instead, they are replaced with letters. Businesses often need algebra to predict their sales.

It’s true that most high school students don’t like math. However, that doesn’t mean it can’t be a fun and compelling subject. In the following section, you will find plenty of enthralling mathematical topics for your paper.

If you’re struggling to start working on your essay, we have some fun and cool math topics to offer. They will definitely engage you and make the writing process enjoyable. Besides, fun math topics can show everyone that even math can be entertaining or even a bit silly.

  • The link between mathematics and art – analyzing the Golden Ratio in Renaissance-era paintings.
  • An evaluation of Georg Cantor’s set theory.
  • The best approaches to learning math facts and developing number sense.
  • Different approaches to probability as explored through analyzing card tricks.
  • Chess and checkers – the use of mathematics in recreational activities.
  • The five types of math used in computer science .
  • Real-life applications of the Pythagorean Theorem .
  • A study of the different theories of mathematical logic .
  • The use of game theory in social science.
  • Mathematical definitions of infinity and how to measure it.
  • What is the logic behind unsolvable math problems?
  • An explanation of mean, mode, and median using classroom math grades.
  • The properties and geometry of a Möbius strip.
  • Using truth tables to present the logical validity of a propositional expression.
  • The relationship between Pascal’s Triangle and The Binomial Theorem.
  • The use of different number types: the history.
  • The application of differential geometry in modern architecture.
  • A mathematical approach to the solution of a Rubik’s Cube.
  • Comparison of predictive and prescriptive statistical analyses.
  • Explaining the iterations of the Koch snowflake.
  • The importance of limits in calculus.
  • Hexagons as the most balanced shape in the universe.
  • The emergence of patterns in chaos theory.
  • What were Euclid’s contributions to the field of mathematics?
  • The difference between universal algebra and abstract algebra.

🏫 Math Essay Topics for High School

When writing a math paper, you want to demonstrate that you understand a concept. It can be helpful if you need to prepare for an exam. Choose a topic from this section and decide what you want to discuss.

  • Explain what we need Pythagoras’ theorem for. 
  • What is a hyperbola? 
  • Describe the difference between algebra and arithmetic. 
  • When is it unnecessary to use a calculator ? 
  • Find a connection between math and the arts. 
  • How do you solve a linear equation? 
  • Discuss how to determine the probability of rolling two dice. 
  • Is there a link between philosophy and math? 
  • What types of math do you use in your everyday life? 
  • What is the numerical data? 
  • Explain how to use the binomial theorem. 
  • What is the distributive property of multiplication? 
  • Discuss the major concepts in ancient Egyptian mathematics . 
  • Why do so many students dislike math? 
  • Should math be required in school? 
  • How do you do an equivalent transformation? 
  • Why do we need imaginary numbers? 
  • How can you calculate the slope of a curve? 
  • What is the difference between sine, cosine, and tangent? 
  • How do you define the cross product of two vectors? 
  • What do we use differential equations for? 
  • Investigate how to calculate the mean value. 
  • Define linear growth. 
  • Give examples of different number types. 
  • How can you solve a matrix? 

🎓 College Math Topics for a Paper

Sometimes you need more than just formulas to explain a complex idea. That’s why knowing how to express yourself is crucial. It is especially true for college-level mathematics. Consider the following ideas for your next research project:

  • What do we need n-dimensional spaces for?
  • Explain how card counting works.
  • Discuss the difference between a discrete and a continuous probability distribution .
  • How does encryption work?
  • Describe extremal problems in discrete geometry.
  • What can make a math problem unsolvable?
  • Examine the topology of a Möbius strip.

Three main types of geometry.

  • What is K-theory?  
  • Discuss the core problems of computational geometry. 
  • Explain the use of set theory . 
  • What do we need Boolean functions for? 
  • Describe the main topological concepts in modern mathematics. 
  • Investigate the properties of a rotation matrix. 
  • Analyze the practical applications of game theory.  
  • How can you solve a Rubik’s cube mathematically? 
  • Explain the math behind the Koch snowflake. 
  • Describe the paradox of Gabriel’s Horn. 
  • How do fractals form? 
  • Find a way to solve Sudoku using math. 
  • Why is the Riemann hypothesis still unsolved? 
  • Discuss the Millennium Prize Problems. 
  • How can you divide complex numbers? 
  • Analyze the degrees in polynomial functions. 
  • What are the most important concepts in number theory? 
  • Compare the different types of statistical methods . 

🤔 Advanced Topics in Math to Write a Paper on

Once you have passed the trials of basic math, you can move on to the advanced section. This area includes topology, combinatorics, logic, and computational mathematics. Check out the list below for enticing topics to write about:

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  • What is an abelian group?
  • Explain the orbit-stabilizer theorem.
  • Discuss what makes the Burnside problem influential.
  • What fundamental properties do holomorphic functions have?
  • How does Cauchy’s integral theorem lead to Cauchy’s integral formula?
  • How do the two Picard theorems relate to each other?
  • When is a trigonometric series called a Fourier series?
  • Give an example of an algorithm used for machine learning .
  • Compare the different types of knapsack problems.
  • What is the minimum overlap problem?
  • Describe the Bernoulli scheme.
  • Give a formal definition of the Chinese restaurant process.
  • Discuss the logistic map in relation to chaos .
  • What do we need the Feigenbaum constants for?
  • Define a difference equation.
  • Explain the uses of the Fibonacci sequence.
  • What is an oblivious transfer?
  • Compare the Riemann and the Ruelle zeta functions.
  • How can you use elementary embeddings in model theory?
  • Analyze the problem with the wholeness axiom and Kunen’s inconsistency theorem.
  • How is Lie algebra used in physics ?
  • Define various cases of algebraic cycles.
  • Why do we need étale cohomology groups to calculate algebraic curves?
  • What does non-Euclidean geometry consist of?
  • How can two lines be ultraparallel?

📚 Math Research Topics for a Paper

Choosing the right topic is crucial for a successful research paper in math. It should be hard enough to be compelling, but not exceeding your level of competence. If possible, stick to your area of knowledge. This way your task will become more manageable. Here are some ideas:

  • Write about the history of calculus.
  • Why are unsolved math problems significant?
  • Find reasons for the gender gap in math students.
  • What are the toughest mathematical questions asked today?
  • Examine the notion of operator spaces.
  • How can we design a train schedule for a whole country?
  • What makes a number big?

Mathematical writing should be well-structured, precise, and easy readable

  • How can infinities have various sizes?
  • What is the best mathematical strategy to win a game of Go?
  • Analyze natural occurrences of random walks in biology.
  • Explain what kind of mathematics was used in ancient Persia.
  • Discuss how the Iwasawa theory relates to modular forms.
  • What role do prime numbers play in encryption ?
  • How did the study of mathematics evolve?
  • Investigate the different Tower of Hanoi solutions.
  • Research Napier’s bones. How can you use them?
  • What is the best mathematical way to find someone who is lost in a maze?
  • Examine the Traveling Salesman Problem. Can you find a new strategy?
  • Describe how barcodes function.
  • Study some real-life examples of chaos theory. How do you define them mathematically?
  • Compare the impact of various ground-breaking mathematical equations .
  • Research the Seven Bridges of Königsberg. Relate the problem to the city of your choice.
  • Discuss Fisher’s fundamental theorem of natural selection.
  • How does quantum computing work?
  • Pick an unsolved math problem and say what makes it so difficult.

✏️ Math Education Research Topics

For many teachers, the hardest part is to keep the students interested. When it comes to math, it can be especially challenging. It’s crucial to make complicated concepts easy to understand. That’s why we need research on math education.

  • Compare traditional methods of teaching math with unconventional ones.
  • How can you improve mathematical education in the U.S.?
  • Describe ways of encouraging girls to pursue careers in STEM fields.
  • Should computer programming be taught in high school?
  • Define the goals of mathematics education .
  • Research how to make math more accessible to students with learning disabilities .
  • At what age should children begin to practice simple equations?
  • Investigate the effectiveness of gamification in algebra classes.
  • What do students gain from taking part in mathematics competitions?
  • What are the benefits of moving away from standardized testing ?
  • Describe the causes of “ math anxiety .” How can you overcome it?
  • Explain the social and political relevance of mathematics education.
  • Define the most significant issues in public school math teaching.
  • What is the best way to get children interested in geometry?
  • How can students hone their mathematical thinking outside the classroom?
  • Discuss the benefits of using technology in math class.
  • In what way does culture influence your mathematical education?
  • Explore the history of teaching algebra .
  • Compare math education in various countries.

E. T. Bell quote.

  • How does dyscalculia affect a student’s daily life?
  • Into which school subjects can math be integrated?
  • Has a mathematics degree increased in value over the last few years?
  • What are the disadvantages of the Common Core Standards ?
  • What are the advantages of following an integrated curriculum in math?
  • Discuss the benefits of Mathcamp.

🧮 Algebra Topics for a Paper

The elegance of algebra stems from its simplicity. It gives us the ability to express complex problems in short equations. The world was changed forever when Einstein wrote down the simple formula E=mc². Now, if your algebra seminar requires you to write a paper, look no further! Here are some brilliant prompts:

  • Give an example of an induction proof.
  • What are F-algebras used for?
  • What are number problems?
  • Show the importance of abstract algebraic thinking .
  • Investigate the peculiarities of Fermat’s last theorem.
  • What are the essentials of Boolean algebra?
  • Explore the relationship between algebra and geometry.
  • Compare the differences between commutative and noncommutative algebra.
  • Why is Brun’s constant relevant?
  • How do you factor quadratics?
  • Explain Descartes’ Rule of Signs.
  • What is the quadratic formula?
  • Compare the four types of sequences and define them.
  • Explain how partial fractions work.
  • What are logarithms used for?
  • Describe the Gaussian elimination.
  • What does Cramer’s rule state?
  • Explore the difference between eigenvectors and eigenvalues.
  • Analyze the Gram-Schmidt process in two dimensions.
  • Explain what is meant by “range” and “domain” in algebra.
  • What can you do with determinants?
  • Learn about the origin of the distance formula.
  • Find the best way to solve math word problems.
  • Compare the relationships between different systems of equations.
  • Explore how the Rubik’s cube relates to group theory .

📏 Geometry Topics for a Research Paper

Shapes and space are the two staples of geometry. Since its appearance in ancient times, it has evolved into a major field of study. Geometry’s most recent addition, topology, explores what happens to an object if you stretch, shrink, and fold it. Things can get pretty crazy from here! The following list contains 25 interesting geometry topics:

  • What are the Archimedean solids? 
  • Find real-life uses for a rhombicosidodecahedron. 
  • What is studied in projective geometry? 
  • Compare the most common types of transformations. 
  • Explain how acute square triangulation works. 
  • Discuss the Borromean ring configuration. 
  • Investigate the solutions to Buffon’s needle problem. 
  • What is unique about right triangles? 

The role of study of non-Euclidean geometry

  • Describe the notion of Dirac manifolds.
  • Compare the various relationships between lines.
  • What is the Klein bottle?
  • How does geometry translate into other disciplines, such as chemistry and physics?
  • Explore Riemannian manifolds in Euclidean space.
  • How can you prove the angle bisector theorem?
  • Do a research on M.C. Escher’s use of geometry.
  • Find applications for the golden ratio .
  • Describe the importance of circles.
  • Investigate what the ancient Greeks knew about geometry.
  • What does congruency mean?
  • Study the uses of Euler’s formula.
  • How do CT scans relate to geometry?
  • Why do we need n-dimensional vectors?
  • How can you solve Heesch’s problem?
  • What are hypercubes?
  • Analyze the use of geometry in Picasso’s paintings.

➗ Calculus Topics to Write a Paper on

You can describe calculus as a more complicated algebra. It’s a study of change over time that provides useful insights into everyday problems. Applied calculus is required in a variety of fields such as sociology, engineering, or business. Consult this list of compelling topics on a calculus paper:

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  • What are the differences between trigonometry, algebra, and calculus?
  • Explain the concept of limits.
  • Describe the standard formulas needed for derivatives.
  • How can you find critical points in a graph?
  • Evaluate the application of L’Hôpital’s rule.
  • How do you define the area between curves?
  • What is the foundation of calculus?

Calculus was developed by Isaac Newton and Gottfried Leibnitz.

  • How does multivariate calculus work?
  • Discuss the use of Stokes’ theorem.
  • What does Leibniz’s integral rule state?
  • What is the Itô stochastic integral?
  • Explore the influence of nonstandard analysis on probability theory.
  • Research the origins of calculus.
  • Who was Maria Gaetana Agnesi?
  • Define a continuous function.
  • What is the fundamental theorem of calculus?
  • How do you calculate the Taylor series of a function?
  • Discuss the ways to resolve Runge’s phenomenon.
  • Explain the extreme value theorem.
  • What do we need predicate calculus for?
  • What are linear approximations?
  • When does an integral become improper?
  • Describe the Ratio and Root Tests.
  • How does the method of rings work?
  • Where do we apply calculus in real-life situations?

💵 Business Math Topics to Write About

You don’t have to own a company to appreciate business math. Its topics range from credits and loans to insurance, taxes, and investment. Even if you’re not a mathematician, you can use it to handle your finances. Sounds interesting? Then have a look at the following list:

  • What are the essential skills needed for business math?
  • How do you calculate interest rates?
  • Compare business and consumer math.
  • What is a discount factor?
  • How do you know that an investment is reasonable?
  • When does it make sense to pay a loan with another loan?
  • Find useful financing techniques that everyone can use.
  • How does critical path analysis work?
  • Explain how loans work.
  • Which areas of work utilize operations research?
  • How do businesses use statistics?
  • What is the economic lot scheduling problem?
  • Compare the uses of different chart types.
  • What causes a stock market crash?
  • How can you calculate the net present value?
  • Explore the history of revenue management .
  • When do you use multi-period models?
  • Explain the consequences of depreciation.
  • Are annuities a good investment?
  • Would the U.S. financially benefit from discontinuing the penny?
  • What caused the United States housing crash in 2008?
  • How do you calculate sales tax?
  • Describe the notions of markups and markdowns.
  • Investigate the math behind debt amortization.
  • What is the difference between a loan and a mortgage?

With all these ideas, you are perfectly equipped for your next math paper. Good luck!

  • What Is Calculus?: Southern State Community College
  • What Is Mathematics?: Tennessee Tech University
  • What Is Geometry?: University of Waterloo
  • What Is Algebra?: BBC
  • Ten Simple Rules for Mathematical Writing: Ohio State University
  • Practical Algebra Lessons: Purplemath
  • Topics in Geometry: Massachusetts Institute of Technology
  • The Geometry Junkyard: All Topics: Donald Bren School of Information and Computer Sciences
  • Calculus I: Lamar University
  • Business Math for Financial Management: The Balance Small Business
  • What Is Mathematics: Life Science
  • What Is Mathematics Education?: University of California, Berkeley
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Research shows the best ways to learn math.

New Stanford paper says speed drills and timed testing in math can be damaging for students. (Cherries/Shutterstock)

Students learn math best when they approach the subject as something they enjoy. Speed pressure, timed testing and blind memorization pose high hurdles in the pursuit of math, according to Jo Boaler, professor of mathematics education  at Stanford Graduate School of Education and lead author on a new working paper called "Fluency Without Fear."

"There is a common and damaging misconception in mathematics – the idea that strong math students are fast math students," said Boaler, also cofounder of YouCubed at Stanford, which aims to inspire and empower math educators by making accessible in the most practical way the latest research on math learning.

Fortunately, said Boaler , the new national curriculum standards known as the Common Core Standards for K-12 schools de-emphasize the rote memorization of math facts. Maths facts are fundamental assumptions about math, such as the times tables (2 x 2 = 4), for example. Still, the expectation of rote memorization continues in classrooms and households across the United States.

While research shows that knowledge of math facts is important, Boaler said the best way for students to know math facts is by using them regularly and developing understanding of numerical relations. Memorization, speed and test pressure can be damaging, she added.

Number sense is critical

On the other hand, people with "number sense" are those who can use numbers flexibly, she said. For example, when asked to solve the problem of 7 x 8, someone with number sense may have memorized 56, but they would also be able to use a strategy such as working out 10 x 7 and subtracting two 7s (70-14).

"They would not have to rely on a distant memory," Boaler wrote in the paper.

In fact, in one research project the investigators found that the high-achieving students actually used number sense, rather than rote memory, and the low-achieving students did not.

The conclusion was that the low achievers are often low achievers not because they know less but because they don't use numbers flexibly.

"They have been set on the wrong path, often from an early age, of trying to memorize methods instead of interacting with numbers flexibly," she wrote. Number sense is the foundation for all higher-level mathematics, she noted.

Role of the brain

Boaler said that some students will be slower when memorizing, but still possess exceptional mathematics potential.

"Math facts are a very small part of mathematics, but unfortunately students who don't memorize math facts well often come to believe that they can never be successful with math and turn away from the subject," she said.

Prior research found that students who memorized more easily were not higher achieving – in fact, they did not have what the researchers described as more "math ability" or higher IQ scores. Using an MRI scanner, the only brain differences the researchers found were in a brain region called the hippocampus, which is the area in the brain responsible for memorizing facts – the working memory section.

But according to Boaler, when students are stressed – such as when they are solving math questions under time pressure – the working memory becomes blocked and the students cannot as easily recall the math facts they had previously studied. This particularly occurs among higher achieving students and female students, she said.

Some estimates suggest that at least a third of students experience extreme stress or "math anxiety" when they take a timed test, no matter their level of achievement. "When we put students through this anxiety-provoking experience, we lose students from mathematics," she said.

Math treated differently

Boaler contrasts the common approach to teaching math with that of teaching English. In English, a student reads and understands novels or poetry, without needing to memorize the meanings of words through testing. They learn words by using them in many different situations – talking, reading and writing.

"No English student would say or think that learning about English is about the fast memorization and fast recall of words," she added.

Strategies, activities

In the paper, coauthored by Cathy Williams, cofounder of YouCubed, and Amanda Confer, a Stanford graduate student in education, the scholars provide activities for teachers and parents that help students learn math facts at the same time as developing number sense. These include number talks, addition and multiplication activities, and math cards.

Importantly, Boaler said, these activities include a focus on the visual representation of number facts. When students connect visual and symbolic representations of numbers, they are using different pathways in the brain, which deepens their learning, as shown by recent brain research.

"Math fluency" is often misinterpreted, with an over-emphasis on speed and memorization, she said. "I work with a lot of mathematicians, and one thing I notice about them is that they are not particularly fast with numbers; in fact some of them are rather slow. This is not a bad thing; they are slow because they think deeply and carefully about mathematics."

She quotes the famous French mathematician, Laurent Schwartz. He wrote in his autobiography that he often felt stupid in school, as he was one of the slowest math thinkers in class.

Math anxiety and fear play a big role in students dropping out of mathematics, said Boaler.

"When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics," she said. "We have the research knowledge we need to change this and to enable all children to be powerful mathematics learners. Now is the time to use it."

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166 Extraordinary Math Research Topics For Your Papers

math research topics

Math research topics cover various genres from which students can choose. Many people think that a research project on a math topic is dull. However, mathematics can be a wonderful and vivid field. Since it’s a universal language, mathematics can describe anything and everything, from galaxies that orbit each other to music. However, the broad nature of this study field also makes selecting a research paper difficult. That’s because learners want to pick interesting topics that will impress educators to award them top scores. This article lists the best math research paper topics. It’s useful because it inspires students to select or customize topics for their academic essays without much struggle.

What Are The Different Types Of Math?

As hinted, math covers several genres. Here are the primary types of mathematics:

Geometry: It’s a math branch that deals with the shapes, size, and relative position of figures. Many people consider geometry a practical math branch because it examines figures, shapes, sizes, and features of various entities, including parts like solids, lines, surfaces, lines, and angles. Algebra: It assists in solving equations and manipulating symbols. This branch helps students represent unknown quantities with alphabets and use them alongside numbers. Calculus: This area is vital in determining rates of change, such as velocity and acceleration. Arithmetic: Arithmetic is the most common and oldest math branch, encompassing basis number operations. These operations include subtraction, addition, divisions, and multiplications, and some schools shorten it as BODMAS. Statistics and Probability: They help analyze numerical data to make predictions. Probability is about chances, while statistics entails handling different data using various techniques. Trigonometry: It assists in calculating angles and distances between points. It mainly deals with triangles’ relationships, sides, and curves.

Now that you understand the types of mathematics, it’s easier to select a suitable research topic. The following are some of the best topic ideas in math. 

 Undergraduate Math Research Topics

Maybe you’re pursuing your undergraduate studies. However, you have challenges comprehending math topics, yet the professor expects you to write a superior paper. In that case, here’s a list of engaging research topics in math to consider for your essays.

  • An in-depth comprehension of the meaning of discrete random variables in math and their identification
  • Math evolution- Comprehending the Gauss-Markov
  • Primary math theorems- Investigating how they work
  • Continuous stochastic process- Exploring its role in the math process
  • Analyzing the Dempster-Shafer theory
  • The application of the transferable belief model
  • Exploring the use of math in artificial intelligence
  • The application of mathematics in daily life
  • Algebra and its history
  • Math and culture- What’s the relationship?
  • How drawing and painting could help with mathematics
  • Ways to boost math interest among learners
  • The social and political significance of learning mathematics
  • Circles and their relevance in mathematics
  • Challenges to math learning in public schools
  • Prove the use of F-Algebras
  • Understanding the meaning of abstract algebra
  • Discuss geometry and algebra
  • How acute square triangulation works
  • Discuss the essence of right triangles
  • Why non-Euclidean geometry should be compulsory for math students
  • Investigating number problems
  • Discuss the meaning of Dirac manifolds
  • How geometry influences chemistry and physics
  • Riemannian manifolds’ application in the Euclidean space

These are exciting math topics for undergraduate students. Nevertheless, prepare adequate time and resources to investigate any of these titles to draft a winning essay. You might have to provide theoretical and practical assessments when writing your essay.

Math Research Topics for High School Learners

Maybe your high school teacher asked you to write a research paper. Choosing a familiar topic is an excellent way to get a high grade. Here are some of the best math research paper topics for high school.

  • How to draw a chart representing the financial analysis of a prominent company over the last five years
  • How to solve a matrix- The vital principles and formulas to embrace
  • Exploring various techniques for solving finance and mathematical gaps
  • Discount factor- Why it’s crucial for learners and ways to achieve it
  • Calculating the interest rate and its essence in the banking industry
  • Why imaginary numbers are important
  • Investigating the application of math in the workplace
  • Explain why learners hate mathematics teachers
  • What makes math a complex subject?
  • Is making math compulsory in high school a good thing?
  • How to solve a dice question from a probability perspective
  • Understanding the Binomial theorem and its essence
  • Investigating Egyptian mathematics
  • Hyperbola- Understanding it and its use in math
  • When should students use calculators in class?
  • How to solve linear equations
  • Is the Pythagoras theorem important in math?
  • The interdependence between math and art
  • Philosophy’s role in math
  • Numerical data overview

High school learners can pick any of these titles and develop them into an essay. Nevertheless, they should prepare to spend some time investigating their topics to write pieces that will impress their educators. Titles that address math history and its influence on education can also suit high school students. However, learners should select titles that fulfil the academic requirements set by the educators.

Applied Math Research Topics

As a branch, applied math deals with mathematical methods and their real-life applications. These methods are manifest in engineering, finance, medicine, biology, physics, and others. Here are some of the exciting topics in this field.

  • Dimensions for examining fingerprints
  • Computer tomography and its significance
  • Step-stress modelling- What is its importance?
  • Explain the essence of data mining- How does it benefit the banking sector?
  • A detailed examination of nonlinear models
  • How genes discovery helps determine unhealthy and healthy patients
  • Algorithms and their role in probabilistic modelling
  • Mathematicians and their importance in robots’ development
  • Mathematicians’ role in crime prevention and data analysis
  • The essence of Law of Motion by Isaac in real life
  • The importance of math in energy conservation
  • Math and its role in quantum theory
  • Analyzing the Lorentz symmetry features
  • Evaluating the processing of the statistical signal in detail
  • Explain the achievement of Galilean Transformation

These are exciting ideas to explore when writing a research paper in applied math. Nevertheless, take your time to carefully and extensively research your preferred title to write a high-quality essay. Students should also note that some topics in this category require specialized knowledge to write superior papers.

It’s a challenge to write a paper for a high grade. Sometimes every student need a professional help with college paper writing. Therefore, don’t be afraid to hire a writer to complete your assignment. Just write a message “Please, write custom research paper for me” and get time to relax. Contact us today and get a 100% original paper. 

Interesting Math Research Topics

Maybe you’re among the learners that prefer working with exciting ideas. In that case, this category has topics that will interest you.

  • The uses of numerical analysis in machine learning
  • Foundations and philosophical problems
  • Convex versus Concave in geometry
  • Homological algebra- What is its purpose?
  • Is math useful in cryptography
  • Probability theory and random variable
  • Functional analysis- What are its four conditions?
  • Vector calculus versus multivariable
  • Mathematics and logicist definitions
  • Ways to apply the number theory in daily life
  • Studying complex math equations
  • How to calculate mode, median, and mean
  • Understanding the meaning of the Scholz conjecture
  • The definition of the past correspondence problem
  • Computational maths- What are its classes?
  • Multiplication table and its importance
  • What the Boolean satisfiability problem means for a learner
  • Understanding the linear speedup theory in mathematics
  • The Turing machine description
  • Understanding the Markov algorithm
  • Investigating the similarities and differences between Buchi automation and Pushdown automation
  • What is the meaning of Tree automation?
  • Describing the enclosing sphere method and its use in combinations
  • Egyptian pyramids and calculus
  • Analyzing De Finetti theorem in statistics and probability
  • Examining the congruence meaning in math
  • Application and purpose of calculus in the banking industry
  • Jean d’Alembert’s most famous works
  • Boolean algebra- What are its essential elements
  • Isaac Newton- His contribution, life, and time in math
  • Understanding the meaning of Sphericon
  • What is the purpose of Martingales?
  • Gauss times, energy, and contributions to math
  • Jakob Bernoulli- Exploring his famous works
  • A brief history of math

Some learners think writing a math essay is complex and tedious. However, you can find a topic you will enjoy working with throughout the project. These are exciting ideas to explore in research papers. However, prepare to spend sufficient time investigating your chosen title to write a winning paper, although these are generally relaxing titles for math papers and essays.

Math Research Topics for Middle School

Some middle school students worry about the math topics for their research. However, they can choose unique titles that will impress their teachers. Here are some of these ideas.

  • The impacts of standard exam curriculum on math education
  • Why is learning math so tricky?
  • What is the meaning of the commutative ring in algebra?
  • The Artin-Wedderburn theorem and its meaning
  • How monopolists and epimorphisms differ
  • Understanding the Jacobson density theorem
  • How linear approximations work
  • Root and ratio test definition
  • Statistics role in business
  • Economic lot scheduling- What does it mean?
  • Causes of the stock market crash
  • How many traders contribute to the New York Stock Exchange
  • The history of revenue management
  • Financial signs of an excellent investment
  • Depreciation and its odds
  • How a poor currency can benefit a country
  • How math helps with debt amortization
  • Ways to calculate a person’s net worth
  • Distinctions in algebra, trigonometry, and calculus
  • Discussing the beginning of calculus
  • The essence of stochastic in math
  • The meaning of limits in math
  • Ways to identify a critical point in a graph
  • Nonstandard analysis- What does it mean in the probability theory?
  • Continuous function description and meaning
  • Calculus- What are its primary principles?
  • Pythagoras theorem- What are its central tenets?
  • Calculus applications in finance
  • Theorem value in math
  • The application of linear approximations

This list has some of the best titles for middle school learners. But they also require some research to write superior essays. However, finding information on such topics is relatively easy, making them suitable for middle school students.

Math Research Topics for College Students

Maybe you’re pursuing college studies and need a title for a math research paper. In that case, here are exciting titles to consider for your essay.

  • What is the purpose of n-dimensional spaces?
  • Card counting- How does it work?
  • How continuous probability and discrete distribution differ
  • Understanding encryption- How Does it work?
  • Extremal problems- Investigating them in discrete geometry
  • The Mobius strip- Examining the topology
  • Why can a math problem be unsolvable?
  • Comparing different statistical methods
  • Explain the vital number theory concepts
  • Analyzing the polynomial functions’ degrees
  • Ways to divide complex numbers
  • Describe the prize problems with the millennium
  • The reasons for the unsolved Riemann hypothesis
  • Methods of solving Sudoku with math
  • Explain the fractals formation
  • Describe the evolution of math
  • Explore different types of Tower of Hanoi solutions
  • Discuss the uses of Napier’s bones
  • With examples, explain the chaos theory
  • Why are mathematical equations important all the time?
  • Fisher’s fundamental theorem and natural selection- Why are they important?

College professors expect students to draft papers with relevant and valuable information. These are relevant titles for college students. However, they require extensive research to write winning papers.

Cool Math Topics to Research

Maybe you don’t need a complex topic for your research paper. In that case, consider any of these ideas for your essay. If you have a problem writing even with these topics and you’re thinking: “solve my math for me,” you can always reach out to our service.

  • How contemporary architectural designs use geometry
  • What makes some math equations complex?
  • Ways to solve the Rubik’s cube
  • Discuss the meaning of prescriptive statistical and predictive analysis
  • Understanding the purpose of the chaos theory
  • What limits calculus?- Provide relevant examples
  • A comparison of universal and abstract algebra- How do they differ?
  • The relationship between probability and card tricks
  • Pascal’s Triangle- What does it mean?
  • Mobius strip- What are its features in geometry?
  • Multiple probability ideas- A brief overview
  • Discuss the meaning of the Golden Ration in Renaissance period paintings
  • How checkers and chess matter in understanding mathematics
  • Ways to measure infinity
  • Evaluating the Georg Contor theory
  • Are hexagons the most balanced shapes in the world?
  • The Koch snowflake- Explain the iterations
  • The history of various number types and their use
  • Game theory use in social science
  • Five math types with significant benefits in computer science

These are some of the most excellent math education research topics. However, they also require extensive research to write high-quality papers.

Enlist the Best College Research Paper Writing Service

Perhaps, you have a topic for your paper but not the time to write a winning piece. Maybe you’re not confident in your research, analytical, and writing skills. Thus, you’re unsure that you can write an essay that will compel your educator to award you the highest grade in your class. Well, you’re not the only one. Many students seek cheap research papers due to varied reasons. Whether it’s limited time and resources or a lack of the necessary skills and experience in academic paper writing, our crew can help you. We offer affordable college paper writing services and help in various math branches. Our experts can assist you if you need help with math research topics for high school students, college, or undergraduates. We are a professional team with a reputation for providing the best-rated academic writing assistance. Whether in university, college, or high school, our crew will offer the service you need to excel academically. Contact us now for cheap and reliable help with your academic essays.

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210 Brilliant Math Research Topics and Ideas for Students

Table of Contents

Do you have to submit a math research paper? Are you looking for the best math research topics? Well, in this blog post, we have shared a list of 150+ interesting math research topics to consider for assignments and academic projects. If you are a student who is pursuing a degree in mathematics, then you can very well use the topic ideas suggested here. Also, you can check this blog post and get to know the important steps for writing a brilliant math research paper.

Math Research Topics

What is Mathematics?

Mathematics is a broad academic discipline that focuses on numbers, structures, spaces, and shapes. This subject contains many analysis and calculation methods. Especially in the real world, math is considered an effective problem-solving tool. By using math, you can find solutions for both simple and complex problems.

Basically, mathematics is an integrated language that is widely used in several fields such as engineering, physics, medicine, finance, computer, business, and biology. Apart from the complex scientific fields, even math plays a vital role in the basic cost and time calculation in our everyday life.

Different Branches of Mathematics

Listed below are some popular branches of mathematics.

Arithmetic: It is a basic branch of math that focuses on numbers and their associated operations such as addition, subtraction, multiplication, and division.

Algebra: When the numbers are unknown, algebra steps in. Generally, along with numbers, algebra uses the letters such as A, B, X, and Y to represent unknown quantities. Mainly, businesses depend on algebra concepts to predict their sales.

Geometry: It is a popular branch of mathematics that deals with shapes, sizes, and figures. The concept commonly revolves around lines, points, solids, angles, and surfaces.

Apart from all these common branches, mathematics also includes more advanced types such as calculus, trigonometry, statistics, topology, probability, etc.

How to Write a Math Research Paper?

In general, a math research paper is an academic paper that is prepared to explain a mathematical concept with proper results. For writing a math research paper, first, you must have a good research topic from any branch of mathematics. As math is a vast discipline, you can easily search and find plenty of research topics from it. But when you have many topics, then it will be more tedious to identify one perfect topic out of them all.

Right now, are you searching for a perfect math research topic? Well, then this is what you should do during the topic selection process to spot the right topic.

Topic Selection

Whenever you are asked to come up with a research paper topic on your own, initially, restrict yourself to the research area that you have strong knowledge of and are passionate about. Next, in that research area, explore and identify one great topic that has a broad scope to evaluate and express your ideas.

Remember, the topic you select should be comfortable for you to perform research and write about. Never pick a topic with less or no research scope. The topic should support the research method of your choice. Most importantly, give preference to the topic that has wide research information, references, and evidence. Also, before finalizing the topic, check whether your topic satisfies your instructor’s guidelines.

Research Paper Writing

After you have found a good math research topic, you can proceed to write the research paper. The research paper you write should follow a proper format and structure. So, in the math research paper, make sure to include the following essential sections.

Introduction

Implications.

In the introduction section, you should first give brief background information about your topic to familiarize your readers. Here, mainly you should explain the primary concepts along with the history of its terms. Also, you should state the basic research problem and discuss the symbols and principles that you are going to use in the essay.

The body of your research paper should elaborate on all your findings. Particularly, in the body paragraphs, you should talk about the formulas, theories, and mathematical analysis methods you have used to find solutions for the research problem.

The implication is the last or closing part of your research paper. Here, you should share your research insights with the readers. Also, you should include a brief summary of all the important points that you have discussed in the entire essay.

List of the Best Math Research Topics

Are you struggling to come up with a good math research paper topic for your assignment? No worries! Here we have shared a list of top-rated math research topic ideas on various branches of mathematics.

Math Research Topics

Explore them all and find a topic that suits you perfectly.

Simple and Easy Math Topics

  • Explain the working of Partial fractions.
  • Discuss the application of Mathematics in daily life.
  • What is the basis of Cramer’s rule?
  • How to solve Heesch’s problem?
  • Explain the history of calculus .
  • What is Euler’s formula?
  • Explain the working of Logarithms.
  • What are the different types of sequences?
  • Explain the different types of Transformations.
  • Define Brun’s constant.
  • What are the methods of factoring quadratics?
  • Examine Archimedean solids.
  • Explain Gaussian elimination.
  • Write about encryption and prime numbers.
  • How does Hypercube work?
  • Analyze Pygaoethores Theorem
  • Describe the logicist definitions of mathematics
  • Describe the purpose of homological algebra
  • Compare and contrast Concave and Convex in geometry
  • The study and contributions of Blaise Pascal to Probability
  • Explain the Fibonacci series briefly
  • How the Ancient Greek architecture influenced by mathematics?
  • Discuss the ancient Egyptian mathematical applications and accomplishments
  • Discuss the easiest ways to memorize algebraic expressions
  • Algebra is an exposition on the invariants of matrices – Explain

Basic Math Topics for Middle School Students

  • Define the Artin-Wedderburn theorem.
  • How to calculate net worth?
  • How to identify critical points in graphs?
  • What is the role of statistics in business?
  • Describe the principles of the Pythagoras theorem.
  • What are the applications of finance in math?
  • What do limits in math mean?
  • Explain the ratio and root test.
  • Define Jacobson’s density theorem.
  • What are the principles of calculus?

Interesting Math Topics for High School Students

  • What are the different number types? Explain with examples.
  • Explain the need for imaginary numbers.
  • How to calculate the interest rate?
  • How to solve a matrix?
  • How to prepare a chart of a company’s financial analysis?
  • When to use a calculator in class?
  • Explain the importance of the Binomial theorem.
  • Write about Egyptian mathematics.
  • Describe the applications of math in the workplace.
  • How to solve linear equations?
  • Describe the usage of hyperbola in math.
  • Why do so many students hate math?
  • What is the difference between algebra and arithmetic?
  • How to calculate the mean value?
  • What is the numerical data?

Math Research Paper Topics for Undergraduate Students

  • Explain the different theories of mathematical logic.
  • Discuss the origins of Greek symbols in mathematics.
  • Explain the significance of circles.
  • Analyze predictive models.
  • Explain the emergence of patterns in chaos theory.
  • Define abstract algebra.
  • What is a continuous stochastic process?
  • Write about the history of algebra.
  • Analyze Monte Carlo methods for inverse problems.
  • What are the goals of standardized testing?
  • Define the Pentagonal number theorem.
  • Discuss the Lorentz–FitzGerald contraction hypothesis in relativity.
  • How to solve simultaneous equations.
  • How do supercomputers solve complex mathematical problems?
  • What is a parabola in geometry?

Math Research Topics

Math Research Topics for College Students

  • Explain the Fibonacci sequence.
  • What are the core problems of computational geometry?
  • Discuss the practical applications of game theory.
  • What is the Traveling Salesman Problem?
  • Describe the Influence of math in biology.
  • Analyze the meaning of fractals.
  • Discuss the origin and evolution of mathematics.
  • What is quantum computing?
  • Explain Einstein’s field equation theory.
  • What is the influence of math on chemistry?
  • How to solve a Rubik’s cube mathematically?
  • How to do complex numbers division?
  • Explain the use of Boolean functions.
  • Analyze the degrees in polynomial functions.
  • How to solve Sudoku using mathematics?
  • Explain the use of set theory.
  • Explain the math behind the Koch snowflake.
  • Explore the varieties of the Tower of Hanoi solutions.
  • What is the difference between a discrete and a continuous probability distribution?
  • How does encryption work?

Applied Math Research Topics

  • What is the role of algorithms in probabilistic modeling?
  • Explain the significance of step-stress modeling.
  • Describe Newton’s laws of motion.
  • What dimensions are used to examine fingerprints?
  • Analyze statistical signal processing.
  • How to do Galilean transformation?
  • What is the role of mathematicians in crime data analysis and prevention?
  • Explain the uncertainty principle.
  • Discuss Liouville’s theorem in Hamiltonian mechanics.
  • Analyze the perpendicular axes rule.

Business Math Research Topics

  • What is the difference between a loan and a mortgage?
  • How to calculate sales tax?
  • Explore the math behind debt amortization.
  • How do businesses use statistics?
  • What is the economic lot scheduling problem?
  • Explain how loans work.
  • Discuss the significance of business math in real life.
  • Define discount factor.
  • What are the major causes of a stock market crash?
  • Compare the uses of different types of charts.
  • Describe the notions of markups and markdowns.
  • How does critical path analysis work?
  • What are the pros and cons of annuities?
  • When to use multi-period models?
  • Compare business and consumer math.

Advanced Math Research Paper Topics

  • What is an oblivious transfer?
  • Compare the Riemann and the Ruelle zeta functions.
  • What are the different types of knapsack problems?
  • Define an abelian group.
  • What are the algorithms used for machine learning?
  • Define various cases of algebraic cycles.
  • When a trigonometric series is called a Fourier series?
  • What is the minimum overlap problem?
  • What are the basic properties of holomorphic functions?
  • Describe the Bernoulli scheme.

Complex Math Research Topics

  • Write about Napier’s bones.
  • What makes a number big?
  • Examine the notion of operator spaces.
  • How do barcodes function?
  • Define Fisher’s fundamental theorem of natural selection.
  • What are the peculiarities of Borel’s paradox?
  • How to design a train schedule for a whole country?
  • Describe a hyperboloid in 3D geometry.
  • What is an orthodiagonal quadrilateral?
  • Explain how the Iwasawa theory relates to modular forms.

Math Research Ideas on Probability and Statistics

  • Roll two dice and calculate a probability.
  • Write about the Factorial moment in the Theory of Probability.
  • Explain the principle of maximum entropy.
  • Compare and contrast Cochran’s C test and his Q test.
  • Discuss Skorokhod’s representation theorem in random variables
  • How to apply the ANOVA method to rank.
  • Analyze the De Moivre-Laplace theorem.
  • What is the autoregressive conditional duration?
  • Explain a negative probability.
  • Discuss the practical applications of the Bates distribution.

Algebra Research Topics

  • Explain Descartes’ Rule of Signs.
  • How to factor quadratics?
  • What is the use of F-algebras?
  • Discuss the differential equation.
  • What is the difference between eigenvectors and eigenvalues?
  • What are the properties of a binary operation in algebra?
  • What is a commutative ring in algebra?
  • Discuss the origin of the distance formula.
  • Explain the quadratic formula.
  • Analyze the unary operator.
  • Define range and domain in algebra.
  • Describe the Noetherian ring.
  • Discuss the Morita duality in algebraic structures.
  • Define the Abel–Ruffini theorem.
  • What is the use of determinants?

Math Research Paper Topics on Geometry

  • Research the real-life uses of a rhombicosidodecahedron.
  • Find out the solutions to Buffon’s needle problem.
  • What is unique about right triangles?
  • What is the Klein bottle?
  • What are the Archimedean solids?
  • What does congruency mean?
  • Discuss the role of trigonometry in computer graphics.
  • What is the need for n-dimensional vectors?
  • Explain the Japanese theorem for concyclic polygons.
  • Prove the angle bisector theorem.
  • Identify the applications for the golden ratio.
  • Explain the Heronian tetrahedron.
  • Describe the notion of Dirac manifolds.
  • What is the use of geometry in Picasso’s paintings?
  • How do CT scans relate to geometry?

Calculus Research Topics

  • How to calculate the Taylor series of a function?
  • What is the role of calculus in real life?
  • Discuss the Leibniz integral rule
  • Discuss and analyze linear approximations.
  • What is the use of predicate calculus?
  • What is the foundation of calculus?
  • How to calculate the area between curves?
  • Describe the standard formulas needed for derivatives.
  • Explain the working of multivariate calculus.
  • Define the fundamental theorem of calculus.

Outstanding Math Research Topics

  • What is a sphericon?
  • What is the role of Mathematics in Artificial Intelligence?
  • Define De Finetti’s theorem in probability and statistics.
  • How to calculate the slope of a curve?
  • Discuss the Stern-Brocot tree.
  • Explain Pascal’s Triangle.
  • Analyze the Georg Cantor set theory.
  • How to measure infinity?
  • Explain the Scholz conjecture.
  • How is geometry used in contemporary architectural designs?
  • How to solve the Suslin problem?
  • What is a tree automaton?
  • Explain the working of the Back-and-forth method.
  • What is a Turing machine?
  • Discuss the linear speedup theorem.
  • Discuss the benefits of using truth tables to present the logical validity of a propositional expression
  • Critical analysis of the major concepts in ancient Egyptian mathematics
  • Discuss the similarities and differences between a continuous and a discrete probability distribution
  • Analysis of the problem with the wholeness axiom and Kunen’s inconsistency theorem
  • Develop a study focusing on the Seven Bridges of Königsberg and relate the problem to the city or state of your choice

Latest Math Research Topics

  • What does point zero reflect on a graph where the vertical and horizontal lines meet?
  • How to recognize adjacent angles easily without any trouble?
  • Compare the differential vs. analytic geometry by citing relevant examples.
  • Explain how to use a graphics system for solving various types of equations.
  • How to divide the feasible and non-feasible regions in linear programming?
  • What are confidence intervals and how it helps in statistical math?
  • How to differentiate the effect of a magnetic field on a given point of the circle by using appropriate differential formula?
  • What are the different types of identities that are used in trigonometric functions?
  • Why polynomials are difficult to solve as compared to monomials? Give examples.
  • Explain radical expressions and their significance with examples.

Final Words

We hope you have identified an ideal topic from the list of math research topics and ideas recommended above. If you haven’t found a unique research topic or need assistance to complete your math research paper, then contact us.

In our team, we have PhD-certified academic writers to offer you math assignment help online . Based on the specifications you send us, our math assignment help experts will guide you with academic paper topic selection, writing, and editing. Note that, the solutions that our math tutors provide would be accurate and simple to understand. Moreover, by utilizing the math research paper help service from our scholars, you can complete your tasks ahead of the deadline and get top scores.

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New tool will make math-heavy research papers easier to view online

T he complex formulas in physics, math and engineering papers might be intimidatingly difficult reading matter for some, but there are many people who have trouble merely seeing them in the first place. The National Institute of Standards and Technology (NIST) has created a tool that makes these papers easier on the eyes for those with visual disabilities, and it's about to be adopted in a major way.

The tool, which converts one commonly used format for displaying math formulas into another, could help make the latest and greatest research papers accessible to all. Most new research papers are distributed as PDF files, which many people in the research community have difficulty reading.

According to the World Health Organization, more than a quarter of the world's population has a diagnosed vision impairment, and Yale's Center for Dyslexia and Creativity reports that in the United States 20% of people have dyslexia. In a recent study of scientific papers distributed as PDFs, researchers found that only 2.4% of the documents they sampled satisfied their accessibility criteria.

"If you're not someone who has been struggling to publish math papers all your life, you might wonder why this is a problem," said NIST's Bruce Miller, a physicist by training who specializes in math software. "PDFs look great on the printed page. But if you want math formulas to be read out loud, or be legible on a different-sized screen, like a tablet or a phone, the mismatch can be painful. You can't easily repurpose PDFs for other media."

How are PDFs typically generated? A scientist creating a paper manuscript that uses many formulas will generally use the language LaTeX (pronounced "lay-tech") or one of its close relatives to render the formulas. LaTeX has been in use since the 1980s and is widely respected for the high-quality typesetting that it creates, but it is designed to produce printed pages in static form.

Since the 1990s, webpage creators have used HTML, which makes it possible to adjust the look, behavior and layout of the displayed text depending on its context. If you've ever dragged a webpage into a different size and watched its text smoothly reposition itself to fit within the new rectangle's boundaries, you are seeing a feature that readers with vision disabilities want.

Modern HTML includes extensions that not only permit this ability to "re-flow" type, but also allow the math formulas to be read aloud by machine for those who can't read the text themselves. These features make HTML ideal for creating accessible text, but for years there was no effective way to convert LaTeX into HTML. This presented a problem to Miller when he needed a way to bring the more than 1,000 pages of NIST's venerable Handbook of Mathematical Functions into the digital realm.

"At the time, some programs purported to convert LaTeX to webpages, but none worked well enough," he said. "I figured, let's try to make our own."

The resulting NIST tool was LaTeXML , which reads a LaTeX source file and builds a representation of the document that it can turn into HTML. LaTeXML was the key to creating the online Digital Library of Mathematical Functions, and several years later the managers of a major online resource realized it could help them too.

This resource is arXiv (pronounced "archive"), a repository of scholarly articles that have yet to be published in scientific journals. Maintained by Cornell University, arXiv currently hosts more than 2 million articles that are free to view and download as PDFs. The server has become a prominent way station, where authors can post findings and discuss them with their peers before formally announcing them.

"Per a survey arXiv conducted in 2022, only 30% of users who rely on assistive technology can access all of the research they need without help. The same survey found that PDF formatting is the biggest barrier," said Shamsi Brinn, lead researcher on arXiv 's accessibility report and manager of the HTML papers project.

That will change with arXiv 's use of the LaTeXML converter, Brinn said. The server will generate HTML versions of papers and include the HTML version next to the link to download a PDF.

The arXiv repository will convert papers on a rolling basis, offering the first in December 2023. The move follows a broader trend of requiring accessible web and electronic information, according to Joe Zesski, assistant director of the Northeast ADA Center. Not only will the change help the scientific community adhere to the White House's updated policy on making federally-funded research freely available, but it will also make information accessible to young scientists, who have grown up using electronic resources.

"There is a growing reliance on the web and electronic information in education alongside a growing expectation of equal access by and for young people with disabilities," Zesski said. "Taking steps to make the information those students will need to access accessible and usable to them is important."

Provided by National Institute of Standards and Technology

A schematic for creating the SciA11y HTML render from a paper PDF. Starting with the raw two-column PDF on the left, S2ORC [24] is used to extract the title, authors, abstract, section headers, body text, and references. S2ORC also identifies links between inline citations and references to figures and table objects. DeepFigures [43] is used to extract figures and tables, along with their captions. The output of these two models is merged with metadata from the Semantic Scholar API. Heuristics are used to construct a table of contents, insert figures and tables in the appropriate places in the text, and repair broken URLs. We add HTML headers as illustrated (header tags for sections, paragraph tags for body text, and figure tags for figures and tables); highlighted components (table of contents and links in references) are not in the PDF and novel navigational features that we introduce to the HTML render. An example HTML render of parts of a paper document is shown to the right (the actual render is a single column, which is split here for presentation). Credit: https://arxiv.org/pdf/2105.00076.pdf

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Article paid for by: Ocasio Media The news and editorial staffs of the Bay Area News Group had no role in this post’s preparation.

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ChatGPT vs. Microsoft Copilot vs. Gemini: Which is the best AI chatbot?

maria-diaz

Artificial intelligence (AI) has transformed how we work and play  in recent months, giving almost anyone the ability to write code , create art , and even make investments . 

Special Feature

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The Rise of Generative AI

A new wave of AI tools has taken the world by storm and given us a vision for a new way of working and finding the information that can streamline our work and our lives. We show you the ways tools like ChatGPT and other generational AI software are making impacts on the world, how to harness their power, as well as potential risks.

For professional and hobbyist users alike, generative AI tools, such as  ChatGPT , offer advanced capabilities to create decent-quality content from a simple prompt given by the user. 

Keeping up with all the latest AI tools can get confusing, especially as Microsoft added  GPT-4 to Bing  and renamed it to Copilot,  OpenAI added new capabilities to ChatGPT , and Bard got plugged into the Google ecosystem  and rebranded to Gemini.

Also: Microsoft Copilot Pro vs. OpenAI's ChatGPT Plus: Which is worth your $20 a month?

Knowing which of the three most popular AI chatbots is best to write code , generate text , or help build resumes is challenging, so we'll break down the biggest differences so you can choose one that fits your needs. 

Testing ChatGPT vs. Microsoft Copilot vs. Gemini

To help determine which AI chatbot gives more accurate answers, I'm going to use a simple prompt to compare the three: 

"I have 5 oranges today, I ate 3 oranges last week. How many oranges do I have left?"

The answer should be five, as the number of oranges I ate last week doesn't affect the number of oranges I have today, which is what we're asking the three bots. First up, ChatGPT.

You should use ChatGPT if...

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1. You want to try the most popular AI chatbot

ChatGPT was created by OpenAI and released for a widespread preview in November 2022. Since then, the AI chatbot quickly gained over 100 million users, with the website alone seeing 1.8 billion visitors a month. It's been at the center of controversies , especially as people uncover its potential to do schoolwork and replace some workers.

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The free version of ChatGPT, which runs on the default GPT-3.5 model, gave the wrong answer to our question.

I've been testing ChatGPT almost daily since its release. Its user interface has remained simple, but minor changes have improved it greatly, like the addition of a copy button, an edit option, Custom Instructions , and easy access to your account. 

Also: How to use ChatGPT

Though ChatGPT has proven itself as a valuable AI tool, it can be prone to misinformation . Like other large language models (LLMs), GPT-3.5 is imperfect, as it is trained on human-created data up to January 2022. It also often fails to comprehend nuances, like it did with our math question example, which it answered incorrectly by saying we have two oranges left when it should be five. 

2. You're willing to pay extra for an upgrade

OpenAI lets users access ChatGPT -- powered by the GPT-3.5 model -- for free with a registered account. But if you're willing to pay for the Plus version, you can access GPT-4 and many more features for $20 per month.

Also: How to write better ChatGPT prompts for the best generative AI results

GPT-4 is the largest LLM available for use when compared to all other AI chatbots and is trained with data up to April 2023 and can also access the internet, powered by Microsoft Bing. GPT-4 is said to have over 100 trillion parameters; GPT-3.5 has 175 billion parameters. More parameters essentially mean that the model is trained on more data, which makes it more likely to answer questions accurately and less prone to hallucinations.

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ChatGPT Plus, which runs using the GPT-4 model, did answer the question correctly. 

As an example, you can see the GPT-4 model, available through a ChatGPT Plus subscription , answered the math question correctly, as it understood the full context of the problem from beginning to end.

Also: I tried Microsoft Copilot's new AI image-generating feature, and it solves a real problem

Next up, let's consider Microsoft Copilot (formerly Bing chat) , which is a great way to access GPT-4 for free, as it's integrated into its new Bing format. 

You should use Microsoft Copilot if...

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1. You want more up-to-date information

In contrast to the free version of ChatGPT, which is limited to being an AI tool that generates text in a conversational style with information leading up to early 2022, Copilot can access the internet to deliver more current information, complete with links for sources. 

Also: How to use Copilot (formerly called Bing Chat)

There are other benefits, too. Copilot is powered by GPT-4, OpenAI's LLM, and it's completely free to use. Unfortunately, you are limited to five responses on a single conversation, and can only enter up to 2,000 characters in each prompt. 

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Copilot's Precise conversation style answered the question accurately, though other styles fumbled.

Copilot's user interface isn't as straightforward as that of ChatGPT, but it's easy to navigate. Though Bing Chat can access the internet to give you more up-to-date results compared to ChatGPT, I've found it is more prone to stall at replying and altogether miss prompts than its competitor. 

2. You prefer more visual features

Through a series of upgrades to its platform, Microsoft added visual features to Copilot, formerly Bing Chat. At this point, you can ask Copilot questions like, 'What is a Tasmanian devil?' and get an information card in response, complete with photos, lifespan, diet, and more for a more scannable result that is easier to digest than a wall of text. 

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All about the Tasmanian devil on Microsoft Copilot.

When you use Copilot, you can also ask it to create an image for you. Give Copilot the description of what you want the image to look like, and have the chatbot generate four images for you to choose from. 

Also: How to use Image Creator from Microsoft Designer (formerly Bing Image Creator)

Microsoft Copilot also features different conversational styles when you interact with the chatbot, including Creative, Balanced, and Precise, which alter how light or straightforward the interactions are. 

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Both the Balanced and Creative conversation styles in Microsoft Copilot answered my question inaccurately.

Finally, let's turn to Google's Gemini, formerly known as Bard, which uses a different LLM and has received some considerable upgrades in the past few months.

You should use Gemini if...

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1. You want a fast, almost unlimited experience

In my time testing different AI chatbots, I saw  Google Bard catch a lot of flack for different shortcomings . While I'm not going to say they're unjustified, I will say that Google's AI chatbot, now named Gemini, has improved greatly, inside and out.

Also: How to use Gemini (formerly Google Bard): Everything you should know

Gemini is speedy with its answers, which have gotten more accurate over time. It's not faster than ChatGPT Plus, but it can be faster at giving responses than Copilot at times and faster than the free GPT-3.5 version of ChatGPT, though your mileage may vary. 

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Gemini answered accurately, like GPT-4 and Copilot's Precise conversation style.

The previous Bard used to make the same mistake as other bots on my example math problem, by incorrectly using the 5 - 3 = 2 formula, but Gemini, powered by Google's new Gemini Pro, the company's largest and latest LLM. Now, Gemini answers the question accurately.

Also: Apple's new AI model edits photos according to text prompts from users

Gemini is also not limited to a set amount of responses like Microsoft Copilot is. You can have long conversations with Google's Gemini, but Bing is limited to 30 replies in one conversation. Even ChatGPT Plus limits users to 40 messages every three hours. 

2. You want the full Google experience

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Google also incorporated more visual elements into its Gemini platform than those currently available on Copilot. Users can also use Gemini to generate images, can upload photos through an integration with Google Lens , and enjoy Kayak, OpenTable, Instacart, and Wolfram Alpha plugins.

Also: 6 AI tools to supercharge your work and everyday life

But Gemini is slowly becoming a full Google experience thanks to Extensions folding the wide range of Google applications into Gemini. Gemini users can add extensions for Google Workspace, YouTube, Google Maps, Google Flights, and Google Hotels, giving them a more personalized and extensive experience.

Artificial Intelligence

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ChatGPT vs. Copilot: Which AI chatbot is better for you?

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What is Copilot (formerly Bing Chat)? Here's everything you need to know

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What is Google's Gemini AI tool (formerly Bard)? Everything you need to know

Bring on the AI guardrails!

Ziad Obermeyer to Senate panel: Here's how AI in health care can do more good than harm

  • By Sheila Kaplan
  • 7 min. read ▪ Published February 14
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Ziad Obermeyer believes that artificial intelligence can help doctors and others in the healthcare system make better decisions, improving health and reducing cost. He also thinks that without strong oversight, much could go wrong.

On February 8,  Obermeyer, Blue Cross Distinguished Associate Professor of Health Policy and Management at Berkeley Public Health, warned the U.S. Senate Finance Committee about some of AI’s potential hazards within the healthcare field, and offered ways to ensure that AI systems are safe, unbiased and useful.

The hearing, “Artificial Intelligence and Health Care: Promise and Pitfalls,” explored the growing use of AI in medicine, and by federal health care agencies.

“Throughout my ten years of practicing medicine, I have agonized over missed diagnoses, futile treatments, unnecessary tests and more,” Obermeyer said. “The collective weight of these errors, in my view, is a major driver of the dual crisis in our healthcare system: suboptimal outcomes at very high cost. AI holds tremendous promise as a solution to both problems.”

Obermeyer, a physician and researcher, studies how machine learning can help doctors make better decisions (like whom to test for heart attack ), and help researchers make new discoveries—by “seeing” the world the way algorithms do (like finding new causes of pain that doctors miss , or linking individual body temperature set points to health outcomes). He has also shown how widely-used algorithms affecting millions of patients automate and scale up racial bias . That work has impacted how many organizations build and use algorithms , and how lawmakers and regulators hold AI accountable.

Obermeyer is a co-PI of a lab, joint between Berkeley and U Chicago, that builds algorithmic tools to improve decision-making and deepen understanding in health. He is the co-founder of Nightingale Open Science, a non-profit that makes massive new medical imaging datasets available for research; and Dandelion, a data platform to jump-start AI innovation in health. He is also a Chan Zuckerberg Biohub Investigator, a Faculty Research Fellow at the National Bureau of Economic Research, and was named an Emerging Leader by the the National Academy of Medicine.

Obermeyer told the panel that one area where AI is already used to improve patient care, is in helping doctors predict which patients are at high risk for potential arrhythmias that cause sudden death.

“In the U.S. alone, 300,000 people experience sudden cardiac death every year,” Obermeyer said. “What makes these events so tragic is that many of them are preventable: had we known a patient was at high risk, we would have implanted a defibrillator in her heart, to terminate the potential arrhythmias that cause sudden death, and save her life. Unfortunately, we are very bad at knowing who is at high risk.”

Obermeyer worked with a team of colleagues in the U.S. and Sweden to train an AI system to predict the risk of sudden cardiac death using just the waveform of a patient’s electrocardiogram.

“It performs far better than our current prediction technologies, based largely on human judgment,” he said. “This means we have the potential to both save more lives and reduce waste, by ensuring that precious defibrillators are implanted in the right patients. It’s rare to have an opportunity to both improve quality and reduce cost; normally we must choose. AI is a transformative new way for us to sidestep this dilemma entirely, and rebuild our health care system on a foundation of data-driven decision making.”

This principle—better human decisions through AI-driven predictions—can apply to many areas of medicine, Obermeyer said. But despite his optimism, Obermeyer worries that without concerted effort from researchers, the private sector, and government, “AI may be on a path to do more harm than good in health care.”

To make this case, Obermeyer walked the senators through a study he led five years ago that showed how a group of poorly designed AI algorithms, built and used in both public and private sectors, perpetuated large-scale racial bias.

The algorithm’s goal was to identify patients with high future health needs. But, Obermeyer said, AI is extremely literal. Absent a data set called future health needs, the AI developers chose to predict a proxy variable that is present in health datasets: future healthcare costs.

It seemed reasonable. But because of discrimination and barriers to access, underserved patients who need health care often don’t get it, Obermeyer said.

“This means Black patients, and also poorer patients, rural patients, less-educated patients, and all those who face barriers to accessing health care when they need it—get less spent on their healthcare than their better-served counterparts, even though they have the same underlying health conditions. Low costs do not necessarily mean low needs.”

The AI ignored those facts, and predicted that Black patients would generate lower costs; and thus deprioritized them for access to help with their health.

“The result,” Obermeyer said, “was racial bias that affected important decisions for hundreds of millions of patients every year.”

“Many of the biased algorithms we studied remain in use today,” he said. When questioned by members of the panel, he added, “unfortunately as AI learns to basically replicate our current system, it’s going to replicate all of the inequalities in our current system.”

Fortunately, Obermeyer said, “there are a number of specific things that programs under this committee’s jurisdiction can do to ensure that AI produces the social value we all want.”

Obermeyer said that Medicare, Medicaid, and other programs under the finance committee’s jurisdiction can realize enormous benefits from well-designed AI products to improve quality of service and reduce costs.

“These programs should be willing to pay for AI—but they should not simply accept the flawed products that the market often produces,” Obermeyer said. “Rather, they should take advantage of their market power to articulate clear criteria for what they will pay for, and how much.”

He also called for transparency by AI businesses.

“We need more accountability in the form of evaluating those algorithms in new data sets and by third parties,” he said, “so that we don’t have to take an algorithm developer’s word that the AI is working well and equitably across groups.”

Claudia Williams, UC Berkeley School of Public Health’s inaugural chief social impact officer, said, “Dr. Obermeyer points out that AI is a policy unicorn. It has the potential to improve health and reduce costs. But it won’t achieve these outcomes without the policy guardrails he recommends.”

Other witnesses at the hearing were Michelle M. Mello, professor of health policy and of law at Stanford University; Peter Shen of Siemens Healthineers; Dr. Mark Sendak of Health AI Partnership; and Katherine Baicker, provost of the University of Chicago.

People of BPH found in this article include:

  • Ziad Obermeyer Blue Cross of California Distinguished Associate Professor, Health Policy and Management

More in category “School News”:

Best of berkeley public health 2023, claudia williams joins uc berkeley school of public health as inaugural chief social impact officer, meet our new faculty: misbath daouda, doctoral candidate iemaan rana named to forbes 30 under 30 list.

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  27. Bring on the AI guardrails!

    Obermeyer worked with a team of colleagues in the U.S. and Sweden to train an AI system to predict the risk of sudden cardiac death using just the waveform of a patient's electrocardiogram. "It performs far better than our current prediction technologies, based largely on human judgment," he said. "This means we have the potential to ...