research questions on mathematics

251+ Math Research Topics [2024 Updated]

Mathematics, often dubbed as the language of the universe, holds immense significance in shaping our understanding of the world around us. It’s not just about crunching numbers or solving equations; it’s about unraveling mysteries, making predictions, and creating innovative solutions to complex problems. In this blog, we embark on a journey into the realm of math research topics, exploring various branches of mathematics and their real-world applications.

How Do You Write A Math Research Topic?

Writing a math research topic involves several steps to ensure clarity, relevance, and feasibility. Here’s a guide to help you craft a compelling math research topic:

• Identify Your Interests: Start by exploring areas of mathematics that interest you. Whether it’s pure mathematics, applied mathematics, or interdisciplinary topics, choose a field that aligns with your passion and expertise.
• Narrow Down Your Focus: Mathematics is a broad field, so it’s essential to narrow down your focus to a specific area or problem. Consider the scope of your research and choose a topic that is manageable within your resources and time frame.
• Review Existing Literature: Conduct a thorough literature review to understand the current state of research in your chosen area. Identify gaps, controversies, or unanswered questions that could form the basis of your research topic.
• Formulate a Research Question: Based on your exploration and literature review, formulate a clear and concise research question. Your research question should be specific, measurable, achievable, relevant, and time-bound (SMART).
• Consider Feasibility: Assess the feasibility of your research topic in terms of available resources, data availability, and research methodologies. Ensure that your topic is realistic and achievable within the constraints of your project.
• Consult with Experts: Seek feedback from mentors, advisors, or experts in the field to validate your research topic and refine your ideas. Their insights can help you identify potential challenges and opportunities for improvement.
• Refine and Iterate: Refine your research topic based on feedback and further reflection. Iterate on your ideas to ensure clarity, coherence, and relevance to the broader context of mathematics research.
• Craft a Title: Once you have finalized your research topic, craft a compelling title that succinctly summarizes the essence of your research. Your title should be descriptive, engaging, and reflective of the key themes of your study.
• Write a Research Proposal: Develop a comprehensive research proposal outlining the background, objectives, methodology, and expected outcomes of your research. Your research proposal should provide a clear roadmap for your study and justify the significance of your research topic.

By following these steps, you can effectively write a math research topic that is well-defined, relevant, and poised to make a meaningful contribution to the field of mathematics.

251+ Math Research Topics: Beginners To Advanced

• Prime Number Distribution in Arithmetic Progressions
• Diophantine Equations and their Solutions
• Applications of Modular Arithmetic in Cryptography
• The Riemann Hypothesis and its Implications
• Graph Theory: Exploring Connectivity and Coloring Problems
• Knot Theory: Unraveling the Mathematics of Knots and Links
• Fractal Geometry: Understanding Self-Similarity and Dimensionality
• Differential Equations: Modeling Physical Phenomena and Dynamical Systems
• Chaos Theory: Investigating Deterministic Chaos and Strange Attractors
• Combinatorial Optimization: Algorithms for Solving Optimization Problems
• Computational Complexity: Analyzing the Complexity of Algorithms
• Game Theory: Mathematical Models of Strategic Interactions
• Number Theory: Exploring Properties of Integers and Primes
• Algebraic Topology: Studying Topological Invariants and Homotopy Theory
• Analytic Number Theory: Investigating Properties of Prime Numbers
• Algebraic Geometry: Geometry Arising from Algebraic Equations
• Galois Theory: Understanding Field Extensions and Solvability of Equations
• Representation Theory: Studying Symmetry in Linear Spaces
• Harmonic Analysis: Analyzing Functions on Groups and Manifolds
• Mathematical Logic: Foundations of Mathematics and Formal Systems
• Set Theory: Exploring Infinite Sets and Cardinal Numbers
• Real Analysis: Rigorous Study of Real Numbers and Functions
• Complex Analysis: Analytic Functions and Complex Integration
• Measure Theory: Foundations of Lebesgue Integration and Probability
• Topological Groups: Investigating Topological Structures on Groups
• Lie Groups and Lie Algebras: Geometry of Continuous Symmetry
• Differential Geometry: Curvature and Topology of Smooth Manifolds
• Algebraic Combinatorics: Enumerative and Algebraic Aspects of Combinatorics
• Ramsey Theory: Investigating Structure in Large Discrete Structures
• Analytic Geometry: Studying Geometry Using Analytic Methods
• Hyperbolic Geometry: Non-Euclidean Geometry of Curved Spaces
• Nonlinear Dynamics: Chaos, Bifurcations, and Strange Attractors
• Homological Algebra: Studying Homology and Cohomology of Algebraic Structures
• Topological Vector Spaces: Vector Spaces with Topological Structure
• Representation Theory of Finite Groups: Decomposition of Group Representations
• Category Theory: Abstract Structures and Universal Properties
• Operator Theory: Spectral Theory and Functional Analysis of Operators
• Algebraic Number Theory: Study of Algebraic Structures in Number Fields
• Cryptanalysis: Breaking Cryptographic Systems Using Mathematical Methods
• Discrete Mathematics: Combinatorics, Graph Theory, and Number Theory
• Mathematical Biology: Modeling Biological Systems Using Mathematical Tools
• Population Dynamics: Mathematical Models of Population Growth and Interaction
• Epidemiology: Mathematical Modeling of Disease Spread and Control
• Mathematical Ecology: Dynamics of Ecological Systems and Food Webs
• Evolutionary Game Theory: Evolutionary Dynamics and Strategic Behavior
• Mathematical Neuroscience: Modeling Brain Dynamics and Neural Networks
• Mathematical Physics: Mathematical Models in Physical Sciences
• Quantum Mechanics: Foundations and Applications of Quantum Theory
• Statistical Mechanics: Statistical Methods in Physics and Thermodynamics
• Fluid Dynamics: Modeling Flow of Fluids Using Partial Differential Equations
• Mathematical Finance: Stochastic Models in Finance and Risk Management
• Option Pricing Models: Black-Scholes Model and Beyond
• Portfolio Optimization: Maximizing Returns and Minimizing Risk
• Stochastic Calculus: Calculus of Stochastic Processes and Itô Calculus
• Financial Time Series Analysis: Modeling and Forecasting Financial Data
• Operations Research: Optimization of Decision-Making Processes
• Linear Programming: Optimization Problems with Linear Constraints
• Integer Programming: Optimization Problems with Integer Solutions
• Network Flow Optimization: Modeling and Solving Flow Network Problems
• Combinatorial Game Theory: Analysis of Games with Perfect Information
• Algorithmic Game Theory: Computational Aspects of Game-Theoretic Problems
• Fair Division: Methods for Fairly Allocating Resources Among Parties
• Auction Theory: Modeling Auction Mechanisms and Bidding Strategies
• Voting Theory: Mathematical Models of Voting Systems and Social Choice
• Social Network Analysis: Mathematical Analysis of Social Networks
• Algorithm Analysis: Complexity Analysis of Algorithms and Data Structures
• Machine Learning: Statistical Learning Algorithms and Data Mining
• Deep Learning: Neural Network Models with Multiple Layers
• Reinforcement Learning: Learning by Interaction and Feedback
• Natural Language Processing: Statistical and Computational Analysis of Language
• Computer Vision: Mathematical Models for Image Analysis and Recognition
• Computational Geometry: Algorithms for Geometric Problems
• Symbolic Computation: Manipulation of Mathematical Expressions
• Numerical Analysis: Algorithms for Solving Numerical Problems
• Finite Element Method: Numerical Solution of Partial Differential Equations
• Monte Carlo Methods: Statistical Simulation Techniques
• High-Performance Computing: Parallel and Distributed Computing Techniques
• Quantum Computing: Quantum Algorithms and Quantum Information Theory
• Quantum Information Theory: Study of Quantum Communication and Computation
• Quantum Error Correction: Methods for Protecting Quantum Information from Errors
• Topological Quantum Computing: Using Topological Properties for Quantum Computation
• Quantum Algorithms: Efficient Algorithms for Quantum Computers
• Quantum Cryptography: Secure Communication Using Quantum Key Distribution
• Topological Data Analysis: Analyzing Shape and Structure of Data Sets
• Persistent Homology: Topological Invariants for Data Analysis
• Mapper Algorithm: Method for Visualization and Analysis of High-Dimensional Data
• Algebraic Statistics: Statistical Methods Based on Algebraic Geometry
• Tropical Geometry: Geometric Methods for Studying Polynomial Equations
• Model Theory: Study of Mathematical Structures and Their Interpretations
• Descriptive Set Theory: Study of Borel and Analytic Sets
• Ergodic Theory: Study of Measure-Preserving Transformations
• Combinatorial Number Theory: Intersection of Combinatorics and Number Theory
• Additive Combinatorics: Study of Additive Properties of Sets
• Arithmetic Geometry: Interplay Between Number Theory and Algebraic Geometry
• Proof Theory: Study of Formal Proofs and Logical Inference
• Reverse Mathematics: Study of Logical Strength of Mathematical Theorems
• Nonstandard Analysis: Alternative Approach to Analysis Using Infinitesimals
• Computable Analysis: Study of Computable Functions and Real Numbers
• Graph Theory: Study of Graphs and Networks
• Random Graphs: Probabilistic Models of Graphs and Connectivity
• Spectral Graph Theory: Analysis of Graphs Using Eigenvalues and Eigenvectors
• Algebraic Graph Theory: Study of Algebraic Structures in Graphs
• Metric Geometry: Study of Geometric Structures Using Metrics
• Geometric Measure Theory: Study of Measures on Geometric Spaces
• Discrete Differential Geometry: Study of Differential Geometry on Discrete Spaces
• Algebraic Coding Theory: Study of Error-Correcting Codes
• Information Theory: Study of Information and Communication
• Coding Theory: Study of Error-Correcting Codes
• Cryptography: Study of Secure Communication and Encryption
• Finite Fields: Study of Fields with Finite Number of Elements
• Elliptic Curves: Study of Curves Defined by Cubic Equations
• Hyperelliptic Curves: Study of Curves Defined by Higher-Degree Equations
• Modular Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Number Theory
• Zeta Functions: Analytic Functions with Special Properties
• Analytic Number Theory: Study of Number Theoretic Functions Using Analysis
• Dirichlet Series: Analytic Functions Represented by Infinite Series
• Euler Products: Product Representations of Analytic Functions
• Arithmetic Dynamics: Study of Iterative Processes on Algebraic Structures
• Dynamics of Rational Maps: Study of Dynamical Systems Defined by Rational Functions
• Julia Sets: Fractal Sets Associated with Dynamical Systems
• Mandelbrot Set: Fractal Set Associated with Iterations of Complex Quadratic Polynomials
• Arithmetic Geometry: Study of Algebraic Geometry Over Number Fields
• Diophantine Geometry: Study of Solutions of Diophantine Equations Using Geometry
• Arithmetic of Elliptic Curves: Study of Elliptic Curves Over Number Fields
• Rational Points on Curves: Study of Rational Solutions of Algebraic Equations
• Galois Representations: Study of Representations of Galois Groups
• Automorphic Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Automorphic Forms
• Selberg Trace Formula: Tool for Studying Spectral Theory and Automorphic Forms
• Langlands Program: Program to Unify Number Theory and Representation Theory
• Hodge Theory: Study of Harmonic Forms on Complex Manifolds
• Riemann Surfaces: One-dimensional Complex Manifolds
• Shimura Varieties: Algebraic Varieties Associated with Automorphic Forms
• Modular Curves: Algebraic Curves Associated with Modular Forms
• Hyperbolic Manifolds: Manifolds with Constant Negative Curvature
• Teichmüller Theory: Study of Moduli Spaces of Riemann Surfaces
• Mirror Symmetry: Duality Between Calabi-Yau Manifolds
• Kähler Geometry: Study of Hermitian Manifolds with Special Symmetries
• Algebraic Groups: Linear Algebraic Groups and Their Representations
• Lie Algebras: Study of Algebraic Structures Arising from Lie Groups
• Representation Theory of Lie Algebras: Study of Representations of Lie Algebras
• Quantum Groups: Deformation of Lie Groups and Lie Algebras
• Algebraic Topology: Study of Topological Spaces Using Algebraic Methods
• Homotopy Theory: Study of Continuous Deformations of Spaces
• Homology Theory: Study of Algebraic Invariants of Topological Spaces
• Cohomology Theory: Study of Dual Concepts to Homology Theory
• Singular Homology: Homology Theory Defined Using Simplicial Complexes
• Sheaf Theory: Study of Sheaves and Their Cohomology
• Differential Forms: Study of Multilinear Differential Forms
• De Rham Cohomology: Cohomology Theory Defined Using Differential Forms
• Morse Theory: Study of Critical Points of Smooth Functions
• Symplectic Geometry: Study of Symplectic Manifolds and Their Geometry
• Floer Homology: Study of Symplectic Manifolds Using Pseudoholomorphic Curves
• Gromov-Witten Invariants: Invariants of Symplectic Manifolds Associated with Pseudoholomorphic Curves
• Mirror Symmetry: Duality Between Symplectic and Complex Geometry
• Calabi-Yau Manifolds: Ricci-Flat Complex Manifolds
• Moduli Spaces: Spaces Parameterizing Geometric Objects
• Donaldson-Thomas Invariants: Invariants Counting Sheaves on Calabi-Yau Manifolds
• Algebraic K-Theory: Study of Algebraic Invariants of Rings and Modules
• Homological Algebra: Study of Homology and Cohomology of Algebraic Structures
• Derived Categories: Categories Arising from Homological Algebra
• Stable Homotopy Theory: Homotopy Theory with Stable Homotopy Groups
• Model Categories: Categories with Certain Homotopical Properties
• Higher Category Theory: Study of Higher Categories and Homotopy Theory
• Higher Topos Theory: Study of Higher Categorical Structures
• Higher Algebra: Study of Higher Categorical Structures in Algebra
• Higher Algebraic Geometry: Study of Higher Categorical Structures in Algebraic Geometry
• Higher Representation Theory: Study of Higher Categorical Structures in Representation Theory
• Higher Category Theory: Study of Higher Categorical Structures
• Homotopical Algebra: Study of Algebraic Structures in Homotopy Theory
• Homotopical Groups: Study of Groups with Homotopical Structure
• Homotopical Categories: Study of Categories with Homotopical Structure
• Homotopy Groups: Algebraic Invariants of Topological Spaces
• Homotopy Type Theory: Study of Foundations of Mathematics Using Homotopy Theory

In conclusion, the world of mathematics is vast and multifaceted, offering endless opportunities for exploration and discovery. Whether delving into the abstract realms of pure mathematics or applying mathematical principles to solve real-world problems, mathematicians play a vital role in advancing human knowledge and shaping the future of our world.

By embracing diverse math research topics and interdisciplinary collaborations, we can unlock new possibilities and harness the power of mathematics to address the challenges of today and tomorrow. So, let’s embark on this journey together as we unravel the mysteries of numbers and explore the boundless horizons of mathematical inquiry.

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181 Mathematics Research Topics From PhD Experts

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

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

Our Newest Research Topics in Math

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

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

Cool Math Topics to Research

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

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

Undergraduate Math Research Topics

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

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

Number Theory Topics to Research

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

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

Math Research Topics for High School

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

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

Complex Math Topics

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

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

Easy Math Research Paper Topics

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

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

Logic Topics to Research

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

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

Algebra Topics for a Research Paper

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

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

Math Education Research Topics

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

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

Computability Theory Topics to Research

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

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

Interesting Math Research Topics

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

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

Applied Math Research Topics

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

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

Geometry Topics for a Research Paper

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

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

Math Research Topics for Middle School

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

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

Math Research Topics for College Students

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

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

Calculus Topics for a Research Paper

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

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

Simple Math Research Paper Topics for High School

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

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

Business Math Topics

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

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

Probability and Statistics Topics for Research

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

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

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Mathematics Research Paper Topics

See our list of mathematics research paper topics . Mathematics is the science that deals with the measurement, properties, and relationships of quantities, as expressed in either numbers or symbols. For example, a farmer might decide to fence in a field and plant oats there. He would have to use mathematics to measure the size of the field, to calculate the amount of fencing needed for the field, to determine how much seed he would have to buy, and to compute the cost of that seed. Mathematics is an essential part of every aspect of life—from determining the correct tip to leave for a waiter to calculating the speed of a space probe as it leaves Earth’s atmosphere.

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• Boolean algebra
• Chaos theory
• Complex numbers
• Correlation
• Fraction, common
• Game theory
• Graphs and graphing
• Imaginary number
• Multiplication
• Natural numbers
• Number theory
• Numeration systems
• Probability theory
• Proof (mathematics)
• Pythagorean theorem
• Trigonometry

Mathematics undoubtedly began as an entirely practical activity— measuring fields, determining the volume of liquids, counting out coins, and the like. During the golden era of Greek science, between about the sixth and third centuries B.C., however, mathematicians introduced a new concept to their study of numbers. They began to realize that numbers could be considered as abstract concepts. The number 2, for example, did not necessarily have to mean 2 cows, 2 coins, 2 women, or 2 ships. It could also represent the idea of “two-ness.” Modern mathematics, then, deals both with problems involving specific, concrete, and practical number concepts (25,000 trucks, for example) and with properties of numbers themselves, separate from any practical meaning they may have (the square root of 2 is 1.4142135, for example).

Fields of Mathematics

Mathematics can be subdivided into a number of special categories, each of which can be further subdivided. Probably the oldest branch of mathematics is arithmetic, the study of numbers themselves. Some of the most fascinating questions in modern mathematics involve number theory. For example, how many prime numbers are there? (A prime number is a number that can be divided only by 1 and itself.) That question has fascinated mathematicians for hundreds of years. It doesn’t have any particular practical significance, but it’s an intriguing brainteaser in number theory.

Geometry, a second branch of mathematics, deals with shapes and spatial relationships. It also was established very early in human history because of its obvious connection with practical problems. Anyone who wants to know the distance around a circle, square, or triangle, or the space contained within a cube or a sphere has to use the techniques of geometry.

Algebra was established as mathematicians recognized the fact that real numbers (such as 4 and 5.35) can be represented by letters. It became a way of generalizing specific numerical problems to more general situations.

Analytic geometry was founded in the early 1600s as mathematicians learned to combine algebra and geometry. Analytic geometry uses algebraic equations to represent geometric figures and is, therefore, a way of using one field of mathematics to analyze problems in a second field of mathematics.

Over time, the methods used in analytic geometry were generalized to other fields of mathematics. That general approach is now referred to as analysis, a large and growing subdivision of mathematics. One of the most powerful forms of analysis—calculus—was created almost simultaneously in the early 1700s by English physicist and mathematician Isaac Newton (1642–1727) and German mathematician Gottfried Wilhelm Leibniz (1646–1716). Calculus is a method for analyzing changing systems, such as the changes that take place as a planet, star, or space probe moves across the sky.

Statistics is a field of mathematics that grew in significance throughout the twentieth century. During that time, scientists gradually came to realize that most of the physical phenomena they study can be expressed not in terms of certainty (“A always causes B”), but in terms of probability (“A is likely to cause B with a probability of XX%”). In order to analyze these phenomena, then, they needed to use statistics, the field of mathematics that analyzes the probability with which certain events will occur.

Each field of mathematics can be further subdivided into more specific specialties. For example, topology is the study of figures that are twisted into all kinds of bizarre shapes. It examines the properties of those figures that are retained after they have been deformed.

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Posing Researchable Questions in Mathematics and Science Education: Purposefully Questioning the Questions for Investigation

• Published: 07 April 2020
• Volume 18 , pages 1–7, ( 2020 )

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• Rachel Mamlok-Naaman 2  

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Perhaps the most obvious example is the set of 23 influential mathematical problems posed by David Hilbert that inspired a great deal of progress in the discipline of mathematics (Hilbert, 1901 -1902). Einstein and Infeld ( 1938 ) claimed that “to raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science” (p. 95). Both Cantor and Klamkin recommended that, in mathematics, the art of posing a question be held as high or higher in value than solving it. Similarly, in the history of science, formulating precise, answerable questions not only advances new discoveries but also gives scientists intellectual excitement (Kennedy, 2005 ; Mosteller, 1980 ).

In research related to mathematics and science education, there is no shortage of evidence for the impact of posing important and researchable questions: Posing new, researchable questions marks real advances in mathematics and science education (Cai et al., 2019a ). Although research in mathematics and science education begins with researchable questions, only recently have researchers begun to purposefully and systematically discuss the nature of researchable questions. To conduct research, we must have researchable questions, but what defines a researchable question? What are the sources of researchable questions? How can we purposefully discuss researchable questions?

This special issue marks effort for the field’s discussion of researchable questions. As the field of mathematics and science education matures, it is necessary to reflect on the field at such a metalevel (Inglis & Foster, 2018 ). Although the authors in this special issue discuss researchable questions from different angles, they all refer to researchable questions as those that can be investigated empirically. For any empirical study, one can discuss its design, its conduct, and how it can be written up for publication. Therefore, researchable questions in mathematics and science education can be discussed with respect to study design, the conduct of research, and the dissemination of that research.

Even though there are many lines of inquiry that we can explore with respect to researchable questions, each exploring the topic from a different angle, we have decided to focus on the following three aspects to introduce this special issue: (1) criteria for selecting researchable questions, (2) sources of researchable questions, and (3) alignment of researchable questions with the conceptual framework as well as appropriate research methods.

Criteria for Selecting Researchable Questions

It is clear that not all researchable questions are worth the effort to investigate. According to Cai et al. ( 2020 ), of all researchable questions in mathematics and science education, priority is given to those that are significant. Research questions are significant if they can advance the fields’ knowledge and understanding about the teaching and learning of science and mathematics. Through an analysis of peer reviews for a research journal, Cai et al. ( 2020 ) provide a window into the field’s frontiers related to significant researchable questions. In an earlier article, Cai et al. ( 2019a ) argued that

The significance of a research question cannot be determined just by reading it. Rather, its significance stands in relation to the knowledge of the field. The justification for the research question itself—why it is a significant question to investigate—must therefore be made clear through an explicit argument that ties the research question to what is and is not already known. (p. 118)

In their analysis, Cai et al. ( 2020 ) provide evidence that many reviews that highlighted issues with the research questions in rejected manuscripts specifically called for authors to make an argument to motivate the research questions, whereas none of the manuscripts that were ultimately accepted (pending revisions) received this kind of comment. Cai et al. ( 2020 ) provide a framework not only for analyzing peer reviews about research questions but also for how to communicate researchable questions in journal manuscript preparations.

Whereas Cai and his colleagues, as editors of a journal, discuss significant research questions from the perspective of peer review and publication, King, Ochsendorf, Solomon, and Sloane ( 2020 ), as program directors at the Directorate for Education and Human Resources at the U.S. National Science Foundation (NSF), discuss fundable research questions for research in mathematics and science education. King et al. ( 2020 ) situate their discussion of fundable research questions in the context of writing successful educational research grant proposals. For them, fundable research questions must be transformative and significant with specific and clear constructs. In addition, they present examples of STEM education research questions from different types of research (Institute of Education Sciences [IES] & NSF, 2013 ) and how the questions themselves direct specific design choices, methodologies, measures, study samples, and analytical models as well as how they can reflect the disciplinary orientations of the researchers.

Hjalmarson and Baker ( 2020 ) take a quite different approach to discussing researchable questions for teacher professional development. They argue for the need to include mathematics specialists (e.g. mathematics coaches or mathematics teacher leaders) for studying teacher learning and development. To Hjalmarson and Baker ( 2020 ), researchable questions related to teacher professional learning should be selected by including mathematics specialists because of their role in connecting research and practice.

Sanchez ( 2020 ) discuss, in particular, the importance of replication studies in mathematics and the kinds of researchable questions that would be productive to explore within this category. With the increased acknowledgement of the importance of replication studies (Cai et al., 2018 ), Sanchez Aguilar has provided a useful typology of fundamental questions that can guide a replication study in mathematics (and science) education.

Schoenfeld ( 2020 ) is very direct in suggesting that researchable questions must advance the field and that these research questions must be meaningful and generative: “What is meant by meaningful is that the answer to the questions posed should matter to either practice or theory in some important way. What is meant by generative is that working on the problem, whether it is ‘solved’ or not, is likely to provide valuable insights” (pp. XX). Schoenfeld calls for researchers to establish research programs—that is, one not only selects meaningful research questions to investigate but also continues in that area of research to produce ongoing insights and further meaningful questions.

Stylianides and Stylianides ( 2020 ) argue that, collectively, researchers can and need to pose new researchable questions. The new researchable questions are worth investigating if they reflect the field’s growing understanding of the web of potentially influential factors surrounding the investigation of a particular area. The argument that Stylianides and Stylianides ( 2020 ) use is very similar to Schoenfeld’s ( 2020 ) generative criteria, but Stylianides and Stylianides ( 2020 ) explicitly emphasize the collective nature of the field’s growing understanding of a particular phenomenon.

Sources of Researchable Questions

Research questions in science and mathematics education arise from multiple sources, including problems of practice, extensions of theory, and lacunae in existing areas of research. Therefore, through a research question’s connections to prior research, it should be clear how answering the question extends the field’s knowledge (Cai et al., 2019a ). Across the papers in this special issue lies a common theme that researchable questions arise from understanding the area under study. Cai et al. ( 2020 ) take the position that the significance of researchable questions must be justified in the context of prior research. In particular, reviewers of manuscripts submitted for publication will evaluate if the study is adequately motivated. In fact, posing significant researchable questions is an iterative process beginning with some broader, general sense of an idea which is potentially fruitful and leading, eventually, to a well-specified, stated research question (Cai et al., 2019a ). Similarly, King et al. ( 2020 ) argue that fundable research questions should be grounded in prior research and make explicit connections to what is known or not known in the given area of study.

Sanchez ( 2020 ) suggest that it is time for the field of mathematics and science education research to seriously consider replication studies. Researchable questions related to replication studies might arise from the examination of the following two questions: (1) Do the results of the original study hold true beyond the context in which it was developed? (2) Are there alternative ways to study and explain an identified phenomenon or finding? Similarly, Hjalmarson and Baker ( 2020 ) specifically suggest two needs related to mathematics specialists in studies of professional development that drive researchable questions: (1) defining practices and hidden players involved in systematic school change and (2) identifying the unit of analysis and scaling up professional development.

Schoenfeld ( 2020 ) uses various examples to illustrate the origin of researchable questions. One of his (perhaps most familiar) examples is his decade-long research on mathematical problem solving. He elaborates on how answering one specific research question leads to another and another. In the context of research on mathematical proof, Stylianides and Stylianides ( 2020 ) also illustrate how researchable questions arise from existing research in the area leading to new researchable questions in the dynamic process of educational research. The arguments and examples in both Schoenfeld ( 2020 ) and Stylianides and Stylianides ( 2020 ) are quite powerful in the sense that this source of researchable questions facilitates the accumulation of knowledge for the given areas of study.

A related source of researchable questions is not discussed in this set of papers—unexpected findings. A potentially powerful source of research questions is the discovery of an unexpected finding when conducting research (Cai et al., 2019b ). Many important advances in scientific research have their origins in serendipitous, unexpected findings. Researchers are often faced with unexpected and perhaps surprising results, even when they have developed a carefully crafted theoretical framework, posed research questions tightly connected to this framework, presented hypotheses about expected outcomes, and selected methods that should help answer the research questions. Indeed, unexpected findings can be the most interesting and valuable products of the study and a source of further researchable questions (Cai et al., 2019b ).

Of course, researchable questions can also arise from established scholars in a given field—those who are most familiar with the scope of the research that has been done. For example, in 2005, in celebrating the 125th anniversary of the publication of Science ’s first issue, the journal invited researchers from around the world to propose the 125 most important research questions in the scientific enterprise (Kennedy, 2005 ). A list of unanswered questions like this is a great source for researchable questions in science, just as the 23 great questions in mathematics by Hilbert ( 1901 -1902) spurred the field for decades. In mathematics and science education, one can look to research handbooks and compendiums. These volumes often include lists of unanswered research questions in the hopes of prompting further research in various areas (e.g. Cai, 2017 ; Clements, Bishop, Keitel, Kilpatrick, & Leung, 2013 ; Talbot-Smith, 2013 ).

Alignment of Researchable Questions with the Conceptual Framework and Appropriate Research Methods

Cai et al. ( 2020 ) and King et al. ( 2020 ) explicitly discuss the alignment of researchable questions with the conceptual framework and appropriate research methods. In writing journal publications or grant proposals, it is extremely important to justify the significance of the researchable questions based on the chosen theoretical framework and then determine robust methods to answer the research questions. According to Cai et al. ( 2019a ), justification for the significance of the research questions depends on a theoretical framework: “The theoretical framework shapes the researcher’s conception of the phenomenon of interest, provides insight into it, and defines the kinds of questions that can be asked about it” (p. 119). It is true that the notion of a theoretical framework can remain somewhat mysterious and confusing for researchers. However, it is clear that the theoretical framework links research questions to existing knowledge, thus helping to establish their significance; provides guidance and justification for methodological choices; and provides support for the coherence that is needed between research questions, methods, results, and interpretations of findings (Cai & Hwang, 2019 ; Cai et al., 2019c ).

Analyzing reviews for a research journal in mathematics education, Cai et al. ( 2020 ) found that the reviewers wanted manuscripts to be explicit about how the research questions, the theoretical framework, the methods, and the findings were connected. Even for manuscripts that were accepted (pending revisions), making explicit connections across all parts of the manuscript was a challenging proposition. Thus, in preparing manuscripts for publication, it is essential to communicate the significance of a study by developing a coherent chain of justification connecting researchable questions, the theoretical framework, and the research methods chosen to address the research questions.

The Long Journey Has Just Begun with a First Step

As the field of mathematics and science education matures, there is a need to take a step back and reflect on what has been done so that the field can continue to grow. This special issue represents a first step by reflecting on the posing of significant researchable questions to advance research in mathematics and science education. Such reflection is useful and necessary not only for the design of studies but also for the writing of research reports for publication. Most importantly, working on significant researchable questions cannot only contribute to theory generation about the teaching and learning of mathematics and science but also contribute to improving the impact of research on practice in mathematics and science classrooms.

To conclude, we want to draw readers’ attention to a parallel between this reflection on research in our field and a line of research that investigates the development of school students’ problem-posing and questioning skills in mathematics and science (Blonder, Rapp, Mamlok-Naaman, & Hofstein, 2015 ; Cai, Hwang, Jiang, & Silber, 2015 ; Cuccio-Schirripa & Steiner, 2000 ; Hofstein, Navon, Kipnis, & Mamlok-Naaman, 2005 ; Silver, 1994 ; Singer, Ellerton, & Cai, 2015 ). Posing researchable questions is critical for advancing research in mathematics and science education. Similarly, providing students opportunities to pose problems is critical for the development of their thinking and learning. With the first step in this journey made, perhaps we can dream of something bigger further on down the road.

Change history

15 may 2020.

The original version of this article unfortunately contains correction.

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Cai, J., Mamlok-Naaman, R. Posing Researchable Questions in Mathematics and Science Education: Purposefully Questioning the Questions for Investigation. Int J of Sci and Math Educ 18 (Suppl 1), 1–7 (2020). https://doi.org/10.1007/s10763-020-10079-5

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Exploring Best Math Research Topics That Push the Boundaries

Mathematics is a vast and fascinating field that encompasses a wide range of topics and research areas. Whether you are an undergraduate student, graduate student, or a professional mathematician, engaging in math research opens doors to exploration, discovery, and the advancement of knowledge. The world of math research is filled with exciting challenges, unsolved problems, and groundbreaking ideas waiting to be explored.

In this guide, we will delve into the realm of math research topics, providing you with a glimpse into the diverse areas of mathematical inquiry. From pure mathematics to applied mathematics, this guide will present a variety of research areas that span different branches and interdisciplinary intersections. Whether you are interested in algebra, analysis, geometry, number theory, statistics, or computational mathematics, there is a wealth of captivating topics to consider.

Math research topics are not only intellectually stimulating but also have significant real-world applications. Mathematical discoveries and advancements underpin various fields such as engineering, physics, computer science, finance, cryptography, and data analysis. By immersing yourself in math research, you have the opportunity to contribute to the development of these applications and make a meaningful impact on society.

Throughout this guide, we will explore different research areas, discuss their significance, and provide insights into potential research questions and directions. However, keep in mind that this is not an exhaustive list, and there are countless other exciting topics awaiting exploration.

Embarking on a math research journey requires dedication, perseverance, and a passion for discovery. As you dive into the world of math research, embrace the challenges, seek guidance from mentors and experts, hire a math tutor , and foster a curious and open mindset.. Math research is a dynamic and ever-evolving field, and by engaging in it, you become part of a vibrant community of mathematicians pushing the boundaries of knowledge.

So, let us embark on this exploration of math research topics together, where new ideas, connections, and insights await. Prepare to unravel the mysteries of numbers, patterns, and structures, and embrace the thrill of contributing to the ever-expanding tapestry of mathematical understanding.

What is math research?

Table of Contents

Math research is the process of investigating new mathematical problems and developing new mathematical theories. It is a vital part of mathematics, as it helps to expand our understanding of the world and to develop new mathematical tools that can be used in other fields, such as science, engineering, and technology.

Math research is a challenging but rewarding endeavor. It requires a deep understanding of mathematics and a strong ability to think logically and creatively. Math researchers must be able to identify new problems, develop new ideas, and prove their ideas correct.

There are many different ways to get involved in math research. One way is to attend a math research conference. Another way is to join a math research group. You can also get involved in math research by working on a math research project with a mentor.

Math Research Topics

A few examples of math research topics:

Number theory

Number theory is a branch of mathematics that studies the properties of integers and other related objects. It is a vast and active field of research, with many open problems that have yet to be solved. Some of the current research topics in number theory include:

The Riemann hypothesis

This is one of the most important unsolved problems in mathematics. It states that the non-trivial zeros of the Riemann zeta function have real part 1/2.

The Birch and Swinnerton-Dyer conjecture

This conjecture relates the zeta function of an elliptic curve to the behavior of its rational points.

The Langlands program

This is a vast program in number theory that seeks to unify many different areas of the field.

The classification of finite simple groups

This is a complete classification of all finite simple groups, which are the building blocks of all other finite groups.

The study of cryptography

Number theory is used in many cryptographic algorithms, such as RSA and Diffie-Hellman.

The study of prime numbers

Prime numbers are fundamental to number theory, and there are many open problems related to them, such as the Goldbach conjecture and the twin prime conjecture.

The study of algebraic number theory

This is a branch of number theory that studies the properties of algebraic numbers, which are roots of polynomials with integer coefficients.

The study of combinatoric number theory

This is a branch of number theory that uses tools from combinatorics to study problems in number theory.

The study of computational number theory

This is a branch of number theory that uses computers to solve problems in number theory.

These are just a few of the many research topics in number theory. The field is constantly evolving, and new problems are being discovered all the time.

Topology is a branch of mathematics that studies the properties of spaces that are preserved under continuous deformations. Some of the most important research topics in topology include:

Algebraic topology

This branch of topology studies topological spaces using algebraic tools, such as homology and cohomology. Algebraic topology has been used to great effect in the study of knot theory, 3-manifolds, and other important topological spaces.

Geometric topology

This branch of topology studies topological spaces using geometric tools, such as triangulations and manifolds. Geometric topology has been used to great effect in the study of surfaces, 3-manifolds, and other important topological spaces.

Differential topology

This branch of topology studies topological spaces using differential geometry. Differential topology has been used to great effect in the study of manifolds, including the study of their smooth structures and their underlying topological structures.

Knot theory

This branch of topology studies knots, which are closed curves in 3-space. Knot theory has applications in many other areas of mathematics, including physics, chemistry, and computer science.

Low-dimensional topology

This branch of topology studies topological spaces of low dimension, such as surfaces and 3-manifolds. Low-dimensional topology has been used to great effect in the study of knot theory, 3-manifolds, and other important topological spaces.

Topological quantum field theory

This branch of mathematics studies the relationship between topology and quantum field theory. Topological quantum field theory has applications in many areas of physics, including string theory and quantum gravity.

Topological data analysis

This branch of mathematics studies the use of topological methods to analyze data. Topological data analysis has applications in many areas, including machine learning, computer vision, and bioinformatics.

These are just a few of the many research topics in topology. Topology is a vast and growing field, and there are many exciting new directions for research.

Differential geometry research topics

Differential geometry is a branch of mathematics that studies the geometry of smooth manifolds. Some of the most important research topics in differential geometry include:

Riemannian geometry

This branch of differential geometry studies Riemannian manifolds, which are smooth manifolds equipped with a Riemannian metric. Riemannian geometry has applications in many areas of mathematics, including physics, chemistry, and computer science.

Complex geometry

This branch of differential geometry studies complex manifolds, which are smooth manifolds that are holomorphically equivalent to a complex vector space. Complex geometry has applications in many areas of mathematics, including physics, chemistry, and computer science.

Geometric analysis

This branch of differential geometry studies the interplay between differential geometry and analysis. Geometric analysis has applications in many areas of mathematics, including physics, chemistry, and computer science.

Mathematical physics

This branch of mathematics uses differential geometry to study physical systems. Mathematical physics has applications in many areas of physics, including general relativity, quantum field theory, and string theory.

Computer graphics

This field of computer science uses differential geometry to create realistic images and animations. Computer graphics has applications in many areas, including video games, movies, and simulations.

Medical imaging

This field of medicine uses differential geometry to create images of the human body. Medical imaging has applications in many areas, including diagnosis, treatment, and research.

These are just a few of the many research topics in differential geometry. Differential geometry is a vast and growing field, and there are many exciting new directions for research.

Algebraic geometry research topics

Algebraic geometry is a branch of mathematics that studies geometric objects using the tools of abstract algebra. Some of the most important research topics in algebraic geometry include:

Algebraic curves

This branch of algebraic geometry studies curves, which are one-dimensional algebraic varieties. Algebraic curves have applications in many areas of mathematics, including number theory, representation theory, and mathematical physics.

Algebraic surfaces

This branch of algebraic geometry studies surfaces, which are two-dimensional algebraic varieties. Algebraic surfaces have applications in many areas of mathematics, including topology, differential geometry, and number theory.

Algebraic threefolds

This branch of algebraic geometry studies threefolds, which are three-dimensional algebraic varieties. Algebraic threefolds have applications in many areas of mathematics, including topology, differential geometry, and number theory.

Algebraic varieties

This branch of algebraic geometry studies varieties, which are arbitrary-dimensional algebraic sets. Algebraic varieties have applications in many areas of mathematics, including topology, differential geometry, and number theory.

Algebraic groups

This branch of algebraic geometry studies groups that are also algebraic varieties. Algebraic groups have applications in many areas of mathematics, including number theory, representation theory, and mathematical physics.

Moduli spaces

This branch of algebraic geometry studies moduli spaces, which are spaces that parameterize objects of a certain type. Moduli spaces have applications in many areas of mathematics, including number theory, representation theory, and mathematical physics.

Arithmetic geometry

This branch of algebraic geometry studies the intersection of algebraic geometry and number theory. Arithmetic geometry has applications in many areas of mathematics, including number theory, representation theory, and mathematical physics.

Complex algebraic geometry

This branch of algebraic geometry studies algebraic varieties over the complex numbers. Complex algebraic geometry has applications in many areas of mathematics, including topology, differential geometry, and mathematical physics.

Algebraic combinatorics

This branch of algebraic geometry studies the intersection of algebraic geometry and combinatorics. Algebraic combinatorics has applications in many areas of mathematics, including combinatorics, computer science, and mathematical physics.

These are just a few of the many research topics in algebraic geometry. Algebraic geometry is a vast and growing field, and there are many exciting new directions for research.

Mathematical physics research topics

Mathematical physics is a field of study that uses the tools of mathematics to study physical systems. Some of the most important research topics in mathematical physics include:

Quantum mechanics

This branch of physics studies the behavior of matter and energy at the atomic and subatomic level. Quantum mechanics has applications in many areas of physics, including chemistry, biology, and engineering.

This branch of physics studies the relationship between space and time. Relativity has applications in many areas of physics, including cosmology, astrophysics, and nuclear physics.

Statistical mechanics

This branch of physics studies the behavior of systems of many particles. Statistical mechanics has applications in many areas of physics, including thermodynamics, chemistry, and biology.

Chaos theory

This branch of physics studies the behavior of systems that are sensitive to initial conditions. Chaos theory has applications in many areas of physics, including meteorology, economics, and biology.

Mathematical finance

This field of mathematics uses the tools of mathematics to study financial markets. Mathematical finance has applications in many areas of finance, including investment banking, insurance, and risk management.

Computational physics

This field of mathematics uses the tools of mathematics to solve physical problems. Computational physics has applications in many areas of physics, including materials science, engineering, and medicine.

Mathematical biology

This field of mathematics uses the tools of mathematics to study biological systems. Mathematical biology has applications in many areas of biology, including genetics, ecology, and evolution.

Mathematical chemistry

This field of mathematics uses the tools of mathematics to study chemical systems. Mathematical chemistry has applications in many areas of chemistry, including materials science, biochemistry, and pharmacology.

Mathematical engineering

This field of mathematics uses the tools of mathematics to study engineering systems. Mathematical engineering has applications in many areas of engineering, including civil engineering, mechanical engineering, and electrical engineering.

These are just a few of the many research topics in mathematical physics. Mathematical physics is a vast and growing field, and there are many exciting new directions for research.

Mathematical biology research topics

Mathematical biology is a field of study that uses the tools of mathematics to study biological systems. Some of the most important research topics in mathematical biology include:

Modeling of biological systems

This branch of mathematical biology uses mathematical models to study the behavior of biological systems. Mathematical models can be used to understand the dynamics of biological systems, to predict how they will respond to changes in their environment, and to design new interventions to improve their health.

Computational biology

This field of mathematical biology uses computational methods to study biological systems. Computational methods can be used to analyze large amounts of biological data, to simulate biological systems, and to design new experiments.

Biostatistics

This field of mathematical biology uses statistical methods to study biological data. Biostatistical methods can be used to identify patterns in biological data, to test hypotheses about biological systems, and to design clinical trials.

Mathematical epidemiology

This field of mathematical biology uses mathematical models to study the spread of diseases. Mathematical models can be used to predict the course of an epidemic, to design public health interventions, and to assess the effectiveness of those interventions.

Mathematical ecology

This field of mathematical biology uses mathematical models to study the interactions between species in an ecosystem. Mathematical models can be used to predict how ecosystems will respond to changes in their environment, to design conservation strategies, and to assess the effectiveness of those strategies.

Mathematical neuroscience

This field of mathematical biology uses mathematical models to study the nervous system. Mathematical models can be used to understand how the nervous system works, to design new treatments for neurological disorders, and to assess the effectiveness of those treatments.

Mathematical genetics

This field of mathematical biology uses mathematical models to study genetics. Mathematical models can be used to understand how genes work, to design new treatments for genetic disorders, and to assess the effectiveness of those treatments.

Mathematical evolution

This field of mathematical biology uses mathematical models to study evolution. Mathematical models can be used to understand how evolution works, to design new conservation strategies, and to assess the effectiveness of those strategies.

These are just a few of the many research topics in mathematical biology. Mathematical biology is a vast and growing field, and there are many exciting new directions for research.

Mathematical finance research topics

Mathematical finance is a field of study that uses the tools of mathematics to study financial markets. Some of the most important research topics in mathematical finance include:

Asset pricing

This branch of mathematical finance studies the prices of assets, such as stocks, bonds, and options. Asset pricing models are used to price new financial products, to manage risk, and to make investment decisions.

Portfolio optimization

This branch of mathematical finance studies how to allocate money between different assets in a portfolio. Portfolio optimization models are used to maximize returns, to minimize risk, and to achieve other investment goals.

Derivative pricing

This branch of mathematical finance studies the prices of derivatives, such as options and futures. Derivatives are used to hedge risk, to speculate on future prices, and to generate income.

Risk management

This branch of mathematical finance studies how to measure and manage risk. Risk management models are used to identify and quantify risks, to develop strategies to mitigate risks, and to comply with regulations.

Market microstructure

This branch of mathematical finance studies the structure and dynamics of financial markets. Market microstructure models are used to understand how markets work, to design new trading systems, and to improve market efficiency.

Financial econometrics

This branch of mathematical finance uses statistical methods to study financial data. Financial econometrics models are used to identify patterns in financial data, to test hypotheses about financial markets, and to forecast future prices.

Computational finance

This field of mathematical finance uses computational methods to solve financial problems. Computational finance methods are used to price financial products, to manage risk, and to simulate financial markets.

Mathematical finance and machine learning

This field of mathematical finance uses machine learning methods to study financial markets and to make financial predictions. Machine learning methods are used to identify patterns in financial data, to predict future prices, and to develop new trading strategies.

These are just a few of the many research topics in mathematical finance. Mathematical finance is a vast and growing field, and there are many exciting new directions for research.

Numerical analysis research topics

Numerical analysis is a branch of mathematics that deals with the approximation of functions and solutions to differential equations using numerical methods. Some of the most important research topics in numerical analysis include:

Error analysis

This branch of numerical analysis studies the errors that are introduced when approximate solutions are used to represent exact solutions. Error analysis is used to design numerical methods that are accurate and efficient.

Stability analysis

This branch of numerical analysis studies the stability of numerical methods. Stability analysis is used to design numerical methods that are guaranteed to converge to the correct solution.

Convergence analysis

This branch of numerical analysis studies the convergence of numerical methods. Convergence analysis is used to design numerical methods that will converge to the correct solution in a finite number of steps.

Adaptive methods

This branch of numerical analysis studies adaptive methods. Adaptive methods are numerical methods that can automatically adjust their step size or mesh size to improve accuracy.

Parallel methods

This branch of numerical analysis studies parallel methods. Parallel methods are numerical methods that can be used to solve problems on multiple processors.

Heterogeneous computing

This branch of numerical analysis studies heterogeneous computing. Heterogeneous computing is the use of multiple processors with different architectures to solve problems.

Nonlinear problems

This branch of numerical analysis studies nonlinear problems. Nonlinear problems are problems that cannot be solved using linear methods.

Optimization

This branch of numerical analysis studies methods for finding the best solution to a problem. Optimization methods are used to find the best parameters for a numerical method, to find the best solution to a problem, and to find the best way to solve a problem.

Scientific computing

This branch of numerical analysis studies the use of numerical methods to solve problems in science and engineering. Scientific computing is used to solve problems in areas such as physics, chemistry, biology, and engineering.

This branch of numerical analysis studies the use of numerical methods to solve problems in physics. Computational physics is used to solve problems in areas such as fluid dynamics, solid mechanics, and quantum mechanics.

Computational chemistry

This branch of numerical analysis studies the use of numerical methods to solve problems in chemistry. Computational chemistry is used to solve problems in areas such as molecular dynamics, quantum chemistry, and materials science.

This branch of numerical analysis studies the use of numerical methods to solve problems in biology. Computational biology is used to solve problems in areas such as genetics, molecular biology, and neuroscience.

These are just a few of the many research topics in numerical analysis. Numerical analysis is a vast and growing field, and there are many exciting new directions for research.

Probability research topics

Probability is a branch of mathematics that deals with the analysis of random phenomena. Some of the most important research topics in probability include:

Foundations of probability

This branch of probability studies the axioms and foundations of probability theory. Foundations of probability is important for understanding the basic concepts of probability and for developing new probability theories.

Stochastic processes

This branch of probability studies the evolution of random phenomena over time. Stochastic processes are used to model a wide variety of phenomena, such as stock prices, traffic patterns, and disease outbreaks.

Random graphs

This branch of probability studies graphs whose vertices and edges are chosen randomly. Random graphs are used to model a wide variety of networks, such as social networks, computer networks, and biological networks.

Markov chains

This branch of probability studies stochastic processes whose future state depends only on its current state. Markov chains are used to model a wide variety of phenomena, such as queuing systems, genetics, and epidemiology.

Queueing theory

This branch of probability studies the behavior of queues. Queues are used to model a wide variety of systems, such as call centers, hospitals, and traffic systems.

Optimal stopping theory

This branch of probability studies the problem of choosing when to stop a stochastic process. Optimal stopping theory is used to make decisions in a wide variety of situations, such as gambling, investing, and medical diagnosis.

Information theory

This branch of probability studies the quantification and manipulation of information. Information theory is used in a wide variety of fields, such as communication, cryptography, and machine learning.

Computational probability

This branch of probability studies the use of computers to solve probability problems. Computational probability is used to solve a wide variety of problems, such as simulating random phenomena, computing probabilities, and designing algorithms .

Applied probability

This branch of probability studies the use of probability in other fields, such as physics, chemistry, biology, and economics. Applied probability is used to solve a wide variety of problems in these fields.

These are just a few of the many research topics in probability. Probability is a vast and growing field, and there are many exciting new directions for research.

Statistics research topics

Statistics is a field of study that deals with the collection, analysis, interpretation, presentation, and organization of data. Some of the most important research topics in statistics include:

This branch of statistics studies the analysis of large and complex datasets. Big data is used in a wide variety of fields, such as business, finance, healthcare, and government.

Machine learning

This branch of statistics studies the development of algorithms that can learn from data without being explicitly programmed. Machine learning is used in a wide variety of fields, such as natural language processing, computer vision, and fraud detection.

Data mining

This branch of statistics studies the extraction of knowledge from data. Data mining is used in a wide variety of fields, such as marketing, customer relationship management, and fraud detection.

Bayesian statistics

This branch of statistics uses Bayes’ theorem to update beliefs in the face of new evidence. Bayesian statistics is used in a wide variety of fields, such as medical diagnosis, finance, and weather forecasting.

Nonparametric statistics

This branch of statistics uses methods that do not make assumptions about the distribution of the data. Nonparametric statistics is used in a wide variety of fields, such as social science, medical research, and environmental science.

Multivariate statistics

This branch of statistics studies the analysis of data that has multiple variables. Multivariate statistics is used in a wide variety of fields, such as marketing, finance, and environmental science.

Time series analysis

This branch of statistics studies the analysis of data that changes over time. Time series analysis is used in a wide variety of fields, such as economics, finance, and meteorology.

Survival analysis

This branch of statistics studies the analysis of data that records the time until an event occurs. Survival analysis is used in a wide variety of fields, such as medical research, epidemiology, and finance.

Quality control

This branch of statistics studies the methods used to ensure that products or services meet a certain level of quality. Quality control is used in a wide variety of fields, such as manufacturing, healthcare, and government.

These are just a few of the many research topics in statistics. Statistics is a vast and growing field, and there are many exciting new directions for research.

How to find math research topics

Here are some tips on how to find math research topics:

Talk to your professors and advisors

They will be able to give you insights into current research in your area of interest and help you identify potential topics.

Read math journals and conferences

This will help you stay up-to-date on the latest research and identify areas where you could make a contribution.

Attend math conferences and workshops

This is a great way to meet other mathematicians and learn about their research.

Think about your own interests and passions

What are you curious about? What do you want to learn more about? These can be great starting points for research topics.

Don’t be afraid to ask for help. If you’re struggling to find a research topic, talk to your professors, advisors, or other mathematicians. They will be happy to help you get started.

How to get started with math research

Getting started with math research can be daunting, but it doesn’t have to be. Here are some tips to help you get started:

Find a mentor

A mentor can help you find a research topic, develop your research skills, and navigate the research process. Talk to your professors, advisors, or other mathematicians to find someone who is interested in your research interests.

Do your research

Read articles, books, and papers on your topic. Talk to experts in the field. The more you know about your topic, the better equipped you will be to conduct research.

Develop a research plan

A research plan will help you stay organized and on track. It should include your research goals, methods, and timeline.

Research can be a slow and challenging process. Don’t get discouraged if you don’t make progress immediately. Just keep working hard and you will eventually reach your goals.

Start small

Don’t try to tackle too much at once. Start with a small research project that you can complete in a reasonable amount of time.

Get feedback

Share your work with others and get their feedback. This will help you identify areas where you can improve.

Don’t be afraid to ask for help

If you’re struggling with something, don’t be afraid to ask for help from your mentor, advisor, or other mathematicians.

Research can be a rewarding experience. By following these tips, you can increase your chances of success.

In conclusion, exploring math research topics provides an opportunity to delve into the fascinating world of mathematics and contribute to its advancement.

The wide range of potential research areas ensures that there is something for everyone, whether you are interested in pure mathematics, applied mathematics, or interdisciplinary studies. By engaging in math research, you can deepen your understanding of mathematical principles, develop problem-solving skills, and contribute to the collective knowledge of the field.

Remember to choose a research topic that aligns with your interests and goals, and seek guidance from mentors and experts in the field to maximize your research potential. Embrace the challenge, curiosity, and creativity that math research offers, and embark on a journey that can lead to exciting discoveries and breakthroughs in the realm of mathematics.

Frequently Asked Question

How do i choose a math research topic.

When choosing a math research topic, consider your interests, background knowledge, and future goals. Explore various branches of mathematics and identify areas that intrigue you. Additionally, consult with professors, mentors, and professionals in the field for guidance and suggestions.

Can I pursue research in math as an undergraduate student?

Yes, many universities and research institutions offer opportunities for undergraduate students to engage in math research. Reach out to your professors or department advisors to inquire about available research programs or projects suitable for undergraduates.

What are some emerging areas in math research?

Math research is a constantly evolving field. Some emerging areas include computational mathematics, data science, cryptography, mathematical biology, quantum computing, and mathematical physics. Staying updated with current research trends and attending conferences or seminars can help you identify new and exciting research avenues.

How can I conduct math research effectively?

Effective math research involves a systematic approach. Start by thoroughly understanding the existing literature on your chosen topic. Develop clear research questions and hypotheses, and apply appropriate mathematical techniques and methodologies.

Can math research have real-world applications?

Absolutely! Math research has numerous real-world applications in fields such as engineering, finance, computer science, cryptography, data analysis, and physics. Mathematical models and algorithms play a crucial role in solving complex problems and optimizing various processes in diverse industries.

What resources can I use for math research?

Utilize academic journals, online databases, research papers, books, and mathematical software to access relevant information and tools. Libraries, online platforms, and research institutions also provide access to valuable resources and databases specific to mathematical research.

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Analysis & pde, applied math, combinatorics, financial math, number theory, probability, representation theory, symplectic geometry & topology.

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.

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.

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:

• 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.

• 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:

• 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?

• 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.

• 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?

• 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:

• 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?

• 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!

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Examining purposeful researchable questions in mathematics education

Journal of Honai Math

Often general, and frequently involving scholarly concepts, research questions are the cornerstone of studies. Thus, from their precise wording to their context, research questions play a vital role in uncovering information, determining answers given by participants, and drawing conclusions. However, poorly structured research questions and misalignments to purpose within theoretical and empirical studies can lead to miscommunication and unanswered queries. To this point, this paper discusses the importance of asking purposeful researchable questions in mathematics education and examines what purposeful questioning in mathematics education using quantitative and qualitative research designs entails. An extensive review of literature, is presented with the purpose of identifying strategies for asking purposeful questions, exploring various criteria for judging researchable questions in mathematics education, and discussing the importance of aligning research questions to methodology...

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With the development of qualitative methodologies, interviewing has become one of the main tools in mathematics education research. As the first step in analyzing interviewing in mathematics education we focus here on the stage of planning, specifically, on designing the interview questions. We attempt to outline several features of interview questions and understand what guides researchers in choosing the interview questions. Our observations and conclusions are based on examining research in mathematics education that uses interviews as a data-collection tool and on interviews with practicing researchers reflecting on their practice.

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CITATION: Stinson, D. W., & Walshaw, M. (2017). “Theory at the crossroads”: Mapping moments of mathematics education research onto paradigms of inquiry. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1407–1414). Hoosier Association of Mathematics Teachers Educators: Indianapolis, IN. ABSTRACT: In this essay, traveling through the past half-century, the authors illustrate how mathematics education research shifted, theoretical, beyond its psychological and mathematical roots. Mapping four historical moments of mathematics education research onto broader paradigms of inquiry, the authors make a case for the field to take up a theoretical " identity " that refutes closure and keeps the possibilities of mathematics teaching and learning open to multiple and uncertain interpretations and analyses.

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In this essay, traveling through the past half-century, the authors illustrate how mathematics education research shifted, theoretical, beyond its psychological and mathematical roots. Mapping four historical moments of mathematics education research onto broader paradigms of inquiry, the authors make a case for the field to take up a theoretical “identity” that refutes closure and keeps the possibilities of mathematics teaching and learning open to multiple and uncertain interpretations and analyses.

Edrine Mutebi CFA

Mutebi Edrine

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Niroj Dahal

My inquiry portrayed the existing classroom practices in mathematics pedagogy on understanding and uses of questioning by mathematics teachers. For this, narrative inquiry approach has been used to focus on experiences of six mathematics teachers working in schools in Kathmandu Valley, Nepal, by using criterion-based selection strategy to choose my participants to be involved in this research (Roulston, 2010). It aims to examine the complexities of experiences by gaining insight into how understanding and uses of questioning in mathematics classroom are embedded in mathematics teachers’ multiple and uniquely situated experiences, and in doing so, this inquiry views from various theoretical lens, namely, sociological perspectives, behaviorists to constructivists approaches, categories of questioning as per expertise, critical pedagogical perspectives and algorithmic and daily life practices, for analysis how interlock to create unequal power relations in mathematics classroom exist while questioning from teachers' view. With those issues in mind, this study was designed to explore the following research question: How do teachers narrate their experience of understanding and usage of questioning in relation to mathematics pedagogy? Subscribing to a narrative inquiry for meaning-making, my study foregrounds the six mathematics teachers voices and experiences, power relationship about whose experiences are valued and whose voice can be heard in their mathematics classroom while questioning the students. In keeping with narrative inquiry approaches, I use a more personal voice to reflect on mathematics teachers' understanding and uses of questioning which is an ethical challenge involved in the research process, namely: Issues of representation, as well as struggles relating to voice, is at the core of the study and reflexively considered throughout. Near to final, in conclusion of my study, the majority of the mathematics teachers seem to be conformist mathematics teacher at the beginning of their teaching career but later on, they were nonconformist by being flexible enough in questioning. Further, the majority of my research participants asked more questions within the simple to complex level and highly focusing on simple (low level) questioning, claiming to encourage students in mathematical discussion. Finally, it attempts to be an example of ethical and respectful research and claims to increase understanding of how mathematics teacher understanding and uses of questioning in the mathematics classrooms in Kathmandu Valley, Nepal.

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all the numbers are changing, but what doesn't change is the relationship between x and y: y is always one more than twice x. That is, y=2x+1. Finding what doesn't change "tames" the situation. So, you have tamed this problem! Yay. And if you want a fancy mathematical name for things that don’t vary, we call these things "invariants." The number of messed-up recruits is invariant, even though they are all wiggling back and forth, trying to figure out which way is right!

3) Encourage generalizations

So, of course, the next question that comes to my mind is how to generalize what you’ve already discovered: there are 15 ways that 2 mistakes can be arranged in a line of 6 recruits. What about a different number of mistakes? Or a different number of recruits? Is there some way to predict? Or, alternatively, is there some way to predict how these 15 ways of making mistakes will play out as the recruits try to settle themselves down? Which direction interests you?

4) Inquire about reasoning and rigor

The students were looking at the number of ways the recruits could line up with 2 out of n faced the wrong way: Anyway, I had a question of my own. It looks like the number of possibilities increases pretty fast, as the number of recruits increases. For example, I counted 15 possibilities in your last set (the line of six). What I wonder is this: when the numbers get that large, how you can possibly know that you've found all the possibilities? (For example, I noticed that >>>><< is missing.) The question "How do I know I've counted 'em all?" is actually quite a big deal in mathematics, as mathematicians are often called upon to find ways of counting things that nobody has ever listed (exactly like the example you are working on).

The students responded by finding a pattern for generating the lineups in a meaningful order: The way that we can prove that we have all the possibilities is that we can just add the number of places that the second wrong person could be in. For example, if 2 are wrong in a line of 6, then the first one doesn’t move and you count the space in which the second one can move in. So for the line of six, it would be 5+4+3+2+1=15. That is the way to make sure that we have all the ways. Thanks so much for giving challenges. We enjoyed thinking!

5) Work towards proof

a) The group wrote the following: When we found out that 6 recruits had 15 different starting arrangements, we needed more information. We needed to figure out how many starting positions are there for a different number of recruits.

By drawing out the arrangements for 5 recruits and 7 recruits we found out that the number of starting arrangements for the recruit number before plus that recruit number before it would equal the number of starting arrangements for that number of recruits.

We also found out that if you divide the starting arrangements by the number of recruits there is a pattern.

To which the mentor replied: Wow! I don't think (in all the years I've been hanging around mathematics) I've ever seen anyone describe this particular pattern before! Really nice! If you already knew me, you'd be able to predict what I'm about to ask, but you don't, so I have to ask it: "But why?" That is, why is this pattern (the 6, 10, 15, 21, 28…) the pattern that you find for this circumstance (two recruits wrong in lines of lengths, 4, 5, 6, 7, 8…)? Answering that—explaining why you should get those numbers and why the pattern must continue for longer lines—is doing the kind of thing that mathematics is really about.

b) Responding to students studying a circular variation of raw recruits that never settled down: This is a really interesting conclusion! How can you show that it will always continue forever and that it doesn’t matter what the original arrangement was? Have you got a reason or did you try all the cases or…? I look forward to hearing more from you.

6) Distinguish between examples and reasons

a) You have very thoroughly dealt with finding the answer to the problem you posed—it really does seem, as you put it, "safe to say" how many there will be. Is there a way that you can show that that pattern must continue? I guess I’d look for some reason why adding the new recruit adds exactly the number of additional cases that you predict. If you could say how the addition of one new recruit depends on how long the line already is, you’d have a complete proof. Want to give that a try?

b) A student, working on Amida Kuji and having provided an example, wrote the following as part of a proof: In like manner, to be given each relationship of objects in an arrangement, you can generate the arrangement itself, for no two different arrangements can have the same object relationships. The mentor response points out the gap and offers ways to structure the process of extrapolating from the specific to the general: This statement is the same as your conjecture, but this is not a proof. You repeat your claim and suggest that the example serves as a model for a proof. If that is so, it is up to you to make the connections explicit. How might you prove that a set of ordered pairs, one per pair of objects forces a unique arrangement for the entire list? Try thinking about a given object (e.g., C) and what each of its ordered pairs tells us? Try to generalize from your example. What must be true for the set of ordered pairs? Are all sets of n C2 ordered pairs legal? How many sets of n C2 ordered pairs are there? Do they all lead to a particular arrangement? Your answers to these questions should help you work toward a proof of your conjecture.

9) Encourage extensions

What you’ve done—finding the pattern, but far more important, finding the explanation (and stating it so clearly)—is really great! (Perhaps I should say "finding and stating explanations like this is real mathematics"!) Yet it almost sounded as if you put it down at the very end, when you concluded "making our project mostly an interesting coincidence." This is a truly nice piece of work!

The question, now, is "What next?" You really have completely solved the problem you set out to solve: found the answer, and proved that you’re right!

I began looking back at the examples you gave, and noticed patterns in them that I had never seen before. At first, I started coloring parts red, because they just "stuck out" as noticeable and I wanted to see them better. Then, it occurred to me that I was coloring the recruits that were back-to-back, and that maybe I should be paying attention to the ones who were facing each other, as they were "where the action was," so I started coloring them pink. (In one case, I recopied your example to do the pinks.) To be honest, I’m not sure what I’m looking for, but there was such a clear pattern of the "action spot" moving around that I thought it might tell me something new. Anything come to your minds?

10) Build a Mathematical Community

I just went back to another paper and then came back to yours to look again. There's another pattern in the table. Add the recruits and the corresponding starting arrangements (for example, add 6 and 15) and you get the next number of starting arrangements. I don't know whether this, or your 1.5, 2, 2.5, 3, 3.5… pattern will help you find out why 6, 10, 15… make sense as answers, but they might. Maybe you can work with [your classmates] who made the other observation to try to develop a complete understanding of the problem.

11) Highlight Connections

Your rule—the (n-1)+(n-2)+(n-3)+… +3+2+1 part—is interesting all by itself, as it counts the number of dots in a triangle of dots. See how?

12) Wrap Up

This is really a very nice and complete piece of work: you've stated a problem, found a solution, and given a proof (complete explanation of why that solution must be correct). To wrap it up and give it the polish of a good piece of mathematical research, I'd suggest two things.

The first thing is to extend the idea to account for all but two mistakes and the (slightly trivial) one mistake and all but one mistake. (If you felt like looking at 3 and all but 3, that'd be nice, too, but it's more work—though not a ton—and the ones that I suggested are really not more work.)

The second thing I'd suggest is to write it all up in a way that would be understandable by someone who did not know the problem or your class: clear statement of the problem, the solution, what you did to get the solution, and the proof.

I look forward to seeing your masterpiece!

Advice for Keeping a Formal Mathematics Research Logbook

As part of your mathematics research experience, you will keep a mathematics research logbook. In this logbook, keep a record of everything you do and everything you read that relates to this work. Write down questions that you have as you are reading or working on the project. Experiment. Make conjectures. Try to prove your conjectures. Your journal will become a record of your entire mathematics research experience. Don’t worry if your writing is not always perfect. Often journal pages look rough, with notes to yourself, false starts, and partial solutions. However, be sure that you can read your own notes later and try to organize your writing in ways that will facilitate your thinking. Your logbook will serve as a record of where you are in your work at any moment and will be an invaluable tool when you write reports about your research.

Ideally, your mathematics research logbook should have pre-numbered pages. You can often find numbered graph paper science logs at office supply stores. If you can not find a notebook that has the pages already numbered, then the first thing you should do is go through the entire book putting numbers on each page using pen.

• Date each entry.

• Work in pen.

• Don’t erase or white out mistakes. Instead, draw a single line through what you would like ignored. There are many reasons for using this approach:

– Your notebook will look a lot nicer if it doesn’t have scribbled messes in it.

– You can still see what you wrote at a later date if you decide that it wasn’t a mistake after all.

– It is sometimes useful to be able to go back and see where you ran into difficulties.

– You’ll be able to go back and see if you already tried something so you won’t spend time trying that same approach again if it didn’t work.

• When you do research using existing sources, be sure to list the bibliographic information at the start of each section of notes you take. It is a lot easier to write down the citation while it is in front of you than it is to try to find it at a later date.

• Never tear a page out of your notebook. The idea is to keep a record of everything you have done. One reason for pre-numbering the pages is to show that nothing has been removed.

• If you find an interesting article or picture that you would like to include in your notebook, you can staple or tape it onto a page.

Advice for Keeping a Loose-Leaf Mathematics Research Logbook

Get yourself a good loose-leaf binder, some lined paper for notes, some graph paper for graphs and some blank paper for pictures and diagrams. Be sure to keep everything that is related to your project in your binder.

– Your notebook will look a lot nicer if it does not have scribbled messes in it.

• Be sure to keep everything related to your project. The idea is to keep a record of everything you have done.

• If you find an interesting article or picture that you would like to include in your notebook, punch holes in it and insert it in an appropriate section in your binder.

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Department members engage in cutting-edge research on a wide variety of topics in mathematics and its applications. Topics continually evolve to reflect emerging interests and developments, but can roughly grouped into the following areas.

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The diversity of this group is reflected in its research interests: numerical analysis of partial differential equations , adaptive methods for scientific computing, computational methods of fluid dynamics and turbulence, numerical solution of nonlinear problems arising from porous media flow and transport, optimal control, and simulation of stochastic reaction diffusion systems.

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Getting smart collective, impact update, talking math: 100 questions that help promote mathematical discourse.

Update 2022: For a recent article on how to talk about math, click here. 

Think about the questions that you ask in your math classroom. Can they be answered with a simple “yes” or “no,” or do they open a door for students to really share their knowledge in a way that highlights their true understanding and uncovers their misunderstandings? Asking better questions can open new doors for students, helping to promote mathematical thinking and encouraging classroom discourse. Such questions help students:

• Work together to make sense of mathematics.
• Rely more on themselves to determine whether something is mathematically correct.
• Learn to reason mathematically.
• Evaluate their own processes and engage in productive peer interaction.
• Discover and seek help with problems in their comprehension.
• Learn to conjecture, invent and solve problems.
• Learn to connect mathematics, its ideas and its applications.
• Focus on the mathematical skills embedded within activities.

Dr. Gladis Kersaint

Help students work together to make sense of mathematics

• What strategy did you use?
• Do you agree?
• Do you disagree?
• Would you ask the rest of the class that question?
• Could you share your method with the class?
• What part of what he said do you understand?
• Would someone like to share ___?
• Can you convince the rest of us that that makes sense?
• What do others think about what [student] said?
• Can someone retell or restate [student]’s explanation?
• Did you work together? In what way?
• Would anyone like to add to this?
• Have you discussed this with your group? With others?
• Did anyone get a different answer?
• Where would you go for help?
• Did everybody get a fair chance to talk, to use the manipulatives, or to be recorded?
• How could you help another student without telling the answer?
• How would you explain ___ to someone who missed class today?

Refer questions raised by students back to the class.

Help students rely more on themselves to determine whether something is mathematically correct

• Is this a reasonable answer?
• Does that make sense?
• Why do you think that? Why is that true?
• Can you draw a picture or make a model to show that?
• How did you reach that conclusion?
• Does anyone want to revise his or her answer?
• How were you sure your answer was right?

Help students learn to reason mathematically

• How did you begin to think about this problem?
• What is another way you could solve this problem?
• How could you prove that?
• Can you explain how your answer is different from or the same as [student]’s?
• Let’s see if we can break it down. What would the parts be?
• Can you explain this part more specifically?
• Does that always work?
• Is that true for all cases?
• How did you organize your information? Your thinking?

Help students evaluate their own processes and engage in productive peer interaction

• What do you need to do next?
• What have you accomplished?
• What are your strengths and weaknesses?
• Was your group participation appropriate and helpful?

Help students with problem comprehension

• What is this problem about? What can you tell me about it?
• Do you need to define or set limits for the problem?
• How would you interpret that?
• Would you please reword that in simpler terms?
• Is there something that can be eliminated or that is missing?
• Would you please explain that in your own words?
• What assumptions do you have to make?
• What do you know about this part?
• Which words were most important? Why?

Help students learn to conjecture, invent and solve problems

• What would happen if ___? What if not?
• Do you see a pattern?
• What are some possibilities here?
• Where could you find the information you need?
• How would you check your steps or your answer?
• What did not work?
• How is your solution method the same as or different from [student]’s?
• Other than retracing your steps, how can you determine if your answers are appropriate?
• What decision do you think he or she should make?
• How did you organize the information? Do you have a record?
• How could you solve this using (tables, trees, lists, diagrams, etc.)?
• What have you tried? What steps did you take?
• How would it look if you used these materials?
• How would you draw a diagram or make a sketch to solve the problem?
• Is there another possible answer? If so, explain.
• How would you research that?
• Is there anything you’ve overlooked?
• How did you think about the problem?
• What was your estimate or prediction?
• How confident are you in your answer?
• What else would you like to know?
• What do you think comes next?
• Is the solution reasonable, considering the context?
• Did you have a system? Explain it.
• Did you have a strategy? Explain it.
• Did you have a design? Explain it.

Help students learn to connect mathematics, its ideas and its application

• What is the relationship of this to that?
• Have we ever solved a problem like this before?
• What uses of mathematics did you find in the newspaper last night?
• What is the same?
• What is different?
• Did you use skills or build on concepts that were not necessarily mathematical?
• Which skills or concepts did you use?
• What ideas have we explored before that were useful in solving this problem?
• Is there a pattern?
• Where else would this strategy be useful?
• How does this relate to ___?
• Is there a general rule?
• Is there a real-life situation where this could be used?
• How would your method work with other problems?
• What other problem does this seem to lead to?

Help students persevere

• Have you tried making a guess?
• What else have you tried?
• Would another recording method work as well or better?
• Is there another way to (draw, explain, say) that?
• Give me another related problem. Is there an easier problem?
• How would you explain what you know right now?

Help students focus on the mathematics from activities

• What was one thing you learned (or two, or more)?
• Where would this problem fit on our mathematics chart?
• How many kinds of mathematics were used in this investigation?
• What were the mathematical ideas in this problem?
• What is the mathematically different about these two situations?
• What are the variables in this problem? What stays constant?

Facilitating student engagement in mathematical discourse begins with the decisions teachers make when they plan classroom instruction. In the next and final blog in this series, we will dive into the specific strategies that teachers can use to foster meaningful conversations about what students are thinking, doing and learning.

This blog is part of a three-post series on the importance of mathematical discourse from Curriculum Associates   and Dr. Gladis Kersaint, the author of the recently published whitepaper Orchestrating Mathematical Discourse to Enhance Student Learning . Download your free copy here . For more on mathematical discourse and Curriculum Associates, check out:

• Talking Math: How to Engage Students in Mathematical Discourse
• Talking Math: 6 Strategies for Getting Students to Engage in Mathematical Discourse
• Curriculum Associates: Leveraging For-profit Power With a Nonprofit Purpose

Dr. Gladis Kersaint is a Professor of Mathematics Education at the University of Connecticut.

Stay in-the-know with all things EdTech and innovations in learning by signing up to receive the weekly Smart Update .  This post includes mentions of a Getting Smart partner. For a full list of partners, affiliate organizations and all other disclosures please see our Partner page .

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I love how you have the questions categorized by outcome goal. The infographic is one that I will be printing and using very often next year in my middle school classroom.

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It's an amazing application and approach in addressing math issues.Shall use them in my class .

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Articles on Mathematics

Displaying 1 - 20 of 553 articles.

Maths makes finding bat roosts much easier, our research shows

Thomas Woolley , Cardiff University and Fiona Mathews , University of Sussex

Why expanding access to algebra is a matter of civil rights

Liza Bondurant , Mississippi State University

What toilet paper and game shows can teach us about the spread of epidemics

Matthew Ryan , CSIRO ; Emily Brindal , CSIRO , and Roslyn Hickson , James Cook University

How the 18th-century ‘probability revolution’ fueled the casino gambling craze

John Eglin , University of Montana

Australian teenagers are curious but have some of the most disruptive maths classes in the OECD

Lisa De Bortoli , Australian Council for Educational Research

‘Dancing’ raisins − a simple kitchen experiment reveals how objects can extract energy from their environment and come to life

Saverio Eric Spagnolie , University of Wisconsin-Madison

Why are algorithms called algorithms? A brief history of the Persian polymath you’ve likely never heard of

Debbie Passey , The University of Melbourne

Homeschooled kids face unique college challenges − here are 3 ways they can be overcome

Kenneth V. Anthony , Mississippi State University and Mark E. Wildmon , Mississippi State University

Maths degrees are becoming less accessible – and this is a problem for business, government and innovation

Neil Saunders , City, University of London

Supercomputers can take months to simulate the climate – but my new algorithm can do it ten times faster

Samar Khatiwala , University of Oxford

Mind-bending maths could stop quantum hackers, but few understand it

Nalini Joshi , University of Sydney

The luck of the puck in the Stanley Cup – why chance plays such a big role in hockey

Mark Robert Rank , Arts & Sciences at Washington University in St. Louis

From thousands to millions to billions to trillions to quadrillions and beyond: Do numbers ever end?

Manil Suri , University of Maryland, Baltimore County

It’s common to ‘stream’ maths classes. But grouping students by ability can lead to ‘massive disadvantage’

Elena Prieto-Rodriguez , University of Newcastle

South Africa is short of academic statisticians: why and what can be done

Inger Fabris-Rotelli , University of Pretoria ; Ansie Smit , University of Pretoria ; Danielle Jade Roberts , University of KwaZulu-Natal ; Daniel Maposa , University of Limpopo ; Fabio Mathias Correa , University of the Free State ; Michael Johan von Maltitz , University of the Free State , and Sonali Das , University of Pretoria

Dali hit Key Bridge with the force of 66 heavy trucks at highway speed

Amanda Bao , Rochester Institute of Technology

How do we solve the maths teacher shortage? We can start by training more existing teachers to teach maths

Ian Gordon , The University of Melbourne ; Mary P. Coupland , University of Technology Sydney , and Merrilyn Goos , University of the Sunshine Coast

How AI and a popular card game can help engineers predict catastrophic failure – by finding the absence of a pattern

John Edward McCarthy , Arts & Sciences at Washington University in St. Louis

The ‘average’ revolutionized scientific research, but overreliance on it has led to discrimination and injury

Zachary del Rosario , Olin College of Engineering

Understanding how the brain works can transform how school students learn maths

Colin Foster , Loughborough University

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Merging AI and human efforts to tackle complex mathematical problems

by Ingrid Fadelli , Phys.org

By rapidly analyzing large amounts of data and making accurate predictions, artificial intelligence (AI) tools could help to answer many long-standing research questions. For instance, they could help to identify new materials to fabricate electronics or the patterns in brain activity associated with specific human behaviors.

One area in which AI has so far been rarely applied is number theory, a branch of mathematics focusing on the study of integers and arithmetic functions. Most research questions in this field are solved by human mathematicians, often years or decades after their initial introduction.

Researchers at the Israel Institute of Technology (Technion) recently set out to explore the possibility of tackling long-standing problems in number theory using state-of-the-art computational models.

In a recent paper , published in the Proceedings of the National Academy of Sciences , they demonstrated that such a computational approach can support the work of mathematicians, helping them to make new exciting discoveries.

"Computer algorithms are increasingly dominant in scientific research , a practice now broadly called 'AI for Science,'" Rotem Elimelech and Ido Kaminer, authors of the paper, told Phys.org.

"However, in fields like number theory, advances are often attributed to creativity or human intuition. In these fields, questions can remain unresolved for hundreds of years, and while finding an answer can be as simple as discovering the correct formula, there is no clear path for doing so."

Elimelech, Kaminer and their colleagues have been exploring the possibility that computer algorithms could automate or augment mathematical intuition. This inspired them to establish the Ramanujan Machine research group, a new collaborative effort aimed at developing algorithms to accelerate mathematical research.

Their research group for this study also included Ofir David, Carlos de la Cruz Mengual, Rotem Kalisch, Wolfram Berndt, Michael Shalyt, Mark Silberstein, and Yaron Hadad.

"On a philosophical level, our work explores the interplay between algorithms and mathematicians," Elimelech and Kaminer explained. "Our new paper indeed shows that algorithms can provide the necessary data to inspire creative insights, leading to discoveries of new formulas and new connections between mathematical constants."

The first objective of the recent study by Elimelech, Kaminer and their colleagues was to make new discoveries about mathematical constants. While working toward this goal, they also set out to test and promote alternative approaches for conducting research in pure mathematics.

"The 'conservative matrix field' is a structure analogous to the conservative vector field that every math or physics student learns about in first year of undergrad," Elimelech and Kaminer explained. "In a conservative vector field, such as the electric field created by a charged particle, we can calculate the change in potential using line integrals.

"Similarly, in conservative matrix fields, we define a potential over a discrete space and calculate it through matrix multiplications rather than using line integrals. Traveling between two points is equivalent to calculating the change in the potential and it involves a series of matrix multiplications."

In contrast with the conservative vector field, the so-called conservative matrix field is a new discovery. An important advantage of this structure is that it can generalize the formulas of each mathematical constant, generating infinitely many new formulas of the same kind.

"The way by which the conservative matrix field creates a formula is by traveling between two points (or actually, traveling from one point all the way to infinity inside its discrete space)," Elimelech and Kaminer said. "Finding non-trivial matrix fields that are also conservative is challenging."

As part of their study, Elimelech, Kaminer and their colleagues used large-scale distributed computing, which entails the use of multiple interconnected nodes working together to solve complex problems. This approach allowed them to discover new rational sequences that converge to fundamental constants (i.e., formulas for these constants).

"Each sequence represents a path hidden in the conservative matrix field," Elimelech and Kaminer explained. "From the variety of such paths, we reverse-engineered the conservative matrix field. Our algorithms were distributed using BOINC , an infrastructure for volunteer computing. We are grateful to the contribution by hundreds of users worldwide who donated computation time over the past two and a half years, making this discovery possible."

The recent work by the research team at the Technion demonstrates that mathematicians can benefit more broadly from the use of computational tools and algorithms to provide them with a "virtual lab." Such labs provide an opportunity to try ideas experimentally in a computer, resembling the real experiments available in physics and in other fields of science. Specifically, algorithms can carry out mathematical experiments providing formulas that can be used to formulate new mathematical hypotheses.

"Such hypotheses, or conjectures, are what drives mathematical research forward," Elimelech and Kaminer said. "The more examples supporting a hypothesis, the stronger it becomes, increasing the likelihood to be correct. Algorithms can also discover anomalies, pointing to phenomena that are the building-blocks for new hypotheses. Such discoveries would not be possible without large-scale mathematical experiments that use distributed computing."

Another interesting aspect of this recent study is that it demonstrates the advantages of building communities to tackle problems. In fact, the researchers published their code online from their project's early days and relied on contributions by a large network of volunteers.

"Our study shows that scientific research can be conducted without exclusive access to supercomputers, taking a substantial step toward the democratization of scientific research," Elimelech and Kaminer said. "We regularly post unproven hypotheses generated by our algorithms, challenging other math enthusiasts to try proving these hypotheses, which when validated are posted on our project website . This happened on several occasions so far. One of the community contributors, Wolfgang Berndt, got so involved that he is now part of our core team and a co-author on the paper."

The collaborative and open nature of this study allowed Elimelech, Kaminer and the rest of the team to establish new collaborations with other mathematicians worldwide. In addition, their work attracted the interest of some children and young people, showing them how algorithms and mathematics can be combined in fascinating ways.

In their next studies, the researchers plan to further develop the theory of conservative matrix fields. These matrix fields are a highly powerful tool for generating irrationality proofs for fundamental constants, which Elimelech, Kaminer and the team plan to continue experimenting with.

"Our current aim is to address questions regarding the irrationality of famous constants whose irrationality is unknown, sometimes remaining an open question for over a hundred years, like in the case of the Catalan constant ," Elimelech and Kaminer said.

"Another example is the Riemann zeta function, central in number theory , with its zeros at the heart of the Riemann hypothesis, which is perhaps the most important unsolved problem in pure mathematics. There are many open questions about the values of this function, including the irrationality of its values. Specifically, whether ζ(5) is irrational is an open question that attracts the efforts of great mathematicians."

The ultimate goal of this team of researchers is to successfully use their experimental mathematics approach to prove the irrationality of one of these constants. In the future, they also hope to systematically apply their approach to a broader range of problems in mathematics and physics. Their physics-inspired hands-on research style arises from the interdisciplinary nature of the team, which combines people specialized in CS, EE, math, and physics.

"Our Ramanujan Machine group can help other researchers create search algorithms for their important problems and then use distributed computing to search over large spaces that cannot be attempted otherwise," Elimelech and Kaminer added. "Each such algorithm, if successful, will help point to new phenomena and eventually new hypotheses in mathematics, helping to choose promising research directions. We are now considering pushing forward this strategy by setting up a virtual user facility for experimental mathematics," inspired by the long history and impact of user facilities for experimental physics.

Journal information: Proceedings of the National Academy of Sciences

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How to help kids manage math anxiety

The consequences of math anxiety can be daunting, but caregivers and educators can take steps to help kids manage their fear of math

• Schools and Classrooms

From buying an area rug to planning for retirement, math is a fact of life. But for people with math anxiety, such tasks can bring a feeling of dread.

Math anxiety is a worry or fear that occurs when people try to solve math problems, take tests, or even think about numbers. “For adults who don’t have it, it can be strange to imagine someone being so scared of mathematics,” said Pooja Sidney, PhD, a developmental cognitive psychologist at the University of Kentucky who studies math anxiety. “But a child with math anxiety can’t just get over it because an adult tells them math is easy.”

Causes and consequences

For the math anxious, dealing with numbers is literally painful. Researchers used fMRI to scan the brains of people with math anxiety and found that just the anticipation of doing math increased activity in brain regions associated with detecting threats and experiencing pain ( Lyons, I. M., & Beilock, S. L., PLOS ONE , Vol. 7, No. 10, 2012 ).

Math anxiety can begin as early as elementary school. For others, it shows up around middle school, when math classes start getting harder and more complex. Estimates suggest about 20% to 25% of kids experience at least some math anxiety, and it often lasts into adulthood.

Math anxiety is higher in women and girls than it is in men and boys. That is not because boys are naturally better at crunching numbers, experts explain. “There’s this societal stereotype that math is a masculine domain, and a lot of women incorporate that idea,” said Molly Jameson, PhD, an educational psychologist at the University of Northern Colorado. That belief can fuel math anxiety. Her research shows that one big factor in math anxiety is a child’s “math self-concept”—in other words, their confidence in their ability to tackle a math problem ( Journal of Experimental Education , Vol. 82, No. 4, 2014 ).

Some people do develop math anxiety because they have not mastered the foundational skills. It is hard to master fractions, for example, if you do not yet have a grasp on whole numbers. But even those who do well in the subject can develop it. “People with high math anxiety aren’t necessarily lacking knowledge,” Jameson said. High anxiety might prevent them from tapping into that knowledge, however. “It’s an overreaction to math that takes up resources in their brains and prevents them from accessing the skills they need to solve the problem,” she added.

That overreaction can affect people’s academic achievement and future choices. People with high math anxiety are less likely to finish graduate school or go into science, technology, engineering, and math (STEM) careers ( Hart, S. A., & Ganley, C. M., Journal of Numerical Cognition , Vol. 5, No. 2, 2019 ). Fear of numbers can also hold people back in their daily lives. “It can lead them to avoid everyday situations using math, including money management and salary negotiation,” said Sian Beilock, PhD, a cognitive scientist who studies math anxiety and is president of Dartmouth College.

Managing math anxiety

The consequences of math anxiety can be daunting. But caregivers and educators can take steps to help kids manage their fear of math.

Spot the signs. If your child is struggling, talk to their teachers. Instructors may recommend you get an evaluation for learning disabilities such as dyscalculia, a condition that impairs math ability. It is worth ruling out, but math anxiety is more common than dyscalculia. If you suspect anxiety, talk to your child. Are they nervous when studying, or just frustrated? Kids with math anxiety tend to use negative statements (“I’m so dumb at math”) and go out of their way to avoid math, like staying home sick on test days.

Promote mental health. Math anxiety is more common in people who experience general anxiety. If your child seems anxious in other situations, treating overall anxiety can help improve math-specific anxiety.

Watch your language. Messages kids hear about math can communicate that math is something to avoid. An offhand comment like “Let’s put away our math books now and do something fun” suggests that math is anything but. Be careful not to talk about math in negative terms. That is especially true if you have math anxiety of your own. “Kids learn from the adults around them,” Beilock said.

Help kids grow. When it comes to math, a lot of people have a “fixed mindset”—they think math skills are something you either have or you don’t. “But anybody can learn math,” Jameson said. Help kids develop a “growth mindset”—a belief that abilities can improve over time. One way to do that is to focus on effort, not results: Instead of making a big deal about the grade they got on their math test, praise them for how hard they worked to learn a tricky new math skill.

Improve study skills. Beilock’s research shows that math anxiety can lead kids to avoid studying. They can end up less prepared for tests, and even more anxious. Math is cumulative, with one lesson building on another. If they are too anxious to learn, they can fall behind and have trouble catching up. To avoid that outcome, help kids practice math, keep up with their work as they move through the curriculum, and develop good overall study skills. “We can teach them better study habits and help them confront the problems that are making them feel anxious,” she said.

Math does not have to be the enemy. Helping kids face their fears can set them up for success in school and in life.

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• Artificial Intelligence /

Perplexity’s ‘Pro Search’ AI upgrade makes it better at math and research

The ai search startup facing questions about its ethics says its pro search is capable enough to ‘pinpoint case laws for attorneys.’.

By Emma Roth , a news writer who covers the streaming wars, consumer tech, crypto, social media, and much more. Previously, she was a writer and editor at MUO.

Share this story

Perplexity has launched a major upgrade to its Pro Search AI tool, which it says “understands when a question requires planning, works through goals step-by-step, and synthesizes in-depth answers with greater efficiency.”

Examples on Perplexity’s website of what Pro Search can do include a query asking the best time to see the northern lights in Iceland or Finland. It breaks down its research process into three searches: the best times to see the northern lights in Iceland and Finland; the top viewing locations in Iceland; and the top viewing locations in Finland. It then provides a detailed answer addressing all aspects of the question, including when to view the northern lights in either country and where.

• Perplexity’s grand theft AI 

Perplexity’s AI search tool can also generate a detailed report based on a prompt with a feature called Pages. But recent reports from  Wired   and  Forbes  have accused Perplexity of committing plagiarism, with a report from  Wired  calling the self-proclaimed “answer engine” a “Bullshit Machine,” with animations that misrepresent what it’s doing and data scrapers that bypass rules in robots.txt files .

Aside from conducting more thorough research, Perplexity says Search Pro now comes with more advanced math and programming capabilities that improve its ability to analyze data, debug code, and generate content. If you want to try it out for yourself, you can get five searches per day with a free account. Perplexity also offers 600 Pro searches per day with its $20 per month subscription.

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Young Research Fellow: Philipp Reiser

We are delighted that Dr. Philipp Reiser joins our Cluster as a new Young Research Fellow. He is motivated by the long-standing question of how the topology and geometry of Riemannian manifolds interact. During his stay at Mathematics Münster from 1st to 26th July 2024, he is especially collaborating with Masoumeh Zarei.   Together, they aim to study Riemannian manifolds of non-negative intermediate Ricci curvature. Intermediate Ricci curvature is a family of curvature conditions that interpolates between the classical notions of sectional and Ricci curvature. In particular, they will explore the question of whether there is a bound on the Betti numbers of a closed Riemannian manifold of non-negative intermediate Ricci curvature, which was famously shown by Gromov in the case of sectional curvature. A key ingredient in his proof is Toponogov's angle comparison theorem, which is not expected to hold in the setting of intermediate Ricci curvature. Their strategy includes convergence arguments and understanding the local structure of such spaces by analyzing the distance functions in the context of partial convexity.

Philipp Reiser received his PhD from the Karlsruhe Institute of Technology (KIT) in 2022. Subsequently, he held postdoctoral positions at the Department of Mathematics at the University of Fribourg in Switzerland.

Link: Mathematics Münster's programme for "Young Research Fellows"

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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• Computational Science & Numerical Analysis
• Theoretical Computer Science
• Mathematics of Data

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