mathematics topics on research

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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Research Areas

Analysis and PDE are a major strength of Stanford’s Department of Mathematics, with strong connections to geometry and applied mathematics (since PDE describe fundamental aspects...

Applied mathematics at the Stanford Department of Mathematics focuses, very broadly, on the areas of scientific computing, stochastic modeling, and applied analysis. Some of the more specific...

Combinatorics concerns the study of discrete objects. It has applications to diverse areas of mathematics and science, and has played a particularly important role in...

Currently research in financial mathematics at Stanford is in two broad areas. One is on mathematical problems arising from the analysis of financial data; it involves...

Modern geometry takes many different guises, ranging from geometric topology and algebraic geometry and symplectic geometry to geometric analysis (which has a significant overlap with...

Contemporary number theory is developing rapidly through its interactions with many other areas of mathematics. Insights from ergodic theory have led to dramatic progress in...

The probability group at Stanford is engaged in numerous research activities, including problems from statistical mechanics, analysis of Markov chains, mathematical finance, problems at the...

Representation theory is fundamental in the study of objects with symmetry. It arises in contexts as diverse as card shuffling and quantum mechanics. An early...

Symplectic topology is at the crossroads of several mathematical disciplines such as low-dimensional topology, algebraic geometry, representation theory, Hamiltonian dynamics, integrable systems, mirror symmetry, and string theory. It...

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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.

Back to Science Research Paper Topics .

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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.

Algebra, Combinatorics, and Geometry

Algebra, combinatorics, and geometry are areas of very active research at the University of Pittsburgh.

Analysis and Partial Differential Equations

The research of the analysis group covers functional analysis, harmonic analysis, several complex variables, partial differential equations, and analysis on metric and Carnot-Caratheodory spaces.

Applied Analysis

The department is a leader in the analysis of systems of nonlinear differential equations and dynamical systems  that arise in modeling a variety of physical phenomena. They include problems in biology, chemistry, phase transitions, fluid flow, flame propagation, diffusion processes, and pattern formation in nonlinear stochastic partial differential equations.

Mathematical Biology

The biological world stands as the next great frontier for mathematical modeling and analysis. This group studies complex systems and dynamics arising in various biological phenomena.

Mathematical Finance

A rapidly growing area of mathematical finance is Quantitative Behavioral Finance. The high-tech boom and bust of the late 1990s followed by the housing and financial upheavals of 2008 have made a convincing case for the necessity of adopting broader assumptions in finance.

Numerical Analysis and Scientific Computing

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.

Topology and Differential Geometry

Research in analytic topology continues in the broad area of generalized metric spaces. This group studies relativity theory and differential geometry, with emphasis on twistor methods, as well as geometric and topological aspects of quantum field theory, string theory, and M-theory.

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Applied mathematics articles from across Nature Portfolio

Applied mathematics is the application of mathematical techniques to describe real-world systems and solve technologically relevant problems. This can include the mechanics of a moving body, the statistics governing the atoms in a gas or developing more efficient algorithms for computational analysis. These ideas are closely linked with those of theoretical physics.

Latest Research and Reviews

An air door opening and closing time identification and stage division method based on the wind speed data of a single sensor

• Wentian Shang

Integrated optimization of scheduling for unmanned follow-me cars on airport surface

• Dezhou Yuan
• Xinping Zhu

New ridge parameter estimators for the quasi-Poisson ridge regression model

• Aamir Shahzad
• Muhammad Amin
• Muhammad Faisal

Existence of reservoir with finite-dimensional output for universal reservoir computing

• Shuhei Sugiura
• Ryo Ariizumi
• Shun-ichi Azuma

Response times are affected by mispredictions in a stochastic game

• Paulo Roberto Cabral-Passos
• Antonio Galves
• Claudia D. Vargas

A tripe diffusion bioconvective model for thixotropic nanofluid with applications of induced magnetic field

• Mohammed A. Albedah
• Zhixiong Li
• Iskander Tlili

News and Comment

The curious case of the test set AUROC

The area under the receiver operating characteristic curve (AUROC) of the test set is used throughout machine learning (ML) for assessing a model’s performance. However, when concordance is not the only ambition, this gives only a partial insight into performance, masking distribution shifts of model outputs and model instability.

• Michael Roberts
• Carola-Bibiane Schönlieb

The role of computational science in digital twins

Digital twins hold immense promise in accelerating scientific discovery, but the publicity currently outweighs the evidence base of success. We summarize key research opportunities in the computational sciences to enable digital twin technologies, as identified by a recent National Academies of Sciences, Engineering, and Medicine consensus study report.

• Karen Willcox
• Brittany Segundo

Distilling data into code

One of the greatest limitations of deep neural networks is the difficulty of interpreting what they learn from the data. Deep distilling addresses this problem by extracting human-comprehensible and executable code from observations.

• Joseph Bakarji

Why even specialists struggle with black hole proofs

Mathematical proofs of black hole physics are becoming too complex even for specialists.

• Alejandro Penuela Diaz

Packing finite numbers of spheres efficiently

A paper in Nature Communications reports experiments and simulations of spherical particles that help show how finite numbers of spheres pack in practice.

• Zoe Budrikis

Graph theory captures hard-core exclusion

Physical networks, composed of nodes and links that occupy a spatial volume, are hard to study with conventional techniques. A meta-graph approach that elucidates the impact of physicality on network structure has now been introduced.

• Zoltán Toroczkai

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Applied Mathematics

Faculty and students interested in the applications of mathematics are an integral part of the Department of Mathematics; there is no formal separation between pure and applied mathematics, and the Department takes pride in the many ways in which they enrich each other. We also benefit tremendously from close collaborations with faculty and students in other departments at UC Berkeley as well as scientists at  Lawrence Berkeley National Laboratory  and visitors to the  Mathematical Sciences Research Institute .

The Department regularly offers courses in ordinary and partial differential equations and their numerical solution, discrete applied mathematics, the methods of mathematical physics, mathematical biology, the mathematical aspects of fluid and solid mechanics, approximation theory, scientific computing, numerical linear algebra, and mathematical aspects of computer science. Courses in probability theory, stochastic processes, data analysis and bioinformatics are offered by the Department of Statistics, while courses in combinatorial and convex optimization are offered by the Department of Industrial Engineering and Operations Research. Our students are encouraged to take courses of mathematical interest in these and many other departments. Topics explored intensively by our faculty and students in recent years include scientific computation and the mathematical aspects of quantum theory, computational genomics, image processing and medical imaging, inverse problems, combinatorial optimization, control, robotics, shape optimization, turbulence, hurricanes, microchip failure, MEMS, biodemography, population genetics, phylogenetics, and computational approaches to historical linguistics. Within the department we also have a  Laboratory for Mathematical and Computational Biology .

Chair:   Per-Olof Persson

Applied Mathematics Faculty, Courses, Dissertations

Senate faculty, graduate students, visiting faculty, meet our faculty, mina aganagic, david aldous, robert m. anderson, sunčica čanić, jennifer chayes, alexandre j. chorin, paul concus, james w. demmel, l. craig evans, steven n. evans, f. alberto grünbaum, venkatesan guruswami, ole h. hald, william m. kahan, richard karp, michael j. klass, hendrik w. lenstra, jr., lin lin (林霖), michael j. lindsey, c. keith miller, john c. neu, beresford n. parlett.

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260 Interesting Math Topics for Essays & Research Papers

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

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

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

🔝 Top 10 Interesting Math Topics

✅ branches of mathematics, ✨ fun math topics.

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

🔍 References

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

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

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

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

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

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

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

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

🏫 Math Essay Topics for High School

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

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

🎓 College Math Topics for a Paper

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

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

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

🤔 Advanced Topics in Math to Write a Paper on

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

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

📚 Math Research Topics for a Paper

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

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

• 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:

Get an originally-written paper according to your instructions!

• 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!

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

Table of Contents

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

What is Mathematics?

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

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

Different Branches of Mathematics

Listed below are some popular branches of mathematics.

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

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

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

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

How to Write a Math Research Paper?

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

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

Topic Selection

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

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

Research Paper Writing

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

Introduction

Implications.

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

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

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

List of the Best Math Research Topics

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

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

Simple and Easy Math Topics

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

Basic Math Topics for Middle School Students

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

Interesting Math Topics for High School Students

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

Math Research Paper Topics for Undergraduate Students

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

Math Research Topics for College Students

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

Applied Math Research Topics

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

Business Math Research Topics

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

Advanced Math Research Paper Topics

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

Complex Math Research Topics

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

Math Research Ideas on Probability and Statistics

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

Algebra Research Topics

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

Math Research Paper Topics on Geometry

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

Calculus Research Topics

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

Outstanding Math Research Topics

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

Latest Math Research Topics

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

Final Words

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

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

Just place your order and earn the necessary academic benefits.

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Future themes of mathematics education research: an international survey before and during the pandemic

• Open access
• Published: 06 April 2021
• Volume 107 , pages 1–24, ( 2021 )

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• Arthur Bakker   ORCID: orcid.org/0000-0002-9604-3448 1 ,
• Jinfa Cai   ORCID: orcid.org/0000-0002-0501-3826 2 &
• Linda Zenger 1  

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Before the pandemic (2019), we asked: On what themes should research in mathematics education focus in the coming decade? The 229 responses from 44 countries led to eight themes plus considerations about mathematics education research itself. The themes can be summarized as teaching approaches, goals, relations to practices outside mathematics education, teacher professional development, technology, affect, equity, and assessment. During the pandemic (November 2020), we asked respondents: Has the pandemic changed your view on the themes of mathematics education research for the coming decade? If so, how? Many of the 108 respondents saw the importance of their original themes reinforced (45), specified their initial responses (43), and/or added themes (35) (these categories were not mutually exclusive). Overall, they seemed to agree that the pandemic functions as a magnifying glass on issues that were already known, and several respondents pointed to the need to think ahead on how to organize education when it does not need to be online anymore. We end with a list of research challenges that are informed by the themes and respondents’ reflections on mathematics education research.

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Avoid common mistakes on your manuscript.

1 An international survey in two rounds

Around the time when Educational Studies in Mathematics (ESM) and the Journal for Research in Mathematics Education (JRME) were celebrating their 50th anniversaries, Arthur Bakker (editor of ESM) and Jinfa Cai (editor of JRME) saw a need to raise the following future-oriented question for the field of mathematics education research:

Q2019: On what themes should research in mathematics education focus in the coming decade?

To that end, we administered a survey with just this one question between June 17 and October 16, 2019.

When we were almost ready with the analysis, the COVID-19 pandemic broke out, and we were not able to present the results at the conferences we had planned to attend (NCTM and ICME in 2020). Moreover, with the world shaken up by the crisis, we wondered if colleagues in our field might think differently about the themes formulated for the future due to the pandemic. Hence, on November 26, 2020, we asked a follow-up question to those respondents who in 2019 had given us permission to approach them for elaboration by email:

Q2020: Has the pandemic changed your view on the themes of mathematics education research for the coming decade? If so, how?

In this paper, we summarize the responses to these two questions. Similar to Sfard’s ( 2005 ) approach, we start by synthesizing the voices of the respondents before formulating our own views. Some colleagues put forward the idea of formulating a list of key themes or questions, similar to the 23 unsolved mathematical problems that David Hilbert published around 1900 (cf. Schoenfeld, 1999 ). However, mathematics and mathematics education are very different disciplines, and very few people share Hilbert’s formalist view on mathematics; hence, we do not want to suggest that we could capture the key themes of mathematics education in a similar way. Rather, our overview of themes drawn from the survey responses is intended to summarize what is valued in our global community at the time of the surveys. Reasoning from these themes, we end with a list of research challenges that we see worth addressing in the future (cf. Stephan et al., 2015 ).

2 Methodological approach

2.1 themes for the coming decade (2019).

We administered the 1-question survey through email lists that we were aware of (e.g., Becker, ICME, PME) and asked mathematics education researchers to spread it in their national networks. By October 16, 2019, we had received 229 responses from 44 countries across 6 continents (Table 1 ). Although we were happy with the larger response than Sfard ( 2005 ) received (74, with 28 from Europe), we do not know how well we have reached particular regions, and if potential respondents might have faced language or other barriers. We did offer a few Chinese respondents the option to write in Chinese because the second author offered to translate their emails into English. We also received responses in Spanish, which were translated for us.

Ethical approval was given by the Ethical Review Board of the Faculties of Science and Geo-science of Utrecht University (Bèta L-19247). We asked respondents to indicate if they were willing to be quoted by name and if we were allowed to approach them for subsequent information. If they preferred to be named, we mention their name and country; otherwise, we write “anonymous.” In our selection of quotes, we have focused on content, not on where the response came from. On March 2, 2021, we approached all respondents who were quoted to double-check if they agreed to be quoted and named. One colleague preferred the quote and name to be deleted; three suggested small changes in wording; the others approved.

On September 20, 2019, the three authors met physically at Utrecht University to analyze the responses. After each individual proposal, we settled on a joint list of seven main themes (the first seven in Table 2 ), which were neither mutually exclusive nor exhaustive. The third author (Zenger, then still a student in educational science) next color coded all parts of responses belonging to a category. These formed the basis for the frequencies and percentages presented in the tables and text. The first author (Bakker) then read all responses categorized by a particular code to identify and synthesize the main topics addressed within each code. The second author (Cai) read all of the survey responses and the response categories, and commented. After the initial round of analysis, we realized it was useful to add an eighth theme: assessment (including evaluation).

Moreover, given that a large number of respondents made comments about mathematics education research itself, we decided to summarize these separately. For analyzing this category of research, we used the following four labels to distinguish types of comments on our discipline of mathematics education research: theory, methodology, self-reflection (including ethical considerations), interdisciplinarity, and transdisciplinarity. We then summarized the responses per type of comment.

It has been a daunting and humbling experience to study the huge coverage and diversity of topics that our colleagues care about. Any categorization felt like a reduction of the wealth of ideas, and we are aware of the risks of “sorting things out” (Bowker & Star, 2000 ), which come with foregrounding particular challenges rather than others (Stephan et al., 2015 ). Yet the best way to summarize the bigger picture seemed by means of clustering themes and pointing to their relationships. As we identified these eight themes of mathematics education research for the future, a recurring question during the analysis was how to represent them. A list such as Table 2 does not do justice to the interrelations between the themes. Some relationships are very clear, for example, educational approaches (theme 2) working toward educational or societal goals (theme 1). Some themes are pervasive; for example, equity and (positive) affect are both things that educators want to achieve but also phenomena that are at stake during every single moment of learning and teaching. Diagrams we considered to represent such interrelationships were either too specific (limiting the many relevant options, e.g., a star with eight vertices that only link pairs of themes) or not specific enough (e.g., a Venn diagram with eight leaves such as the iPhone symbol for photos). In the end, we decided to use an image and collaborated with Elisabeth Angerer (student assistant in an educational sciences program), who eventually made the drawing in Fig. 1 to capture themes in their relationships.

Artistic impression of the future themes

2.2 Has the pandemic changed your view? (2020)

On November 26, 2020, we sent an email to the colleagues who responded to the initial question and who gave permission to be approached by email. We cited their initial response and asked: “Has the pandemic changed your view on the themes of mathematics education research for the coming decade? If so, how?” We received 108 responses by January 12, 2021. The countries from which the responses came included China, Italy, and other places that were hit early by the COVID-19 virus. The length of responses varied from a single word response (“no”) to elaborate texts of up to 2215 words. Some people attached relevant publications. The median length of the responses was 87 words, with a mean length of 148 words and SD = 242. Zenger and Bakker classified them as “no changes” (9 responses) or “clearly different views” (8); the rest of the responses saw the importance of their initial themes reinforced (45), specified their initial responses (43), or added new questions or themes (35). These last categories were not mutually exclusive, because respondents could first state that they thought the initial themes were even more relevant than before and provide additional, more specified themes. We then used the same themes that had been identified in the first round and identified what was stressed or added in the 2020 responses.

3 The themes

The most frequently mentioned theme was what we labeled approaches to teaching (64% of the respondents, see Table 2 ). Next was the theme of goals of mathematics education on which research should shed more light in the coming decade (54%). These goals ranged from specific educational goals to very broad societal ones. Many colleagues referred to mathematics education’s relationships with other practices (communities, institutions…) such as home, continuing education, and work. Teacher professional development is a key area for research in which the other themes return (what should students learn, how, how to assess that, how to use technology and ensure that students are interested?). Technology constitutes its own theme but also plays a key role in many other themes, just like affect. Another theme permeating other ones is what can be summarized as equity, diversity, and inclusion (also social justice, anti-racism, democratic values, and several other values were mentioned). These values are not just societal and educational goals but also drivers for redesigning teaching approaches, using technology, working on more just assessment, and helping learners gain access, become confident, develop interest, or even love for mathematics. To evaluate if approaches are successful and if goals have been achieved, assessment (including evaluation) is also mentioned as a key topic of research.

In the 2020 responses, many wise and general remarks were made. The general gist is that the pandemic (like earlier crises such as the economic crisis around 2008–2010) functioned as a magnifying glass on themes that were already considered important. Due to the pandemic, however, systemic societal and educational problems were said to have become better visible to a wider community, and urge us to think about the potential of a “new normal.”

3.1 Approaches to teaching

We distinguish specific teaching strategies from broader curricular topics.

3.1.1 Teaching strategies

There is a widely recognized need to further design and evaluate various teaching approaches. Among the teaching strategies and types of learning to be promoted that were mentioned in the survey responses are collaborative learning, critical mathematics education, dialogic teaching, modeling, personalized learning, problem-based learning, cross-curricular themes addressing the bigger themes in the world, embodied design, visualization, and interleaved learning. Note, however, that students can also enhance their mathematical knowledge independently from teachers or parents through web tutorials and YouTube videos.

Many respondents emphasized that teaching approaches should do more than promote cognitive development. How can teaching be entertaining or engaging? How can it contribute to the broader educational goals of developing students’ identity, contribute to their empowerment, and help them see the value of mathematics in their everyday life and work? We return to affect in Section 3.7 .

In the 2020 responses, we saw more emphasis on approaches that address modeling, critical thinking, and mathematical or statistical literacy. Moreover, respondents stressed the importance of promoting interaction, collaboration, and higher order thinking, which are generally considered to be more challenging in distance education. One approach worth highlighting is challenge-based education (cf. Johnson et al. 2009 ), because it takes big societal challenges as mentioned in the previous section as its motivation and orientation.

3.1.2 Curriculum

Approaches by which mathematics education can contribute to the aforementioned goals can be distinguished at various levels. Several respondents mentioned challenges around developing a coherent mathematics curriculum, smoothing transitions to higher school levels, and balancing topics, and also the typical overload of topics, the influence of assessment on what is taught, and what teachers can teach. For example, it was mentioned that mathematics teachers are often not prepared to teach statistics. There seems to be little research that helps curriculum authors tackle some of these hard questions as well as how to monitor reform (cf. Shimizu & Vithal, 2019 ). Textbook analysis is mentioned as a necessary research endeavor. But even if curricula within one educational system are reasonably coherent, how can continuity between educational systems be ensured (cf. Jansen et al., 2012 )?

In the 2020 responses, some respondents called for free high-quality curriculum resources. In several countries where Internet access is a problem in rural areas, a shift can be observed from online resources to other types of media such as radio and TV.

3.2 Goals of mathematics education

The theme of approaches is closely linked to that of the theme of goals. For example, as Fulvia Furinghetti (Italy) wrote: “It is widely recognized that critical thinking is a fundamental goal in math teaching. Nevertheless it is still not clear how it is pursued in practice.” We distinguish broad societal and more specific educational goals. These are often related, as Jane Watson (Australia) wrote: “If Education is to solve the social, cultural, economic, and environmental problems of today’s data-driven world, attention must be given to preparing students to interpret the data that are presented to them in these fields.”

3.2.1 Societal goals

Respondents alluded to the need for students to learn to function in the economy and in society more broadly. Apart from instrumental goals of mathematics education, some emphasized goals related to developing as a human being, for instance learning to see the mathematics in the world and develop a relation with the world. Mathematics education in these views should empower students to combat anti-expertise and post-fact tendencies. Several respondents mentioned even larger societal goals such as avoiding extinction as a human species and toxic nationalism, resolving climate change, and building a sustainable future.

In the second round of responses (2020), we saw much more emphasis on these bigger societal issues. The urgency to orient mathematics education (and its research) toward resolving these seemed to be felt more than before. In short, it was stressed that our planet needs to be saved. The big question is what role mathematics education can play in meeting these challenges.

3.2.2 Educational goals

Several respondents expressed a concern that the current goals of mathematics education do not reflect humanity’s and societies’ needs and interests well. Educational goals to be stressed more were mathematical literacy, numeracy, critical, and creative thinking—often with reference to the changing world and the planet being at risk. In particular, the impact of technology was frequently stressed, as this may have an impact on what people need to learn (cf. Gravemeijer et al., 2017 ). If computers can do particular things much better than people, what is it that students need to learn?

Among the most frequently mentioned educational goals for mathematics education were statistical literacy, computational and algorithmic thinking, artificial intelligence, modeling, and data science. More generally, respondents expressed that mathematics education should help learners deploy evidence, reasoning, argumentation, and proof. For example, Michelle Stephan (USA) asked:

What mathematics content should be taught today to prepare students for jobs of the future, especially given growth of the digital world and its impact on a global economy? All of the mathematics content in K-12 can be accomplished by computers, so what mathematical procedures become less important and what domains need to be explored more fully (e.g., statistics and big data, spatial geometry, functional reasoning, etc.)?

One challenge for research is that there is no clear methodology to arrive at relevant and feasible learning goals. Yet there is a need to choose and formulate such goals on the basis of research (cf. Van den Heuvel-Panhuizen, 2005 ).

Several of the 2020 responses mentioned the sometimes problematic way in which numbers, data, and graphs are used in the public sphere (e.g., Ernest, 2020 ; Kwon et al., 2021 ; Yoon et al., 2021 ). Many respondents saw their emphasis on relevant educational goals reinforced, for example, statistical and data literacy, modeling, critical thinking, and public communication. A few pandemic-specific topics were mentioned, such as exponential growth.

3.3 Relation of mathematics education to other practices

Many responses can be characterized as highlighting boundary crossing (Akkerman & Bakker, 2011 ) with disciplines or communities outside mathematics education, such as in science, technology, engineering, art, and mathematics education (STEM or STEAM); parents or families; the workplace; and leisure (e.g., drama, music, sports). An interesting example was the educational potential of mathematical memes—“humorous digital objects created by web users copying an existing image and overlaying a personal caption” (Bini et al., 2020 , p. 2). These boundary crossing-related responses thus emphasize the movements and connections between mathematics education and other practices.

In the 2020 responses, we saw that during the pandemic, the relationship between school and home has become much more important, because most students were (and perhaps still are) learning at home. Earlier research on parental involvement and homework (Civil & Bernier, 2006 ; de Abreu et al., 2006 ; Jackson, 2011 ) proves relevant in the current situation where many countries are still or again in lockdown. Respondents pointed to the need to monitor students and their work and to promote self-regulation. They also put more stress on the political, economic, and financial contexts in which mathematics education functions (or malfunctions, in many respondents’ views).

3.4 Teacher professional development

Respondents explicitly mentioned teacher professional development as an important domain of mathematics education research (including teacher educators’ development). For example, Loide Kapenda (Namibia) wrote, “I am supporting UNESCO whose idea is to focus on how we prepare teachers for the future we want.” (e.g., UNESCO, 2015 ) And, Francisco Rojas (Chile) wrote:

Although the field of mathematics education is broad and each time faced with new challenges (socio-political demands, new intercultural contexts, digital environments, etc.), all of them will be handled at school by the mathematics teacher, both in primary as well as in secondary education. Therefore, from my point of view, pre-service teacher education is one of the most relevant fields of research for the next decade, especially in developing countries.

It is evident from the responses that teaching mathematics is done by a large variety of people, not only by people who are trained as primary school teachers, secondary school mathematics teachers, or mathematicians but also parents, out-of-field teachers, and scientists whose primary discipline is not mathematics but who do use mathematics or statistics. How teachers of mathematics are trained varies accordingly. Respondents frequently pointed to the importance of subject-matter knowledge and particularly noted that many teachers seem ill-prepared to teach statistics (e.g., Lonneke Boels, the Netherlands).

Key questions were raised by several colleagues: “How to train mathematics teachers with a solid foundation in mathematics, positive attitudes towards mathematics teaching and learning, and wide knowledge base linking to STEM?” (anonymous); “What professional development, particularly at the post-secondary level, motivates changes in teaching practices in order to provide students the opportunities to engage with mathematics and be successful?” (Laura Watkins, USA); “How can mathematics educators equip students for sustainable, equitable citizenship? And how can mathematics education equip teachers to support students in this?” (David Wagner, Canada)

In the 2020 responses, it was clear that teachers are incredibly important, especially in the pandemic era. The sudden change to online teaching means that

higher requirements are put forward for teachers’ educational and teaching ability, especially the ability to carry out education and teaching by using information technology should be strengthened. Secondly, teachers’ ability to communicate and cooperate has been injected with new connotation. (Guangming Wang, China)

It is broadly assumed that education will stay partly online, though more so in higher levels of education than in primary education. This has implications for teachers, for instance, they will have to think through how they intend to coordinate teaching on location and online. Hence, one important focus for professional development is the use of technology.

3.5 Technology

Technology deserves to be called a theme in itself, but we want to emphasize that it ran through most of the other themes. First of all, some respondents argued that, due to technological advances in society, the societal and educational goals of mathematics education need to be changed (e.g., computational thinking to ensure employability in a technological society). Second, responses indicated that the changed goals have implications for the approaches in mathematics education. Consider the required curriculum reform and the digital tools to be used in it. Students do not only need to learn to use technology; the technology can also be used to learn mathematics (e.g., visualization, embodied design, statistical thinking). New technologies such as 3D printing, photo math, and augmented and virtual reality offer new opportunities for learning. Society has changed very fast in this respect. Third, technology is suggested to assist in establishing connections with other practices , such as between school and home, or vocational education and work, even though there is a great disparity in how successful these connections are.

In the 2020 responses, there was great concern about the current digital divide (cf. Hodgen et al., 2020 ). The COVID-19 pandemic has thus given cause for mathematics education research to understand better how connections across educational and other practices can be improved with the help of technology. Given the unequal distribution of help by parents or guardians, it becomes all the more important to think through how teachers can use videos and quizzes, how they can monitor their students, how they can assess them (while respecting privacy), and how one can compensate for the lack of social, gestural, and embodied interaction that is possible when being together physically.

Where mobile technology was considered very innovative before 2010, smartphones have become central devices in mathematics education in the pandemic with its reliance on distance learning. Our direct experience showed that phone applications such as WhatsApp and WeChat have become key tools in teaching and learning mathematics in many rural areas in various continents where few people have computers (for a report on podcasts distributed through WhatsApp, community loudspeakers, and local radio stations in Colombia, see Saenz et al., 2020 ).

3.6 Equity, diversity, and inclusion

Another cross-cutting theme can be labeled “equity, diversity, and inclusion.” We use this triplet to cover any topic that highlights these and related human values such as equality, social and racial justice, social emancipation, and democracy that were also mentioned by respondents (cf. Dobie & Sherin, 2021 ). In terms of educational goals , many respondents stressed that mathematics education should be for all students, including those who have special needs, who live in poverty, who are learning the instruction language, who have a migration background, who consider themselves LGBTQ+, have a traumatic or violent history, or are in whatever way marginalized. There is broad consensus that everyone should have access to high-quality mathematics education. However, as Niral Shah (USA) notes, less attention has been paid to “how phenomena related to social markers (e.g., race, class, gender) interact with phenomena related to the teaching and learning of mathematical content.”

In terms of teaching approaches , mathematics education is characterized by some respondents from particular countries as predominantly a white space where some groups feel or are excluded (cf. Battey, 2013 ). There is a general concern that current practices of teaching mathematics may perpetuate inequality, in particular in the current pandemic. In terms of assessment , mathematics is too often used or experienced as a gatekeeper rather than as a powerful resource (cf. Martin et al., 2010 ). Steve Lerman (UK) “indicates that understanding how educational opportunities are distributed inequitably, and in particular how that manifests in each end every classroom, is a prerequisite to making changes that can make some impact on redistribution.” A key research aim therefore is to understand what excludes students from learning mathematics and what would make mathematics education more inclusive (cf. Roos, 2019 ). And, what does professional development of teachers that promotes equity look like?

In 2020, many respondents saw their emphasis on equity and related values reinforced in the current pandemic with its risks of a digital divide, unequal access to high-quality mathematics education, and unfair distribution of resources. A related future research theme is how the so-called widening achievement gaps can be remedied (cf. Bawa, 2020 ). However, warnings were also formulated that thinking in such deficit terms can perpetuate inequality (cf. Svensson et al., 2014 ). A question raised by Dor Abrahamson (USA) is, “What roles could digital technology play, and in what forms, in restoring justice and celebrating diversity?”

Though entangled with many other themes, affect is also worth highlighting as a theme in itself. We use the term affect in a very broad sense to point to psychological-social phenomena such as emotion, love, belief, attitudes, interest, curiosity, fun, engagement, joy, involvement, motivation, self-esteem, identity, anxiety, alienation, and feeling of safety (cf. Cobb et al., 2009 ; Darragh, 2016 ; Hannula, 2019 ; Schukajlow et al., 2017 ). Many respondents emphasized the importance of studying these constructs in relation to (and not separate from) what is characterized as cognition. Some respondents pointed out that affect is not just an individual but also a social phenomenon, just like learning (cf. Chronaki, 2019 ; de Freitas et al., 2019 ; Schindler & Bakker, 2020 ).

Among the educational goals of mathematics education, several participants mentioned the need to generate and foster interest in mathematics. In terms of approaches , much emphasis was put on the need to avoid anxiety and alienation and to engage students in mathematical activity.

In the 2020 responses, more emphasis was put on the concern about alienation, which seems to be of special concern when students are socially distanced from peers and teachers as to when teaching takes place only through technology . What was reiterated in the 2020 responses was the importance of students’ sense of belonging in a mathematics classroom (cf. Horn, 2017 )—a topic closely related to the theme of equity, diversity, and inclusion discussed before.

3.8 Assessment

Assessment and evaluation were not often mentioned explicitly, but they do not seem less important than the other related themes. A key challenge is to assess what we value rather than valuing what we assess. In previous research, the assessment of individual students has received much attention, but what seems to be neglected is the evaluation of curricula. As Chongyang Wang (China) wrote, “How to evaluate the curriculum reforms. When we pay much energy in reforming our education and curriculum, do we imagine how to ensure it will work and there will be pieces of evidence found after the new curricula are carried out? How to prove the reforms work and matter?” (cf. Shimizu & Vithal, 2019 )

In the 2020 responses, there was an emphasis on assessment at a distance. Distance education generally is faced with the challenge of evaluating student work, both formatively and summatively. We predict that so-called e-assessment, along with its privacy challenges, will generate much research interest in the near future (cf. Bickerton & Sangwin, 2020 ).

4 Mathematics education research itself

Although we only asked for future themes, many respondents made interesting comments about research in mathematics education and its connections with other disciplines and practices (such as educational practice, policy, home settings). We have grouped these considerations under the subheadings of theory, methodology, reflection on our discipline, and interdisciplinarity and transdisciplinarity. As with the previous categorization into themes, we stress that these four types are not mutually exclusive as theoretical and methodological considerations can be intricately intertwined (Radford, 2008 ).

Several respondents expressed their concern about the fragmentation and diversity of theories used in mathematics education research (cf. Bikner-Ahsbahs & Prediger, 2014 ). The question was raised how mathematics educators can “work together to obtain valid, reliable, replicable, and useful findings in our field” and “How, as a discipline, can we encourage sustained research on core questions using commensurable perspectives and methods?” (Keith Weber, USA). One wish was “comparing theoretical perspectives for explanatory power” (K. Subramaniam, India). At the same time, it was stressed that “we cannot continue to pretend that there is just one culture in the field of mathematics education, that all the theoretical framework may be applied in whichever culture and that results are universal” (Mariolina Bartolini Bussi, Italy). In addition, the wish was expressed to deepen theoretical notions such as numeracy, equity, and justice as they play out in mathematics education.

4.2 Methodology

Many methodological approaches were mentioned as potentially useful in mathematics education research: randomized studies, experimental studies, replication, case studies, and so forth. Particular attention was paid to “complementary methodologies that bridge the ‘gap’ between mathematics education research and research on mathematical cognition” (Christian Bokhove, UK), as, for example, done in Gilmore et al. ( 2018 ). Also, approaches were mentioned that intend to bridge the so-called gap between educational practice and research, such as lesson study and design research. For example, Kay Owens (Australia) pointed to the challenge of studying cultural context and identity: “Such research requires a multi-faceted research methodology that may need to be further teased out from our current qualitative (e.g., ethnographic) and quantitative approaches (‘paper and pencil’ (including computing) testing). Design research may provide further possibilities.”

Francisco Rojas (Chile) highlighted the need for more longitudinal and cross-sectional research, in particular in the context of teacher professional development:

It is not enough to investigate what happens in pre-service teacher education but understand what effects this training has in the first years of the professional career of the new teachers of mathematics, both in primary and secondary education. Therefore, increasingly more longitudinal and cross-sectional studies will be required to understand the complexity of the practice of mathematics teachers, how the professional knowledge that articulates the practice evolves, and what effects have the practice of teachers on the students’ learning of mathematics.

4.3 Reflection on our discipline

Calls were made for critical reflection on our discipline. One anonymous appeal was for more self-criticism and scientific modesty: Is research delivering, or is it drawing away good teachers from teaching? Do we do research primarily to help improve mathematics education or to better understand phenomena? (cf. Proulx & Maheux, 2019 ) The general gist of the responses was a sincere wish to be of value to the world and mathematics education more specifically and not only do “research for the sake of research” (Zahra Gooya, Iran). David Bowers (USA) expressed several reflection-inviting views about the nature of our discipline, for example:

We must normalize (and expect) the full taking up the philosophical and theoretical underpinnings of all of our work (even work that is not considered “philosophical”). Not doing so leads to uncritical analysis and implications.

We must develop norms wherein it is considered embarrassing to do “uncritical” research.

There is no such thing as “neutral.” Amongst other things, this means that we should be cultivating norms that recognize the inherent political nature of all work, and norms that acknowledge how superficially “neutral” work tends to empower the oppressor.

We must recognize the existence of but not cater to the fragility of privilege.

In terms of what is studied, some respondents felt that the mathematics education research “literature has been moving away from the original goals of mathematics education. We seem to have been investigating everything but the actual learning of important mathematics topics.” (Lyn English, Australia) In terms of the nature of our discipline, Taro Fujita (UK) argued that our discipline can be characterized as a design science, with designing mathematical learning environments as the core of research activities (cf. Wittmann, 1995 ).

A tension that we observe in different views is the following: On the one hand, mathematics education research has its origin in helping teachers teach particular content better. The need for such so-called didactical, topic-specific research is not less important today but perhaps less fashionable for funding schemes that promote innovative, ground-breaking research. On the other hand, over time it has become clear that mathematics education is a multi-faceted socio-cultural and political endeavor under the influence of many local and global powers. It is therefore not surprising that the field of mathematics education research has expanded so as to include an increasingly wide scope of themes that are at stake, such as the marginalization of particular groups. We therefore highlight Niral Shah’s (USA) response that “historically, these domains of research [content-specific vs socio-political] have been decoupled. The field would get closer to understanding the experiences of minoritized students if we could connect these lines of inquiry.”

Another interesting reflective theme was raised by Nouzha El Yacoubi (Morocco): To what extent can we transpose “research questions from developed to developing countries”? As members of the plenary panel at PME 2019 (e.g., Kazima, 2019 ; Kim, 2019 ; Li, 2019 ) conveyed well, adopting interventions that were successful in one place in another place is far from trivial (cf. Gorard, 2020 ).

Juan L. Piñeiro (Spain in 2019, Chile in 2020) highlighted that “mathematical concepts and processes have different natures. Therefore, can it be characterized using the same theoretical and methodological tools?” More generally, one may ask if our theories and methodologies—often borrowed from other disciplines—are well suited to the ontology of our own discipline. A discussion started by Niss ( 2019 ) on the nature of our discipline, responded to by Bakker ( 2019 ) and Cai and Hwang ( 2019 ), seems worth continuing.

An important question raised in several comments is how close research should be to existing curricula. One respondent (Benjamin Rott, Germany) noted that research on problem posing often does “not fit into school curricula.” This makes the application of research ideas and findings problematic. However, one could argue that research need not always be tied to existing (local) educational contexts. It can also be inspirational, seeking principles of what is possible (and how) with a longer-term view on how curricula may change in the future. One option is, as Simon Zell (Germany) suggests, to test designs that cover a longer timeframe than typically done. Another way to bridge these two extremes is “collaboration between teachers and researchers in designing and publishing research” (K. Subramaniam, India) as is promoted by facilitating teachers to do PhD research (Bakx et al., 2016 ).

One of the responding teacher-researchers (Lonneke Boels, the Netherlands) expressed the wish that research would become available “in a more accessible form.” This wish raises the more general questions of whose responsibility it is to do such translation work and how to communicate with non-researchers. Do we need a particular type of communication research within mathematics education to learn how to convey particular key ideas or solid findings? (cf. Bosch et al., 2017 )

4.4 Interdisciplinarity and transdisciplinarity

Many respondents mentioned disciplines which mathematics education research can learn from or should collaborate with (cf. Suazo-Flores et al., 2021 ). Examples are history, mathematics, philosophy, psychology, psychometry, pedagogy, educational science, value education (social, emotional), race theory, urban education, neuroscience/brain research, cognitive science, and computer science didactics. “A big challenge here is how to make diverse experts approach and talk to one another in a productive way.” (David Gómez, Chile)

One of the most frequently mentioned disciplines in relation to our field is history. It is a common complaint in, for instance, the history of medicine that historians accuse medical experts of not knowing historical research and that medical experts accuse historians of not understanding the medical discipline well enough (Beckers & Beckers, 2019 ). This tension raises the question who does and should do research into the history of mathematics or of mathematics education and to what broader purpose.

Some responses go beyond interdisciplinarity, because resolving the bigger issues such as climate change and a more equitable society require collaboration with non-researchers (transdisciplinarity). A typical example is the involvement of educational practice and policy when improving mathematics education (e.g., Potari et al., 2019 ).

Let us end this section with a word of hope, from an anonymous respondent: “I still believe (or hope?) that the pandemic, with this making-inequities-explicit, would help mathematics educators to look at persistent and systemic inequalities more consistently in the coming years.” Having learned so much in the past year could indeed provide an opportunity to establish a more equitable “new normal,” rather than a reversion to the old normal, which one reviewer worried about.

5 The themes in their coherence: an artistic impression

As described above, we identified eight themes of mathematics education research for the future, which we discussed one by one. The disadvantage of this list-wise discussion is that the entanglement of the themes is backgrounded. To compensate for that drawback, we here render a brief interpretation of the drawing of Fig. 1 . While doing so, we invite readers to use their own creative imagination and perhaps use the drawing for other purposes (e.g., ask researchers, students, or teachers: Where would you like to be in this landscape? What mathematical ideas do you spot?). The drawing mainly focuses on the themes that emerged from the first round of responses but also hints at experiences from the time of the pandemic, for instance distance education. In Appendix 1 , we specify more of the details in the drawing and we provide a link to an annotated image (available at https://www.fisme.science.uu.nl/toepassingen/28937/ ).

The boat on the river aims to represent teaching approaches. The hand drawing of the boat hints at the importance of educational design: A particular approach is being worked out. On the boat, a teacher and students work together toward educational and societal goals, further down the river. The graduation bridge is an intermediate educational goal to pass, after which there are many paths leading to other goals such as higher education, citizenship, and work in society. Relations to practices outside mathematics education are also shown. In the left bottom corner, the house and parents working and playing with children represent the link of education with the home situation and leisure activity.

The teacher, represented by the captain in the foreground of the ship, is engaged in professional development, consulting a book, but also learning by doing (cf. Bakkenes et al., 2010 , on experimenting, using resources, etc.). Apart from graduation, there are other types of goals for teachers and students alike, such as equity, positive affect, and fluent use of technology. During their journey (and partially at home, shown in the left bottom corner), students learn to orient themselves in the world mathematically (e.g., fractal tree, elliptical lake, a parabolic mountain, and various platonic solids). On their way toward various goals, both teacher and students use particular technology (e.g., compass, binoculars, tablet, laptop). The magnifying glass (representing research) zooms in on a laptop screen that portrays distance education, hinting at the consensus that the pandemic magnifies some issues that education was already facing (e.g., the digital divide).

Equity, diversity, and inclusion are represented with the rainbow, overarching everything. On the boat, students are treated equally and the sailing practice is inclusive in the sense that all perform at their own level—getting the support they need while contributing meaningfully to the shared activity. This is at least what we read into the image. Affect is visible in various ways. First of all, the weather represents moods in general (rainy and dark side on the left; sunny bright side on the right). Second, the individual students (e.g., in the crow’s nest) are interested in, anxious about, and attentive to the things coming up during their journey. They are motivated to engage in all kinds of tasks (handling the sails, playing a game of chance with a die, standing guard in the crow’s nest, etc.). On the bridge, the graduates’ pride and happiness hints at positive affect as an educational goal but also represents the exam part of the assessment. The assessment also happens in terms of checks and feedback on the boat. The two people next to the house (one with a camera, one measuring) can be seen as assessors or researchers observing and evaluating the progress on the ship or the ship’s progress.

More generally, the three types of boats in the drawing represent three different spaces, which Hannah Arendt ( 1958 ) would characterize as private (paper-folded boat near the boy and a small toy boat next to the girl with her father at home), public/political (ships at the horizon), and the in-between space of education (the boat with the teacher and students). The students and teacher on the boat illustrate school as a special pedagogic form. Masschelein and Simons ( 2019 ) argue that the ancient Greek idea behind school (σχολή, scholè , free time) is that students should all be treated as equal and should all get equal opportunities. At school, their descent does not matter. At school, there is time to study, to make mistakes, without having to work for a living. At school, they learn to collaborate with others from diverse backgrounds, in preparation for future life in the public space. One challenge of the lockdown situation as a consequence of the pandemic is how to organize this in-between space in a way that upholds its special pedagogic form.

6 Research challenges

Based on the eight themes and considerations about mathematics education research itself, we formulate a set of research challenges that strike us as deserving further discussion (cf. Stephan et al., 2015 ). We do not intend to suggest these are more important than others or that some other themes are less worthy of investigation, nor do we suggest that they entail a research agenda (cf. English, 2008 ).

6.1 Aligning new goals, curricula, and teaching approaches

There seems to be relatively little attention within mathematics education research for curricular issues, including topics such as learning goals, curriculum standards, syllabi, learning progressions, textbook analysis, curricular coherence, and alignment with other curricula. Yet we feel that we as mathematics education researchers should care about these topics as they may not necessarily be covered by other disciplines. For example, judging from Deng’s ( 2018 ) complaint about the trends in the discipline of curriculum studies, we cannot assume scholars in that field to address issues specific to the mathematics-focused curriculum (e.g., the Journal of Curriculum Studies and Curriculum Inquiry have published only a limited number of studies on mathematics curricula).

Learning goals form an important element of curricula or standards. It is relatively easy to formulate important goals in general terms (e.g., critical thinking or problem solving). As a specific example, consider mathematical problem posing (Cai & Leikin, 2020 ), which curriculum standards have specifically pointed out as an important educational goal—developing students’ problem-posing skills. Students should be provided opportunities to formulate their own problems based on situations. However, there are few problem-posing activities in current mathematics textbooks and classroom instruction (Cai & Jiang, 2017 ). A similar observation can be made about problem solving in Dutch primary textbooks (Kolovou et al., 2009 ). Hence, there is a need for researchers and educators to align problem posing in curriculum standards, textbooks, classroom instruction, and students’ learning.

The challenge we see for mathematics education researchers is to collaborate with scholars from other disciplines (interdisciplinarity) and with non-researchers (transdisciplinarity) in figuring out how the desired societal and educational goals can be shaped in mathematics education. Our discipline has developed several methodological approaches that may help in formulating learning goals and accompanying teaching approaches (cf. Van den Heuvel-Panhuizen, 2005 ), including epistemological analyses (Sierpinska, 1990 ), historical and didactical phenomenology (Bakker & Gravemeijer, 2006 ; Freudenthal, 1986 ), and workplace studies (Bessot & Ridgway, 2000 ; Hoyles et al., 2001 ). However, how should the outcomes of such research approaches be weighed against each other and combined to formulate learning goals for a balanced, coherent curriculum? What is the role of mathematics education researchers in relation to teachers, policymakers, and other stakeholders (Potari et al., 2019 )? In our discipline, we seem to lack a research-informed way of arriving at the formulation of suitable educational goals without overloading the curricula.

6.2 Researching mathematics education across contexts

Though methodologically and theoretically challenging, it is of great importance to study learning and teaching mathematics across contexts. After all, students do not just learn at school; they can also participate in informal settings (Nemirovsky et al., 2017 ), online forums, or affinity networks (Ito et al., 2018 ) where they may share for instance mathematical memes (Bini et al., 2020 ). Moreover, teachers are not the only ones teaching mathematics: Private tutors, friends, parents, siblings, or other relatives can also be involved in helping children with their mathematics. Mathematics learning could also be situated on streets or in museums, homes, and other informal settings. This was already acknowledged before 2020, but the pandemic has scattered learners and teachers away from the typical central school locations and thus shifted the distribution of labor.

In particular, physical and virtual spaces of learning have been reconfigured due to the pandemic. Issues of timing also work differently online, for example, if students can watch online lectures or videos whenever they like (asynchronously). Such reconfigurations of space and time also have an effect on the rhythm of education and hence on people’s energy levels (cf. Lefebvre, 2004 ). More specifically, the reconfiguration of the situation has affected many students’ levels of motivation and concentration (e.g., Meeter et al., 2020 ). As Engelbrecht et al. ( 2020 ) acknowledged, the pandemic has drastically changed the teaching and learning model as we knew it. It is quite possible that some existing theories about teaching and learning no longer apply in the same way. An interesting question is whether and how existing theoretical frameworks can be adjusted or whether new theoretical orientations need to be developed to better understand and promote productive ways of blended or online teaching, across contexts.

6.3 Focusing teacher professional development

Professional development of teachers and teacher educators stands out from the survey as being in need of serious investment. How can teachers be prepared for the unpredictable, both in terms of beliefs and actions? During the pandemic, teachers have been under enormous pressure to make quick decisions in redesigning their courses, to learn to use new technological tools, to invent creative ways of assessment, and to do what was within their capacity to provide opportunities to their students for learning mathematics—even if technological tools were limited (e.g., if students had little or no computer or internet access at home). The pressure required both emotional adaption and instructional adjustment. Teachers quickly needed to find useful information, which raises questions about the accessibility of research insights. Given the new situation, limited resources, and the uncertain unfolding of education after lockdowns, focusing teacher professional development on necessary and useful topics will need much attention. In particular, there is a need for longitudinal studies to investigate how teachers’ learning actually affects teachers’ classroom instruction and students’ learning.

In the surveys, respondents mainly referred to teachers as K-12 school mathematics teachers, but some also stressed the importance of mathematics teacher educators (MTEs). In addition to conducting research in mathematics education, MTEs are acting in both the role of teacher educators and of mathematics teachers. There has been increased research on MTEs as requiring professional development (Goos & Beswick, 2021 ). Within the field of mathematics education, there is an emerging need and interest in how mathematics teacher educators themselves learn and develop. In fact, the changing situation also provides an opportunity to scrutinize our habitual ways of thinking and become aware of what Jullien ( 2018 ) calls the “un-thought”: What is it that we as educators and researchers have not seen or thought about so much about that the sudden reconfiguration of education forces us to reflect upon?

6.4 Using low-tech resources

Particular strands of research focus on innovative tools and their applications in education, even if they are at the time too expensive (even too labor intensive) to use at large scale. Such future-oriented studies can be very interesting given the rapid advances in technology and attractive to funding bodies focusing on innovation. Digital technology has become ubiquitous, both in schools and in everyday life, and there is already a significant body of work capitalizing on aspects of technology for research and practice in mathematics education.

However, as Cai et al. ( 2020 ) indicated, technology advances so quickly that addressing research problems may not depend so much on developing a new technological capability as on helping researchers and practitioners learn about new technologies and imagine effective ways to use them. Moreover, given the millions of students in rural areas who during the pandemic have only had access to low-tech resources such as podcasts, radio, TV, and perhaps WhatsApp through their parents’ phones, we would like to see more research on what learning, teaching, and assessing mathematics through limited tools such as Whatsapp or WeChat look like and how they can be improved. In fact, in China, a series of WeChat-based mini-lessons has been developed and delivered through the WeChat video function during the pandemic. Even when the pandemic is under control, mini-lessons are still developed and circulated through WeChat. We therefore think it is important to study the use and influence of low-tech resources in mathematics education.

6.5 Staying in touch online

With the majority of students learning at home, a major ongoing challenge for everyone has been how to stay in touch with each other and with mathematics. With less social interaction, without joint attention in the same physical space and at the same time, and with the collective only mediated by technology, becoming and staying motivated to learn has been a widely felt challenge. It is generally expected that in the higher levels of education, more blended or distant learning elements will be built into education. Careful research on the affective, embodied, and collective aspects of learning and teaching mathematics is required to overcome eventually the distance and alienation so widely experienced in online education. That is, we not only need to rethink social interactions between students and/or teachers in different settings but must also rethink how to engage and motivate students in online settings.

6.6 Studying and improving equity without perpetuating inequality

Several colleagues have warned, for a long time, that one risk of studying achievement gaps, differences between majority and minority groups, and so forth can also perpetuate inequity. Admittedly, pinpointing injustice and the need to invest in particular less privileged parts of education is necessary to redirect policymakers’ and teachers’ attention and gain funding. However, how can one reorient resources without stigmatizing? For example, Svensson et al. ( 2014 ) pointed out that research findings can fuel political debates about groups of people (e.g., parents with a migration background), who then may feel insecure about their own capacities. A challenge that we see is to identify and understand problematic situations without legitimizing problematic stereotyping (Hilt, 2015 ).

Furthermore, the field of mathematics education research does not have a consistent conceptualization of equity. There also seem to be regional differences: It struck us that equity is the more common term in the responses from the Americas, whereas inclusion and diversity were more often mentioned in the European responses. Future research will need to focus on both the conceptualization of equity and on improving equity and related values such as inclusion.

6.7 Assessing online

A key challenge is how to assess online and to do so more effectively. This challenge is related to both privacy, ethics, and performance issues. It is clear that online assessment may have significant advantages to assess student mathematics learning, such as more flexibility in test-taking and fast scoring. However, many teachers have faced privacy concerns, and we also have the impression that in an online environment it is even more challenging to successfully assess what we value rather than merely assessing what is relatively easy to assess. In particular, we need to systematically investigate any possible effect of administering assessments online as researchers have found a differential effect of online assessment versus paper-and-pencil assessment (Backes & Cowan, 2019 ). What further deserves careful ethical attention is what happens to learning analytics data that can and are collected when students work online.

6.8 Doing and publishing interdisciplinary research

When analyzing the responses, we were struck by a discrepancy between what respondents care about and what is typically researched and published in our monodisciplinary journals. Most of the challenges mentioned in this section require interdisciplinary or even transdisciplinary approaches (see also Burkhardt, 2019 ).

An overarching key question is: What role does mathematics education research play in addressing the bigger and more general challenges mentioned by our respondents? The importance of interdisciplinarity also raises a question about the scope of journals that focus on mathematics education research. Do we need to broaden the scope of monodisciplinary journals so that they can publish important research that combines mathematics education research with another disciplinary perspective? As editors, we see a place for interdisciplinary studies as long as there is one strong anchor in mathematics education research. In fact, there are many researchers who do not identify themselves as mathematics education researchers but who are currently doing high-quality work related to mathematics education in fields such as educational psychology and the cognitive and learning sciences. Encouraging the reporting of high-quality mathematics education research from a broader spectrum of researchers would serve to increase the impact of the mathematics education research journals in the wider educational arena. This, in turn, would serve to encourage further collaboration around mathematics education issues from various disciplines. Ultimately, mathematics education research journals could act as a hub for interdisciplinary collaboration to address the pressing questions of how mathematics is learned and taught.

7 Concluding remarks

In this paper, based on a survey conducted before and during the pandemic, we have examined how scholars in the field of mathematics education view the future of mathematics education research. On the one hand, there are no major surprises about the areas we need to focus on in the future; the themes are not new. On the other hand, the responses also show that the areas we have highlighted still persist and need further investigation (cf. OECD, 2020 ). But, there are a few areas, based on both the responses of the scholars and our own discussions and views, that stand out as requiring more attention. For example, we hope that these survey results will serve as propelling conversation about mathematics education research regarding online assessment and pedagogical considerations for virtual teaching.

The survey results are limited in two ways. The set of respondents to the survey is probably not representative of all mathematics education researchers in the world. In that regard, perhaps scholars in each country could use the same survey questions to survey representative samples within each country to understand how the scholars in that country view future research with respect to regional needs. The second limitation is related to the fact that mathematics education is a very culturally dependent field. Cultural differences in the teaching and learning of mathematics are well documented. Given the small numbers of responses from some continents, we did not break down the analysis for regional comparison. Representative samples from each country would help us see how scholars from different countries view research in mathematics education; they will add another layer of insights about mathematics education research to complement the results of the survey presented here. Nevertheless, we sincerely hope that the findings from the surveys will serve as a discussion point for the field of mathematics education to pursue continuous improvement.

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Acknowledgments

We thank Anna Sfard for her advice on the survey, based on her own survey published in Sfard ( 2005 ). We are grateful for Stephen Hwang’s careful copyediting for an earlier version of the manuscript. Thanks also to Elisabeth Angerer, Elske de Waal, Paul Ernest, Vilma Mesa, Michelle Stephan, David Wagner, and anonymous reviewers for their feedback on earlier drafts.

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Bakker, A., Cai, J. & Zenger, L. Future themes of mathematics education research: an international survey before and during the pandemic. Educ Stud Math 107 , 1–24 (2021). https://doi.org/10.1007/s10649-021-10049-w

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202 Math Research Topics: List To Vary Your Ideas

Mathematics is an exceptional field of study dealing primarily with numbers. It also deals with structures, formulas, shapes, spaces, and quantities of where they are contained.

Maths encompasses different types of computations that are applied in the real world.

Math requires a lot of analysis. This is why there are different types of maths. They all encompass different subjects and deal with different things. What are the types of maths?

Arithmetic This is perhaps the commonest type or branch of maths. It is one of the oldest and it encompasses basic numbers operations. These are addition, subtraction, multiplication, and divisions; in some schools, the short word for it is BODMAS. This is known as the Bracket of Division, Multiplication, Addition, and Subtraction. Algebra This is where unknown quantities are represented by alphabets and used along with numbers. The letters these unknown quantities are represented by are usually A, B, X, and Y, and they could also be symbols. Geometry This is considered one of the practical branches of maths as it examines sizes, shapes, figures, and the features of these entities. The most common parts of geometry are lines, points, solids, surfaces, and angles.

There are many other types but the above are the most popular. Others are trigonometry, topology, mathematical analysis, calculus, probability, statistics, and a few others.

As many students find it hard to develop maths research topics on their own, this is a chance for you. It’s okay to be worked up when you can’t find undergraduate math research topics that fit your project, essay or paper choices. This article will provide custom maths education research topics for your use. Before that, how do you structure your math essay or paper?

How to Structure Your Math Essay or Paper

Structuring your essay or paper may require that you’ve been reading critical math journals. Reading them could have made it easy to understand how to structure your paper. However, you don’t have to worry if you haven’t. Structuring your paper as expected is an essential part of writing and you’ll know about it in this section. Before you learn that, how do you choose a topic?

Choosing a Topic to Discuss

One of the difficult yet significant parts of any math essay or paper is choosing your topic. This is because you need to solve a problem or engage in a subject that has got less attention. You also need to understand the background to the subject you want to discuss as you can’t write blindly.

You must also be able to articulate your thoughts well as you must show visible knowledge before you commence the research and writing. How do you go about this? You can consider reading existing research. You can even take notes during classes to see the areas you think more work needs to be done.

After choosing your topic, conduct your research to see if you can investigate the sphere. If you can, you need to structure your research thus:

The Background This includes the discussion on what the essay is about. Depending on what you’re writing about, you need to discuss the primary concepts, including the history of some terms, where essential, in this section. This section is more like general information about the subject you want to discuss with your paper. This helps your readers familiarize themselves with your intended discussion. The Introduction This is where the main ideas behind your essays (and the solutions you hope to proffer) are tended to the readers. This is where you also explain the symbols you’ll use and the principles which are required in your essay. Aside from this, you need to state the basic issues, the solutions you could offer, the laws which are essential to discuss to make your work comprehensible. The Main Body This is where you elaborate on your findings. You need to state the research problem, the formulas, the theories you’ll use in tackling the problem, and many other things. You also need to introduce different sections of maths into the main body which is divided by paragraphs and/or chapters as well as mathematical analysis where needed. Implications This is the last part of your essay or paper. This is where you share the insights of your research with your readers. You offer short explanations of the things you have discussed. If you have treated a subject in applied mathematics, this is where you give summaries of how math is connected to human life and the strategic importance of these to people.

By adhering to this structure, you would have crafted the best rated and high-quality maths paper. Furthermore, remember you always have an option to get help with dissertations and save your time. Since it is sometimes challenging to choose cool maths topics to research on your own, these are some for you:

Research Topics in Math

Math is a broad subject. There is a study of the history of math as well as its influence in education, amongst many other sub-sections. If you’d like to create stunning research, you may choose to discuss any of these research topics in math to fulfill one of your academic requirements:

• What are the distinctions between commutative and noncommutative algebra?
• Discuss the methods of factoring quadratics
• Types of sequences and your understanding of them
• Partial fractions: what are they and how do they work?
• Logarithms: what are they and how do they work?
• An overview of Gaussian elimination
• An overview of Brun’s constant relevant
• A description of the effect of dyscalculia on daily student lives
• Describe Descartes’s Dukes of Signs and their application
• Greeks and geometry: discuss
• Describe Euler’s formula
• The progression in the study of math
• Congruence meaning and methods
• Describe the correlation of CT scans to geometry
• Hypercubes and how they work
• The basis of Cramer’s rule
• The examination of Archimedean solids
• Projective geometry and why it’s studied
• Types of Transformations and the available types
• Picasso’s works and the application of geometry
• Difference between the conventional and unconventional approaches to teaching
• Math education and the process of Improvement in the US
• Rhombicosidodecahedron and how it operates in real life
• What are the STEM career fields and why are they important?
• Why women are needed in STEM
• The goals of teaching maths
• How to teach maths to special students
• The correlation between maths and accounting
• The distinction between computer programming and applied maths
• Applied maths and its dynamics
• Processes of solving Heesch’s problem
• Why should kids learn equations?
• History of calculus
• Why there is a need for math camps in schools
• The need for more maths competition in the US
• Methods of draining flight schedule for a country
• Why are some math problems unsolved?
• Discuss the consequences of the gender gap in math students
• Encryption and prime numbers: how do they apply?
• The significance of maths in day to day living.

Undergraduate Math Research Topics

As an undergraduate, you may also have a difficult time wrapping your head around math research topics. You may need to offer both practical and theoretical assessments while writing your paper or essay. The following are undergraduate math research topics:

• Show the proofs of what F-algebras are used
• Abstract algebra, what does it mean?
• Algebra and geometry: discuss
• Acute square triangulation: how it works
• Right triangles: discuss their importance
• Discuss number problems
• Why every math student should study non-Euclidean geometry
• Dirac manifolds and what it means
• Influence of geometry in physics, chemistry, and others
• The application of Riemannian manifolds in the Euclidean space
• How to improve your mathematical thinking ability
• Technology in maths: how is it used?
• Studies of maths in Europe
• Math anxiety and what it truly means
• Standardized testing and the goals of such
• Challenges of learning maths from public schools
• The significance of circles in maths
• The political and social significance of learning maths
• Research into how to increase student interest in maths
• How painting and drawing could help with maths
• Relationship of culture and maths
• History of algebra
• Role of maths in everyday life
• How math is used in Artificial intelligence
• The transferable belief model and its application
• An analysis of the Dempster-Shafer theory
• The role of continuous stochastic process in mathematics
• The major math theorems: discuss how they work
• Understanding the Gauss-Markov: The Evolution of maths
• Discrete random variable: an in-depth understanding of what it means in math and how to identify one.

Math Research Topics for High School Students

As a high school student writing a research paper, one way to get high grades is to write what you know. If you know any math research paper topics for high school, they are the topics you should pick. You can consider:

• Hyperbola: what it is and how it’s used in math
• When to use a calculator in class
• How to find solutions to linear equations
• The need for Pythagoras theorem in maths
• The role of art in maths and vice versa
• Role of philosophy in maths
• An overview of numerical data
• Egyptian mathematics explained
• Binomial theorem and its importance
• Probability, and how to solve a question on dice
• Why is math made compulsory in schools?
• Why do students hate maths?
• Why do students hate math teachers?
• How is math applied in the workplace?
• What are imaginary numbers and why are they needed
• How to calculate the interest rate and what is their importance in the banking sector?
• Discount factor: how is it achieved and why is it important for students?
• Types of techniques to be used while finding solutions to mathematical and finance gaps
• Solving a matrix: what are the important formulas and principles to embrace?
• How to create a chart on a company’s financial analysis for the past 5 years.

Interesting Math Research Topics

Writing a mathematical essay may seem complex to you if you can’t find simple topics to write about. There are many easy topics which are also general in maths. If you want to choose a relaxing topic for your math essay or paper, you can write about the following:

• The basic elements of Boolean algebra
• The life, time, and contribution of Isaac Newton to maths
• Sphericon and what it means
• Martingales and what they mean
• Hyperboloid and importance in geometry
• Describe the life, times, and contribution of Gauss to maths
• The most famous work of Jakob Bernoulli
• The most famous work of Jean d’Alembert
• Meaning and application of calculus in the banking field
• The meaning of congruence in math
• Analysis of De Finetti theorem in probability and statistics
• Describe Egyptian pyramids in concert with calculus
• Describe the enclosing sphere technique as used in combinatorics
• Tree automation meaning
• Pushdown automaton and Buchi automaton: differences and similarities
• What is the Markov algorithm?
• Describe what a Turing machine is
• What is the linear speedup theory in math?
• The Boolean satisfiability problem and what it means for students
• Why is the multiplication table important?
• Computational maths and its classes
• What does the post correspondence problem mean?
• What does the Scholz conjecture mean?
• How to calculate mean, median, and mode
• A study of the most difficult equations in math.

Cool Math Topics to Research

As a student of any level, you may love to create math topics that are not exactly complex. These are topics that lean on the history of maths, math education research topics, and others. Consider these math research topics for college students for your next essay or paper:

• Discuss what the Golden Ratio means in the paintings of the Renaissance period
• How to learn math
• An overview of the multiple ideas to probability
• How chess and checkers is essential in understanding mathematics
• How Pythagorean theorem is applied in real-life maths
• How to measure infinity
• The features of Mobius strip in geometry
• Describe what is meant by the Pascal’s Triangle
• Evaluate the Georg Cantor set theory
• What is the history of the number types?
• How does probability relate to card tricks?
• Compare and contrast abstract and universal algebra
• Describe Euclid’s role in the evolution of maths
• Evaluate the role of Indians in maths
• Explain the limits of calculus
• Discuss what predictive and prescriptive statistical analysis means
• What does chaos theory mean?
• Explain how to solve the Rubik’s Cube
• Why are some math equations so complex?
• How is geometry used in contemporary architectural designs?

Math Research Topics for Middle School

It’s okay to be worried about math topics for your research as a middle school student. There are still different unique topics that are rebranded from existing ones. You can find some of the right math research paper topics for you here:

• The role of statistics in business
• Definition of economic lot scheduling
• Why stock market crash
• The contribution of many traders in the New York Stock Exchange
• Revenue management and its history
• What are the financial indicators of a good investment?
• What are the odds of depreciation?
• How can any country benefit from the poor currency?
• Describe debt amortization and how math helps
• How to calculate net worth
• Distinctions in calculus, trigonometry, and algebra
• How did calculus start?
• How did trigonometry start?
• Why is Ito stochastic important in math?
• What do limits in math mean?
• How to know critical points in graphs
• What does nonstandard analysis in the probability theory mean?
• Describe continuous function
• The main principles of calculus
• The main principles of Pythagoras theorem
• Application of calculus in finance
• Value theorem in math
• Ratio and root test definition
• Linear approximations and how they work
• What is the Jacobson density theorem?
• Similarities and differences between epimorphisms and monopolists
• What does the Artin-Wedderburn theorem mean?
• Commutative ring and its meaning in algebra
• How difficult is it to teach maths?
• How standards examination curriculum affects math education.

Applied Math Research Topics

Applied math is a branch which deals with the application of mathematical methods in real life. These are manifested by applications in finance, physics, engineering, biology, medicine, and others. Through specialized knowledge, applied math is made possible. These are some topics for you in this area:

• How discovering genes can help determine healthy and unhealthy patients
• Role of algorithms in probabilistic modeling
• The need for mathematicians in developing robots
• The role of mathematicians in crime data analysis and prevention
• How did Isaac’s Laws of Motion help in real life?
• How math helped with energy conservation
• The role of math in quantum theory
• Analyze the features of the Lorentz symmetry
• Evaluate statistical signal processing in details
• Discuss how Galilean Transformation was achieved
• Examine nonlinear models
• Elucidate on the importance of data mining in banking
• The importance of step-stress modeling
• The significance of computer tomography
• What are the dimensions used in examining fingerprints?

Math Research Topics for College Students

As college students, you are at a critical level. You need maths topics for your essays and paper. You may also need them to prepare for your exams. These are some math research topics for you:

• Evolution of mathematics
• Explore the varieties of the Tower of Hanoi solutions
• Discuss how to use Napier’s bones
• Give examples of chaos theory and explain
• Discuss the important mathematical equations of all times
• Examine the nitty-gritty of barcodes
• What is the Traveling Salesman Problem?
• Natural selection and Fisher’s fundamental theorem of understanding it
• The Influence of math in biology
• The Influence of math in chemistry
• What is quantum computing?
• How to solve extremal problems in maths
• Analyze the meaning of fractals
• Discuss Einstein’s field equation theory
• Who created computer vision and object recognition?
• Five formulas and how they are applied
• Give three approaches to understanding maths
• Explain the origin and importance of algebra
• What do you know about the Fibonacci sequence?
• Trace the origin of math
• How does math help in geography?
• What does the operator spaces notion mean?

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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, 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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Articles on Mathematics

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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

Anyone can play Tetris, but architects, engineers and animators alike use the math concepts underlying the game

Leah McCoy , Wake Forest University

GOP primary elections use flawed math to pick nominees

Ismar Volić , Wellesley College

What are ‘multiplication facts’? Why are they essential to your child’s success in maths?

Bronwyn Reid O'Connor , University of Sydney and Ben Zunica , University of Sydney

How bats ‘leapfrog’ their way home at night – new research

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

Orbital resonance − the striking gravitational dance done by planets with aligning orbits

Chris Impey , University of Arizona

Spreadsheet errors can have disastrous consequences – yet we keep making the same mistakes

Simon Thorne , Cardiff Metropolitan University

I wrote a play for children about integrating the arts into STEM fields − here’s what I learned about encouraging creative, interdisciplinary thinking

Rob Roznowski , Michigan State University

Here’s why you should (almost) never use a pie chart for your data

Adrian Barnett , Queensland University of Technology and Victor Oguoma , The University of Queensland

AI is our ‘Promethean fire’: using it wisely means knowing its true nature – and our own minds

Randolph Grace , University of Canterbury

How counting by 10 helps children learn about the meaning of numbers

Helena Osana , Concordia University ; Jairo A. Navarrete-Ulloa , Universidad de O’Higgins (Chile) , and Vera Wagner , Concordia University

How many people need to be in a room for two to share a birthday? It’s fewer than you think. Here’s why

Ben Zunica , University of Sydney

Rishi Sunak wants more maths at school – but finding the teachers will be hard when university departments are closing

Neil Saunders , University of Greenwich

Wales’s Pisa school test results have declined – but it’s not a true reflection of an education system

Alma Harris , Cardiff Metropolitan University

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Laureate Professor of Mathematics, University of Newcastle

PhD; Lawrence Berkeley Laboratory (retired) and Research Fellow, University of California, Davis

Emeritus Professor, University of Nottingham, (currently CEO and Senior Vice President of the PETRA Group), University of Nottingham

Professor of Mathematics, University of Florida

Senior Lecturer in Mathematical Biology, University of Bath

PhD Candidate, Australian National University

Professor of Mathematics, Toronto Metropolitan University

Mathematician, University of Portsmouth

Professor of Mathematics and Statistics, University of Maryland, Baltimore County

Lillian Gilbreth Postdoctoral Fellow, Purdue University

Lecturer, London School of Hygiene & Tropical Medicine

Acting Director of Research, Reader in Global Development, Newcastle University

Senior lecturer, UNSW Sydney

Lecturer, School of Biological Sciences, Monash University

Associate Professor, Department of Physiology, Monash University

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166 Extraordinary Math Research Topics For Your Papers

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

What Are The Different Types Of Math?

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

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

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

 Undergraduate Math Research Topics

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

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

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

Math Research Topics for High School Learners

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

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

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

Applied Math Research Topics

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

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

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

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

Interesting Math Research Topics

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

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

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

Math Research Topics for Middle School

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

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

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

Math Research Topics for College Students

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

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

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

Cool Math Topics to Research

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

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

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

Enlist the Best College Research Paper Writing Service

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

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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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100+ Amazing Algebra Topics for Research Papers

Many students seek algebra topics when writing research papers in this mathematical field. Algebra is the study field that entails studying mathematical symbols and rules for their manipulation. Algebra is the unifying thread for most mathematics, including solving elementary equations to learning abstractions like rings, groups, and fields.

In most cases, people use algebra when unsure about the exact numbers. Therefore, they replace those numbers with letters. In business, algebra helps with sales prediction. While many students dislike mathematics, avoiding algebra research paper topics is almost impossible at an advanced study level.

Therefore, this article lists topics to consider when writing a research paper in this academic field. It’s helpful because many learners struggle to find suitable topics when writing research papers in this field.

How to Write Theses on Advanced Algebra Topics

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

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

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

College Algebra Topics

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

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

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

Cool Topics in Algebra

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

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

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

Linear Algebra Topics

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

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

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

Pre Algebra Topics

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

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

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

Intermediate Algebra Topics for Research

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

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

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

Algebra Topics High School Students Can Explore

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

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

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

Advanced Topics in Algebra and Geometry

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

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

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

Abstract Algebra Topics

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

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

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

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Computer scientists show the way: AI models need not be SO power hungry

The development of AI models is an overlooked climate culprit. Computer scientists at the University of Copenhagen have created a recipe book for designing AI models that use much less energy without compromising performance. They argue that a model's energy consumption and carbon footprint should be a fixed criterion when designing and training AI models.

The fact that colossal amounts of energy are needed to Google away, talk to Siri, ask ChatGPT to get something done, or use AI in any sense, has gradually become common knowledge. One study estimates that by 2027, AI servers will consume as much energy as Argentina or Sweden. Indeed, a single ChatGPT prompt is estimated to consume, on average, as much energy as forty mobile phone charges. But the research community and the industry have yet to make the development of AI models that are energy efficient and thus more climate friendly the focus, computer science researchers at the University of Copenhagen point out.

"Today, developers are narrowly focused on building AI models that are effective in terms of the accuracy of their results. It's like saying that a car is effective because it gets you to your destination quickly, without considering the amount of fuel it uses. As a result, AI models are often inefficient in terms of energy consumption," says Assistant Professor Raghavendra Selvan from the Department of Computer Science, whose research looks in to possibilities for reducing AI's carbon footprint.

But the new study, of which he and computer science student Pedram Bakhtiarifard are two of the authors, demonstrates that it is easy to curb a great deal of CO 2 e without compromising the precision of an AI model. Doing so demands keeping climate costs in mind from the design and training phases of AI models.

"If you put together a model that is energy efficient from the get-go, you reduce the carbon footprint in each phase of the model's 'life cycle'. This applies both to the model's training, which is a particularly energy-intensive process that often takes weeks or months, as well as to its application," says Selvan.

Recipe book for the AI industry

In their study, the researchers calculated how much energy it takes to train more than 400,000 convolutional neural network type AI models -- this was done without actually training all these models. Among other things, convolutional neural networks are used to analyse medical imagery, for language translation and for object and face recognition -- a function you might know from the camera app on your smartphone.

Based on the calculations, the researchers present a benchmark collection of AI models that use less energy to solve a given task, but which perform at approximately the same level. The study shows that by opting for other types of models or by adjusting models, 70-80% energy savings can be achieved during the training and deployment phase, with only a 1% or less decrease in performance. And according to the researchers, this is a conservative estimate.

"Consider our results as a recipe book for the AI professionals. The recipes don't just describe the performance of different algorithms, but how energy efficient they are. And that by swapping one ingredient with another in the design of a model, one can often achieve the same result. So now, the practitioners can choose the model they want based on both performance and energy consumption, and without needing to train each model first," says Pedram Bakhtiarifard, who continues:

"Oftentimes, many models are trained before finding the one that is suspected of being the most suitable for solving a particular task. This makes the development of AI extremely energy-intensive. Therefore, it would be more climate-friendly to choose the right model from the outset, while choosing one that does not consume too much power during the training phase."

The researchers stress that in some fields, like self-driving cars or certain areas of medicine, model precision can be critical for safety. Here, it is important not to compromise on performance. However, this shouldn't be a deterrence to striving for high energy efficiency in other domains.

"AI has amazing potential. But if we are to ensure sustainable and responsible AI development, we need a more holistic approach that not only has model performance in mind, but also climate impact. Here, we show that it is possible to find a better trade-off. When AI models are developed for different tasks, energy efficiency ought to be a fixed criterion -- just as it is standard in many other industries," concludes Raghavendra Selvan.

The "recipe book" put together in this work is available as an open-source dataset for other researchers to experiment with. The information about all these 423,000 architectures is published on Github which AI practitioners can access using simple Python scripts.

The UCPH researchers estimated how much energy it takes to train 429,000 of the AI subtype models known as convolutional neural networks in this dataset. Among other things, these are used for object detection, language translation and medical image analysis.

It is estimated that the training alone of the 429,000 neural networks the study looked at would require 263,000 kWh. This equals the amount of energy that an average Danish citizen consumes over 46 years. And it would take one computer about 100 years to do the training. The authors in this work did not actually train these models themselves but estimated these using another AI model, and thus saving 99% of the energy it would have taken.

Training AI models consumes a lot of energy, and thereby emits a lot of CO2e. This is due to the intensive computations performed while training a model, typically run on powerful computers. This is especially true for large models, like the language model behind ChatGPT. AI tasks are often processed in data centers, which demand significant amounts of power to keep computers running and cool. The energy source for these centers, which may rely on fossil fuels, influences their carbon footprint.

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Materials provided by University of Copenhagen - Faculty of Science . Note: Content may be edited for style and length.

Journal Reference :

• Pedram Bakhtiarifard, Christian Igel, Raghavendra Selvan. EC-NAS: Energy Consumption Aware Tabular Benchmarks for Neural Architecture Search . , DOI: 10.1109/ICASSP48485.2024.10448303

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Changing partisan coalitions in a politically divided nation, party identification among registered voters, 1994-2023.

Pew Research Center conducted this analysis to explore partisan identification among U.S. registered voters across major demographic groups and how voters’ partisan affiliation has shifted over time. It also explores the changing composition of voters overall and the partisan coalitions.

For this analysis, we used annual totals of data from Pew Research Center telephone surveys (1994-2018) and online surveys (2019-2023) among registered voters. All telephone survey data was adjusted to account for differences in how people respond to surveys on the telephone compared with online surveys (refer to Appendix A for details).

All online survey data is from the Center’s nationally representative American Trends Panel . The surveys were conducted in both English and Spanish. Each survey is weighted to be representative of the U.S. adult population by gender, age, education, race and ethnicity and other categories. Read more about the ATP’s methodology , as well as how Pew Research Center measures many of the demographic categories used in this report .

The contours of the 2024 political landscape are the result of long-standing patterns of partisanship, combined with the profound demographic changes that have reshaped the United States over the past three decades.

Many of the factors long associated with voters’ partisanship remain firmly in place. For decades, gender, race and ethnicity, and religious affiliation have been important dividing lines in politics. This continues to be the case today.

Yet there also have been profound changes – in some cases as a result of demographic change, in others because of dramatic shifts in the partisan allegiances of key groups.

The combined effects of change and continuity have left the country’s two major parties at virtual parity: About half of registered voters (49%) identify as Democrats or lean toward the Democratic Party, while 48% identify as Republicans or lean Republican.

In recent decades, neither party has had a sizable advantage, but the Democratic Party has lost the edge it maintained from 2017 to 2021. (Explore this further in Chapter 1 . )

Pew Research Center’s comprehensive analysis of party identification among registered voters – based on hundreds of thousands of interviews conducted over the past three decades – tracks the changes in the country and the parties since 1994. Among the major findings:

The partisan coalitions are increasingly different. Both parties are more racially and ethnically diverse than in the past. However, this has had a far greater impact on the composition of the Democratic Party than the Republican Party.

The share of voters who are Hispanic has roughly tripled since the mid-1990s; the share who are Asian has increased sixfold over the same period. Today, 44% of Democratic and Democratic-leaning voters are Hispanic, Black, Asian, another race or multiracial, compared with 20% of Republicans and Republican leaners. However, the Democratic Party’s advantages among Black and Hispanic voters, in particular, have narrowed somewhat in recent years. (Explore this further in Chapter 8 .)

Education and partisanship: The share of voters with a four-year bachelor’s degree keeps increasing, reaching 40% in 2023. And the gap in partisanship between voters with and without a college degree continues to grow, especially among White voters. More than six-in-ten White voters who do not have a four-year degree (63%) associate with the Republican Party, which is up substantially over the past 15 years. White college graduates are closely divided; this was not the case in the 1990s and early 2000s, when they mostly aligned with the GOP. (Explore this further in Chapter 2 .)

Beyond the gender gap: By a modest margin, women voters continue to align with the Democratic Party (by 51% to 44%), while nearly the reverse is true among men (52% align with the Republican Party, 46% with the Democratic Party). The gender gap is about as wide among married men and women. The gap is wider among men and women who have never married; while both groups are majority Democratic, 37% of never-married men identify as Republicans or lean toward the GOP, compared with 24% of never-married women. (Explore this further in Chapter 3 .)

A divide between old and young: Today, each younger age cohort is somewhat more Democratic-oriented than the one before it. The youngest voters (those ages 18 to 24) align with the Democrats by nearly two-to-one (66% to 34% Republican or lean GOP); majorities of older voters (those in their mid-60s and older) identify as Republicans or lean Republican. While there have been wide age divides in American politics over the last two decades, this wasn’t always the case; in the 1990s there were only very modest age differences in partisanship. (Explore this further in Chapter 4 .)

Education and family income: Voters without a college degree differ substantially by income in their party affiliation. Those with middle, upper-middle and upper family incomes tend to align with the GOP. A majority with lower and lower-middle incomes identify as Democrats or lean Democratic. There are no meaningful differences in partisanship among voters with at least a four-year bachelor’s degree; across income categories, majorities of college graduate voters align with the Democratic Party. (Explore this further in Chapter 6 .)

Rural voters move toward the GOP, while the suburbs remain divided: In 2008, when Barack Obama sought his first term as president, voters in rural counties were evenly split in their partisan loyalties. Today, Republicans hold a 25 percentage point advantage among rural residents (60% to 35%). There has been less change among voters in urban counties, who are mostly Democratic by a nearly identical margin (60% to 37%). The suburbs – perennially a political battleground – remain about evenly divided. (Explore this further in Chapter 7 . )

Growing differences among religious groups: Mirroring movement in the population overall, the share of voters who are religiously unaffiliated has grown dramatically over the past 15 years. These voters, who have long aligned with the Democratic Party, have become even more Democratic over time: Today 70% identify as Democrats or lean Democratic. In contrast, Republicans have made gains among several groups of religiously affiliated voters, particularly White Catholics and White evangelical Protestants. White evangelical Protestants now align with the Republican Party by about a 70-point margin (85% to 14%). (Explore this further in Chapter 5 .)

What this report tells us – and what it doesn’t

In most cases, the partisan allegiances of voters do not change a great deal from year to year. Yet as this study shows, the long-term shifts in party identification are substantial and say a great deal about how the country – and its political parties – have changed since the 1990s.

The steadily growing alignment between demographics and partisanship reveals an important aspect of steadily growing partisan polarization. Republicans and Democrats do not just hold different beliefs and opinions about major issues , they are much more different racially, ethnically, geographically and in educational attainment than they used to be.

Yet over this period, there have been only modest shifts in overall partisan identification. Voters remain evenly divided, even as the two parties have grown further apart. The continuing close division in partisan identification among voters is consistent with the relatively narrow margins in the popular votes in most national elections over the past three decades.

Partisan identification provides a broad portrait of voters’ affinities and loyalties. But while it is indicative of voters’ preferences, it does not perfectly predict how people intend to vote in elections, or whether they will vote. In the coming months, Pew Research Center will release reports analyzing voters’ preferences in the presidential election, their engagement with the election and the factors behind candidate support.

Next year, we will release a detailed study of the 2024 election, based on validated voters from the Center’s American Trends Panel. It will examine the demographic composition and vote choices of the 2024 electorate and will provide comparisons to the 2020 and 2016 validated voter studies.

The partisan identification study is based on annual totals from surveys conducted on the Center’s American Trends Panel from 2019 to 2023 and telephone surveys conducted from 1994 to 2018. The survey data was adjusted to account for differences in how the surveys were conducted. For more information, refer to Appendix A .

Previous Pew Research Center analyses of voters’ party identification relied on telephone survey data. This report, for the first time, combines data collected in telephone surveys with data from online surveys conducted on the Center’s nationally representative American Trends Panel.

Directly comparing answers from online and telephone surveys is complex because there are differences in how questions are asked of respondents and in how respondents answer those questions. Together these differences are known as “mode effects.”

As a result of mode effects, it was necessary to adjust telephone trends for leaned party identification in order to allow for direct comparisons over time.

In this report, telephone survey data from 1994 to 2018 is adjusted to align it with online survey responses. In 2014, Pew Research Center randomly assigned respondents to answer a survey by telephone or online. The party identification data from this survey was used to calculate an adjustment for differences between survey mode, which is applied to all telephone survey data in this report.

Please refer to Appendix A for more details.

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Table of contents, behind biden’s 2020 victory, a voter data resource: detailed demographic tables about verified voters in 2016, 2018, what the 2020 electorate looks like by party, race and ethnicity, age, education and religion, interactive map: the changing racial and ethnic makeup of the u.s. electorate, in changing u.s. electorate, race and education remain stark dividing lines, most popular.

About Pew Research Center Pew Research Center is a nonpartisan fact tank that informs the public about the issues, attitudes and trends shaping the world. It conducts public opinion polling, demographic research, media content analysis and other empirical social science research. Pew Research Center does not take policy positions. It is a subsidiary of The Pew Charitable Trusts .

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Two key brain systems are central to psychosis, Stanford Medicine-led study finds

When the brain has trouble filtering incoming information and predicting what’s likely to happen, psychosis can result, Stanford Medicine-led research shows.

April 11, 2024 - By Erin Digitale

People with psychosis have trouble filtering relevant information (mesh funnel) and predicting rewarding events (broken crystal ball), creating a complex inner world. Emily Moskal

Inside the brains of people with psychosis, two key systems are malfunctioning: a “filter” that directs attention toward important external events and internal thoughts, and a “predictor” composed of pathways that anticipate rewards.

Dysfunction of these systems makes it difficult to know what’s real, manifesting as hallucinations and delusions. 

The findings come from a Stanford Medicine-led study , published April 11 in  Molecular Psychiatry , that used brain scan data from children, teens and young adults with psychosis. The results confirm an existing theory of how breaks with reality occur.

“This work provides a good model for understanding the development and progression of schizophrenia, which is a challenging problem,” said lead author  Kaustubh Supekar , PhD, clinical associate professor of psychiatry and behavioral sciences.

The findings, observed in individuals with a rare genetic disease called 22q11.2 deletion syndrome who experience psychosis as well as in those with psychosis of unknown origin, advance scientists’ understanding of the underlying brain mechanisms and theoretical frameworks related to psychosis.

During psychosis, patients experience hallucinations, such as hearing voices, and hold delusional beliefs, such as thinking that people who are not real exist. Psychosis can occur on its own and isa hallmark of certain serious mental illnesses, including bipolar disorder and schizophrenia. Schizophrenia is also characterized by social withdrawal, disorganized thinking and speech, and a reduction in energy and motivation.

It is challenging to study how schizophrenia begins in the brain. The condition usually emerges in teens or young adults, most of whom soon begin taking antipsychotic medications to ease their symptoms. When researchers analyze brain scans from people with established schizophrenia, they cannot distinguish the effects of the disease from the effects of the medications. They also do not know how schizophrenia changes the brain as the disease progresses. 

To get an early view of the disease process, the Stanford Medicine team studied young people aged 6 to 39 with 22q11.2 deletion syndrome, a genetic condition with a 30% risk for psychosis, schizophrenia or both. 

Kaustubh Supekar

Brain function in 22q11.2 patients who have psychosis is similar to that in people with psychosis of unknown origin, they found. And these brain patterns matched what the researchers had previously theorized was generating psychosis symptoms.

“The brain patterns we identified support our theoretical models of how cognitive control systems malfunction in psychosis,” said senior study author  Vinod Menon , PhD, the Rachael L. and Walter F. Nichols, MD, Professor; a professor of psychiatry and behavioral sciences; and director of the  Stanford Cognitive and Systems Neuroscience Laboratory .

Thoughts that are not linked to reality can capture the brain’s cognitive control networks, he said. “This process derails the normal functioning of cognitive control, allowing intrusive thoughts to dominate, culminating in symptoms we recognize as psychosis.”

Cerebral sorting  

Normally, the brain’s cognitive filtering system — aka the salience network — works behind the scenes to selectively direct our attention to important internal thoughts and external events. With its help, we can dismiss irrational thoughts and unimportant events and focus on what’s real and meaningful to us, such as paying attention to traffic so we avoid a collision.

The ventral striatum, a small brain region, and associated brain pathways driven by dopamine, play an important role in predicting what will be rewarding or important. 

For the study, the researchers assembled as much functional MRI brain-scan data as possible from young people with 22q11.2 deletion syndrome, totaling 101 individuals scanned at three different universities. (The study also included brain scans from several comparison groups without 22q11.2 deletion syndrome: 120 people with early idiopathic psychosis, 101 people with autism, 123 with attention deficit/hyperactivity disorder and 411 healthy controls.) 

The genetic condition, characterized by deletion of part of the 22nd chromosome, affects 1 in every 2,000 to 4,000 people. In addition to the 30% risk of schizophrenia or psychosis, people with the syndrome can also have autism or attention deficit hyperactivity disorder, which is why these conditions were included in the comparison groups.

The researchers used a type of machine learning algorithm called a spatiotemporal deep neural network to characterize patterns of brain function in all patients with 22q11.2 deletion syndrome compared with healthy subjects. With a cohort of patients whose brains were scanned at the University of California, Los Angeles, they developed an algorithmic model that distinguished brain scans from people with 22q11.2 deletion syndrome versus those without it. The model predicted the syndrome with greater than 94% accuracy. They validated the model in additional groups of people with or without the genetic syndrome who had received brain scans at UC Davis and Pontificia Universidad Católica de Chile, showing that in these independent groups, the model sorted brain scans with 84% to 90% accuracy.

The researchers then used the model to investigate which brain features play the biggest role in psychosis. Prior studies of psychosis had not given consistent results, likely because their sample sizes were too small. 

Vinod Menon

Comparing brain scans from 22q11.2 deletion syndrome patients who had and did not have psychosis, the researchers showed that the brain areas contributing most to psychosis are the anterior insula (a key part of the salience network or “filter”) and the ventral striatum (the “reward predictor”); this was true for different cohorts of patients.

In comparing the brain features of people with 22q11.2 deletion syndrome and psychosis against people with psychosis of unknown origin, the model found significant overlap, indicating that these brain features are characteristic of psychosis in general.

A second mathematical model, trained to distinguish all subjects with 22q11.2 deletion syndrome and psychosis from those who have the genetic syndrome but without psychosis, selected brain scans from people with idiopathic psychosis with 77.5% accuracy, again supporting the idea that the brain’s filtering and predicting centers are key to psychosis.

Furthermore, this model was specific to psychosis: It could not classify people with idiopathic autism or ADHD.

“It was quite exciting to trace our steps back to our initial question — ‘What are the dysfunctional brain systems in schizophrenia?’ — and to discover similar patterns in this context,” Menon said. “At the neural level, the characteristics differentiating individuals with psychosis in 22q11.2 deletion syndrome are mirroring the pathways we’ve pinpointed in schizophrenia. This parallel reinforces our understanding of psychosis as a condition with identifiable and consistent brain signatures.” However, these brain signatures were not seen in people with the genetic syndrome but no psychosis, holding clues to future directions for research, he added.

Applications for treatment or prevention

In addition to supporting the scientists’ theory about how psychosis occurs, the findings have implications for understanding the condition — and possibly preventing it.

“One of my goals is to prevent or delay development of schizophrenia,” Supekar said. The fact that the new findings are consistent with the team’s prior research on which brain centers contribute most to schizophrenia in adults suggests there may be a way to prevent it, he said. “In schizophrenia, by the time of diagnosis, a lot of damage has already occurred in the brain, and it can be very difficult to change the course of the disease.”

“What we saw is that, early on, functional interactions among brain regions within the same brain systems are abnormal,” he added. “The abnormalities do not start when you are in your 20s; they are evident even when you are 7 or 8.”

Our discoveries underscore the importance of approaching people with psychosis with compassion.

The researchers plan to use existing treatments, such as transcranial magnetic stimulation or focused ultrasound, targeted at these brain centers in young people at risk of psychosis, such as those with 22q11.2 deletion syndrome or with two parents who have schizophrenia, to see if they prevent or delay the onset of the condition or lessen symptoms once they appear. 

The results also suggest that using functional MRI to monitor brain activity at the key centers could help scientists investigate how existing antipsychotic medications are working. 

Although it’s still puzzling why someone becomes untethered from reality — given how risky it seems for one’s well-being — the “how” is now understandable, Supekar said. “From a mechanistic point of view, it makes sense,” he said.

“Our discoveries underscore the importance of approaching people with psychosis with compassion,” Menon said, adding that his team hopes their work not only advances scientific understanding but also inspires a cultural shift toward empathy and support for those experiencing psychosis. 

“I recently had the privilege of engaging with individuals from our department’s early psychosis treatment group,” he said. “Their message was a clear and powerful: ‘We share more similarities than differences. Like anyone, we experience our own highs and lows.’ Their words were a heartfelt appeal for greater empathy and understanding toward those living with this condition. It was a call to view psychosis through a lens of empathy and solidarity.”

Researchers contributed to the study from UCLA, Clinica Alemana Universidad del Desarrollo, Pontificia Universidad Católica de Chile, the University of Oxford and UC Davis.

The study was funded by the Stanford Maternal and Child Health Research Institute’s Uytengsu-Hamilton 22q11 Neuropsychiatry Research Program, FONDEYCT (the National Fund for Scientific and Technological Development of the government of Chile), ANID-Chile (the Chilean National Agency for Research and Development) and the U.S. National Institutes of Health (grants AG072114, MH121069, MH085953 and MH101779).

About Stanford Medicine

Stanford Medicine is an integrated academic health system comprising the Stanford School of Medicine and adult and pediatric health care delivery systems. Together, they harness the full potential of biomedicine through collaborative research, education and clinical care for patients. For more information, please visit med.stanford.edu .

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The Research-Backed Benefits of Daily Rituals

• Michael I. Norton

A survey of more than 130 HBR readers asked how they use rituals to start their days, psych themselves up for stressful challenges, and transition when the workday is done.

While some may cringe at forced corporate rituals, research shows that personal and team rituals can actually benefit the way we work. The authors’ expertise on the topic over the past decade, plus a survey of nearly 140 HBR readers, explores the ways rituals can set us up for success before work, get us psyched up for important presentations, foster a strong team culture, and help us wind down at the end of the day.

“Give me a W ! Give me an A ! Give me an L ! Give me a squiggly! Give me an M ! Give me an A ! Give me an R ! Give me a T !”

• Michael I. Norton is the Harold M. Brierley Professor of Business Administration at the Harvard Business School. He is the author of The Ritual Effect and co-author of Happy Money: The Science of Happier Spending . His research focuses on happiness, well-being, rituals, and inequality. See his faculty page here .

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