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Pure mathematics articles from across Nature Portfolio

Pure mathematics uses mathematics to explore abstract ideas, mathematics that does not necessarily describe a real physical system. This can include developing the fundamental tools used by mathematicians, such as algebra and calculus, describing multi-dimensional space, or better understanding the philosophical meaning of mathematics and numbers themselves.

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On uniform stability and numerical simulations of complex valued neural networks involving generalized Caputo fractional order

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A comprehensive study for selecting optimal treatment modalities for blood cancer in a Fermatean fuzzy dynamic environment

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Novel waves structures for the nonclassical Sobolev-type equation in unipolar semiconductor with its stability analysis

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Mathematical discoveries from program search with large language models

FunSearch makes discoveries in established open problems using large language models by searching for programs describing how to solve a problem, rather than what the solution is.

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The real value of numbers

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Machine learning to guide mathematicians

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A different perspective on the history of the proof of Hall conductance quantization

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e is everywhere

From determining the compound interest on borrowed money to gauging chances at the roulette wheel in Monte Carlo, Stefanie Reichert explains that there’s no way around Euler’s number.

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Imagination captured

Imaginary numbers have a chequered history, and a sparse — if devoted — following. Abigail Klopper looks at why a concept as beautiful as i gets such a bad rap.

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Prime interference

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

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Robert Gerver

  • Table of Contents

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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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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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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When should AI tools be used in university labs?

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

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Applied Mathematics Research

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.

Applied Mathematics Fields

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

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Math-Vision (Math-V) dataset is a meticulously curated collection of 3,040 high-quality mathematical problems with visual contexts sourced from real math competitions. Spanning 16 distinct mathematical disciplines and graded across 5 levels of difficulty, our dataset provides a comprehensive and diverse set of challenges for evaluating the mathematical reasoning abilities of LMMs.

Through extensive experimentation, we unveil a notable performance gap between current LMMs and human performance on Math-Vision, underscoring the imperative for further advancements in LMMs. Moreover, our detailed categorization allows for a thorough error analysis of LMMs, offering valuable insights to guide future research and development.

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Article Contents

  • 1 Introduction
  • 2 Preliminaries
  • 3 Causal Order on Metric Spaces
  • 4 Magnitude Homotopy Type
  • 5 Gluing and Magnitude Homotopy Type
  • Acknowledgments
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Causal Order Complex and Magnitude Homotopy Type of Metric Spaces

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Yu Tajima, Masahiko Yoshinaga, Causal Order Complex and Magnitude Homotopy Type of Metric Spaces, International Mathematics Research Notices , Volume 2024, Issue 4, February 2024, Pages 3176–3222, https://doi.org/10.1093/imrn/rnad124

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In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space |$X$| and a real parameter |$\ell \geq 0$|⁠ . This space is roughly consisting of all paths of length |$\ell $| and has the reduced homology group that is isomorphic to the magnitude homology group of |$X$|⁠ . To construct the magnitude homotopy type, we consider the poset structure on the spacetime |$X\times \mathbb{R}$| defined by causal (time- or light-like) relations. The magnitude homotopy type is defined as the quotient of the order complex of an intervals on |$X\times \mathbb{R}$| by a certain subcomplex. The magnitude homotopy type gives a covariant functor from the category of metric spaces with |$1$| -Lipschitz maps to the category of pointed topological spaces. The magnitude homotopy type also has a “path integral” like expression for certain metric spaces. By applying discrete Morse theory to the magnitude homotopy type, we obtain a new proof of the Mayer–Vietoris-type theorem and several new results including the invariance of the magnitude under sycamore twist of finite metric spaces.

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Phys.org

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

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."

More information: Steve Murphy et al, A scoping review of research into mathematics classroom practices and affect, Teaching and Teacher Education (2023). DOI: 10.1016/j.tate.2023.104235

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

Credit: Unsplash/CC0 Public Domain

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best math research papers

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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100+ Amazing Algebra Topics for Research Papers

Algebra Topics

Many students seek algebra topics when writing research papers in this mathematical field. Algebra is the study field that entails studying mathematical symbols and rules for their manipulation. Algebra is the unifying thread for most mathematics, including solving elementary equations to learning abstractions like rings, groups, and fields. In most cases, people use algebra when unsure about the exact numbers. Therefore, they replace those numbers with letters. In business, algebra helps with sales prediction. While many students dislike mathematics, avoiding algebra research paper topics is almost impossible at an advanced study level. Therefore, this article lists topics to consider when writing a research paper in this academic field. It’s helpful because many learners struggle to find suitable topics when writing research papers in this field.

How to Write Theses on Advanced Algebra Topics

A thesis on an algebra topic is an individual project that the learner writes after investigating and studying a specific idea. Here’s a step-by-step guide for writing a thesis on an algebra topic.

Pick a topic: Start by selecting a title for your algebra thesis. Your topic should relate to your research interests and your supervisor’s guidelines. Investigate your topic: Once you’ve chosen a topic, research it extensively to know the relevant theories, formulas, and texts. Your thesis should be an extension of a particular topic’s analysis and a report on your research. Write the thesis: Once you’ve explored the topic extensively, start writing your paper. Your dissertation should have an abstract, an introduction, the body, and a conclusion.

The abstract should summarise your thesis’ aims, scope, and conclusions. The introduction should introduce the topic, size, and significance while providing relevant literature and outlining the logical structure. The body should have several chapters with details and proofs of numerical implementations, while the conclusion should restate your main arguments and tell readers the effects. Also, it should suggest future work.

College Algebra Topics

You may need topics to consider if you’re in college and want to write an algebra research paper. Here’s a list of titles worth considering for your essay.

  • Exploring the relationship between Rubik’s cube and the group theory
  • Comparing the relationship between various equation systems
  • Finding the most appropriate way to solve mathematical word problems
  • Investigating the distance formula and its origin
  • Exploring the things you can achieve with determinants
  • Explaining what “domain” and “range” mean in algebra
  • A two-dimension analysis of the Gram-Schmidt process
  • Exploring the differences between eigenvalues and eigenvectors
  • What the Cramer’s rule states, and why does it matter
  • Describing the Gaussian elimination
  • Provide an induction-proof example
  • Describe the uses of F-algebras
  • Understanding the number problems in algebra
  • What’s the essence of abstract algebra?
  • Investigating Fermat’s last theorem peculiarities
  • Exploring the algebra essentials
  • Investigating the relationship between geometry and algebra

These are exciting topics in college algebra. However, writing a winning paper about any of them requires careful research and analysis. Therefore, prepare to spend sufficient time working on any of these titles.

Cool Topics in Algebra

Perhaps, you want to write about an excellent topic in this mathematical field. If so, consider the following ideas for your algebra paper.

  • Discussing a differential equation with illustrations
  • Describing and analysing the Noetherian ring
  • Explain the commutative ring from an algebra viewpoint
  • Describe the Artin-Weddderburn theorem
  • Studying the Jacobson density theorem
  • Describe the four properties of any binary operation from an algebra viewpoint
  • A detailed analysis of the unary operator
  • Analysing the Abel-Ruffini theorem
  • Monomorphisms versus Epimorphisms: Contrast and comparison
  • Discus Morita duality with algebraic structures in mind
  • Nilpotent versus Idempotent in Ring theory

Pick any idea from this list and develop it into a research topic. Your educator will love your paper and award you a good grade if you research it and write an informative essay.

Linear Algebra Topics

Linear algebra covers vector spaces and the linear mapping between them. Linear equation systems have unknowns, and mathematicians use vectors and matrices to represent them. Here are exciting topics in linear algebra to consider for your research paper.

  • Decomposition of singular value
  • Investigating linear independence and dependence
  • Exploring projections in linear algebra
  • What are linear transformations in linear algebra?
  • Describe positive definite matrices
  • What are orthogonal matrices?
  • Describe Euclidean vector spaces with examples
  • Explain how you can solve equation systems with matrices
  • Determinants versus matrix inverses
  • Describe mathematical operations using matrices
  • Functional analysis of linear algebra
  • Exploring linear algebra and its fundamentals

These are some of the exciting project topics in linear algebra. Nevertheless, prepare sufficient resources and time to investigate any of these titles to write a winning paper.

Pre Algebra Topics

Are you interested in a pre-algebra research topic? If so, this category has some of the most exciting ideas to explore.

  • Investigating the importance of pre-algebra
  • The best way to start pre-algebra for a beginner
  • Pre-algebra and algebra- Which is the hardest and why?
  • Core lessons in pre-algebra
  • What follows pre-algebra?
  • The first things to learn in pre-algebra
  • Investigating the standard form in pre-algebra
  • Provide pre-algebra examples using the basic rules to evaluate expressions
  • Differentiate pre-algebra and algebra
  • Describe five pre-algebra formulas

Consider exploring any of these ideas if you’re interested in pre-algebra. Nevertheless, choose a title you’re comfortable with to develop a winning paper.

Intermediate Algebra Topics for Research

Perhaps, you’re interested in intermediate algebra. If so, consider any of these ideas for your research paper.

  • Reviewing absolute value and real numbers
  • Investigating real numbers’ operations
  • Exploring the cube and square roots of real numbers
  • Analysing algebraic formulas and expressions
  • What are the rules of scientific notation and exponents?
  • How to solve a linear inequality with a single variable
  • Exploring relations, functions, and graphics from an algebraic viewpoint
  • Investigating linear systems with two variables and solutions
  • How to solve a linear system with two variables
  • Exploring linear systems applications with two variables
  • How to solve a linear system with three variables
  • Gaussian elimination and matrices
  • How to simplify a radical expression
  • How to add and subtract a radical expression
  • How to multiply and divide a radical expression
  • How to extract a square root and complete the square
  • Investigating quadratic functions and graphs
  • How to solve a polynomial and rational inequality
  • How to solve logarithmic and exponential equations
  • Exploring arithmetic series and sequences

These are exciting topics in intermediate algebra to consider for research papers. Nevertheless, learners should prepare to solve equations in their work.

Algebra Topics High School Students Can Explore

Are you in high school and want to explore algebra? If yes, consider these topics for your research, they could be a great coursework help to you.

  • Crucial principles and formulas to embrace when solving a matrix
  • Ways to create charts on a firm’s financial analysis for the past five years
  • How to find solutions to finance and mathematical gaps
  • Ways to solve linear equations
  • What is a linear equation- Provide examples
  • Describe the substitution and elimination methods for solving equations
  • How to solve logarithmic equations
  • What are partial fractions?
  • Describe linear inequalities with examples
  • How to solve a quadratic equation by factoring
  • How to solve a quadratic equation by formula
  • How to solve a quadratic equation with a square completion method
  • How to frame a worksheet for a quadratic equation
  • Explain the relationship between roots and coefficients
  • Describe rational expressions and ways to simplify them
  • Describe a cubic equation roots
  • What is the greatest common factor- Provide examples
  • What is the least common multiple- Provide examples
  • Describe the remainder theorem with examples

Explore any of these titles for your high school paper. However, pick a title you’re comfortable working with from the beginning to the end to make your work easier.

Advanced Topics in Algebra and Geometry

Maybe you want to explore something more advanced in your paper. In that case, the following list has advanced topics in geometry and algebra worth considering.

  • Arithmetical structures and their algorithmic aspects
  • Fractional thermoentropy spaces in topological quantum fields
  • Fractional thermoentripy spaces in large-scale systems
  • Eigenpoints configurations
  • Investigating the higher dimension aperiodic domino problem
  • Exploring math anxiety, executive functions, and math performance
  • Coherent quantiles and lifting elements
  • Absolute values extension on two subfields
  • Reviewing the laws of form and Majorana fermions
  • Studying the specialisation and rational maps degree
  • Investigating mathematical-pedagogical knowledge of prospective teachers in ECD programs
  • The adeles I model theory
  • Exploring logarithmic vector fields, arrangements, and divisors’ freeness
  • How to reconstruct curves from Hodge classes
  • Investigating Eigen points configuration

These are advanced topics in algebra and geometry worth investigating. However, please prepare to explore your topic extensively to write a strong essay.

Abstract Algebra Topics

Most people study abstract algebra in college. If you’re interested in research in this area, consider these topics for your project.

  • Describe abstract algebra applications
  • Why is abstract algebra essential?
  • Describe ring theory and its application
  • What is group theory, and why does it matter?
  • Describe the critical conceptual algebra levels
  • Describe the fundamental theorem of the finite Abelian groups
  • Describe Sylow’s theorems
  • What is Polya counting?
  • Describe the RSA algorithm
  • What are the homomorphisms and ideals of Rings?
  • Describe integral domains and factorisation
  • Describe Boolean algebra and its importance
  • State and explain Cauchy’s Theorem- Why is it important?

This algebra topics list is not exhaustive. You can find more ideas worth exploring in your project. Nevertheless, pick an idea you will work with comfortably to deliver a winning paper.

Get Professional Math Homework Help!

Perhaps, you don’t have the time to find accurate algebra homework solutions. Maybe you need math thesis help from an expert. If so, you’ve no reason to search further. Our thesis writing services in USA can help you write a winning assignment. We offer custom help with math assignments at cheap prices. If you want to get a quality algebra dissertation without sweating, place an order with us. We’re an online team providing homework help to students across educational levels. We guarantee you a top-notch service once you approach us, saying, “Please do my math assignment.” We’re fast and can beat even a tight deadline without compromising quality. And whether you’re in high school, university, or college, we will write a paper that will compel your teacher to award you the best grade in your class. Contact us now!

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The Cybersecurity Award is presented to authors whose work represent outstanding and groundbreaking research in all essential aspects of cybersecurity. The award will be bestowed upon three distinguished papers focused on the following perspectives:

Track A--- Best Theoretical Research Paper

Track B--- Best Practical Research Paper

Track C--- Best Machine Learning and Security Paper

The award carries a USD $1500 prize for every winning paper and comes with statue and certificate to commemorate.

Any paper by any author written in the area of cybersecurity is eligible for nomination. Please note that your paper must have appeared in a refereed journal, conference, or workshop with proceedings published in the period from January 1, 2023 until December 31, 2023.

Your paper shall cover at least one of the following aspects:

• Cryptography and its applications

• Network and critical infrastructure security

• Hardware security

• Software and system security

• Cybersecurity data analytics

• Data-driven security and measurement studies

• Adversarial reasoning

• Malware analysis

• Privacy-enhancing technologies and anonymity

• IoT Security

• AI Security

Submission Guidelines

Your nomination shall be directly sent to [email protected] with the following contents:

• Full paper (Word or PDF format preferred)

• Official publication site of the paper

• The track you prefer (only one track can be chosen per paper)

• Anything else you feel necessary to note

February    Call for nomination opened

*All nominations must be submitted by May 10, 2024

June-July    Review period

August- September    Award recipients are notified; selections announced

Till the end of 2024  ‘ Cybersecurity ’s salon event’

Reviewing Process and Instructions

Recipient selection will be administered through the Awards Committee and up to three nominated papers will be selected.

ONLY authors of the awarded papers will be notified via email. Subsequently, these authors will be invited to present seminar in ‘ Cybersecurity ’s salon event’. Besides, an extended paper of your winning paper will be invited to publish in journal Cybersecurity .

Previous Winner

In previous 2023 nomination, the award was given to the following two papers. Let’s celebrate the winners mention below to get a taste of what to expect during Cybersecurity Award 2024:

Best Practical Paper--- Kaihua Qin; Liyi Zhou; Arthur Gervais, "Quantifying Blockchain Extractable Value: How dark is the forest?", 2022 IEEE Symposium on Security and Privacy (SP), San Francisco, CA, USA, 2022, pp. 198-214.

Best Machine Learning and Security Paper--- Hammond Pearce; Baleegh Ahmad; Benjamin Tan; Brendan Dolan-Gavitt; Ramesh Karri, "Asleep at the Keyboard? Assessing the Security of GitHub Copilot’s Code Contributions", 2022 IEEE Symposium on Security and Privacy (SP), San Francisco, CA, USA, 2022, pp. 754-768.

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CISCE ISC Math Exam 2024: Students say paper was tricky and lengthy

Students from various schools share mixed reactions on isc mathematics papers, ranging from application-based to tricky questions..

The ISC (Class 12) students who appeared in the mathematics examination on Tuesday found the question paper to be moderately difficult, tricky and tad lengthy too.

CISCE ISC Math Exam 2024: Students say paper was tricky and lengthy(Mourya/ Hindustan Times)

In Lucknow, students found the multiple-choice questions a bit tricky but section C was comparatively easy. The questions of section A covered the whole syllabus and was based on the specimen paper provided by the council.

Kashvi Pandey, a class 12 student of La Martiniere Girls College said, “The paper was very much application based. Despite general apprehension, we were able to give our best. Our pre board preparation helped us to understand and solve the questions quicker.”

Vaidehi Baranwal from LMGC said, “The paper was completely based on higher order thinking and practical application of the formulae. It was surely lengthy, additionally, I did find a few questions to be tricky but nonetheless the standard of ISC was maintained. The objective questions took longer than expected due to which I couldn't manage my time well but at the end I was able to complet all the questions.”

Chhavi, a student of City Montessori School, station road branch said: “The paper was mainly Application based and it required in-depth knowledge to solve certain graphical MCQs. It showcased a wide range of mathematical concepts at moderate difficulty level. A well-balanced distribution of topics contributed to a fair and thorough assessment.”

Dev Chaturvedi, another student from the same school said, “ The medium difficulty Mathematics paper provided a solid evaluation of fundamental concepts without overwhelming complexity. The statements were too complex to understand.”

Anshika Gupta, another student said, “Paper was moderate to difficult. 1 marks questions were application based and could not be solved until some 4-mark questions were from the Previous Year Question papers.”

Pranam Goyal, a student said: “The paper was of moderate level. The questions were application based. The in-depth knowledge was tested from MCQs and multiple concepts were questioned through various questions. Few questions were however lengthy. Overall the paper needed a thoughtful approach and problem solving skills.”

Piyush Tripathi, a student said: “The paper was very different from previous year papers. The paper was application based which made it a bit difficult. On a conclusion basis in depth knowledge was required for the paper.”

Shaurya Jaiswal said, “The paper was application based and involved usage of in-depth knowledge. There were many case studies in the paper which needed proper analysis. The paper was quite lengthy but doable. On a complete basis the paper was of moderate level.” Srishti Singh said, “The questions this year were very twisted and not direct. Somewhat deceptive language was used. The change in the pattern of the paper and its application-based nature was new to students and children are still adapting to this.”

Tanishka Sharma of St Joseph College said, “The ISC mathematics paper was overall moderate but a bit lengthy. The mcqs very conceptual and to the point but apart from that the rest of the paper was really calculative.”

Aarav Shukla of St Joseph College said, “The paper was more challenging compared to the previous year.It had a total of 22 questions divided into three sections.Out of the last two optional sections, Section C was easier. All questions were from the prescribed syllabus.”

Vasu, a student of City Montessori School, LDA branch said, “There were a few tricky questions." Another science Student Varnit said, “The paper was neither tough nor easy and I enjoyed solving it.” Jay said, “The paper needed better time management." Vineeta Kamran, the principal, praised the efforts made by the teachers that made students comfortable.

By and large the students found paper moderate and a few students are expecting full marks. However, the question on the topic Probability was tricky. The paper was lengthy but the students managed to finish the paper on time, said Tanmay Pandey and Anant Anand of Class XII here at CMS RDSO Campus.

Students of Strawberry Fields High School in Sector 26, Chandigarh overall felt that the mathematics exam had gone as per their expectations. Vithal said that the exam wasn't hard but the way that the questions tested their understanding and application of the concepts made the exam a bit lengthy. Sashit Sapra added that the exam was well balanced, and all topics were covered almost equally from Chandigarh.

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Rajeev Mullick is a Special Correspondent, he writes on education, telecom and heads city bureau at Lucknow. Love travelling ...view detail

best math research papers

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  28. Cybersecurity Best Paper 2024

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  29. CISCE ISC Math Exam 2024: Students say paper was tricky and lengthy

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